Mathematical Modeling, Simulations, and AI for Emergent Pandemic Diseases: Lessons Learned From COVID-19 0323950647, 9780323950640

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Mathematical Modeling, Simulations, and AI for Emergent Pandemic Dis-  eases  Lessons Learned from COVID-19 

First Edition 

Esteban A. Hernandez-Vargas  Department of Mathematics and Statistical Science, University of Idaho, Mos-  cow, ID, United States  Jorge X. Velasco-Hernández  Instituto de Matemáticas, UNAM Juriquilla, Juriquilla, Queretaro, Mexico  Series editor  Dr. Edgar Sanchez 

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Table of Contents  Cover image  Title page  Copyright  Contributors  Acknowledgments  1: Modeling during an unprecedented pandemic  Abstract 

1: Modeling epidemics 

2: Book overview 

References 

2: Global epidemiology and impact of the SARS-CoV-2 pandemic  Abstract 

1: Introduction 

2: Global epidemiology 

3: Epidemiological parameters of SARS-COV-2 

4: Mitigation strategies 

5: Reinfections 

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6: SARS-COV-2 variants

7: Lessons learned from COVID-19 

Appendix 

References 

3: Analysis of an ongoing epidemic: Advantages and limitations of COVID-  19 modeling  Abstract 

Acknowledgments 

1: Introduction 

2: The beginning of the pandemic 

3: Implementation and relaxation of nonpharmaceutical interventions 

4: Estimating the total number of COVID-19-infected people 

5: Vaccination 

6: Lessons learned from COVID-19 

Competing interests 

References 

4: On spatial heterogeneity of COVID-19 using shape analysis of pandemic  curves☆

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Abstract    1: Introduction    2: Methodology    3: Experimental results    4: Lessons learned from COVID-19    References    5: Pandemic response: Isolationism or solidarity?: An evolutionary perspec-  tive    An evolutionary perspective    Abstract    Acknowledgments    1: Introduction    2: Background    3: Methods and results    4: Lessons learned from COVID-19    Appendix A. Supplementary data    References   

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6: Optimizing contact tracing: Leveraging contact network structure    Abstract    Acknowledgments    1: Introduction    2: Methods    3: Results    4: Lessons learned from COVID-19    References    7: Applications of deep learning in forecasting COVID-19 pandemic and  county-level risk warning    Abstract    Acknowledgments    1: Introduction    2: Applications of DL in predicting COVID-19 pandemic    3: Spatial-temporal analysis    4: Epidemiological model-driven DL    5: Lessons learned from COVID-19    References 

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8: COVID-19 population dynamics neural control from a complex network  perspective    Abstract    1: Introduction    2: MSEIR model    3: Inverse optimal impulsive control    4: Results    5: Lessons learned from COVID-19    References    Further reading    9: An agent-based model for COVID-19 and its interventions and impact in  different social phenomena    Abstract    1: Introduction    2: Nonpharmaceutical interventions assessment    3: An ABM to evaluate NPI to support postpandemic work activities and  the well-being of workers    4: Sensitivity analysis    5: Assumptions and scenario 

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6: Computational analysis    7: Lessons learned from COVID-19    References    10: Implementation of mitigation measures and modeling of in-hospital  dynamics depending on the COVID-19 infection status    Abstract    Acknowledgments    1: Introduction    2: Dynamic model of quarantine scenarios and hospital overload    3: Stochastic extension for in-hospital dynamics    4: Conclusions    Code availability    Appendix    References    11: A mathematical model for the reopening of schools in Mexico    Abstract    Acknowledgments    1: Introduction 

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2: Mathematical model for reopening schools    3: Reopening schools in the city of Queretaro    4: Epidemic scenarios    5: Risk level of being in contact with an infected individual at school    6: Lessons learned in COVID-19    References    12: Mathematical assessment of the role of vaccination against COVID-19 in  the United States    Abstract    Acknowledgments    1: Introduction    2: Basic vaccination model    3: Basic vaccination model with waning immunity    4: Modeling dynamics and impact of SARS-CoV-2 variants    5: Lessons learned from COVID-19    References    13: Ascertainment and biased testing rates in surveillance of emerging infec-  tious diseases   

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Abstract    Acknowledgment    1: Biases in the epidemiological analysis of emerging infectious dis-  eases    2: Temporal variation of the confirmed case fatality rate reflects the  societal response to outbreaks    3: Lessons learned from COVID-19    References    14: Dynamical study of SARS-CoV-2 mathematical models under antiviral  treatments    Abstract    1: Introduction    2: Review of the target-cell-limited model for SARS-CoV-2 infection    3: Antiviral treatment effectiveness    4: Inclusion of the PK of antiviral treatment    5: Control strategy to tailor therapies    6: Conclusions and future works    Appendix A: Stability theory    Appendix B: Behavior of the terminal healthy cells count  10

References    15: Statistical modeling to understand the COVID-19 pandemic    Abstract    Acknowledgment    1: Introduction    2: Epidemic curves via censorship    3: COVID-19 footprint and mortality    4: Lessons learned from COVID-19    References    16: After COVID-19: Mathematical models, epidemic preparedness, and  external factors in epidemic management    Abstract    Acknowledgments    1: Introduction    2: Data    3: Confronting the challenge    4: Vaccination    5: Some features of the epidemic in Mexico 

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6: On public trust in science    7: Lessons learned in COVID-19    References    Index 

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Contributors   Manuel A. Acuña-Zegarra     Department of Mathematics, University of  Sonora, Hermosillo, Sonora, Mexico  Alma Y. Alanis     CUCEI, Universidad de Guadalajara, Guadalajara, Jalis-  co, Mexico  Francisco Aleman     Cinvestav Guadalajara, Zapopan, Jalisco, Mexico  Guillermo de Anda-Jáuregui  Computational Genomics Division, National Institute of Genomic Medicine  Center for Complexity Sciences, Universidad Nacional Autónoma de Méx-  ico, Mexico City, CDMX  Researcher for Mexico (Formerly CONACYT Research Fellow—Cátedras  CONACYT), National Council for Science and Technology. Mexico City,  Mexico  Sofia Bernal-Silva     School of Medicine, Autonomous University of San  Luis Potosi, San Luis Potosi, Mexico  Pablo Castañeda     Academic Department of Mathematics, ITAM, Mexico  City, Mexico  Gerardo Chowell     Department of Population Health Sciences, School of  Public Health, Georgia State University, Atlanta, GA, United States  Alexandre Colato     Department of Natural Sciences, Mathematics and  Education, Federal University of Sao Carlos, Araras, SP, Brazil  Andreu Comas-García     School of Medicine, Autonomous University of  San Luis Potosi, San Luis Potosi, Mexico  Ruth Corona-Moreno     Institute of Mathematics, National Autonomous  University of Mexico (UNAM), Juriquilla, Queretaro, Mexico  Agustina D’Jorge     Institute of Technological Development for the  Chemical Industry (INTEC), CONICET-UNL, Santa Fe, Argentina  Liliana Durán-Polanco     Cinvestav Guadalajara, Zapopan, Jalisco, Mex-  ico  Philip J. Gerrish  Department of Biology, University of New Mexico, Albuquerque, NM  School of Biological Sciences, University of Michigan, Ann Arbor, MI, Unit-  ed States  Academic Area of Mathematics and Physics, Autonomous University of the 

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State of Hidalgo (UAEH), Pachuca, Mexico  Alejandro H. González     Institute of Technological Development for the  Chemical Industry (INTEC), CONICET-UNL, Santa Fe, Argentina  Abba B. Gumel  Department of Mathematics, University of Maryland, College Park, MD,  United States  Department of Mathematics and Applied Mathematics, University of Pre-  toria, Pretoria, South Africa  Shuai Han  Xidian-FIAS International Joint Research Center  Institute for Theoretical Physics, Goethe University Frankfurt, Frankfurt am  Main, Germany  Mauricio Hernández-Ávila     Mexican Social Security Institute, Mexico  City, Mexico  Enrique Hernández-Lemus  Computational Genomics Division, National Institute of Genomic Medicine  Center for Complexity Sciences, Universidad Nacional Autónoma de Méx-  ico, Mexico City, CDMX, Mexico  Esteban A. Hernandez-Vargas  Department of Mathematics and Statistical Science  Institute for Modeling Collaboration and Innovation, University of Idaho,  Moscow, ID, United States  Yin Jiang  Department of Physics, Beihang University, Beijing  Beihang Hangzhou Innovation Institute, Yuhang, Hangzhou, China  Gabriel Martinez-Soltero     CUCEI, Universidad de Guadalajara, Guadala-  jara, Jalisco, Mexico  Ramsés H. Mena     Department of Probability and Statistics, IIMAS-  UNAM, Mexico City, Mexico  Rafael Meza     Department of Epidemiology, University of Michigan  School of Public Health, Ann Arbor, MI, United States  Calistus N. Ngonghala  Department of Mathematics  Emerging Pathogens Institute, University of Florida, Gainesville, FL, United  States  Mayra Núñez-López     Academic Department of Mathematics, ITAM, 

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Mexico City, Mexico  T.Y. Okosun     Department of Justice Studies, Northeastern Illinois Univ-  ersity, Chicago, IL, United States  Ryosuke Omori     Division of Bioinformatics, International Institute for  Zoonosis Control, Hokkaido University, Sapporo, Japan  Gamaliel A. Palomo-Briones     Cinvestav Guadalajara, Zapopan, Jalisco,  Mexico  Nancy F. Ramirez     CUCSUR, Universidad de Guadalajara, Guadalajara,  Jalisco, Mexico  Daniel Ríos-Rivera     CUCEI, Universidad de Guadalajara, Guadalajara,  Jalisco, Mexico  Carlos E. Rodríguez     Department of Probability and Statistics, IIMAS-  UNAM, Mexico City, Mexico  Erika E. Rodriguez Torres     Academic Area of Mathematics and Physics,  Autonomous University of the State of Hidalgo (UAEH), Pachuca, Mexico  Fernando Saldaña     Institute of Mathematics, National Autonomous  University of Mexico (UNAM), Juriquilla, Queretaro, Mexico  Ignacio J. Sánchez     Institute of Technological Development for the  Chemical Industry (INTEC), CONICET-UNL, Santa Fe, Argentina  Mario Santana-Cibrian     Escuela Nacional de Estudios Superiores  Unidad Juriquilla, Universidad Nacional Autonoma de Mexico, Queretaro,  Mexico  Mario Siller     Cinvestav Guadalajara, Zapopan, Jalisco, Mexico  Sarah Skolnick     Department of Epidemiology, University of Michigan  School of Public Health, Ann Arbor, MI, United States  Anuj Srivastava     Department of Statistics, Florida State University,  Tallahassee, FL, United States  Horst Stoecker  Frankfurt Institute for Advanced Studies  Institute for Theoretical Physics, Goethe University Frankfurt, Frankfurt am  Main, Germany  Mayra R. Tocto-Erazo     Department of Mathematics, University of Sono-  ra, Hermosillo, Sonora, Mexico  Angélica Torres-Díaz  Division of Higher Studies for Equity, School of Medicine, Autonomous  University of San Luis Potosi 

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Organization to Restore the Environment and Social Harmony (ORMA,  A.C.), San Luis Potosi, Mexico  Jorge X. Velasco-Hernández     Institute of Mathematics, National Auton-  omous University of Mexico (UNAM), Juriquilla, Queretaro, Mexico  Lingxiao Wang  Frankfurt Institute for Advanced Studies  Xidian-FIAS International Joint Research Center, Frankfurt am Main, Ger-  many  Rodrigo Zepeda-Tello     Mexican Social Security Institute, Mexico City,  Mexico  Kai Zhou     Frankfurt Institute for Advanced Studies, Frankfurt am Main,  Germany 

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Acknowledgments  Mathematical and computational models have been the cornerstone for the  development, planning, and implementation of sound public health policies  for pandemic containment and mitigation. The scientific community has  played the most important role in the understanding of the dynamics of the  disease and the complex socioeconomic system in which it is embedded.  The National Autonomous University of Mexico has been at the forefront  of this effort, particularly in the area of mathematical modeling. We acknowl-  edge the support that UNAM provided through grants DGAPA-PAPIIT  IV100220 and IA102521. We also thank the Instituto de Matemáticas, Insti-  tuto de Física, ENES Juriquilla, and Facultad de Ciencias UNAM under  which the epidemiological analysis and modeling of the pandemic in Mex-  ico have been pursued. Furthermore, among the many people who made  possible this project, we particularly thank William Lee Alardín, Ramsés  Mena Chavez, Dwight Dyer, Rosaura Ruiz, Carlos Arredondo, Alejandro  Diaz-Barriga, Guillermo Ramírez Santiago, José Seade Kuri, Raúl Gerardo Paredes, Hortensia Galeana, Marco Angulo, Natalia Ramírez, Santiago Es-  pinoza, Isaac Perez, Denis Boyer, David Sanders, Luis Benet, and Alfredo  Varela for their support and confidence in our work.  This project was also supported by the National Institute of General Med-  ical Sciences of the National Institutes of Health under Award Number  P20GM104420. The content is solely the responsibility of the authors and  does not necessarily represent the official views of the National Institutes of  Health.  Last but not least, we are truly fortunate to have had the opportunity to  work with all the authors who elaborated fantastic chapters for this book.  Thanks to all of them. We thank the Elsevier team composed of Chris Kat-  saropoulos and Rafael G. Trombaco for the great support and guidance  throughout the publishing process. We are also grateful to all the anony-  mous referees for carefully reviewing and selecting the final chapters of this  book. 

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  1: Modeling during an unprecedented pandemic   

a,b a Esteban A. Hernandez-Vargas      Department of Mathematics and  Statistical Science, University of Idaho, Moscow, ID, United States  b Institute for Modeling Collaboration and Innovation, University of Idaho,  Moscow, ID, United States 

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Abstract  Emerging and reemerging pathogens are latent threats in our society  with the risk of killing millions of people worldwide, without forgetting  the severe economic and educational backlogs. The spread of  pathogens from one individual to another can be abstract in mathe-  matical terms, helping public health agencies with predictions of the  pandemic and evaluating possible scenarios. Nevertheless, modeling  disease transmission is a central vexation for the scientific commu-  nity—this involves several complex and dynamic processes. The  emerging pathogen SARS-CoV-2 has led us to long terms of confine-  ment and social isolation. Many lessons have been learned during  COVID-19; a very important one is that mathematics is a central tool to  follow uncharted territories during pandemics.  This chapter aims to put in perspective the subsequent chapters pre-  sented in this book, which are a collection of mathematical models,  computational simulations, and artificial intelligence approaches that  were employed during the COVID-19 pandemic.   

Keywords  Mathematical modeling; AI; Simulations; Emergent diseases; COVID-19 

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1: Modeling epidemics  Infectious diseases can cause the invasion of an individual by pathogens  whose activities can highly harm the host's tissues while transmitting to  other individuals. Infectious diseases can be divided into acute or chronic  infections. Examples of infectious diseases in history are many; for instance,  the year 1918 saw the deadliest pandemic reported in history named “the  1918 flu pandemic,” killing about 50 million people. In 2009, the H1N1 in-  fluenza pandemic globally caused between 100,000 and 400,000 deaths in  the first year. The magnitude of the threat represented by emerging virus diseases is immense; for example, HIV/AIDS has resulted in more than 30  million deaths from all socioeconomic backgrounds [1]. Furthermore,  reemerging viruses like Ebola in 2014 and Zika in 2016 have baffled us with  their threat to humans and healthcare systems around the globe. Nowadays,  we all experience the consequences of the COVID-19 pandemic, which has  spread across borders, affecting several nations simultaneously and having  a considerable economic and social impact worldwide.  An emerging infectious disease is defined as a disease that has newly ap-  peared in a population or has existed but is rapidly increasing in incidence [2]. The National Institute of Allergy and Infectious Diseases (NIAID) has  defined three categories of priority [2]. Pathogens in the first category “A”  are those that pose the highest risk to national security and public health be-  cause they can be quickly disseminated or transmitted from person to per-  son, for example, Yersinia pestis (plague), Bacillus anthracis (anthrax),  dengue, and ebola [2]. The second category “B” is composed of pathogens  that are moderately easy to disseminate and may result in moderate mor-  bidity rates and low mortality rates; examples are Hepatitis A, Salmonella,  Yellow fever virus, Chikungunya virus, and Zika virus, among many others.  The third category “C” includes emerging pathogens that could be engi-  neered for mass dissemination in the future because of the availability and  ease of production and dissemination; examples are tuberculosis, influenza  viruses, SARS-CoV, and Rabies virus, among others [2].  The causes and sources of disease are very difficult to identify at the early  stages of an outbreak. Surveillance, monitoring, and preparedness remain  the main counterattack against infections—and consequently require con-  stant improvements at the local, regional, and national levels [3]. In the past  60 years, in particular the last months, the study of infectious diseases has 

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matured into a multidisciplinary field at the intersection of epidemiology,  mathematics, artificial intelligence, ecology, sociology, immunology, and  public health.  Mathematical modeling is an abstraction of a system based on mathe-  matical terms to study the effects of different components and consequently to make predictions [4]. Mathematical models for the spread of diseases  have played a central role in epidemics, providing a cost-effective way of as-  sessing disease transmission, targets for preventing disease, and control [5].  Mathematical modeling was pivotal in suggesting new vaccination strategies  to protect against influenza infection [6]; support public health strategies for  containing SARS-CoV-2 [7–9]; or for the use of antiretroviral treatment for  HIV-infected patients as a prevention measure [10].  During the COVID-19 pandemic, the Kermack-McKendrick model of Sus-  ceptible-Infec-tious-Recov-ered [11] (SIR or compartmental model) was used  to derive an overwhelming amount of research in different countries. How-  ever, Kermack-McKendrick’s models often overlook population hetero-  geneity. These models typically assume that the population of interest is well  mixed: every individual has the same chance to get the disease from an in-  fected case [5]. This simplification makes the SIR model analytically  tractable, but at the same time leads to less accurate evaluations of the dis-  ease spread [12]. Classifying the population into homogeneous subgroups con-  cerning the disease of interest can increase the model representativeness  [13] but also quickly increase the model complexity. Nevertheless, even in  these cases, the homogeneous assumption remains [12].  Modeling during an unprecedented pandemic such as COVID-19  waschallenging and chaotic. Uncertainties about the transmission of the  virus were debated in the scientific community and public news. Massive  sampling for detecting infected individuals was too challenging in devel-  oped countries, while in developing countries, it was not even envisaged.  Confinement strategies in different countries were not popular but neces-  sary, resulting in millions of students taking lectures online while millions  of workers doing home office. Regarding the scientific community, thou-  sands of scientists did try to contribute to the pandemic with their respec-  tive expertise in different fields, while many other “scientists” without any  background in infectious diseases were finding the opportunity to become  “famous.” Governmental institutions were not prepared worldwide. Thus, in 

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many countries, public health policies were misleading and confusing their  citizens, which derived in a lack of credibility just after a few months of pan-  demics. Indeed, there are many lessons that we can learn from COVID-19.  As we enter the fourth year of the COVID-19 pandemic, this book brings  back an effort to recapitulate major mathematical modeling efforts from dif-  ferent angles, discussing what was done, what needed to be done, and what  type of efforts should be ready for the next pandemic. 

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2: Book overview  This book consists of 15 chapters that provide an effort with different mathe-  matical techniques to tackle the COVID-19 pandemic. Leveraging different  kinds of data through novel quantitative approaches will guide public health  policies in decision-making in the following decades. More importantly, this  book ends each chapter with the lessons learned from COVID-19. The au-  thors addressed central problems during the COVID-19 pandemic, which  will remain as critical documentation for future pandemics.  Chapter 2 provides a global overview of the COVID-19 pandemic with the  cumulative SARS-CoV-2 incidence rate (weekly cases per 100,000 inhab-  itants) by continent as well as a list of countries with the lower COVID-19  mortality excess rate per 100,000 inhabitants. Other fundamental aspects of  the COVID-19 pandemics are discussed in this chapter, such as reinfections  and the emergence of variations of SARS-CoV-2.  Chapter 3 explores how the evolution of COVID-19 disease has been  changing since the beginning of the pandemic as well as the mathematical  models used to answer questions such as: when the first case will be re-  ported in a certain region? How fast the disease will spread? How local  transmission dynamics could develop? How effective were the mitigation  measures? What would be the best vaccination strategy? How would vacci-  nation impact the transmission dynamics? The authors emphasize the limi-  tations of mathematical modeling that appeared in COVID-19, and whenever  possible, strategies will be proposed to address these limitations.  Chapter 4 aims to consider regional differences and discover dominant  patterns in the profiles of the COVID-19 incidence rate curves across re-  gions. This introduced geometrical methods to analyze the shapes of local  curves and statistically group them into distinct clusters, according to their  shapes. This information derives the so-called shape averages of curves  within these clusters, which represent a better characterization of a pan-  demic's trajectory. The authors apply this methodology to data for two geo-  graphic areas: a state-level analysis within the United States and a country-  level analysis within Europe.  Chapter 5 brings to discussion the tremendous inequality in vaccine sup-  ply; some regions of the world continue to hoard surpluses of the vaccine  while other regions still have very limited access to vaccines. Isolationism  (and

vaccine

nationalism)

is

a

more

obvious

way

to

achieve 

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self-preservation than solidarity (and vaccine globalism). The central ques-  tion addressed in this chapter is which of these two strategies more effec-  tively achieves national self-protection? Simulations show that tightening  borders is extraordinarily costly and ineffective. In addition, the findings  also show that highly granular vaccine disparities promote the evolution of  vaccine escape.  Chapter 6 introduces a network-guided contact tracing strategy based on  the heterogeneity of contact networks in metropolitan areas such as Mexico  City. Two contact tracing scenarios were simulated: one in which secondary  contacts are randomly identified and the other one in which secondary con-  tacts are identified by a biased sampling caused by the heterogeneous net-  work. The results show that this second strategy can identify a larger num-  ber of infected individuals by using the same resources.  Chapter 7 considers deep learning in forecasting the COVID-19 pandemic  and county-level risk warnings. The authors showed that a physics-hybrid  deep learning framework on county-level predictions can predict the preva-  lence of multiscale COVID-19 in all 412 counties in Germany. It can also be  naturally extended to multinational or transnational analyses. Based on this  framework, the authors also discuss the possible extensions of introducing  vaccination rates and virus variants into their methods. This chapter high-  lights the importance of combing epidemiological models with deep learn-  ing methods for the inherent stability and generalization ability.  Chapter 8 introduces an approach that considers the an epidemiological  model, neural networks, control theory, and complex systems. A controller  for the vaccine application strategy is designed using impulse control. The  behavior of the dynamics of the model with this type of control is very inter-  esting since in all the classes the dynamics start faster. The authors observe  that this control design can stabilize the system; with this, we can achieve a  balance in the transmission of the disease.  Chapter 9 discusses the normalization of the home office as a work  scheme that has caused collateral effects on the well-being of workers de-  rived from isolation and lack of socialization. The authors developed an  agent-based model for COVID-19 to include the application of vaccines, so-  cial phenomena related to human interactions, anxiety in workers, and com-  munication technologies to facilitate reintegration. Numerical results  showed that anxiety levels and the basic reproduction number decreased 

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from the scenario without interventions to vaccination, and the activities  carried out increased, allowing workers to reintegrate and face anxiety.  Chapter 10 presents the implementation of mitigation measures and mod-  eling of in-hospital dynamics depending on the COVID-19 infection status.  Patients are considered from their entry until their discharge or death, thus  predicting the total number of individuals in hospital beds and the intensive  care unit for each healthcare unit. Consequently, this model was to establish  which combination of several quarantine scenarios and other policies was  best at what time. Chapter 11 presents a very debatable aspect of COVID-19, the reopening of  schools, which has not been an easy decision to make, because this implies  an increase in mobility, as well as in contact with the population and there-  fore in the number of cases. In this chapter, the authors explored mathe-  matical compartmental models to analyze possible scenarios after the re-  opening of schools, within a campus college in Mexico. The proposed  model is based on the Paltiel model and the Reed Frost model, considering  the daily screening in the population, as well as parameters and probability  densities based on the official public data on COVID-19 in Mexico.  Chapter 12 considers a mathematical assessment of the role of vacci-  nation against COVID-19 in the United States. Simulations show that vac-  cine-de-rived herd immunity can be achieved in the United States using Pfiz-  er or Moderna vaccine if at least 78% of the populace is fully vaccinated. The  herd immunity threshold decreases if the vaccination program is combined  with other NPIs, such as face mask usage (at increased coverage and effec-  tiveness, in comparison to their baseline values during the third wave of the  pandemic). It is shown that although the use of low to moderately effective  face masks (such as cloth and surgical masks) alone might not be sufficient  to contain the COVID-19 pandemic even at full coverage, the use of masks  or the Pfizer or Moderna vaccine alone may be sufficient to lead to the elimi-  nation of the pandemic if the coverage is moderately high enough.  Chapter 13 proposes an approach to ascertainment and biased testing  rates in the surveillance of emerging infectious diseases. Epidemiological  analysis showed that using passive surveillance data and discussing the fol-  lowing can be determined: (i) What kind of biases exists? (ii) How do biases  affect epidemiological analyses? (iii) How to reduce such biases. This chap-  ter also discusses how the contribution of an infected individual to the viral 

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load in wastewater may depend on the severity of the symptoms.  Chapter 14 focuses on SARS-CoV-2 at the within-host level. The authors  fully characterize the dynamical behavior of the target-cell models under  treatment actions. Based on the concept of virus spreadability, antiviral  effectiveness thresholds are determined to establish whether a given treat-  ment can clear the infection without secondary effects. Also, it is shown  how to simultaneously minimize the total fraction of infected cells while  maintaining the virus load under a given level, through an optimal control  strategy.  Chapter 15 presents two novel modeling ideas to describe and understand  the effects and evolution of the COVID-19 pandemic in Mexico. The first  idea is to model epidemic curves assuming that the times when a certain  number of infected individuals are observed have been censored but follow  a known probability distribution; the censorship point is the most recent  date for which a record is available. The second idea exploits the infor-  mation of patients identified as SARS-CoV-2 positive in Mexico to under-  stand the relationship between comorbidities, symptoms, hospitalizations,  and deaths due to the COVID-19 disease.  Chapter 16 wraps up the book by presenting the crucial topics by late  2022, for the mitigation and control of the SARS-CoV-2 pandemic. This  chapter reviews important factors related to the evolution of the epidemic,  such as nonpharmaceutical interventions, vaccination rates, coverage, the  evolution of SARS-CoV-2 variants, and their impact on the control of the epi-  demic. Ultimately, this chapter presents the notable successes in the appli-  cation of mathematical models to understand different factors, scales, and  behaviors of the epidemic, but likewise there have been failures, many aris-  ing from the political use of model outputs but others having their roots on  the incomplete knowledge of the basic theory of the biology of the virus and  the epidemic. 

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References    [1] Murphy F., Nathanson N. The emergence of new virus dis-  eases: an overview. Semin. Virol. 1994;5:87–102.  [2] NIAID. NIAID Emerging Infectious Diseases/Pathogens.  NIH: National Institute of Allergy and Infectious Diseases;  2022.

Available

at:

https://www.niaid.nih.gov/research/ 

emerg-ing-infec-tious-eases-pathogens (Accessed 7 July 2022).  [3] Grundmann O. The current state of bioterrorist attack  surveillance and preparedness in the US. Risk Manag. Healthc.  Policy. 2014;7:177–187.  [4] Hernandez-Vargas E.A. Modeling and Control of Infectious  Diseases: With MATLAB and R. Elsevier Academic Press; 2019.  [5] Heesterbeek H., et al. Modeling infectious disease dynam-  ics in the complex landscape of global health. Science.  2015;347:aaa4339.  [6] Rose M.A., et al. The epidemiological impact of childhood  influenza vaccination using live-attenuated influenza vaccine  (LAIV) in Germany: predictions of a simulation study. BMC In-  fect. Dis. 2014;14:40.  [7] Mejia-Hernandez G., Hernandez-Vargas E.A. When is SARS-  CoV-2 in your shopping list?. Math. Biosci. 2020;328:1–7.  [8] Ravichandran S., et al. Antibody signature induced by SARS-  CoV-2 spike protein immunogens in rabbits. Sci. Transl. Med.  2020;12:1–9.  [9] Weitz J.S., et al. Intervention serology and interaction sub-  stitution: exploring the role of ‘immune shielding’ in reducing  COVID-19 epidemic spread. Nat. Med. 2020;26:849–854.  [10] Tanser F., Baernighausen T., Graspa E., Zaidi J., Newell  M.-L. High coverage of ART associated with. Science.  2013;339:966–972.  [11] Kermack W.O., McKendrick A.G. A contribution to the  mathematical theory of epidemics. Proc. R. Soc. A Math. Phys.  Eng. Sci. 1927;115(772):700–721.  [12] Bansal S., Grenfell B.T., Meyers L.A. When individual  behaviour matters: homogeneous and network models in  epidemiology. J. R. Soc. Interface. 2007;4:879–891. 

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[13] Nishiura H., Oshitani H. Household transmission of in-  fluenza (H1N1-2009) in Japan: age-specificity and reduction of  household transmission risk by zanamivir treatment. J. Int. Med. Res. 2011;39:619–628. 

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2: Global epidemiology and impact of the SARS-CoV-2 pan-  demic    Sofia Bernal-Silvaa; Angélica Torres-Díazb,c; Andreu Comas-Garcíaa    a  School of Medicine, Autonomous University of San Luis Potosi, San Luis Po-  tosi, Mexico  b Division of Higher Studies for Equity, School of Medicine, Autonomous  University of San Luis Potosi, San Luis Potosi, Mexico  c Organization to Restore the Environment and Social Harmony (ORMA,  A.C.), San Luis Potosi, Mexico 

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Abstract  In this chapter, we evaluated the development and impact of the SARS-  CoV-2 pandemic. The evaluation was performed by continent and  country. The structure of the weekly mortality curves on all continents  is very different from the weekly incidence rate. With the data from the  Johns Hopkins University Coronavirus Resource Center and the Insti-  tute of Health Metrics and Evaluation published in The Lancet, we have calculated the COVID-19 directed and associated mortality rate for 181  countries. Despite the structural similarity between the second, third,  and fourth waves being similar, the weekly mortality rate intensity of the  four waves was very asynchronous among all the countries. One of the  primary consequences of this pandemic is orphanhood. Recently, the  Imperial College of London has published a webpage that calculates  the number of orphans generated by the COVID-19 pandemic. Despite  the fact that before the SARS-CoV-2 outbreak there was plenty of scien-  tific evidence about how to handle and/or mitigate a pandemic, almost  every government adopted different policies with different anti-  epidemic effects and impacts.   

Keywords  Epidemiology; SARS-CoV-2; Incidence; Morality; Lethality; Orphanhood;  Public policies 

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1: Introduction  The enveloped RNA viruses with the largest known genome coronaviruses  have a high capacity for genomic recombination infecting mammals, birds,  and humans and causing respiratory, enteric, hepatic, and neurological dis-  eases. One of the most frequent agents that cause common colds are zoonotic viruses whose wide distribution means that the interaction be-  tween humans and wild animals increases the probability of the emergence  of new coronavirus species or strains. In humans, four seasonal coron-  aviruses frequently cause common colds but rarely cause pneumonia  (hCoV-229E, -OC43, -NL63, and -HKU1), while, prior to 2019, only two types  of coronaviruses had caused major outbreaks of severe respiratory infec-  tion: SARS-CoV occurred in 2002–03 in China and MERS-Co occurred in  2012 in the Middle East [1,2].  In late December 2019, a new cluster of patients presenting a severe acute  respiratory syndrome (SARS) of unknown cause was detected in the City of  Wuhan, in the province of Hubei, China. At the beginning of the outbreak, the Chinese Government expressed serious doubts about the possibility of  person-to-person transmission. However, on January 24, 2020, the first re-  port of a family cluster presenting verified person-to-person transmission  was published. By March 12, almost 3 months after the outbreak began,  when the World Health Organization (WHO) declared the SARS-CoV-2 out-  break a pandemic, SARS-CoV-2 was present in 114 countries, with more than  132,496 infections and 4917 deaths recorded. On May 15, the first chain of  multiple human-to-human transmission outside Asia was documented. By  June 1, 2022, 531,567,231 cases had been reported in 191 countries or terri-  tories, 6,297,253 deaths had been registered, and 11,653,556,780 doses of  COVID-19 vaccine had been administered [1–5]. While SARS-CoV-2 infection  mainly occurs via airborne transmission, the infection can also occur via  close interpersonal contact.  The recent finding that the virus can also be detected in feces and waste-  water samples 6,7 has the potential to diversify and improve both the epi-  demiological surveillance measures employed to track the pandemic and  the strategies for controlling/mitigating it. The sampling of wastewater  could be applied on a mass scale as a novel public health strategy that  could enable the genomic surveillance of the virus without the need for  human testing. 

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2: Global epidemiology  The SARS-CoV-2 pandemic has been characterized by repeated waves, ap-  proximately two per year, of cases driven by the emergence of new variants  of concern (VOC). Until now, each new VOC presented enhanced fitness  that affects its growth rate and basic reproduction number (R0) and results  in a higher immune-evasion rate/higher rate of immune escape and a short-  er generation time [8].  In general, while several SARS-CoV-2 waves have been observed in all  continents and countries, the ongoing transmission of the virus has at no  point been arrested. Therefore, rather than waves, this behavior could be  considered, in epidemiological terms, a serial recrudescence phenomenon  comprising transmission, the presentation of cases, and death. None-  theless, in this chapter, we will continue to use the term waves. Fig. 1 shows the weekly incidence rate per continent, while Fig. 2 shows the cumulative  incidence rate for each continent and Table A.1 compares the incidence,  mortality, and lethality of the virus among the continents.   

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Fig. 1 Weekly SARS-CoV-2 incidence rate (weekly cases per 100,000  inhabitants) by continent. Data from Johns Hopkins University of  Medicine, Coronavirus Resource Centre.  https://coronavirus.jhu.edu/map.html.   

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Fig. 2 Cumulative SARS-CoV-2 incidence rate (weekly cases per  100,000 inhabitants) by continent. Data from Johns Hopkins  University of Medicine, Coronavirus Resource Centre.  https://coronavirus.jhu.edu/map.html.    The weekly incidence rate in Africa shows four waves, each of which  presents a trend of continuous increase (Fig. 1), which could be a conse-  quence of not only VOC (beta, delta, and omicron) circulation but also the  increased testing rate. It should be noted that the peak of the fourth wave  was 2.5 times higher than the peak of the third. While the cumulative inci-  dence rate recorded in Africa was 2.8, 3.0, 9.5, and 33.6 times lower than that  observed in America, Asia, Europe, and the rest of the world, respectively, it  was 1.6 times higher than that observed in Oceania (Fig. 2). The intensity of  the weekly incidence rate for the first three waves among all countries was  asynchronous, while the fourth wave occurred almost at the same time  throughout the African continent (Fig. 3A).   

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Fig. 3 Weekly SARS-CoV-2 incidence rate intensity per country. (A)  Africa, (B) America, (C) Asia, (D) Europe, and (E) Oceania. The  incidence rate was adjusted from 0 to 1, and was represented in color.  Data from Johns Hopkins University of Medicine, Coronavirus  Resource Centre. https://coronavirus.jhu.edu/map.html.    America presents five waves, the smallest of which was the first, to which  the next three were very similar, while the fifth was the largest, with a peak  3.7 times higher than the peak of the fourth wave (Fig. 1). The cumulative  incidence rate observed for America was 1.1, 3.5, and 12.3 times lower than  that observed for Asia, Europe, and the rest of the world, respectively, but  2.7 and 4.3 times higher than that observed for Africa and Oceania, respec-  tively (Fig. 2). In the case of America, the first wave was the smallest, the  next three were very similar, and the fourth was the largest. The heat map for  the intensity of the incidence rates (Fig. 3B) shows an asynchrony of inci-  dence patterns between countries, although continuous transmission is  clearly observed for all. In contrast with the first four waves, the fifth is very  short and synchronic among all countries.  Each wave registered in Asia has been larger than its predecessor, al-  though the second wave is hard to identify as it resembles a continuous, as-  cending straight line. The peak of the fifth wave, larger than the previous  four, was 3.9 times higher than the peak of the fourth (Fig. 1). The cumu-  lative incidence of the virus in Asia was 3.2 and 11.2 times lower than that  recorded for Europe and the rest of the world, respectively, and, in contrast,  4.8, 3.0, and 1.1 times higher than that recorded for Oceania, Africa, and America, respectively (Fig. 2). The heat map for Asia presented in Fig. 3C  has a very similar structure to that generated for America, although the  fourth wave of the latter was less synchronic.  37

Although the pandemic originated in Asia, its epicenter is now located in  Europe, which has undergone three waves (Fig. 1). The cumulative inci-  dences of COVID-19 in Europe were 15.0, 9.5, 3.5, and 3.1 times higher than  those observed for Oceania, Africa, America, and Asia, respectively (Fig. 2),  which could be the consequence of the high testing rate, a high population  density, and the great interconnection between European populations. Eu-  rope presented a cumulative incidence that was 3.5 times lower than that  calculated for the rest of the world and was the continent presenting the  lowest rate. The heat map for the weekly incidence for Europe shows that  the first wave was detected in just a few countries (Fig. 3D), while the sec-  ond wave is synchronic for almost all the countries and lasted for a long  time. The third wave is very interesting, because, despite being synchronic  among all countries, the outbreak was long-lasting in some and short in  others.  All the continents presented a higher cumulative incidence rate than  Oceania, which presented a rate that was 53.0, 15, 4.8, 4.3, and 1.6 times  lower than the cumulative incidences observed for the rest of the world, Eu-  rope, Asia, America, and Africa, respectively (Fig. 2). Prior to 2022, cases in  Oceania were scarce and sporadic, aside from in countries such as Papua  New Guinea, which presented continuous transmission since mid-2020  (Fig. 3E). Oceania is the only continent in which the omicron wave was not  synchronic, which can be explained by the isolation of the countries that  comprise it.  The structure of the weekly mortality curves for all continents is very dif-  ferent from that of the weekly incidence rate (Fig. 4), with Africa undergoing  four mortality waves, the second and the fourth of which were very similar  (presenting a sharp peak), while the peak of the third wave was broad. Offi-  cially, the cumulative mortality rate recorded for Africa was 13.1, 7.0, 3.3, and  1.5 times lower than that recorded for the rest of the world, Europe, America,  and Asia, respectively, and 4.1 times higher than that presented by Oceania  (Fig. 5).   

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Fig. 4 Weekly SARS-Cov-2 mortality by continent. Data from Johns  Hopkins University of Medicine, Coronavirus Resource Centre.  https://coronavirus.jhu.edu/map.html.   

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Fig. 5 Cumulative SARS-CoV-2 mortality rate (weekly cases per 100,000  inhabitants) by continent. Data from Johns Hopkins University of  Medicine, Coronavirus Resource Centre.  https://coronavirus.jhu.edu/map.html.    America presented an ascending weekly mortality rate that peaked during  the fourth wave and then began to descend. The cumulative mortality inci-  dence on the continent was 13.6, 3.3, and 2.1 times higher than that recorded  for Oceania, Africa, and Asia, respectively, and 4.0 and 2.1 times lower than  that recorded for the rest of the world and Europe, respectively (Figs. 4 and  5).  The cumulative mortality rate observed for Asia was 6.4 and 1.5 times  higher than that observed for Oceania and Africa, respectively and 8.5, 4.5,  and 2.1 times lower than that recorded for the rest of the world, Europe, and  America, respectively (Figs. 4 and 5).  Europe was the continent presenting the highest cumulative mortality rate, which was 15.0, 9.5, 3.5, and 3.1 times higher than that recorded for  Oceania, Africa, America, and Asia, respectively, and 4.5 times lower than  the global mortality rate. In contrast, the continent presenting the lowest  cumulative mortality rate was Oceania, which was 53, 15, 4.8, 4.3, and 1.6  times lower than that recorded for the rest of the world, Europe, Asia, Amer-  ica, and Africa, respectively (Figs. 4 and 5).  In Africa, the second peak in the weekly mortality rate was the highest,  while this was the fourth peak in America. The highest mortality rate ob-  served in both Asia and Europe was at the end of the third wave and during 

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the last omicron wave in Oceania (Figs. 4 and 5).  The United Nations divides each continent into different subregions. The  highest cumulative estimated excess deaths per capita observed in Africa  was in the Southern and North African subregions, wherein the magnitude  and rate of the excess deaths per capita were at least 4 times higher than  those calculated for East, Central, and West Africa [8].  For the Americas region, the highest cumulative excess deaths per capita  estimated was observed in Central America, followed by South America,  Northern America, and the Caribbean subregions. The excess mortality  recorded for Central America was twice that of the Caribbean subregion [8].  The two subregions of Asia with the highest estimated cumulative excess  deaths per capita were Southern and Western Asia, which presented a mor-  tality twice that of South-Eastern and Central Asia and almost 8 times that  found for Eastern Asia [8].  Of the continent's subregion, Eastern Europe presented the highest  cumulative estimated excess deaths, which is 2.3–2.5 times that of Southern  Europe and 3.5 times that of Northern and Western Europe. In fact, the East-  ern Europe subregion recorded the highest excess deaths in the world. The  cumulative estimated excess deaths per capita for Oceania is the lowest in  the world, namely approximately 20–30 deaths per 100,000 inhabitants [8].  Using data obtained from the Johns Hopkins University Coronavirus Re-  source Center and data obtained from the Institute of Health Metrics and  Evaluation (IHME), which was published in The Lancet, we have calculated  the direct and associated COVID-19 mortality rate for 181 countries (Table  A.1, Appendix) [9]. The top 10 countries with the highest COVID-19 excess  deaths per 100,000 inhabitants are Bangladesh, Bolivia, Bulgaria, Myanmar,  Solomon Island, Eswatini, Norte Macedonia, Belarus, Peru, and Poland. The  10 countries with the lowest COVID-19 excess deaths per 100,000 inhab-  itants are Antigua and Barbuda, Burundi, Republic of Korea, Iceland, Nor-  way, Ireland, Australia, Singapore, New Zealand, and China.  Despite the structural similarity among the second, third, and fourth  waves, the weekly intensity of the mortality rate for the four waves was high-  ly asynchronic among all countries (Fig. 6A). While some countries did not  report deaths at different points of the pandemic (Chad, Central Africa  Republic, and Togo, etc.) others reported a continuous wave of deaths  (Egypt, Tunisia, South Africa, Algeria, Angola, and Libya). Both of the 

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foregoing phenomena may be explained by the lack of testing undertaken in  said countries.   

Fig. 6 Weekly SARS-CoV-2 mortality rate intensity per country. Data  from Johns Hopkins University of Medicine, Coronavirus Resource  Centre. https://coronavirus.jhu.edu/map.html.    Although we have detected five waves of mortality in America, the inten-  sity of the mortality rate shows a high asynchrony among countries (Fig.  7B). While some countries, such as the islands of the Caribbean subregion,  did not report deaths, Peru, Venezuela, Mexico, Argentina, Colombia, Costa  Rica, Guatemala, and Guyana presented a continuous death rate with points  of high intensity occurring during the fourth peak.   

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Fig. 7 Weekly SARS-CoV-2 lethality rate by continent. Data from Johns  Hopkins University of Medicine, Coronavirus Resource Centre.  https://coronavirus.jhu.edu/map.html.    The pattern for the intensity of the weekly mortality rate for Asia was very  similar to that observed for America (Fig. 6C). With some countries not re-  porting deaths (Thailand, Tajikistan, Twain, Bhutan, and Brunei), others pre-  sented a continuous intensity of mortality intensity (Yemen, United Araba  Emirates, Syria, Qatar, Philippines, Lebanon, Kuwait, Jordan, Japan, Iraq,  Iran, Indonesia, India, Fiji, Bangladesh, Bahrain, Armenia, and Afghanistan).  The highest intensity for Asia was observed between the 25th and 42nd  weeks of 2021.  Europe presented the most complete mortality reporting of all the conti-  nents, with only small countries such as San Marino, Monaco, Malta,  43

Liechtenstein, Iceland, the Holy See, and Andorra reporting deaths intermit-  tently (Fig. 7D) and the remaining countries conducting continuous report-  ing. In contrast with other continents, two periods of high mortality intensity were observed, with the first occurring between Week 37 of 2020 and Week  21 of 2021 and the second during Week 36 of 2021 and Week 15 of 2022. The  most synchronic mortality intensity was observed during the second wave,  while countries such as Belarus, Turkey, Serbia, Russia, and Georgia pre-  sented a continuously high mortality intensity.  The weekly mortality intensity observed for Oceania was very similar to  the continent's pattern of intensity incidence, with the highest and longest  mortality intensity observed in Papua New Guinea, followed by Australia, the  Solomon Islands, and Kiribati (Fig. 7E).  In most countries and continents, the lethality rate (deaths per case) de-  creased after the first wave (Fig. 7). The cumulative lethality rate observed  for Africa was 1.16%, while the mean lethality rate was 1.6% (with a min-  imum of 0.0% and a maximum of 6.0%). After Week 51 of 2021, a signif-  icant decrease in the African mortality rate was observed.  While America presented the highest official median lethality rate of 2.2%  (0.2%–7.6%), its cumulative lethality rate was 1.6%. The first wave, which  presented the lowest number of cases, presented a median lethality rate of  4.0%, while that of the other four waves was 2.1%, with, notably, the fifth  wave presenting a rate of 0.8%.  Asia presented two peaks of lethality, the first between weeks 1 and 20 of  2020 and the second between weeks 27 and 30 of 2021. After Week 51 of  2021, a drastic decrease in the lethality rate (with a median of 0.2%) was ob-  served. The cumulative lethality rate was 1.02%, with a median of 1.1% (0.1%–5.4%).  Europe presented the highest lethality rate recorded (31.9%), during Week  9 of 2020, while the median lethality rate was 5.8% during the first wave.  The cumulative lethality rate was 0.78%, with a median of 1.7% (0.2%–  31.9%). After the first wave, the lethality rate remains low and stable (median  of 1.3%). Oceania presented the lowest cumulative and median lethality rate of the  world's continents, 0.14% and 0.6% (0.0%–22.0%), respectively. Due to  the lower and intermittent number of cases and deaths, the continent pre-  sented an erratic lethality curve. 

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The lethality rate for the rest of the world presents a structure very similar  to that of Europe, which could result from the high amount of data regis-  tered for this data cohort, for which the cumulative lethality rate was 0.5%,  with a median of 0.9% (0.1%–14.2%).  As observed in Table 1, Europe shows a higher cumulative incidence rate,  although it is the continent with the second lowest lethality rate. It should be  noted that, while the cumulative incidence reported for America is 2 times  higher than that reported for Africa, the lethality presented by both conti-  nents is almost the same. Oceania is the continent that experienced the low-  est impact of the pandemic, which could be explained by the dispersed and  isolated nature of the countries, many of them island nations, that it com-  prises.    Table 1    Continent  Cumulative incidence  rate  Africa 

Europe 

Official 

adjusted mortality cumulative  ratea  lethality ratea 

472.75 (243.52–1639.32)  196.18  (132.17–322.24) 

America  1021.34  Asia 

Cumulative 

292.30 

1.61%  (1.02%–2.37%)  1.60% 

(4805.61–15,808.27) 

(249.35–457-68)  (1.05%–2.41$%) 

6044.60 

162.82 

1.02% 

(870.70–14,370.96) 

(94.68–312.81) 

(0.36%–1.79%) 

2978.15  (16,102.74–42,652.27) 

450.56  0.78%  (258.91–656.27)  (0.49%–1.35%) 

Oceania  2508.51  (473.62–18,423.27) 

71.86  (13.29–185.48) 

0.14%  (0.07%–0.42%) 

Median rates per 100,000 inhabitants (p25–p75); until 4/25/2022.  a Adjusted according IHME methodology 9.    Data from Johns Hopkins University of Medicine, Coronavirus Resource  Centre. https://coronavirus.jhu.edu/map.html (Last accessed 25 April  2022).   A devastating consequence of the COVID-19 pandemic is orphanhood.  Recently, Imperial College London published a webpage calculating the 

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number of orphans produced by the COVID-19 pandemic (Tables 2 and 3).  In general terms and in absolute numbers, the continent with the highest  number of orphans is Asia, due to the estimated 3,503,700 orphans in India  (Table A.2) 10.    Table 2    Continent 

Estimated number  Estimated number  Estimated number  of orphans 

of deceased  of deceased  primary caregivers  primary or  secondary  caregivers 

Africa 

1,271,980  (130–150,900) 

1,397,250  (140–172,300) 

1,834,290  (179–289,000) 

America 

863,094 

941,631 

1,273,821 

(28–215,800) 

(31–229,800) 

(50–318,900) 

2,520,900 

2,566,150

3,965,245 

(28–2,168,900) 

(31–2.223,200) 

(55–3,488,100) 

Asia 

Europe 

149,745 (57–57,100)  172,079  (62–76,100) 

312,109  (99–144,700) 

Oceania 

2666 (66–1000) 

4600 (120–1800) 

2953 (73–1100) 

Estimated number of orphans (min-max).    Data from Imperial College of London, COVID-19 Orphanhood. https://  imperialcolleglondon.github.io/orphanhood_calculator (Last accessed 24  April 2022).    Table 3    Continent 

Estimated rate of  Estimated rate of  orphanhood  loss of a primary  caregiver 

Estimated rate of  loss of a primary or  secondary caregiver 

Africa 

90.73 (0.01–10.76)  99.66 (0.01–12.29)  130.84  (0.01–20.61) 

America 

61.56  (0.002–15.39) 

67.16  (0.002–16.39) 

90.86  (0.004–22.75) 

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Asia 

14.8 (2.9–34.2) 

16.9 (3.2–33.7) 

20.7 (5.8–48.7) 

Europe 

10.68  (0.004–4.07) 

12.27 (0.004–5.43)  22.26  (0.007–10.32) 

Oceania 

0.19 (0.005–0.07)  0.21 (0.005–0.08)  0.22 (0.009–0.13) 

Estimated cumulative rate of orphanhood (min-max), orphans/100,000  inhabitants.    Data from Imperial College of London, COVID-19 Orphanhood. https://  imperialcolleglondon.github.io/orphanhood_calculator (Last accessed 24  April 2022).    The number of orphans in Asia is 862.0, 12.7, 3.1, and 2.2 times higher than the number in Oceania, Europe, America, and Africa, respectively  (Table 2). After adjusting the number of orphans per 100,000 inhabitants,  the results for each continent changes drastically (Table 2), with Africa  becoming the continent with the highest estimated number of deceased pri-  mary or secondary caregivers (130.84 orphans/100,000 inhabitants), which  is 594.7, 6.3, 5.8, and 1.4 times higher than the rate from Oceania, Asia, Eu-  rope, and America, respectively.  Of the 169 countries, those with the 10 highest estimated number of de-  ceased primary or secondary caregivers are India, Mexico, Brazil, Rwanda,  the United States, South Africa, Ethiopia, Russia, Peru, and Iran. The coun-  tries with the 10 lowest estimated number of deceased primary or secondary  caregivers are Norway, Estonia, New Zealand, Singapore, Finland, Grenada,  Maldives, Barbados, Brunei, and Antigua and Barbados (Table A.2, Appen-  dix).  After adjusting for the number of orphans estimated per 100,000 inhab-  itants, the position of the countries changes dramatically from the foregoing  list (Table A.3, Appendix), with the countries with the 10 highest estimated  rates of orphanhood becoming Namibia, Rwanda, Eswatini, Botswana, Sey-  chelles, Zimbabwe, Sao Tome and Principe, Lesotho, Zambia, and Mauri-  tania. The countries with the 10 lowest estimated rates of orphanhood be-  come Albania, Japan, United Arab Emirates, Norway, New Zealand,  Nicaragua, Republic or Korea, Singapore, Finland, Tajikistan, Uzbekistan,  and China. 

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3: Epidemiological parameters of SARS-COV-2  When a new pathogen emerges, the first epidemiological goal is to collect  and analyze data in order to formulate effective public health policy 11. Via  mathematical models, this data can be used to study the transmission  dynamics of an infectious agent, which, in turn, can be used to formulate  public health policy and strategies aimed at stopping the transmission. To study an outbreak, we need to know or estimate the full distribution and  variability of the incubation and infectious period, the generation time, and  the basic reproductive number.  In epidemiology, two parameters are essential to modeling an epidemic  and designing public health strategies to mitigate an outbreak (Fig. 3). The  first is the basic reproduction number (R ) and the second is the generation  0 time. R0 is a parameter that indicates, in a 100% susceptible population, the  number of secondary cases that one infected subject can generate. For  SARS-CoV-2, R has been estimated to range from two to five (1.8–3.6). R   0 0 of SARS-CoV, 1918 influenza pandemic, and respiratory syncytial virus were  from two to three, while this was 1.5 and 0.9 for the 2009 influenza pan-  demic and MERS-CoV, respectively. Therefore, SARS-CoV-2 is more trans-  missible than influenza, respiratory syncytial virus, and SARS-CoV0; how-  ever, its transmissibility is similar to that of HIV-1 [11–13].  The term generation time refers to the number of days required for an in-  fected person to be able to infect another person. For the SARS-CoV-2 virus  originally emerging in Wuhan, the generation time was 7.4 days and was  very similar to the SARS-CoV virus from 2002 to 2003 (7–8 days) [11–13].  However, a generation time of 5.4 and 4.8, respectively, was calculated for  delta and omicron [14].  R0 and generation time are determined not only by the viral capacity to in-  fect an organism and then replicate within it but also by factors such as  weather, crowding, traveling, school/commercial activities, social distance,  human interaction, and socioeconomic level.  Table 1 presents the comparison undertaken by Petersen et al. among  characteristics of SARS-CoV-2, SARS-CoV, the 1918 pandemic influenza, and  the 2009 pandemic influenza [15]. The information in the table reveals that  SARS-CoV-2 is the most transmissible and difficult to contain the foregoing  viruses, giving it the potential to collapse a nation's health system. 

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4: Mitigation strategies  In 1919, G.A. Soper published, in the journal Science, the lessons learned  from the Spanish Flu pandemic [16], concluding that there were three fac-  tors that affected the strategies used for controlling or mitigating an outbreak: (i) people do not appreciate risk, (ii) the application of rigid isolation  to protect others goes against human nature, and (iii) people often act  unconsciously to endanger themselves and/or others. These three factors  have affected the effectiveness and success of the anti-SARS-CoV-  pandemicdemic strategies applied by each national government.  Despite the fact that, prior to the SARS-CoV-2 outbreak, plenty of scien-  tific evidence was available on how to handle and/or mitigate a pandemic,  almost every national government adopted different policies with different  antiepidemic effects and impacts [17]. Governmental response is the most  important component of managing a disaster such as a pandemic [18]. An  adequate governmental response during a disaster must include the fol-  lowing seven aspects:    1.An optimal allocation of resources.  2.A project schedule.  3.Interaction between the government and industry.  4.An evacuation or quarantine process.  5.A decision-making system.  6.Network analysis.  7.The supply chain.    Given the foregoing, the high variability in incidence rate, mortality rate,  lethality rate, hospitalization rate, and health systems, etc., observed among  countries, depends, in the main, on the abovementioned aspects and the  approach employed by governments in response to the pandemic.  Decision makers and governments need to take into account the fact that  mask use and the application of other NPIs are determined by various factors, such as age group, gender, ethnicity, and the level of trust in the au-  thorities. Females, teens, and seniors are the main groups that correctly fol-  low the recommendations and guidelines for the implementation of NPIs  [19–21].  The first two countries to establish severe lockdown strategies were China  and Italy. With the initial outbreak occurring in the northern regions of Italy, 

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the Italian Government implemented a total lockdown from March to May  2020. However, when the lockdown was lifted, the ensuing increase in  tourism flows, the reopening of schools, and social events caused a second  peak in November 2020 [22], a phenomenon observed in almost all countries.  The initial lockdown strategy applied in Italy failed because, prior to its  implementation, Italian media networks announced that a lockdown would  be imposed, leading a large proportion of the population to abandon the  northern region of the country, spreading the outbreak around the country.  Another social factor in 2020 that contributed to spreading the virus around  Italy, thus triggering outbreaks around Europe, was a series of football  matches that formed part of the Champions and Europa leagues, while a  third factor was the massive social mobilizations organized by the feminist  movement around International Women's Day on March 8.  While the isolation of cases and their contacts is necessary to break the  chain of transmission, this also can negatively affect human health. Social  distancing restrictions, quarantine, and isolation impact health behaviors,  healthcare access, and physical and mental health. People with preexisting  health conditions are more vulnerable to the negative consequences of so-  cial distancing and isolation [23]. Therefore, decision makers and govern-  ments need to find a balance between a strict public health strategy and viral  spread.  Van Bavel et al. summarized various factors related to the degree of com-  pliance, in a population or society, with these massive global public health  strategies [24]: the threat (risk or emotion) perception of the disease; social  prejudice toward and discrimination against either the disease or the pa-  tient; social norms; social inequality; culture; political polarization; con-  spiracy theories; fake news and misinformation; moral decision-making;  political leadership; the degree of trust in politicians and/or scientists; so-  cial isolation; intimate relationships; and, mental health. It is important to  note that, during the first two waves, NPIs were the only interventions avail-  able to reduce both the burden on the healthcare system and deaths caused  by COVID-19 [22].  A mathematical modeling strategy developed by Gurevich et al. [25] pro-  poses that the greater the social and NPI measures applied, the greater the  reduction in disease severity. The most critical parameters for reducing the 

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impact of a pandemic are the timing of the application of the NPI, a higher  testing rate, and strain virulence. Their strategy also showed that a lower  testing rate can cause an increase in false negative results due to viral evolu-  tion.  Mathematical models of COVID-19 based on epidemiological and viro-  logical data can provide useful insights into the impact of the vaccination  campaign, variant spread, and the cost–benefit of NPI strategies. Many  mathematical models that try to understand the dynamics of COVID-19 have  been developed. In public health, the use of mathematical models is crucial.  As SARS-CoV-2 becomes endemic, the use of modeling will be helpful in  identifying the best long-term strategies to contain its spread [22]. 

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5: Reinfections  Reinfections were an unusual event at the beginning of the pandemic, with  the first case of COVID-19 reinfection described in Hong Kong in August  2022, wherein a 33-year-old male presented a second asymptomatic infec-  tion caused by a different SARS-CoV-2 strain [26].  A cohort study conducted in New York followed 1343 suspected or con-  firmed cases of SARS-CoV-2, of whom 47% had a confirmed infection. At  the end of the study, 57% were antibody positive, 5% were weakly positive,  and 39% were negative, while 82% of confirmed cases and 35% of suspected  cases were antibody positive. Approximately 16% of confirmed cases had  lost antibody response [27].  A second study showing that the humoral immune response against  SARS-CoV-2 decays over time was conducted in China, finding that 77% of  hospitalized patients presented decreasing levels of Immunoglobulin (IgG)  4–10 weeks postinfection. It should be noted that IgG titers decreased by  50% 4–10 weeks postinfection [28]. Unfortunately, most studies published  do not evaluate the titers of neutralizing antibodies.  During the first year of the pandemic, it has been reported that between  6% and 17% of fully recovered patients can be reinfected 29,30, with rein-  fection defined via the following criteria 31–33:    1.Previous acute COVID-19 disease with an initial RT-qPCR positive  test.  2.Clinical recovery was confirmed via an RT-qPCR negative test.  3.A second RT-qPCR test (with or without symptoms) at least 28 -  28 - after the negative test.  4.A second RT-qPCR test with a Ct value 0, b < 0, and 0 1 and  (d)no endemic equilibrium otherwise.    Case (a) of Theorem 3 shows that the basic model (1) has a unique en-  demic equilibrium whenever

. Furthermore, Case (c) of Theorem 3 

suggests the possibility of backward bifurcation, a dynamic phenomenon characterized by the coexistence of multiple stable equilibria (a DFE and a  stable endemic equilibrium) when the associated reproduction number of  the model (

) is less than unity [35]. The epidemiological implication of 

the phenomenon of backward bifurcation is that the requirement

, 

while necessary, is not sufficient for disease elimination. In this case, dis-  ease elimination will depend upon the initial sizes of the subpopulations of  the model [35]. To compute the backward bifurcation point of the basic  model (1), when critical value of

, we set the discriminant (denoted by

and solve for the 

) given by 

    (11)    where A1 is as defined in Eq. (2), A7 is as defined in Eqs. (2), (8), and b1 and  b2 are as defined in Eq. (10). It follows from Eq. (11) that the basic model (1)  may undergo a backward bifurcation at

for

. 

Fig. 1 depicts a generic backward bifurcation diagram, showing the region  of coexistence of stable equilibria when the associated reproduction number  of the model is less than unity. It should be stated that the existence of the  phenomenon of backward bifurcation can be rigorously established using 

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center manifold theory [25, 35, 36]. The presence of backward bifurcation in  the transmission dynamics of a disease makes its control or elimination  difficult, since the reproduction number of the model has to be reduced to  be significantly below unity (i.e., to be outside the backward bifurcation re-  gion) to have the possibility of the unique disease-free equilibrium being  globally asymptotically stable [35].   

Fig. 1 Generic backward bifurcation diagram for the basic model ( 1).  BB, EE, and DFE denote backward bifurcation region, endemic  equilibrium, and disease-free equilibrium, respectively.   

2.3: Model fitting and parameter estimation of basic model  The basic model (1) contains 17 parameters. The values of most of these  parameters are known from published literature. For instance, the values of  13 of them (namely Λ, μ, ξ , ɛ , σ , σ , r, ϕ , γ , γ , γ , δ , and  v v e p s a s h s δ ) are known [6, 12], and are tabulated in Table 3. The values of the re-  h maining four parameters of the basic model (1), namely the transmission  rate parameters β , β , β , and β will be estimated by fitting the basic  p a s h model (1) with observed daily COVID-19 mortality for the United States for 

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the period from December 10, 2020 to July 23, 2021, using standard model  fitting techniques, such as the least squares regression method [6, 37, 38]. In  particular, applying the least squares regression method allows for the de-  termination of the best values of the unknown parameters that minimize the  root mean square difference between the predicted daily mortality generated  from the basic model (1) and the observed daily mortality data. Fig. 2 de-  picts the fitting result obtained. The estimated values of the four unknown  parameters obtained from the fitting, together with their associated 95%  confidence intervals, are tabulated in Table 4.   

Fig. 2 Data fitting of the basic model ( 1) using daily COVID-19  mortality data for the United States from December 10, 2020 to July 23,  2021 [1].    It should be mentioned, although numerous notable modeling studies  have demonstrated the fact, prior to the commencement of vaccination pro-  grams, the main drivers of the COVID-19 pandemic were the presymp-  tomatic and asymptomatic infectious individuals (see, for instance, Ref. [11]  and some of the references therein), our fitting in Table 4 suggests that,  during the vaccination era, the main drivers of the COVID-19 pandemic in  the United States are the symptomatic individuals (since the value of βs  378

exceeds those for β , β , and β ). In other words, there has been a  p a h switch in terms of the main generators of new COVID-19 infections based  on the period before and during vaccination. A possible justification for this  switch can be attributed to (a) the high vaccination coverage in the United  States (as of September 5, 2021, at least 75% of the US population has re-  ceived one dose and over half the populace is fully vaccinated) and (b)  therapeutic benefits of the three FDA-EUA vaccines used in the United  States. Specifically, in addition to reducing the likelihood of acquiring infec-  tion by vaccinated susceptible individuals, each of these vaccines reduces  the transmissibility of infected vaccinated individuals (since those with  breakthrough infections exhibit mild or no symptoms at all), in addition to  accelerating recovery, reducing hospitalization and death in breakthrough  infections. As of September 5, 2021, the COVID-19 pandemic in the United  States has essentially been reduced to the epidemic of unvaccinated individuals (being the group that suffers the overwhelming burden, measured in  terms of cases, hospitalization and death, of the pandemic).   

2.4: Numerical simulation of basic model  The basic model (1) will now be simulated, using the fixed and estimated  parameters tabulated in Tables 3 and 4, to assess the population-level im-  pact of the vaccination program implemented against the spread of the  COVID-19 pandemic in the United States. We consider three cases: (a)  when vaccination is the sole intervention implemented, (b) when vacci-  nation is combined with NPIs, such as social distancing, masks usage,  community lockdowns, etc., and (c) when vaccination is combined with  face mask intervention (with various mask types, such as cloth masks, surgical masks, and respirators) [6, 7, 10, 11, 19, 39].    2.4.1: Assessment of impact of vaccination as a sole intervention  Fig. 3 A depicts a contour plot of the reproduction number (

) of the 

basic model, as a function of vaccine coverage (fv) at the disease-free equi-  librium and vaccine efficacy (ɛv), for the case where no other interventions  (other than vaccination) are implemented. This figure shows that if only the  Pfizer or Moderna vaccine (with estimated efficacy of ɛv = 0.95) is used in the United States, at least 78% of the US population needs to be fully vacci-  nated (with either of the two vaccines) to bring the reproduction number to  a value less than unity (so that by Theorems 1 and 2), herd immunity can be 

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achieved, and, subsequently, the pandemic can be eliminated. Hence, using  either the Pfizer or Moderna vaccine alone can lead to the elimination of the  pandemic in the United States if the coverage in their usage is high enough  (i.e., if at least 78% of the US population receive the two doses of either vac-  cine). If, on the other hand, only the Johnson & Johnson vaccine (with effi-  cacy ɛv = 0.7) is used, the contours in Fig. 3 show that, although the  COVID-19 burden can be reduced with increasing vaccination coverage (f ),  v the reproduction number cannot be brought to a value less than unity. In  other words, while the use of the Johnson & Johnson vaccine (as the only  vaccine used against COVID-19 pandemic) will decrease the pandemic bur-  den (with increasing vaccination coverage), it will fail to lead to the elimi-  nation of the pandemic in the United States.   

Fig. 3 Contour plot of the vaccine reproduction number (

) of the 

basic model ( 1), as a function of the vaccine coverage (f ) and efficacy  v (ɛ ). (A) No reduction in community transmission from baseline due  v to the implementation of other NPIs (i.e., cr = 0.0). (B) 20% Reduction  in community transmission due to implementation of other NPIs (i.e.,  cr = 0.20). (C) 40% Reduction in community transmission due to the  implementation of other NPIs (i.e., cr = 0.40). The other parameter  values used in the simulations are as given in Tables 3 and 4.   

2.5: Assessment of combined impact of vaccination and NPIs  The basic model (1) is further simulated to assess the potential impact of a  combined intervention that involves vaccination (with either of the three  vaccines that received FDA EUA in the United States) and NPIs. For simu-  lation purposes, we incorporate the impact of NPIs in the basic model (1) by  reducing the baseline values of the contact rate parameters (β , β , β ,  p s a and βh) by a factor (1 − cr), where 0 ≤ cr ≤ 1 is the increase in the overall  coverage and effectiveness of the NPIs above their baseline values during  380

the period of the fitting of the basic model (December 10, 2020 to July 23,  2021). Fig. 3B depicts a contour plot of the reproduction number (

) of 

the basic model, as a function of vaccine coverage at steady state (fv) and  efficacy (ɛ ), for the case where the overall coverage and effectiveness of  v other NPIs is increased by 20% above their baseline values (i.e., cr = 0.2). This figure shows that the level of coverage needed to achieve herd immu-  nity if either the Pfizer or Moderna vaccine is used reduces to 70% (as  against 78% for the case where cr = 0, depicted in Fig. 3A). For this case,  Fig. 3B shows that disease elimination is feasible using the Johnson & John-  son vaccine if at least 96% of the US population received this vaccine  (which is certainly not practicable). If NPIs are increased by 40% above  their baseline values (i.e., c = 0.4), the simulations depicted in Fig. 3C show  r that the use of the Pfizer or Moderna vaccine alone will lead to pandemic  elimination in the United States if at least 60% of the populace is fully vacci-  nated with either of the two vaccines. The Johnson & Johnson vaccine will  require 80% coverage to achieve elimination in this case. Hence, the simulations in Fig. 3B and C clearly show that combining vaccination with other  NPIs (implemented at a level that increases their coverage and effectiveness  above their baseline values) will reduce the vaccination coverage needed to  achieve herd immunity. In other words, combining vaccination with other  NPIs (at increased coverage and effectiveness levels, in comparison to their  baseline values) greatly enhances the prospect of pandemic elimination in  the United States.  It is worth noting from Fig. 3C that if the vaccine efficacy wanes to about  60% (i.e., ɛ = 0.6), pandemic elimination is still feasible if at least 94% of  v the US population is fully vaccinated. If the vaccine efficacy further de-  creases to, for instance, 57% (i.e., ɛv = 0.57), Fig. 3C shows that elimi-  nation is feasible if the entire US population is fully vaccinated. These high  coverage (94%–100%) are certainly not realistically attainable in a popu-  lation like the United States. Waning vaccine efficacy certainly makes the  prospect of pandemic elimination a lot more difficult, even if the vaccination  program is combined with other NPIs at improved coverage and effec-  tiveness levels above their baseline values (e.g., c = 0.4). Hence, these sim-  r ulations clearly suggest the need to boost the vaccine-derived immunity  when the vaccine efficacy begins to wane (significantly). This supports the  current call in the United States for a third dose of the Pfizer/Moderna 

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vaccine or a second dose for the Johnson & Johnson vaccine.    2.5.1: Assessment of combined impact of vaccination and various mask  types  In this section, the basic model (1) will be simulated to assess the impact of  the combined use of a vaccination program together with a mask-based  strategy on curtailing the COVID-19 pandemic in the United States. The use  of face masks, of various types and efficacies, has generally proven to be a  vital strategy to control the spread of the COVID-19 pandemic [39]. There are  three main categories of masks used in the battle against the COVID-19 pan-  demic, namely, cloth masks, surgical masks, and respirators. While cloth  masks and surgical or procedure masks are considered to be loose fitting  face coverings and are designed to be used as source control (and their pri-  mary purpose is to prevent the spread of disease from infected wearers to  those around them, and not to protect the wearer from acquiring infection  from others), respirators are defined as tight-fitting respiratory protective  devices that meet or exceed the National Institute for Occupational Safety  and Health (NIOSH) N95 standard [40–42]. In contrast to cloth and sur-  gical masks, most respirators are designed to fit tightly to the face and  represent true respiratory protection, protecting the wearer from respiratory  hazards around them [39]. The efficacy estimates of cloth masks range from  0%–50% [10, 42]. Similarly, surgical masks and procedure masks have esti-  mated efficacy ranging from 50% to 85% [10, 43]. Due to their lack of a tight  seal, the real-world filtration efficiency of the surgical or procedure masks  can be much lower [42]. Respirators have efficacy ranging from 95% to  100% [10].  The use of face masks is incorporated into the basic model (1) by multi-  plying the contact rate parameters (βp, βs, βa, and βh) by a factor (1 −  c ɛ ), where 0 < c < 1 is the face mask coverage in the community and  mm m 0 < ɛm ≤ 1 is the overall mask efficacy in preventing infection. For simu-  lation purposes, we set the efficacy of cloth masks, surgical masks, and  respirators to be 30%, 70%, and 95%, respectively. The basic model (1) is  simulated using the fixed and estimated parameters in Tables 3 and 4 and  and ɛ (noting that the transmission rate parameters  m m are now being multiplied by (1 − c ɛ ).  mm ) of the basic  Fig. 4 depicts a contour plot of reproduction number (

various values of c

model (1), as a function of face mask coverage (c ) and vaccine coverage at  m 382

steady state (f ). Specifically, Fig. 4A and B shows that using cloth or sur-  v gical masks alone (i.e., without vaccination, so that ξ = 0), pandemic  v elimination is not feasible even with very high mask coverage (cm). This re-  sult is in line with some of the simulation results presented in Fig. 3). This  shows that, although a mask strategy that relies on the use of cloth masks  (with estimated efficacy ɛm = 0.3) or surgical masks (with estimated effi-  cacy ɛ = 0.7) only can reduce the burden of the pandemic, it will fail to  m to a value less than  eliminate the pandemic (since it is unable to reduce 1). However, pandemic elimination is feasible if respirators only are used  (Fig. 4C). In particular, using respirators alone can lead to pandemic elimi-  nation if the coverage in its usage is at least 78%. On the other hand, Fig. 4  shows that using vaccination alone (i.e., without masking, so that c

= 0),  m pandemic elimination is feasible if the vaccination coverage (fv) is high  enough (in line with some of the simulations depicted in Fig. 3). Further-  more, Fig. 4 shows that combining a vaccination program with a face mask  strategy that relies on the use of cloth masks only (with estimated efficacy  c

= 0.3) can reduce the burden of the pandemic with the possibility of  m elimination. For instance, if the Pfizer or Moderna vaccine (each with esti-  mated efficacy ɛv = 0.95) is used, and half of the US population wears  cloth masks in public consistently, then pandemic elimination is feasible if  73% of the populace is fully vaccinated with either of the two vaccines (Fig.  4A). The vaccination coverage needed to achieve elimination reduces to  63% and 53%, if the vaccination program is combined with a masking strat-  egy that prioritizes surgical masks and respirators, respectively (Fig. 4B and  C). In summary, Fig. 4 shows a promising prospect for pandemic elimi-  nation if high-quality masks, such as surgical masks and respirators, are  used (even in the absence of vaccination) or if a combined vaccination-  masking strategy is used.   

Fig. 4 Contour plot of the vaccine reproduction number (

) of the 

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basic model ( 1), as a function of the vaccine coverage (f ) and mask  v coverage (c ) for (A) cloth masks (with efficacy of 30%), (B) surgical  m masks (with efficacy of 70%), and (C) respirators (with efficacy of  95%). The other parameter values used in the simulations are as given  in Tables 3 and 4.   

2.6: Sensitivity analysis The basic model (1) contains numerous parameters (as stated earlier), and  uncertainties exist in the estimate of these parameters. Hence, it is instruc-  tive to assess the impact of such uncertainties on the overall outcome of the  numerical simulations of the model. Specifically, it is important to deter-  mine the set of parameters that have the most influence on the dynamics of  the disease (as measured in terms of the impact of these parameters on the  value of a chosen response function). In this section, we choose the vacci-  nation reproduction number

as the response function for the basic 

model (1), and we seek to assess the sensitivity of each parameter of the  basic model to the chosen response function. To achieve this objective,  global uncertainty and sensitivity analysis, using Latin hypercube sampling  and partial rank correlation coefficients (PRCCs) [44–46], will be carried out  with respect to the response function

of the basic model. To implement 

this method, the ranges of the known parameters of the basic model (1) are  obtained by taking 20% to the left, and then 20% to the right of their base-  line values (tabulated in Table 3). Further, we assume that each parameter of  the basic model follows a uniform distribution. We split each parameter  range into 1000 equal subintervals. The results obtained, for the uncertainty  and sensitivity analysis, are presented in Fig. 5, with the level of significance  decreasing from the left to the right side of the figure. This figure shows that  the vaccine efficacy (ϵv) and the community transmission by symptomat-  ically infectious individuals (β ) have the most influence on the value of   s , followed by the proportion of presymptomatically infectious individuals,  who exhibit disease symptoms after the incubation period and the commu-  nity transmission by presymptomatically infectious individuals (β ).  p Parameters such as those that represent the rate at which individuals  progress from the exposed to the presymptomatically infectious class (σe)  and the rate at which presymptomatically infectious progress to the asymp-  tomatically infectious or the symptomatically infectious class (σp) have 

384

very low PRCC values (hence, they do not significantly impact the value of  the response function

). 

 

Fig. 5 Partial rank correlation coefficients (PRCCs) illustrating the  sensitivities of parameters of the basic model ( 1) on the chosen  response function (namely, the vaccination reproduction number,

). 

The level of significance of the parameters reduces as we go from left  to the right side of the plot. The baseline values of the parameters are  as given in Tables 3 and 4. 

385

3: Basic vaccination model with waning immunity  In this section, the basic model formulated and analyzed in Section 2 will be  extended to include waning of both vaccine-derived and natural immunity to  COVID-19. Furthermore, we will incorporate the vaccination of recovered  individuals. Hence, in addition to the compartments (Su, Sv, E, Ip, Is, Ia, and  Ih) in Section 2, the extended model will include compartments for suscep-  tible individuals with partial immunity due to the waning of natural immu-  nity (Srp(t)), susceptible individuals with partial immunity due to the wan-  ing of vaccine-derived immunity (S (t)), unvaccinated recovered indi-  vp viduals (R ), and vaccinated recovered individuals (R ). Consequently, in  u v this modeling scenario, the total population at time t, denoted by N(t), is  given by      The extended model is given by the following deterministic system of non-  linear differential equations:   

  (12)    In the extended model (12), 0 < ɛ

rp

≤ 1 is the efficacy of partial immunity to 

386

protect individuals in the S

class from acquiring COVID-19 infection, 0 0, I = 0, and V = 0. In the following, we de-  0 the state con-  note the state vector x(t) = (U(t), I(t), V (t)), and straints set.   

Fig. 1 UIV system.    2.1.1: Typical behavior of in-host infections  In this section, we will study the course of the SARS-CoV-2 infection, which  has a relatively fast evolution that enables us to neglect the regeneration and  death of healthy target cells in the model.  Initially, it will be assumed that the system is in a healthy equilibrium just  before t = 0, that is,

, and at t = 0 a small number of 

virions enter the system, such that U(0) = U , I(0) = 0, and V (0) = V > 0.  0 0 Initially, it can be shown that the solutions of Eq. (1) have nonnegative  values for initial nonnegative conditions, and in particular, the number of  healthy cells is strictly positive, U(t) > 0. Looking at Eq. (1a), it can be noted  that

is nonincreasing, due to the nonnegativity of its factors. Typically,  432

for healthy patients, an exponential growth both in infected cells and viral  load takes place in the early stages of the infection, followed by their marked  decrease due to the limitation of target cells, as it is explained next. Ana-  lyzing the system with a constant infection rate , it can be seen that there is  a turning point of the infection outbreak, that is, where the viral load is max-  imal, corresponds to a time instant where

and

. Note that for 

the typical case where c ≫ δ, the viral and infected cell peaks are virtually  simultaneous, therefore the approximated infected dynamics at the viral  peak time

are maximal

. That corresponds by Eq. (1b) to 

, which is the critical healthy cell number threshold  that limits the viral growth and is given by      (2)    This critical value of U represents the number of healthy cells that guarantee  that the viral load and infected cells number will be decreasing for any  propagation rate lower than .  Moreover, it can be shown that the terminal values of I and V, I∞ :=  lim I(t), V := lim V (t), converge to zero, regardless of the initial  t→∞ ∞ t→∞ values I(t ) and V (t ), while the terminal value of U := lim U(t) de-  0 0 ∞ t→∞ pends on both the initial conditions U(t0), I(t0), V (t0), and the param-  eters. A detailed analysis of the dynamics of Eq. (1) can be found in Ref. [22].  Although we lack an explicit solution for Eq. (1), the terminal value of the  nominal evolution (for the untreated case) can be expressed as   

  (3)    where W(⋅) is the Lambert function [22] and t0 an arbitrary time.  For illustrative purposes, a numerical solution of the target-cell model,  corresponding to a patient denoted Patient A, for an untreated condition is  shown in Fig. 2. Patient A model parameters were estimated by using viral  load data of an RT-PCR COVID-19 positive patient—reported in Ref.  −7 [41]—with β = 1.35 × 10 , δ = 0.61, p = 0.2, and c = 2.4. The initial condi-  tions are given by U = 4 × 10⁸, I = 0, and V = 31. The critical value for the  0 0 0 433

susceptible cells of Patient A (Fig. 2) is given by U* = 5.44 × 10⁷, which  means that the final value of U (if no antiviral treatment is applied; i.e., t =  0 0) is given by U∞ = 2.57 × 10⁵.   

Fig. 2 Numerical simulation of the target-cell model, corresponding to  Patient A.   

2.2: Reduced system model and generic infection evolution  In this section, we will shortly consider a reduced system model which will enable us to gain some insight related to the system behavior and the con-  trol objectives. In order to simplify our analysis, the system (1) can be  deemed dimensionless by an adequate change of variables. Moreover,  under the assumption that the viral dynamics is much faster than the in-  fected cell dynamics, which is typically so in this type of infection, a nondi-  mensional reduced model can be attained, which is useful for illustrative  purposes. The reduced model has the following form [16]:      (4a)   

434

  (4b)    where u and v describe the dimensionless healthy cell count and viral load,  and τ is a generalized time variable. Let y denote the state vector in the  nondimensionalized variables, that is, y := (u, v), with

. 

With the aim of providing some insight into the meaning of such results,  the vector field corresponding to the dynamics of the reduced dimen-  sionless system is shown in Fig. 3. It is evident that the system has infinite  steady states for v = 0; therefore, the horizontal axis is an equilibrium set Note that this model is dependent on a single parameter

. 

, which indi- 

cates the critical threshold for the reduced system. It can also be seen that  for values of u greater than q, the vector field components point away from  the equilibrium set, while the opposite happens for values lower than q.   

Fig. 3 Vector field ( blue [ dark gray in print versions ]) and phase portrait  ( black ) for the reduced system.    In order to get some ideas on the behavior of the system, phase portraits  obtained from solutions to Eq. (4), with initial values very close to the equi-  librium set (very low values of v corresponding to situations in which the  disease develops from a very small number of viral particles entering the  system) are shown in Fig. 3. It can be noted that initial conditions with

 

435

have a final steady state (u define

for

and

∞

for

= lim

u(τ)) with . Therefore, we  τ→∞   , which are shown in green (light gray in 

print versions) and red (dark gray in print versions), respectively. Note that  trajectories with initial states close to the set

remain close to the set

. 

Also, it can be seen that the larger the ratio of initial number of healthy cells  with respect to

, the lower the final healthy cell count. Note that even 

though the initial number of healthy cells U is invariant in the original sys-  0 tem, that value is mapped in the nondimensionalization process by and p  which are dependent on the available therapies. More important it is to note  that higher viral loads at the time when  

, maps to lower u .  ∞

2.2.1: Admissible treatments  In the general context of antiviral treatments, the infection rate will be time  varying, accounting for the antiviral therapy effects. To account for realistic  treatments, it is assumed that β(⋅) ∈ Ωβ, where Ωβ is the set of func-  such that , for t ∈ [t , t ], and , for  tions i f , being 0 < ti < tf < ∞ the starting and ending treatment time (tf  is assumed to be finite since acute infection treatments are always finite-  duration treatments) and

are the minimal and maximal values of the 

infection rate, respectively ( and represent the untreated and fully treated  infection rate, respectively; the case

is not considered since antivirals 

have always a limited effect on the infection rate). The bounds on β(t) and  a possible curve under treatment are shown in Fig. 4.   

436

Fig. 4 Bounds on β ( t ).   

2.3: Equilibria characterization and stability  This section is devoted to the characterization of the equilibrium states of  system (1) and their stability. We will only consider untreated systems in our  analysis, since the treatment is assumed to be time limited and therefore its  effects vanish after a finite time.  We can characterize the set of equilibrium states by equating to zero all  subequations of system (1), resulting in the set

given by 

 

  (5)    A first attempt at determining the stability condition on the states con-  tained in

is not conclusive, as it results that all these are indifferent equi- 

librium point since the associated linearized system of differential equations  has a zero eigenvalue.  From the linearization of the system at a generic equilibrium state the set

can be split into two subsets

and

, 

according to the value of 

the largest eigenvalue of the Jacobian matrix, where  437

  (6)    corresponds to states at which the evaluation of the Jacobian matrix has  eigenvalues lower or equal to 0, while the set   

  (7)    contains equilibrium states that are definitely unstable, and therefore also  the set

itself, because of the existence of a positive eigenvalue.

Nevertheless, if instead of considering the stability of each single equi-  , we analyze the stability of the set

librium state within

as a whole, it 

can be determined that it is stable—regardless of zero-valued eigenvalue of  the Jacobian matrix evaluated at all states in

—and, moreover, it is locally 

AS, in the sense of Definitions 12–15, in Appendix A. Indeed, even when  every equilibrium point in

is ϵ − δ stable, there is no single equilibrium  and

point in it that is attractive. In the following, we will refer to the sets as

and

 

, respectively, accounting for their stability conditions. 

Next, it is shown in Section 2.3.1 that

is the smallest attractive set in

, 

while in Section 2.3.2 it is demonstrated that this set is also the largest lo-  cally ϵ − δ stable set in

. These two results allow us to show, in Section 

2.3.3, a strong results concerning UIV models: is unstable.a  which means that

is the unique AS set in

, 

  2.3.1: Attractivity analysis  In this section, we will study the attractivity of set

. According to Defi- 

nition 12, any set containing an attractive set is also attractive. So, we are  interested in finding the smallest closed attractive set in

. 

  Theorem 1: (Attractivity of

)

Consider system (1), with

replaced by β(t), constrained by

, with 

β(⋅) ∈ Ωβ. Then, the set

is the smallest attractive set in

. 

Furthermore,

is not attractive. 

  Proof 

438

The proof is divided into two parts. First, it is proved that

is an 

attractive set, and then, that it is the smallest one.  is attractive: To prove the attractivity of

in

(and to show that 

is not attractive), we need to prove that U ∈ [0, U*] for any ini-  ∞ for t > tf (with tf finite), U∞ de-  tial conditions. Recalling that pends on U* and the conditions U(tf), V (tf), and I(tf), according to  Eq. (11). The minimum of U

is given by U = 0, and it is reached  ∞ ∞ when U(tf) = 0 (for any value of U*, I(tf), and V (tf)). The maximum 

of U , on the other hand, is given by U = U*, and it is reached  ∞ ∞ only when U(t ) = U* and I(t ) = V (t ) = 0 (for any value of U*), as  f f f it is shown in Lemma 18 in Appendix B. Then, for any  and U* > 0, U ∈ [0, U*], which means that ∞ attractive, and the proof of attractivity is complete. 

is 

is the smallest attractive set: It is clear from the previous analysis,  converges to a state x   that any initial state ∞ = (U∞, 0, 0) with U∞∈ [0, U*]. This means that is not attrac-  . However, to show that

is the smallest 

attractive set, we need to prove that every point

is necessary 

tive for any point in for the attractivity. 

Let us consider Eq. (11) with an initial state of the form (U*, I(t ), 0)  f with I(tf) ≥ 0. Since W is a bijective function from (−1/e, 0) to (−1,  0), then U∞(U*, ⋅, 0) is bijective from (0, +∞) to (0, U*). Hence,  , there exists I(t ) ≥ 0 such that the initial  for every point f state (U*, I(tf), 0) converges to xs. Since every interior point of is  necessary for the attractivity, then the smallest closed attractive set  is

, and the proof is concluded.□ 

  2.3.2: Local stability analysis  In this section, we will study the local ϵ − δ stability of

. Next, a the- 

orem is introduced that shows the ϵ − δ or Lyapunov stability of

. 

  Theorem 2: (Local ϵ − δ stability of Consider system (1), with

) 

replaced by β(t), constrained by

β(⋅) ∈ Ω . Then, the equilibrium set β δ stable. 

, with 

is the largest locally ϵ − 

 

439

Proof  We proceed by analyzing the stability of single equilibrium points  , with (i.e., sider the following Lyapunov function: 

). For each

, let us con- 

 

  (8)    Function J is continuous in and

, is positive for all nonnegative

. Furthermore, for

 

and t ≥ tf, we have 

 

  Note that function

depends on x(t) only through V (t). So, 

independently of the value of

for V (t) ≡ 0. This means 

,

, V (t ) = I(t ) = 0 and so, V (t) = 0, for all t  f f is null for any (i.e., it is not only null for  

that for any single ≥ tf. So,

). 

but for any

On the other hand, for

, function

positive, depending on if the value of than for any

is smaller, equal or greater 

, respectively, and this holds for all ,

(particularly, for

is negative, zero, or  and t ≥ t . So,  f , , 

and t ≥ tf) which means that each is locally ϵ −  , that  δ stable (see Theorem 16 in Appendix A). Finally, when for all

is,

, we define the Lyapunov functional as J(x) = U + I + 

δ/pV and proceed analogously as before. Therefore, since every  state in

is locally ϵ − δ stable and

is compact, by Lemma 14, 

440

the whole set tive in

is locally ϵ − δ stable. Finally, since

, then it is impossible for any

which implies that

is attrac- 

to be ϵ − δ stable, 

is also the largest locally ϵ − δ stable set in 

, which completes the proof.□    2.3.3: Asymptotic stability  In this section, we will study the asymptotic stability (AS) of

. In the next 

theorem, based on the previous results concerning the attractivity and ϵ −  δ stability of

, the AS is formally stated. 

  Theorem 3  Consider system (1), with replaced by β(t), constrained by the posi-  , with β(⋅) ∈ Ω . Then, the set is the unique AS equi-  β . Fur-  librium set, with a domain of attraction (DOA) given by

tive set

thermore,

is unstable. 

  Proof  The proof follows from Theorems 1, which states that smallest attractive set in

, and Eq. (2), which states that

largest locally ϵ − δ stable set in

is the  is the 

.□ 

  Fig. 5 shows phase portrait plots of system (1), corresponding to different  states x(t ) = (U(t ), I(t ), V (t )), for t > t .  f f f f f  

Fig. 5 Phase portrait for virtual Patient A. (A) Case U ( t f ) > U *. States  (in red [ dark gray in print versions ]),  x ( t ) arbitrarily close to f converges to (in green [ light gray in print versions ]). (B) Case U ( t f ) 

441

< U *. States x ( t ) arbitrarily close to , also converges to , but  f without increasing V at any time. Empty circles represent the states at t   f while solid circles represent the states for a large time.   

2.4: Assessment on the severity of infections and the effect of  antiviral treatments  Typically, two metrics are used to assess the severity of an infectious dis-  ease, namely, the IFS and V LP. They describe, respectively, the accumulated  dead cells during the infection and the maximal value of the viral load. High  viral loads typically correlate with the severity of the disease and its infec-  tivity. Also, diseases with high final sizes promote coinfection and have  stronger consequences. These indexes, IFS and V LP, are defined as      (9)      (10)    where U0 denotes the number of healthy (susceptible) cells just before the  outbreak of the disease and U is an explicit function of the system param-  ∞ and the states at tf, (U(tf), I(tf), V (tf)), but not of β(⋅) ∈ Ωβ  eters

(i.e., β(t) affect U∞ only trough (U(tf), I(tf), V (tf))):   

  (11)    Since, as shown in Lemma 18, Appendix B, the supremum of U is  ∞   reached when U(tf) = U*, then, the infimum of IFS is given by .  A metric related to the IFS is

, namely, the area under the 

curve of the viral load, for any β(⋅) ∈ Ωβ. Moreover, it can be shown that  the infima of AUC

VL

and IFS are bound to each other by 

 

442

  This states an important condition for time-limited treatments as, in order  to minimize the IFS (and simultaneously the AUC ), the state should be as  VL close as possible to (U*, 0, 0) by the end of the treatment.  On the other hand, this also states some compromise on the viral load  and the treatment duration, since the minimization of the IFS implies a fixed  value for the AUCVL, which, for a fixed time, requires the viral load to be  above some threshold for some interval. Nevertheless, keeping the VLP  below some adequate threshold while applying a treatment leading to the  infimum IFS are compatible objectives. 

443

3: Antiviral treatment effectiveness  3.1: Inclusion of the PD of antiviral treatment  Next, we will incorporate the PD to system (1). To obtain a controlled sys-  tem, that is, a system with certain control actions—given by the antivirals—  that allows us to (even partially) modify the whole system dynamic accord-  ing to some control objectives, we need to specify how parameter is modi-  fied in Eq. (1) (note that antivirals inhibit the virus infection and replication  rates). Thus, the PD is introduced in system (1) as follows:      (12a)   

  (12b)      (12c)    where

and η(t) ∈ [0, 1) represent the inhibition antiviral ef- 

fects affecting the infection rate β (according to Ref. [16], the effect of an-  tivirals on the replication rate p is analogous to the one on ).   

3.2: Antiviral effectiveness characterization  The treatment initiation time ti is assumed to be between the minimum and  maximum time of V without treatment (as shown in Ref. [22], V (t) has a  minimum at the very beginning of the infection and, then, a unique maximum, at a time ) since otherwise (if

) the treatment has not signif- 

icant effect on the patient. The final time t is assumed in this section large  f enough for the virus to reach an undetectable level of approximately 100 -  −1 100 -copies mL (Section 5, we explicitly consider the effect of a finite  treatment final time), while η(t) is a fixed value in [0, 1) from t to t .  i f Biologically, the antiviral effect η is limited by the inhibitory potential of  the drug (expressed in terms of EC50, or drug concentration for inhibiting  50% of antigen particles, as described in the following section) and its cyto-  toxic effect (expressed in terms of IC50, or drug concentration which causes  death to 50% of susceptible cells) [30, 42]. As the antiviral treatment re-  duces the parameter

in some amount, it quantitatively modifies the virus  444

behavior. Particularly, the virus peak time will be modified from (untreated  patient case) to

(treated patient case). However, given that the treatment 

is initiated when the virus is increasing (i.e., between its minimum and its  maximum), the new peak will occur at the same time or after the treatment  time, that is,

. This way, even when the virus peak will always be small- 

er with a treatment (smaller peaks are obtained for smaller values of ), the  time at which this peak takes place can be smaller or greater than the one  without any treatment. The ability to reduce the time of the virus peak (and  doing so, also its value V LP) could be critical to define whether or not a  given antiviral is able to prevent severe disease. Indeed, in some cases, an-  tivirals significantly delay the virus peaks, largely increasing the area under  the curve of the viral load AUC

. In order to qualitatively assess antiviral  VL effectiveness according to the time of the virus peak, the following classi-  fication is made. 

  Definition 4: (Antiviral treatment effectiveness [22])  Consider system (12), constrained by the positive set , such that  the virus spreadsb in the host from time t = 0, which implies that:  U(0) > U*, I(0) = 0, and V (0) > 0. Consider also that, at time t ,  i , an antiviral treatment is initiated such that η jumps from  with 0 to some value in [0, 1). Then, the treatment is said to be effective  if the virus peaks at a time

, the latter being the virus peak time 

for the untreated viral dynamics. Otherwise, if

, it is said that 

the treatment is ineffective.    Definition 4 is closely related to the capacity of the antiviral drug to clear  the infection (or, the same, cut off its spread) in such a way that an effective  treatment could: (a) decline the viral growth at the treatment time, and, so,  the virus clearance starts when the therapy is initiated or (b) hasten the  virus peak, and, so, even though the virus clearance is not started at treat-  ment time, it begins prior to the untreated case. Note also that Definition 4  accounts for three typical antiviral effect metrics: the virus peak V LP, the  area under the curve of the viral load AUCVL, and the duration of infection  (the time from t = 0 up to a time when the virus is undetectable, V , ap-  det −1 proximately 100 copies mL ) [43], in a direct way, and the viral load peak V  LP [44], in an indirect way.  Now, we will show that the effectiveness of antivirals depends on whether 

445

η is greater or smaller than a specified threshold, which is a function of the  parameters and the time of the treatment initiation, t . The critical value of i η is the one that makes U* = 1, that is:      (13)    From Eq. (13), it can be inferred that η*(ti) is an increasing function of  U(ti). Fig. 6 shows η*(ti) for the COVID-19 Patient A, mentioned in Section 

2.1.1. Note that η*(t ) ≈ 1 − cδ/(U βp) for t → 0 and η*(t ) ≈ 1 −  i 0 i i . Similar results concerning the critical drug effi-  cδ/(U*βp) = 0 for

cacy with respect to the availability of target cells at the treatment time were  obtained in Ref. [45], for acute models, although the authors focused the  analysis on the treatment starting at the beginning of the infection.   

Fig. 6 η *( t i ) corresponding to Patient A. 

 

The following theorem determines the effectiveness of antivirals in terms   

of η*(t ).  i Theorem 5  446

Consider system (1), constrained by the positive set

. Consider 

that the virus spreads in the host for t ≥ 0, which implies that: U(0) >    U*, I(0) = 0, and V (0) > 0. Consider also that, at time ti, with (when the virus is increasing), an antiviral treatment is initiated such  that η(t) jumps from 0 to some value in [0, 1). Then,    (i)If the inhibition effect is such that η > η*(ti), then the max-  imum of V (t) occurs at t , which is smaller than by hypothesis.  i In this case,the antiviral treatment is said to be effective.  (ii)For two antiviral treatment, 1 and 2, with inhibition effects  η¹, η²such that η*(t ) < η¹ < η², there exist a time t* >  i ti(large enough) such that V2(t) < V1(t), for all t ∈ (ti, t*],  being V (t) and V (t) the virus corresponding to treatments η¹  1 2 and η², respectively. In this case, both treatments are effec-  tive, and treatment 2 is more eff icient than treatment 1.  (iii)If the inhibition effect is such that η < η*(t ), then the  i . Furthermore, there  virus reaches a maximum at a time exists a time t = t (η, t )—denoted as early treatment time—  e e i for t ∈ (0, t ]. In this case,the antiviral treat-  such that i e ment is ineffective.  (iv)For two antiviral treatment, 1 and 2, with inhibition effects  η¹, η²such that η¹ < η² < η*(ti) with ti ∈ (0, te(η, ti)],  , being and the virus maximum time corre-  it is sponding to treatments η¹ and η², respectively. In this  case,both treatments are ineffective, but treatment 1 is more  eff icient than treatment 2, which is a rather counterintuitive  fact.    Proof  The proof of Theorem 5 can be seen in Ref. [16].□    Remark 6  Item (iii) of Theorem 5 establishes just the existence of te > 0 such  that for the treatments starting at t ∈ (0, t ], and η < η*(t ), the  i e i new virus peak time is larger than the one corresponding to the 

447

untreated case, that is,

. However, it should be noted that for 

parameters coming from real patient data, t is indeed close to .  e This means that the time period where treatments can be ineffective  is in most of the cases similar to

, that is, comparable with the 

time period where the virus is growing.    A consequence of item (iii) of Theorem 5 is that early treatments, if not  effective, could be worse than late ones. Even more, when early treatment is  not strong enough to avoid the virus spreadability right after t , the highest  i the antiviral effectiveness η is, the longer time the virus remains in the  host, since the maximum time

is delayed, as established in item (iv). As a 

result, antivirals could be worse as treated patients would need to be iso-  lated for larger periods of time than untreated ones.  Note that even when virus peak time can be delayed for some treatments,  the virus peak will be always smaller than the one without any treatment.  Furthermore, the IFS (fraction of dead cells at the end of the infection, IFS)  will be always greater if no antiviral is administrated [16].    Remark 7  This preliminary analysis should be understood just as an addi-  tional description of the antiviral effects, concerning the capability  of reducing the time of the VLP. Indeed, the final time of the treat-  ment t is not considered here (or it is considered large enough).  f So, according to the stability analysis made for time-limited treat-  ments, if the susceptible cells are such that U(tf) > U* for any of the  cases described in Theorem 5, then a rebound of the infection will  take place.    Fig. 7 shows the virus evolution, corresponding to Patient A, when dif-  ferent treatments (different values of η, which is assumed to be constant  from t to a large enough t ) are applied, at a starting time t = 7 days (which  i f i is a time between (0, te)). As it can be seen, for values of η greater than  η* = 0.85 the virus starts to monotonically decrease to zero (i.e., the virus 

does not spread in the host) which means that the treatment is effective  (and more efficient for larger values of η). For values of η smaller than  η*, the treatment is ineffective (according to Definition 4).    448

Fig. 7 Viral load time evolution with treatment initial time given by t =  i 7 days, corresponding to Patient A. Values of η ( t i ) smaller,  approximately equal to and greater than η * are simulated to 

demonstrate the results in Theorem 5. The blue line (dark gray in print  versions) denotes the untreated case (η = 0). 

449

4: Inclusion of the PK of antiviral treatment  Although the effect of the PK of antivirals does not alter the dynamic funda-  mentals of system (1) made in the previous sections, we will briefly discuss  its details, for completeness in the description of the problem. If we define  D as the amount of drug available, then η can be expressed as      (14)    where EC

represents the drug concentration in the blood where the drug  50 ,  is half-maximal. The antiviral effects η(t) are then assumed to be in with

(full antiviral effect is not considered, since that would be an un- 

realistic scenario, as discussed in the previous sections). The PK can be  modeled as a one compartment with an impulsive input action (impulses  account for pills intakes or injections, as described in Refs. [28, 29]):      (15a)      (15b)    where D is the amount of drug available (with D(0) = D = 0), δ is the  0 D drug elimination rate and the antiviral dose uk enters the system impulsively  . Time de-  at times t := kT , with T > 0 being a fixed time interval and k d d notes the instant just before t , that is, . Note that Eq.  k (15) is a continuous-time system impulsively controlled, which shows dis-  continuities of the first kind (jumps) at times t and free responses in t ∈  k [tk, tk+1) (see Ref. [28] for details).  Based on the PK and PD analysis, the complete COVID-19 infection  model, which takes into account an antiviral treatment (the controlled sys-  tem) reads as follows:   

  (16a)   

450

  (16b)      (16c)      (16d)     (16e)   with initial conditions given by x = (U , I , V , D ). Given that D(t) ≥ 0  0 0 0 0 0 . Also, a con-  for all t ≥ 0, the constraint set is enlarged to be , where

straint for the input, u, is defined as represent the maximal antiviral dosage (

is usually determined by the  ), while sets

drug side effects and maximal effectiveness, larged by considering

 

are en- 

. Fig. 8 shows a schematic plot of the 

complete compartmental PK-PD system.   

Fig. 8 PK-PD system.    The form of uk over time is directly related to the constraint Ωβ on β(t)  through the relations

and

. Indeed, the antiviral 

doses u must be bounded from below by zero and from above by some  k umax > 0, which fulfills the condition that, if uk= umax, for tk ∈ [0, ∞),  then

, where

represents the mean value 

operator over Td. In other words, any possible antiviral treatment uk must  fulfill 

451

  (17)    To demonstrate the impulsive control actions describing the effects of an-  −1 tiviral administration, we simulated Patient A with D0 = 0, δD = 2 day   mg. A scenario of 30 days was considered, and a fixed dose of  and uk = 20 mg of antivirals is administered each Td days, starting at ti = 4 days,  with Td = 1. As shown in Fig. 9, the system response shows a ripple because  of the impulsive nature of the inputs. The idea, however, is that regardless  of the time period Td of the doses, β(t) should fulfill condition (17).   

Fig. 9 Time evolution of virtual Patient A, with u = 20 mg of antivirals  and T d = 1 day. 

452

5: Control strategy to tailor therapies  Control objectives in in-host infections can be defined in several ways. The  peak of the virus load uses to be a critical index to minimize (as described  in Section 3) since it is directly related to the severity of the infection and the  ineffective capacity of the host. However, other indexes—usually put in a  second place—are also important. This is the case of the time the infection  lasts in the host over significant levels [16]—including virus rebounds after  reaching a pseudo-steady state, and the total viral load or infected cells at  the end of the infection (i.e., the AUC ). These latter indexes also inform  VL (in a different manner) about the severity of the infection and the time dur-  ing which the host is able to infect other individuals, and are directly deter-  mined by the amount of susceptible cells at the end of the infection.  The following definition formalizes the control objectives of the problem  under study, according to the discussion made in Section 2.4.    Definition 8: (Control objectives)  The control objective for the closed-loop equation (12) consists in  (1) minimizing the IFS, and (2) keeping the V LP under a given  upper bound

(determined by the severity of the disease and the 

host infectiousness), while minimizing—as long as possible—the  total amount of administered antiviral (to avoid drug side effects).   Thus, the control problem consists in finding a function β(⋅) ∈ Ω ac-  β counting for the control objectives from Definition 8. In view of the dynamic  analysis made in the previous sections, we can pose the following optimal  control problem

that—in contrast to many 

other strategies—takes advantage of the stationary/transient distinction for  each of the objectives:   

453

  (18)    where T > tf is a large enough time that covers the whole dynamic of the  forces variable V (τ) to be smaller than the  infection. Conditions externally imposed maximum

at every time t ∈ [0, T], while constraints 

U(T) = U* and V (T) ≤ Vdet, where Vdet is a small detectable virus concen-  tration (usually 100 copies mL−1), are meant to force the system to reach a  quasisteady-state condition (i.e., with V (T) approaching zero) at the end  time T, while U(t) is equal to U*. The key point of Problem

is that the 

clinical objectives (i.e., controlling the IFS and the V LP) are imposed by  constraints while only the objective of minimizing the antiviral drug admi-  nistration is achieved by optimality, avoiding the competition between them.    Remark 9  It can be shown that Problem

is well posed and properly ac- 

counts for the control objectives (Eq. 8). Furthermore, any other optimization problem, that is, the one minimizing V (t) (or U(t) =  U*) along T, without terminal constraints, will necessarily produce a  suboptimal solution, as the control objectives would be competing  in the cost function.   

5.1: Numerical simulations  Here, we resume the simulation of Patient A. Fig. 10 (dashed blue and red  lines [dark gray in print versions]) shows the open-loop system time evolu-  tion, which consists in using

(no treatment is a particular case of 

β(⋅) ∈ Ω ), while Fig. 11 (dashed blue lines [dark gray in print versions])  β shows the phase portrait in the plane U, V. As predicted, U ol is  ∞

454

(significantly) smaller than U* (IFSol = U − U ol = 3.9974 × 10⁸ cells  0 ∞ mm−3), while the peak of V is given by V LPol = 1.5064 × 10⁷ copies mL−1  (the viral load is considered undetectable under the detectable value of Vdet  = 100 copies mL−1).   

Fig. 10 U ( blue [ dark gray in print versions ]), V ( red [ dark gray in print  versions ]), and β ( black ) time evolution ( left ). System with U * =  −3 −1 copies mL .  5.44 × 10 ⁷ cells mm and  

455

Fig. 11 Phase portrait ( right, blue [ dark gray in print versions ]),  corresponding to open loop ( dashed line ) and optimal control ( solid    line ). System with U * = 5.44 × 10 ⁷ cells mm −3 and copies mL −1 .    Fig. 11 (solid blue and red lines [dark gray in print versions]) also shows the  optimal control system time evolution, considering a β(t) (left, solid black  lines) obtained from P . The time horizon T was selected to be 32.8 days,  opt ol (30% of V LP ).  while the maximal allowed VLP is given by As it can be seen, the control objectives are reasonably approached, with  opt −3 ol   IFS = 3.4723 × 10⁸ cells mm (80% of IFS ), −1 copies mL , while the total amount of antiviral is minimized.  Note that β(⋅) separates the control objectives over time: first, it handles  the V LP (from t ≈ 8 to t ≈ 17 days) and, once V cannot further increase, it  tries to reach U(T) ≈ U*, at steady state, to minimize IFS. Although not  simulated here, it can be shown that any other optimization problem—for  instance, the one minimizing the viral load along the time (as it is usually  done in the literature)—systematically produces a suboptimal performance,  avoiding the achievement of the control objectives.    Remark 10 

456

A better description of the control would be obtained by considering  system (16) instead of Eq. (12) in Eq. (18), since this way we include  the PKs and PDs of a particular antiviral, which considers also the  impulsive nature of the doses (pill intakes) that affect parameter β.  However, it is important to note that the optimality concepts intro-  duced in this chapter are not altered. 

457

6: Conclusions and future works  In this work, the dynamical long-term behavior of UIV-type models describ-  ing the SARS-CoV-2 infection in the host has been analyzed, including a PK  and PD study. The effectiveness of antiviral treatments is defined according  to the time of the virus peak and its ability to stop the spread of the virus.  Furthermore, an optimal control is developed that takes explicit advantage  of the closed-loop dynamical analysis, and the characterization of a thresh-  old for the susceptible cells U. The control strategy accounts for three con-  trol objectives simultaneously: minimizes the number of infected (death)  cells at the end of the infection, maintains the virus peak under a maximum,  and minimizes, as long as possible, the total amount of administered antivi-  rals. Future works include the study of more complex control strategies  (mainly model-based control strategies such as model predictive control and similar) to explicitly account for uncertain scenarios. 

458

Appendix A: Stability theory  In this section, some basic definitions and results are given concerning the  AS of sets and Lyapunov theory, in the context of nonlinear continuous-time  systems [20, Appendix B]. All the following definitions are referred to system      (A.1)    where x is the system state constrained to be in

, f is a Lipschitz con- 

tinuous nonlinear function, and ϕ(t;x) is the solution for time t and initial  condition x.    Definition 11: (Equilibrium set)  Consider system (A.1) constrained by librium set if each point

. The set

is an equi- 

is such that f(x) = 0 (this implies that 

ϕ(t;x) = x for all t ≥ 0).    Definition 12: (Attractivity of an equilibrium set)  Consider system (A.1) constrained by

and a set

. A closed 

is attractive in if for all   equilibrium set c . If is a ɛ-neighborhood of for some η > 0, we say that is  locally attractive.   We define the DOA of an attractive set of all initial states x such that

for the system (A.1) to be the set  as t →∞. We use the term region 

of attraction to denote any set of initial states contained in the DOA.  A closed subset of an attractive set (for instance, a single equilibrium  point) is not necessarily attractive. On the other hand, any set containing an  attractive set is attractive, so the significant attractivity concept in a cond strained system is given by the smallest one.     Definition 13: (Local ϵ − δ stability of an equilibrium set)  Consider system (A.1) constrained by

. A closed equilibrium set 

is ϵ − δ locally stable if for all ϵ > 0 there exists δ > 0  such that if then , for all t ≥ 0.    Unlike attractive sets, a set containing a locally ϵ − δ stable equilibrium 

459

set is not necessarily locally ϵ − δ stable. Even more, a closed subset of a  locally ϵ − δ stable equilibrium set (for instance, a single equilibrium  point) is not necessarily locally ϵ − δ stable. However, any (finite) union  of equilibrium sets locally ϵ − δ stable is also locally ϵ − δ stable. So,  the significant stability concept in a constrained system is given by the  largest one.  Although a finite union of equilibrium set locally ϵ − δ stable is also lo-  cally ϵ − δ stable, in general we cannot extend this result to the case of  arbitrary unions of points. Thus, even when every equilibrium point of an  equilibrium set is locally ϵ − δ stable, we cannot assure that the whole set  would be locally ϵ − δ stable. This is due to the fact that given a fixed ϵ  the δ chosen for each point depend on the point and so the infimum of  them could be zero. However, if in addition we also assume that the set is  compact, then the stability of the set can be inherited from the stability of its  points.    Lemma 14  Let

be a compact equilibrium set. If every

ble, then

is ϵ − δ locally sta- 

is ϵ − δ locally stable. 

  Proof  such that if  Given ϵ > 0, there exists δ = δ(xs) > 0 for each then ϕ(t;x) ∈ B (x ) for t ≥ 0. The family of δ-balls  ϵ s form an open cover of . Let V denote the union of this cover, that  . Since

is,

is compact and the complement 

of V is closed, then the distance between them is strictly positive,  . Therefore, the δ* neighborhood of the 

that is, equilibrium set

is contained in V. Thus, if

for t ≥ 0. Therefore,

, then 

is ϵ − δ locally stable.□ 

  Definition 15: (AS of an equilibrium set)  Consider system (A.1) constrained by equilibrium set tractive in

is AS in

and a set

. A closed 

if it is ϵ − δ locally stable and at- 

. 

  Next, the theorem of Lyapunov, which refers to single equilibrium points 

460

and provides sufficient conditions for both, local ϵ − δ stability and AS, is  introduced.    Theorem 16  [Lyapunov's stability theorem [47, Theorem 4.1]] Consider system  and an equilibrium state

(A.1) constrained by

. Let

be a 

such that V (x) >  neighborhood of xsand consider a function , denoted as Lyapunov function.  0 for x≠x , V (x ) = 0, and s s Then, the existence of such a function in a neighborhood of xsimplies  that

is locally ϵ − δ stable in

x≠x , then x is AS in s s

. If in addition

for all 

. 

461

Appendix B: Behavior of the terminal healthy cells count  As described by Eq. (11), U can be expressed as a function of U(t ), I(t ),  ∞ f f and V (tf) (denoted here as U, I, and V, for simplicity) as follows:   

  (B.1)    is the critical susceptible cells threshold. The following 

where  

. 

property specifies how U∞ behaves for different values of Property 17 

, for some 

Consider system (1) with arbitrary conditions  

finite tf ≥ 0. Then: (i)For fixed I > 0, V > 0, if U > U* then U decreases when U  ∞ increases, and U∞ < U*. This means that the closer U is to U*  from above, the closer will be U

to U* from below.  ∞ (ii)For fixed I > 0, V > 0, if U < U* then U increases with U,  ∞ and U∞ < U*. This means that smaller values of U produce  smaller values of U , both below U*.  ∞ (iii)For any fixed U, U∞decreases with I and V, and U∞≤ U*.  (iv)For fixed U = U* and I = V = 0, U∞reaches its maximum  over , and the maximum is given by U* (seeLemma 18).    Proof  It follows directly from Eq. (B.1) and properties of the exponential  and the Lambert W functions.□    Consider now an ɛ ≥ 0 and define a domain of

given by 

 

  (B.2)     

The following lemma specifies the maximum of U on each ∞

. 

462

Lemma 18: (Maximum of the function U )  ∞ Consider the function U given by Eq. (B.1) and for each ɛ ≥ 0 the  ∞ given by Eq. (B.2). Then, the maximum of U∞(U, I, V  domains ) in

is reached in (U*, ɛ, ɛ). In particular, the maximum value 

of U∞ over

is reached in (U*, 0, 0) and is given by U∞(U*, 0, 

0) = U*.    Proof  According to Eq. (B.1), U can be written as  ∞  

  with

. Since − W(−⋅) is an increasing (injective) 

at the  function, then U (U, I, V ) achieves its maximum over ∞ same values as f(U, I, V ). Then, we focus our attention in finding  the maximum (and the maximizing variables) of f(U, I, V ).  , f can be stud-  −(x+y) ied as a function of the form g(x, y) = xe . Note that 

Through the change of variables

and

if and only if x ≥ 0 and y ≥ η where Therefore, to find extremes of f in

. 

it is enough to study the ex- 

treme points of g over . Since ∇g = [(1 −  −(x+y) −(x+y) x)e , −xe ] does not vanish and g tends to 0 when ∥(x,  y)∥ goes to ∞, then the maximum is reached at the boundaries of  . A simple analysis shows that g restricted to the boundary of   achieves its maximum in (1, η). This means that f(U, I, V ) achieves  its maximum at U = U* and I = V = ɛ, and the maximum of U∞ is  given by 

, which is an decreasing function of ϵ (the maximum of U∞ in-  creases for decreasing values of ϵ). In particular, when ɛ = 0, f(U,  I, V ) reaches its maximum at (U*, 0, 0), while the maximum of U   ∞ reads   

463

which concludes the proof.□ 

464

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[24] Hernandez-Vargas E.A., Velasco-Hernandez J.X. In-host  modelling of COVID-19 in humans. Annu. Rev. Control.  2020;50:448–456.  [25] Hernandez-Mejia G., Alanis A.Y., Hernandez-Gonzalez M.,  Findeisen R., Hernandez-Vargas E.A. Passivity-based inverse  optimal impulsive control for influenza treatment in the host.  IEEE Trans. Control Syst. Technol. 2019;28:94–105.  [26] Boianelli A., Sharma-Chawla N., Bruder D., Hernandez-  Vargas E.A. Oseltamivir PK/PD modeling and simulation to  evaluate treatment strategies against influenza-pneumococcus  coinfection. Front. Cell. Infect. Microbiol. 2016;6:60.  [27] Canini L., Perelson A.S. Viral kinetic modeling: state of the  art. J. Pharmacokinet. Pharmacodyn. 2014;41(5):431–443.  [28] Rivadeneira P.S., Ferramosca A., González A.H. Control  strategies for nonzero set-point regulation of linear impulsive  systems. IEEE Trans. Autom. Control. 2018;63(9):2994–3001.  [29] González A.H., Rivadeneira P.S., Ferramosca A., Magde-  laine N., Moog C.H. Impulsive zone MPC for type I diabetic  patients based on a long-term model. IFAC-PapersOnLine.  2017;50(1):14729–14734.  [30] Vergnaud J.-M., Rosca I.-D. Assessing Bioavailablility of  Drug Delivery Systems: Mathematical Modeling. CRC Press;  2005.  [31] Lewis F.L., Vrabie D., Syrmos V.L. Optimal Control. John  Wiley & Sons; 2012.  [32] Alamo T., Ferramosca A., González A.H., Limón D., Od-  loak D. A gradient-based strategy for integrating real time opti-  mizer (RTO) with model predictive control (MPC). IFAC Proc.  2012;45(17):33–38.  [33] Perelson A.S., Kirschner D.E., De Boer R. Dynamics of HIV  infection of CD4+ T cells. Math. Biosci. 1993;114(1):81–125.  [34] Legrand M., Comets E., Aymard G., Tubiana R., Katlama  C., Diquet B. An in vivo pharmacokinetic/pharmacodynamic  model for antiretroviral combination. HIV Clin. Trials.  2003;4(3):170–183.  [35] Perelson A.S., Ribeiro R.M. Modeling the within-host 

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Kelley

J.L.

General

Topology,

Dover

Books

on 

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Mathematics. Dover Publications; 2017.  [47] Khalil H.K., Grizzle J.W. Nonlinear Systems. Upper Saddle River, NJ: Prentice Hall; . 2002;3 vol.   

a

Although similar results are given in Ref. [22], uniqueness was not proved.  b According to Ref. [16], the virus spreads in the host if it shows a positive  derivative in some future time. In other words, the virus does not spread in  the host, if c The notation

for all future time.  indicates the distance from x to the set

.  d Given two different attractive sets in

, that is, 

with the same DOA, one must be 

contained in the other. So, the family of all attractive sets in

with the same 

DOA is a totally ordered set under the set inclusion (nested family). An arbi-  trary (finite, countable, or uncountable) intersection of nested nonempty  closed subsets of a compact space is a nonempty compact set [46]. Then, if  one element of the family is bounded, and therefore compact, the inter-  section of all the family is a nonempty compact set. This set is the smallest  attractive set. 

469

  15: Statistical modeling to understand the COVID-19 pandemic    Carlos E. Rodríguez; Ramsés H. Mena    Department of Probability and  Statistics, IIMAS-UNAM, Mexico City, Mexico 

470

Abstract  We present two modeling ideas that aim at describing and under-  standing the effects and evolution of the COVID-19 pandemic in Mex-  ico. First, a new and straightforward statistical methodology is pro-  posed to model epidemic curves. The key here is to assume that the  times when a certain number of infected individuals are observed have  been censored, but follow a known probability distribution; the censor-  ship point is the most recent date for which a record is available. The  second idea exploits the information of patients identified as SARS-  CoV-2 positive in Mexico to understand the relationship between  comorbidities, symptoms, hospitalizations, and deaths due to the  COVID-19 disease. Using the presence or absence of these latter vari-  ables, a clinical footprint for each patient is created. The proposal con-  siders all possible footprint combinations resulting in a robust model  suitable for Bayesian inference.   

Keywords  Bayesian modeling; Correlation matrix; Density estimation; Multivariate  Bernoulli distribution 

471

Acknowledgment  The authors are grateful for the support of PAPIIT-UNAM projects IV100220  and IG100221. 

472

1: Introduction  COVID-19 has placed an intense spotlight on statistical science. From the  outset of the pandemic, the number of cases, hospitalizations, and deaths  have been reported by governments and media outlets daily around the  world. In particular, the impact of sex, age, and comorbidities on infected  patients was recognized and followed closely. With the vaccination program  moving forward, time series of death and hospitalization rates desegregated  by age groups have been also incorporated into the communication strat-  egy. It would appear that statistics has never enjoyed such recognition as a  critical tool for supporting decision-making and providing public account-  ability.  According to the World Health Organization (WHO), most infected peo-  ple will develop mild to moderate illness and recover without hospital-  ization. The WHO divides COVID-19 symptoms in three groups: most com-  mon symptoms: fever, dry cough, and tiredness; less common symptoms:  aches and pains, sore throat, diarrhea, conjunctivitis, headache, loss of taste  or smell, a rash on skin, or discoloration of fingers or toes; and serious  symptoms: breathing difficulty or shortness of breath, chest pain or pres-  sure, and loss of speech or movement [1].  People of any age who have underlying medical conditions, such as  hypertension and diabetes, have shown worse prognosis [2]. Diabetic pa-  tients have increased morbidity and mortality rates, which have been linked  to more hospitalization and intensive care unit admissions [3]. People with  chronic obstructive pulmonary disease (COPD) or any respiratory illnesses  are also at higher risk for severe illness from COVID-19 [4].  From the first reports from China, a sex imbalance with regard to the  fatality rate of COVID-19 patients has been detected. Case fatality rates re-  ported in China, Italy, Spain, France, Germany, and Switzerland support the  view that a consistent phenomenon is operating, accounting for a higher  case fatality in men. Such observation is independent of country-specific  demographics and testing strategies [5]. Age has also been identified as a  variable with high impact over the mortality rate of COVID-19 cases. All age  groups appear to have significantly higher mortality compared with the im-  mediately younger age group [6].  Many modeling ideas have been used to track, understand, and forecast  several aspects of the COVID-19 pandemic. In Mexico, for example, 

473

Antonio-Villa et al. [7] assessed the impact of individual and municipal-level  social inequalities; Azanza and Hernández-Vargas [8] evaluated different  quarantine lifting strategies aiming to find the best alternative and avoid a  high death toll; Santana-Cibrian et al. [9] used key high transmission dates  for the year 2020 to create scenarios of the evolution of the pandemic in  several states of Mexico for 2021; Mas and Pérez-Vega [10] analyzed spa-  tiotemporal patterns of the epidemic and found that COVID-19 pandemic in  Mexico is characterized by clusters evolving in space and time as parallel  epidemics; and Capistran et al. [11] implemented a SEIRD-type model to pre-  dict hospital occupancy in several metropolitan areas of Mexico, among  many others.  This chapter showcases two simple modeling ideas that aid to describe  and understand the evolution of the COVID-19 pandemic in Mexico. Specif-  ically, Section 2 describes a novel and straightforward statistical method-  ology to model an epidemic curve, without the epidemiological context of a  compartmental model. The central idea is to assume that the times when a  certain number of infected individuals is observed have been censored, but  follow a known probability distribution; the censorship point is the latest  date for which a record is available. In Section 3, information of over 1.6 mil-  lion patients identified as SARS-CoV-2 positive in Mexico is used to under-  stand the relationship between comorbidities, symptoms, hospitalizations,  and deaths due to the COVID-19 disease. Some concluding remarks are de-  ferred to Section 4. 

474

2: Epidemic curves via censorship  An epidemic curve is a statistical chart that displays the course of an epi-  demic by plotting the number of infections according to the time of onset.  Epidemic curves can be constructed in many different ways, namely via  solutions of nonlinear ODEs, as those induced by the classical Kermack-  McKendrick [12] SIR model, simplistic approaches such as the logistic and  Richards curves [13], Bayesian hierarchical models [14], or even with ensem-  ble forecasts of various approaches [15].   

2.1: Censorship idea  During the course of an outbreak or epidemic, data about the number of  daily new infected individuals are collected and carefully monitored. Clearly,  the evolution of the number of infected individuals is partially observed,  namely at most information until the present day is available. Thus, we treat  the last available record's date as censoring threshold.  Let I(t) be the number of infected cases at day t, for

, thus, the 

censorship threshold will be the last infected case at day tk. The total num-  . As time is  ber of infected cases until this latter date is then measured in days, our infection dates dataset can be disaggregated as   

  namely a vector of dimension m1. Infections do not occur in a discrete—  daily—manner as in , but rather in different moments during a given day.  However, infection counts are reported as an aggregation summary of the  whole day. Given this, one could try to distribute the number infections dur-  ing the 24 h of a day, however, doing so neither impact our inference nor re-  solves the uncertainty about the exact infection time.   

2.2: The basic model: First wave  Let

and assume the dates of the infected cases, 

    are a sample from some parametric probability distribution with density 

475

f(t|θ), for t > 0 and some parameter θ. We will denote with F(t|θ) the  corresponding cumulative distribution function (cdf). Furthermore, assume  there is censorship and only

is observed, that is, we do not observe

. 

With these assumptions and following a Bayesian approach, the full joint  distribution is given by [e.g., Ref. 16, p. 176]   

  (1)    where p(θ) and p(m2) are prior distributions for the parameters, θ, and  , but  the number of missing observations, m , respectively. Observe 2 and  for the sake of simplicity in notation we will stick to .   

2.3: Approximating I(t)  To approximate the number of infected cases I(t), we scale the density func-  tion that describes the distribution of dates, that is,      (2)    where

is the area under the curve of observed cases. 

Observe the scaling factor

forces the area under the left tail of the 

density to match the area of the scaled curve of the observed cases. This is  obtained integrating both sides of expression (2), that is,   

476

  The last approximation is valid if and only if

. 

 

2.4: Further waves of the outbreak  It is straightforward to generalize Eq. (1) to describe a second or more  waves of the original outbreak, this can be done via mixture models, that is,      (3)    where ν denotes the number of outbreaks, the mixture weights w =  ν ν {wj}j=1 add up to one, and θ = {θ}j=1 are the parameters identifying  the mixture components.  Hence, the distribution of the times in jth outbreak is modeled via a  scaled density w f(y|θ ), where w represents the weight of such outbreak.  j j j and , are  For inference purposes, two vectors of latent variables, introduced, where zi indicates the outbreak were the observed time yi be-  longs, and d is the same but for the censored x s. The joint distribution of  i i the model is then given by   

  where

,

,

,

 

 

2.5: Informative priors  With an infectious disease like the one at issue, it is pertinent to use infor-  mative priors over the parameters of the model.  As ν is fixed and represents the number of waves of the outbreak, we can  use information about the range of days each wave occurred to set the  ν parameters of the priors over θ = {θ}j=1 . Also, any information available  477

about the current wave is used to set the prior over m , which is a sensitive  2 parameter.   

2.6: Example  A half-truncated normal distribution was used to model the times of each  wave, that is,

where

. Conjugate normal and 

gamma distributions were used as priors over the means and precisions, re-  spectively. To set the hyperparameters, information of each wave was used.  Hence, breaking the symmetry of the posterior distribution and avoiding  identifiability problems. A negative binomial is used as the prior over the  censured observations m . In this case, only the number of cases that be-  2 long to the last wave is used to set the negative binomial parameters. With  these specifications, and following the ideas described later, epidemic  curves to describe the evolution of the COVID-19 pandemic in Japan (v = 5)  a and Mexico (v = 3) were generated. These are shown in Fig. 1.   

478

Fig. 1 Epidemic curves via censured ideas. (A) Japan and (B) Mexico.    In both cases, a good fit is observed. It is important to mention that since  we have analytical expressions for the curve that describes the number of  cases, I(t), it is straightforward to obtain any statistical summary of interest. 

479

3: COVID-19 footprint and mortality  In this section, we use the available data from the COVID-19 pandemic in  Mexico to gain insight into the COVID-19 disease. The objective is to under-  stand the relationship between comorbidities, symptoms, mortality risk, and  hospitalization.   

3.1: Data and variables  The source of information for this analysis is the database of the National  Epidemiological Surveillance System for monitoring possible cases of  COVID-19 in Mexico (SINAVE/SISVER for its acronym in Spanish), coordi-  nated by the Secretary of Health (Spanish: Secretaría de Saludb ).  The SINAVE/SISVER platform considers cases that are suspected of  COVID-19. People who have had flu like symptoms, or that believe to have  been infected with the SARS-CoV-2 virus, are entitled to attend any public or  private health service in Mexico, and after an initial examination is sus-  pected to suffer from the COVID-19 disease, are registered on this database.  The database is updated daily with the suspected cases and 48 variables,  these

data

can

be

accessed

using

the

following

link: 

https://www.gob.mx/salud/documentos/datos-abiertos-152127. Additional  variables

are

available

upon

request

via

the

platform 

http://covid-19.iimas.unam.mx/, as described in Loza et al. [17].  By August 23, 2021, this database had information for a total of 8,143,484  suspected cases and 115 variables. For the relevance of this analysis, we will  use only variables meeting the following:    •Have a positive result for the presence of the SARS-CoV-2 virus in  a blood test. We work with the information of 1,755,985 positive  cases.  •Gender (female or male) and age group (four groups were used  60).  •Symptoms: fever (1), cough (2), ears pain (3), difficulty breathing  (4), irritability (5), diarrhea (6), chest pain (7), chills (8), headache  (9), muscle pain (10), joint pain (11), attack general state (12), nasal  discharge (13), increased respiratory frequency and depth (14),  vomiting (15), abdominal pain (16), conjunctivitis (17), blue color  lack of oxygen (18), and sudden onset of symptoms (19).  •Comorbidities: chronic kidney failure (1), COPD (2), heart disease 

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(3), diabetes (4), immunosuppression (5), hypertension (6), obe-  sity (7), smoking (8), and asthma (9).  •Disease outcomes: hospitalization (1) and death (2).    It is important to observe that almost all relevant variables in this study  are binary and indicate the presence or the absence of a symptom/  comorbidity or whether a patient has died, has been hospitalized, or not.  Hence, we need a model that can handle all these dichotomous variables.  Clinical knowledge about the COVID-19 disease has evolved during the pan-  demic and several countries are implementing vaccination programs. These  factors influence hospitalization and mortality rates of the disease across  time. To have a glance of their impact in Mexico, we have included the onset  date of symptoms in the analysis. We use this variable to select COVID-19  positive patients by month to perform some of the analysis as well as to  have an idea of the evolution through time.   

3.2: The model  Let Yk be a k-dimensional random vector of possibly correlated Bernoulli  random variables and y be a realization of it. A way to model such random  k object is via the multivariate Bernoulli distribution, where its mass proba-  bility function can be written as   

  (4)    where

and

denotes to the lth parameter repre- 

senting the probability of the lth combination of the k binary outcomes.  denotes the matrix with 2k rows of possible outcomes of  Here, the random vector Y and k outcome. For each

as the k dimensional vector of the lth possible 

, the marginal probability of Yj (element of Yk)  is given by the Bernoulli distribution 

    (5)    where 

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  (6)    Furthermore, the conditional distribution of a subset of Y , namely Y for s 1), low risk (r100 < 50%), (III) high transmission (Rt > 1), high risk (r100 >  50%), and (IV) low transmission (Rt < 1), high risk (r100 > 50%). Two dif-  ferent dates of the epidemic are shown 3 weeks apart (October 10 and 31). In  general, the trajectory of the states goes in a counter-clockwise direction  moving from I →II →III →IV although, in high transmission events, the  states may move back and forth between quadrants III and IV. One can see  from the graphs that in 3 weeks, the epidemic in most states has entered re-  gion I with the exception of one (Baja California), which is still in region IV  with high risk although reduced Rt. Since risk depends on the number of active cases and the depletion of these depends on the size of the active cases  pool, then even if R is slightly below 1, high-risk events may still occur.  t Lastly, by late October, the neighboring states of Chiapas and Tabasco in  Southeast Mexico have increased their R with the possibility of experi-  t encing an increase of their risk in the following weeks (passing from II  →III).   

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Fig. 6 The R t -risk plane for two different weeks for all the federal  states of Mexico. (A) October 10 and (B) October 31. The x -axis is the  reproductive number and the y -axis is the risk of infection in a  gathering of 100 persons. The black horizontal and vertical lines divide  the regions at R = 1 and risk = 50 % . The R is the average of two  t t values obtained with different methodologies as described in the text. 

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6: On public trust in science  Science and mathematical models were important and popular keywords of  the governmental discourse during the first quarter of 2020. In the United  States, Kreps and Kriner [13] report results on credibility and public trust in  science, in particular, in mathematical models. Apparently, the perception of  the American people, in the early stages of the epidemic, was framed by the  ideological and political divide of the country (Republican vs. Democrat).  This situation is not directly applicable to the Mexican population given the  different political factors that play within the Mexican context. In Mexico, we  have a particular situation. A new government was elected in 2018 with a  political agenda based on the so-called fourth transformation of the republic  that aims to recover the country from the neoliberal policies of the past  (roughly from 1970 to 2018). In this context, the CONACyT (National Coun-  cil of Science and Technology) has aligned with this view and has publicly  condemned the “neoliberal science” of that same past, that in its scientific  sector was, in fact, lead by the same CONACyT. Regardless of the political  or ideological virtues and failures of this scientific view, the management  and public policies for the pandemic have developed within this framework.  A brief summarized perspective of the main points of that development is  given below:    •In Mexico, there is a very popular president whose political party  has the majority in Congress. The opposition parties are weak,  many in disarray (there is a long historical reason for this). This has  created a political vacuum where practically without organized polit-  ical opposition and counterbalance (except the press) the govern-  ment, not the scientists, had the principal voice when speaking  about mathematical models and epidemiological issues in the first  quarter of 2020 (although the academic, expert criticism from the  scientific quarters has dramatically increased since the second half  of 2020).  •In Mexico, one “official” mathematical model was made public at  the start of the epidemic. From very early on in the pandemic and  until late May, this model was largely used with a political rather  than a scientific aim. The model notoriously failed in its predictions  but notwithstanding this and the opacity of its construction, the  lack of expert scientific revision and analysis of its structure and 

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assumptions, it continued to be used to inform the government  and the nation on the fate of the epidemic. Later, it was clear that  the uncertainty of the data was not the only cause of its failure:  inadequate assumptions, misguided parametrization, and limited  advice in epidemiological dynamics were also key factors.  •Media reporting about scientific issues is still a fringe activity in  Mexico that requires strengthening and growth. Very quickly, it was  sadly obvious that we do not have in the country enough science  media reporters nor enough number of trustworthy media outlets  to expose and divulge what scientists think about the scientific  claims and opinions of the government. “Science” and “scientific”  were words in the vocabulary of the ones in charge of the epidemic,  frequently using these terms not so much to ensure people of the  adequacy of their policies, but rather to dismiss any criticism, thus  indirectly (negatively) qualifying it as “unscientific.”  •At the early stages of the epidemic (before June), science and  mathematical modeling had some authoritative meaning for the  general population and the political class. However, the official  model was not published nor put in an open archive before June, it  was a black box only known to the handful of people making the  predictions. The predictions on peak and epidemic ending were  largely inaccurate but stubbornly embraced by the public health au-  thorities. The government strategy, to counterbalance the growing  criticism of the opacity of the “scientific model,” resorted to give  newspaper and TV interviews to defend the “science” behind the  model and other related issues. Later, it even claimed that the  model could not be publicly released because it was “CONACyT  intellectual property.”  •The strategy did not quite work. By late April and the whole of May,  the newspapers, from the left and the right, became sort of “edito-  rial boards” of epidemic models and model criticism. This triggered  a colorful and very politically charged fight between the contestants.  A handful of models were “published” in the newspapers. This  quickly generated a lot of widespread skepticism in the general pub-  lic toward mathematical models and the “scientific” policies de-  rived from them. However, the main thrust of support of 

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governmental policies regarding the management of the pandemic,  included the mathematical model and other scientific products, has  been largely based on political factors: approval ratings of the pres-  ident have remained above 60% during 2020 and 2021.  •To date, the words “model” and “scientific” have significantly de-  creased its appearance in the political discourse, with infrequent and brief mentions except in the case of vaccines.  •On May 1, 2020, an open letter was published in Science [59] call-  ing for transparency in COVID models. The call pointed out that the  open exchange of knowledge, at a time of crisis, demanded to  openly share knowledge, expertise, tools, and technology. The pan-  demic has been, after all, a national as well as international emer-  gency. This call was largely ignored by the Mexican governmental  scientific group in charge of the “scientific management” of the  pandemic. The “scientific model” remained a black box for the rest  of the month. 

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7: Lessons learned in COVID-19  The academic analysis on the Mexican Government management of the epi-  demic has produced the term and concept of Punt politics described as a  particular strategy of populist democratic leaders to defer, either by omis-  sion or commission the national stewardship of health systems to subna-  tional governments [31]. Although it is undeniable that populism and the  concomitant prioritization of political gain versus evidence-based decisions  have been indeed significant in Mexico and elsewhere, the causes of the  behavior may go beyond populism. The COVID pandemic has thrown the  world and every national government, into deep political, health, and eco-  nomic crisis. Governments, in general, had no clue on how to manage the  very rapid development and changing factors related to the epidemic growth  and impact. In a crisis such as this, uncritical support for the particular  strategy adopted by governments to resolve the crisis may appear, an atti-  tude which, on its own, diminishes the credibility of criticisms from political  adversaries or the media, particularly in polarized societies [60]. Besides,  the pandemic has been dealt with differently in different parts of the world.  In particular, mid- or low-income countries have problems associated with  limited economic capability to implement many of the recommendations that wealthier countries can apply and this naturally lead to the prioritization  of strategies summarized in the dichotomy: public health or economic via-  bility? Important to frame a sound strategy of mitigation and control is also  the past experience of different countries, on epidemic events. The expe-  rience of African countries, in particular, highlights the importance of infor-  mation sharing and collaborative scientific and policy efforts across coun-  tries to combat the spread of infectious diseases and minimize the costs of  pandemics in the region [61].  Well-designed and appropriately parameterized mathematical models are  primary tools for effective policy making and, for this, open data collection  and data sharing is a necessary condition. The wealth of problems, and at-  tempted solutions, that the COVID pandemic has presented everywhere,  highlight the deep differences that exist between different countries in eco-  nomic constraints, social inequalities, access to information and resources,  and research and development opportunities. Quoting a recent report [62]  of the Johns Hopkins University:   

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The COVID-19 pandemic has revealed that a true, end-to-end research and  development […] and response ecosystem—meaning, one that develops, pro-  duces, and delivers needed vaccines to global populations in a rapid and equi-  table fashion—remains an elusive goal. Most LMICs have been unable to ac-  quire and administer a sufficient supply of COVID-19 vaccines, and the dearth of  vaccines and limited capacity to deliver them are prolonging the pandemic and  contributing to destabilizing economies and societies around the world. […] The  consequences of this deeply inequitable global response extend beyond the  COVID-19 pandemic. Global initiatives to prepare for and respond to future  pandemic threats cannot succeed if LMIC governments believe they will be the  last to benefit from vaccines produced as a result of improvements in global dis-  ease surveillance, increased sample sharing, or expedited vaccine R&D.  A mathematical model is an approximation to reality constructed with a  well-defined purpose. The COVID pandemic has put on the forefront of the  epidemic fight the power of mathematical models as strategic tools to  understand the basic science behind its immunological and epidemiological  dynamics, and frame the design of public policies to control and mitigate  the epidemic. In particular, as has been discussed in this chapter, models  have contributed to clarify the role of the NPIs in the initial growth of the  epidemic, the role of superdispersion events in outbreak generation, the  effectiveness of the use of face masks, the dynamic nature of the generation  interval, the impact of habitat and biodiversity destruction in spillover  events, the correct estimation of the incubation period of the disease from  the measurement of viral loads in patients, the selection pressures involved  in the generation of variants of concern and probably vaccine escape mu-  tants. This list is a small (and biased) sample of the important practical and  theoretical problems, where mathematical models have contributed to ex-  plain, clarify, or advance. However, the recent history in the epidemic man-  agement has also confirmed a fundamental fact: the strategic use of mathe-  matical modeling in medicine and public health is a fundamentally multidis-  ciplinary activity that requires critical judgment in its construction and also  when interpreting model results in the meaning of both the multiple param-  eters involved and the validity of projections. There have been notable suc-  cesses in the application of mathematical models to understand different  factors, scales, and behaviors of the epidemic but, likewise, there have been  failures, many arising by the political use of model outputs but others 

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having their roots on the incomplete knowledge of the basic theory of the  biology of the virus and the epidemic, and on an uncritical generalization  and interpretation and, finally, on the abuse of statistics conducive to the  prioritization of parameter estimation and statistical inference to fit models  to data (using an ample variety of methodologies) but forgetting that the  epidemic is essentially a biological phenomenon which, in turn, is the ex-  pression of both the population dynamics of the pathogen and human so-  cial behavior, which imprints high variability, an inherent heterogeneity, and  significant uncertainty to its phenomenological expression in both time and  space. 

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Index    Note: Page numbers followed by f indicate figures and t indicate tables.    A    Admissible treatments 267  Africa     •

SARS-CoV-2  

  •

cumulative incidence rate 10, 10f 

•

cumulative mortality rate in 13, 13f 

•

lethality rate 16f, 17 

•

weekly incidence rate in 8, 9f, 11f 

•

weekly mortality rate 12f 

  Agent-based model (ABM), COVID-19 169–170    •

anxiety, level of 146, 161f 

•

assumptions and scenario 162–163 

•

computational analysis  

  •

epidemic with both PI and NPI 168 

•

epidemic without interventions 164 

•

epidemic with PI 164–168 

  •

nonpharmaceutical interventions assessment 146–147 

  •

deployment model 147–148 

•

postpandemic work activities and workers well-being 147–158 

  •

one-at-a-time sensitivity validation analysis 158–162 

•

technological NPI 146, 152 

 

531

Akaike Information Criterion (AIC) 253–255, 288  Alpha variant 238–239, 242–243  America     •

SARS-CoV-2  

  •

cumulative incidence rate 10, 10f 

•

cumulative mortality rate in 13, 13f 

•

lethality rate 16f, 17 

•

weekly incidence rate in 8, 9f, 11f 

•

weekly mortality rate 12f 

  Antiviral treatments, SARS-CoV-2 mathematical models     •

coinfections 262 

•

disease-free equilibrium 262 

•

effectiveness characterization 272–275 

•

PD, inclusion of 272 

•

PK, inclusion of 275–277 

•

stability theory 281–283 

•

target-cell-limited model 263–272 

  •

equilibria characterization and stability 267–270 

•

infections severity and antiviral treatment effect 271–272 

•

reduced system model and generic infection evolution 266–267 

•

SARS-CoV-2 in-host mathematical model 263–266 

  •

therapies, control strategy to 278–281 

  •

numerical simulations 279–281 

•

terminal healthy cells count, behavior of 283–284 

  Artificial intelligence (AI) 120–121  Artificial neural networks 138–139  Asia    

532

•

SARS-CoV-2  

  •

cumulative incidence rate 10, 10f 

•

cumulative mortality rate in 13, 13f 

•

lethality rate 16f, 17 

•

weekly incidence rate in 8–9, 9f, 11f 

•

weekly mortality rate 12f 

  Astra Zeneca vaccine 210  Asymptotic stability (AS) equilibrium 262, 270  Attractivity analysis 269  Autoregressive integrated moving average (ARIMA) modeling 120    B    Backward bifurcation 228–229  Basal transmission probability (BTP) 113  Basic vaccination model, COVID-19 in United States 222–235    •

combined impact vaccination and NPIs 231–234 

•

disease-free equilibrium, asymptotic stability of  

  •

herd immunity threshold 225 

•

next-generation operator method 223–225 

  •

endemic equilibria, existence of 227–229 

•

model fitting and parameter estimation 229–230 

•

numerical simulation of 230–231 

•

sensitivity analysis 234–235 

•

with waning immunity 235–238 

  Behavior with emotions and norms (BEN) architecture 152  Bernoulli distribution 293  Biases, in epidemiological analysis     •

pathogen, natural history of 252–253, 288–291  533

•

testing system 253–255, 288–289 

  C    CanSino vaccine 210  Case fatality rate (CFR) 255–257, 289, 303–304, 312–314  Case fatality ratio (CFR) 26  Cellular automata sensitive-undiagnosed-infected-removed (CA-SUIR)  model 123–124, 128  Center manifold theory 228–229  Ceteris paribus115  China, lockdown strategy in 21  Chronic obstructive pulmonary disease (COPD) 287  Constant control 136  Contact tracing (CT) 115–116    •

complex networks 110–111 

•

contagion trees 113–114 

•

epidemiological simulation 112–113 

•

heterogeneity characteristics, contact networks 109–110 

•

methods  

  •

fundamentals and assumptions 110–111 

  •

Mexico City, contact network of 111 

•

network epidemiology 110 

  •

dynamic models of 110 

  •

process 114 

•

random and network-guided sampling 115, 116t 

•

superspreading events (SSEs) 109 

  Continuous linear functions 50  Conv2LSTM neural networks 124–126, 125f  Convolutional neural networks (CNNs) 120–121  534

Coronavirus Disease 2019 (COVID-19) pandemic 2    •

Bluetooth-based contact-tracing applications 147 

•

deep learning SeeDeep learning 

•

incidence-rate curves of  

  •

averaging of growth curves within clusters 80–81 

•

country-level analysis in Europe 82–87, 86–87f 

•

methodology 77–81 

•

overview 73–77, 88–89 

•

preprocessing steps 78, 79f 

•

publicly available code 81 

•

shape metric 78–80 

•

shapes of curves 74–77, 75–76f 

•

state-level analysis for United States 81–82, 82–85f 

  Covidestim model 55–57, 207  CT SeeContact tracing (CT)    D    Decision support system (DSS) 147–148  Deep learning (DL), COVID-19 applications 120–121, 127–128    •

dynamical parameters via DL, extraction of 124–126 

•

epidemiological model-driven DL approach 122–127 

•

multiscale risk information 119–120 

•

nation-state-county-level risk warning 126–127 

•

nonpharmaceutical intervention strategies 119–120 

•

prediction tasks 121 

•

simulations via CA-SUIR model, preparation of 123–124 

•

spatial-temporal analysis 121–122 

  Delta variant 64–65, 94, 238–239, 242–243, 311–312  Delta wave 26 

535

Dirichlet distribution 294  Disability-adjusted life years (DALYs) 60–62  Discrete recurrent high-order neural network 138–139  Disease-free equilibrium (DFE) 192, 205–206  Dynamic Programming algorithm 80    E    Effective reproductive number 206  Elastic alignment algorithm 78  Elastic FDA 77  Emergency use authorization (EUA) 221–222  Emerging infectious diseases 257–258, 289–290    •

biases, in epidemiological analysis  

  •

pathogen, natural history of 252–253, 288–291 

•

testing system 253–255, 288–289 

  •

case fatality rate, temporal variation of 255–257, 289 

  Epidemic management 318–320    •

confronting challenge 305–308 

•

COVID-19 302 

  •

data 303–304 

•

prevention 302–303 

  •

features, in Mexico 312–314 

•

mathematical modeling 302–303 

•

prediction 302–303 

•

in science 315–318 

•

vaccination 308–312 

  Epidemics on networks (EoN) 110 

536

Epidemiological traffic light system (ETLS) indicator 153, 164, 169–170  Escape efficacy 96–97, 99  Escape-infection event 96  Escape mutants 94    •

infection with 98–99, 99f 

  Europe     •

country-level analysis in 82–87, 86–87f 

•

SARS-CoV-2  

  •

cumulative incidence rate 10, 10f 

•

cumulative mortality rate in 13, 13f 

•

lethality rate 16f, 17 

•

weekly incidence rate in 9f, 10, 11f 

•

weekly mortality rate 12f 

  European Centre for Disease Prevention and Control 23    F    Food and Drug Administration of the United States (FDA) 58–59  Force of infection 227–228  Friendship Paradox 111  Functional data analysis (FDA) 74    G    Generation time 19    H    Healthy Distance program 48–49 

537

H1N1 influenza 1    I    Impulsive control 136, 142–143, 142f  Infection, COVID-19 status     •

in-hospital dynamics, stochastic extension for  

  •

model description 188–189 

•

model fitting 189–190 

  •

intervention strategies, qualitative analysis of  

  •

model fitting 183–184 

•

scenarios 185–187 

  •

quarantine scenarios and hospital overload, dynamic model  of 176–187 

  •

code availability 191–192 

•

mathematical model 178–183 

•

modeling context 176–178 

  •

reproduction number, isolation time 194–196 

•

SEIR model parameters 180t 

•

without quarantine and isolation effects 192–194 

  Infection final size (IFS) 263  Infectious diseases 1  Institute of Health Metrics and Evaluation (IHME) 14  Inverse optimal impulsive control 136–139  Italy, lockdown strategy in 21    J    538

Jacobian matrix 205–206, 267–268  Johnson & Johnson vaccine 221–222, 226, 236–238    K    Kermack-McKendrick model 45, 50, 55    L    Lambert function 265  Latin hypercube sampling 234–235  Locally asymptotically stable (LAS) 225  Local stability analysis 269–270  Lockdowns 307–308  Long short-term memory (LSTM) 120–121  Low- and middle income countries (LMICs) 301–304  Lyapunov function 262  Lyapunov theory 281    M    Mathematical models 2    •

reopening schools, in Mexico (see Mexico, school reopening) 

  Mexican Institute of Social Security (IMSS) 175  Mexico, COVID-19 disease in     •

beginning of 45–48 

•

epidemic curve in 45, 46f 

•

epidemiological indicators for 57–61f 

•

first nonpharmaceutical interventions 48–52, 49f, 51f 

•

implementation

and

relaxation

of

nonpharmaceutical 

interventions 48–55 

539

•

mathematical model 50–52, 52f 

•

overview 43–45 

•

relaxation mitigation measures 53–55, 54f 

•

total number of infected people in 55–58, 56f 

•

vaccination 58–65 

  •

beginning of 59–64, 62–64f 

•

new variants and booster shots 64–65 

  Mexico, reopening of schools     •

asymptomatic individuals 201 

•

controlled external epidemic 219 

•

COVID-19 infections 199–200, 219 

•

discrete time SEIRS epidemiological model 200–201 

•

effective contact probability, computation of 205–207 

•

epidemic scenarios  

  •

controlled epidemic with sudden peak in cases 216 

•

declining epidemic with sudden increase 214–215 

•

in growing external epidemic 211–213 

•

risk level, contact with infected individual at school 217–218 

  •

false-positive compartment 201 

•

model compartments 202–203 

•

model parameters 203–205 

•

presymptomatic individuals 201 

•

Queretaro city 207–211 

  •

confirmed cases, number of tests according to 210 

•

daily tests, constant number of 209 

•

fixed proportion of people in school, screening according to 210 

•

parameters and probabilities, set up of 210–211 

•

people in school, number of tests according to 209 

  •

Reed-Frost model 199–200 

540

•

true positive individuals 201 

  Moderna vaccine 221–222, 226, 231–232, 236–237, 242–243, 245  MSEIR epidemiological model, for spread of COVID-19 dynamics 135,  135f    •

behavioral dynamics of 140–141f, 141 

•

description of 134t 

•

for five-node network 138f 

•

inverse optimal impulsive control 136–139 

  •

complex network for 137–138 

•

control and identification of epidemic models 136–137 

•

discrete recurrent high-order neural network 138–139 

  •

momentum dynamics of 138 

•

overview 133–134 

•

simulation results 139–143 

  Multidisciplinary Node of Applied Mathematics (NoMMA) 207  Multivariate Bernoulli model 294    N    National Epidemiological Surveillance System 292  National Institute of Allergy and Infectious Diseases (NIAID) 1  New Normality 53  New variants, and booster shots 64–65  Next-generation matrix 205–206  Next-generation operator method 223–225  Nonpharmaceutical intervention (NPI) 145–147, 176, 221–222, 305–307    •

postpandemic work activities and workers well-being  

  •

decision support system (DSS) 147–148 

•

deployment model 147–148 

541

•

epidemiological model 149–151 

•

recreational activities 149 

•

technological NPI model 152 

•

vaccination model 151 

•

worker model 152–158 

•

work schemes 149 

  N-terminal domain 95    O    Oceania     •

SARS-CoV-2  

  •

cumulative incidence rate 10, 10f, 13f 

•

cumulative mortality rate in 13, 13f 

•

lethality rate 16f, 17 

•

weekly incidence rate in 8–9, 9f, 11f 

•

weekly mortality rate 12f 

  Omicron variant 64–65, 94  Ordinary differential equations (ODEs) 133  Oxford-AstraZeneca vaccine 221–222    P    Partial rank correlation coefficients (PRCCs) 234–235  Passive immunity 134  Pfizer vaccine 221–222, 226, 231–232, 236–238, 242–243, 245  Pharmaceutical interventions (PI) 145  Phase-amplitude separation 78  Phylodynamics 25  Phylogenetics 25 

542

Poisson distribution 252, 287  Postpandemic work activities and workers well-being, NPI     •

decision support system (DSS) 147–148 

•

deployment model 147–148 

•

epidemiological model 149–151 

•

recreational activities 149 

•

technological NPI model 152 

•

vaccination model 151 

•

worker model 152–158 

  •

anxiety 152–156, 160f 

•

NPI adoption 156–157 

•

process 157–158, 159f 

•

routine 156 

  •

work schemes 149 

  Punt politics 318    Q    Quarantine scenarios and hospital overload 176–187    •

code availability 191–192 

•

infected symptomatic individuals 181 

•

mathematical model 178–183 

  •

hospital overload 180–181 

•

SEIR model 178, 178f, 180t 

  •

modeling context 176–178 

•

periodic isolation cycles 181–182 

•

susceptible, asymptomatic and exposed individuals, isolation  of 181 

•

time-dependent effect, NPI on infection rate 182–183 

543

R    Receptor-binding domain 95  Recurrent high-order neural networks (RHONNs) 138–139  Recurrent neural network (RNN) 120–121  Reinfections 22  Relaxation mitigation measures 53–55, 54f  Remdesivir 221–222    S    Sampling bias 111  SARS-CoV-2 pandemic     •

case fatality ratio (CFR) 26 

•

continents 18t 

•

cumulative number of 18t 

•

cumulative rate of 18t 

•

epidemiological parameters of 19–20 

•

escape mutants 94 

  •

infection with 98–99, 99f 

  •

excess deaths 27–31t 

•

fundamental assumption and justification 96–97 

•

global epidemiology 8–19, 15f 

•

impact of 25 

•

local epidemics 96, 97f 

•

mathematical models of 21 

•

mitigation strategies 20–21 

•

model parameters 101 

•

orphanhood 31–39t 

•

overview 7–8, 94 

•

reinfections 22 

•

transmission of wildtype virus 97–98 

•

vaccine escape 95 

  544

•

risk indicators 100 

  •

variants 23–25 

  Shape analysis 74–77  Shape metric 78–80  SINAVE/SISVER platform 292  Square root velocity function (SRVF) 80  Statistical modeling, COVID-19 pandemic 296–298    •

epidemic curves via censorship 288–291 

  •

approximate infected cases, number of 289–290 

•

basic model, first wave 289 

•

hyperparameters 290 

•

informative priors 290 

•

negative binomial parameters 290 

•

outbreak, waves of 290 

  •

footprint and mortality 291–296 

  •

Bayesian inference 293–294 

•

COVID-19 footprint 294 

•

data and variables 292 

•

hierarchical clustering algorithm 296 

•

model 293 

  Stay-at-home strategy 305–307  Susceptible-exposed-infected-recovered (SEIR) model 47f, 112–113  Susceptible-infectious-recovered (SIR) model 2, 122–123, 133–134, 143,  175–176  Susceptible-undiagnosed-infected-removed (SUIR) model 122–123    T    Time-warping function 80  545

U    United States, COVID-19 pandemic, incidence-rate curves for 81–82,  82–85f  US Food and Drug Administration (FDA) 221–222    V    Vaccination 58–65  Vaccination, COVID-19 in United States 243–245    •

basic model 222–235 

  •

combined impact vaccination and NPIs 231–234 

•

disease-free equilibrium, asymptotic stability of 223–226 

•

endemic equilibria, existence of 227–229 

•

model fitting and parameter estimation 229–230 

•

numerical simulation of 230–231 

•

sensitivity analysis 234–235 

•

with waning immunity 235–238 

  •

modeling dynamics and SARS-CoV-2 variants 238–243 

  •

resident strain equations, dynamics of 239–240 

•

variant strain equations, dynamics of 240–243 

  •

RNA virus, SARS-CoV-2 221 

  Vaccine efficacy 96–97, 99  Vaccine escape 95–96, 98, 101–103, 102f    •

risk indicators 100 

•

in SARS-CoV-2 95 

  Variants of concern (VOC) 8, 26, 64–65  Variants of interest (VOI) 64–65 

546

Virus load peak (VLP) 263    W    Waning immunity 235–238    •

disease-free equilibrium, asymptotic stability of 236–238 

  •

natural and vaccine-derived immunity 236–238 

  Wildtype virus, transmission of 97–98  World Health Organization (WHO) 7, 43, 146, 261, 287 

547