Contents. Mathematical Epidemiology 1 F. Brauer, P. van den Driessche and J. Wu, editors. Part I Introduction and General Framework

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1 Mathematical Epidemiology 1 F. Brauer, P. van den Driessche and J. Wu, editors Part I Introduction and General Framework 1 A Light Introduction to Modelling Recurrent Epidemics.. 3 David J.D. Earn 1.1 Introduction Plague Measles The SIR Model Solving the Basic SIR Equations SIR with Vital Dynamics Demographic Stochasticity Seasonal Forcing Slow Changes in Susceptible Recruitment Not the Whole Story Take Home Message 16 References 16 2 Compartmental Models in Epidemiology 19 Fred Brauer 2.1 Introduction Simple Epidemie Models The Kermack-McKendrick Model Kermack-McKendrick Models with General Contact Rates Exposed Periods Treatment Models An Epidemie Management (Quarantme-Isolation) Model 40 ix Bibliografische Informationen digitalisiert durch

2 2.1.7 Stochastic Models for Disease Outbreaks Models with Demographic Effects The SIR Model The SIS Model Some Applications Herd Immunity Age at Infection The Interepidemic Period "Epidemie" Approach to the Endemic Equilibrium Disease as Population Control Age of Infection Models The Basic SI*R Model Equilibria The Characteristic Equation The Endemic Equilibrium An SI*S Model An Age of Infection Epidemie Model 76 References 78 An Introduction to Stochastic Epidemie Models 81 Linda J.S. Allen 3.1 Introduction Review of Deterministic SIS and SIR Epidemie Models Formulation of DTMC Epidemie Models SIS Epidemie Model Numerical Example SIR Epidemie Model Numerical Example Formulation of CTMC Epidemie Models SIS Epidemie Model Numerical Example SIR Epidemie Model Formulation of SDE Epidemie Models SIS Epidemie Model Numerical Example SIR Epidemie Model Numerical Example Properties of Stochastic SIS and SIR Epidemie Models Probability of an Outbreak Quasistationary Probability Distribution Final Size of an Epidemie Expected Duration of an Epidemie Epidemie Models with Variable Population Size Numerical Example Other Types of DTMC Epidemie Models 121

3 xi Chain Binomial Epidemie Models Epidemie Branching Processes MatLab Programs 125 References 128 Part II Advanced Modeling and Heterogeneities 4 An Introduction to Networks in Epidemie Modeling 133 Fred Brauer 4.1 Introduction The Probability of a Disease Outbreak Transmissibility The Distribution of Disease Outbreak and Epidemie Sizes Some Examples of Contact Networks Conclusions 145 References Deterministic Compartmental Models: Extensions of Basic Models 147 P. van den Driessche 5.1 Introduction Vertical Transmission Kermack-McKendrick SIR Model SEIR Model Immigration of Infectives General Temporary Immunity 154 References Further Notes on the Basic Reproduction Number 159 P. van den Driessche and James Watmough 6.1 Introduction Compartmental Disease Transmission Models The Basic Reproduction Number Examples The SEIR Model A Variation on the Basic SEIR Model A Simple Treatment Model A Vaccination Model A Vector-Host Model A Model with Two Strains TZ O and the Local Stability of the Disease-Free Equilibrium TZ O and Global Stability of the Disease-Free Equilibrium References 177

4 *" Contents 7 Spatial Structure: Patch Models 179 P. van den Driessche 7.1 Introduction 17g 7.2 Spatial Heterogeneity ISO 7.3 Geographie Spread Effect of Quarantine on Spread of Influenza in Central Canada Tuberculosis in Possums Concluding Remarks 188 References.-~ Spatial Structure: Partial Differential Equations Models Jianhong Wu 8.1 Introduction Model Derivation Case Study I: Spatial Spread of Rabies in Continental Europe Case Study II: Spread Rates of West Nile Virus Remarks 202 References Continuous-Time Age-Structured Models in Population Dynamics and Epidemiology 205 Jia Li and Fred Brauer 9.1 Why Age-Structured Models? Modeling Populations with Age Structure Solutions along Characteristic Lines Equilibria and the Characteristic Equation Age-Structured Integral Equations Models The Renewal Equation Age-Structured Epidemie Models A Simple Age-Structured AIDS Model The Reproduction Number Pair-Formation in Age-Structured Epidemie Models The Semigroup Method Modeling with Discrete Age Groups Examples 223 References Distribution Theory, Stochastic Processes and Infectious Disease Modelling 229 Ping Yan 10.1 Introduction A Review of Some Probability Theory and Stochastic Processes Non-negative Random Variables and Their Distributions 231

5 xiii Some Important Discrete Random Variables Representing Count Numbers Continuous Random Variables Representing Time-to-Event Durations Mixture of Distributions Stochastic Processes Random Graph and Random Graph Process Formulating the Infectious Contact Process The Expressions for RQ and the Distribution of N such that RQ = E[N] Competing Risks, Independence and Homogeneity in the Transmission of Infectious Diseases Some Models Under Stationary Increment Infectious Contact Process {K{x)} Classification of some Epidemics Where N Arises from the Mixed Poisson Processes Tail Properties for N The Invasion and Growth During the Initial Phase of an Outbreak Invasion and the Epidemie Threshold The Risk of a Large Outbreak and Quantities Associated with a Small Outbreak Behaviour of a Large Outbreak in its Initial Phase: The Intrinsic Growth Summary for the Initial Phase of an Outbreak Beyond the Initial Phase: The Final Size of Large Outbreaks Generality of the Mean Final Size Some Cautionary Remarks When the Infectious Contact Process may not Have Stationary Increment The Linear Pure Birth Processes and the Yule Process Parallels to the Preferential Attachment Model in Random Graph Theory Distributions for N when {K(x)} Arises as a Linear Pure Birth Process 288 References 291 Part III Case Studies 11 The Role of Mathematical Models in Explaining Recurrent Outbreaks of Infectious Childhood Diseases Chris T. Bauch 11.1 Introduction The SIR Model with Demographics 300

6 11.3 Historical Development of Compartmental Models Early Models Stochasticity Seasonality Age Structure Alternative Assumptions About Incidence Terms Distribution of Latent and Infectious Period Seasonality Versus Nonseasonality Chaos Transitions Between Outbreak Patterns Spectral Analysis of Incidence Time Series Power Spectra Wavelet Power Spectra Conclusions 314 References Modeling Influenza: Pandemics and Seasonal Epidemics Fred Brauer 12.1 Introduction A Basic Influenza Model Vaccination Antiviral Treatment A More Detailed Model A Model with Heterogeneous Mixing A Numerical Example Extensions and Other Types of Models 345 References Mathematical Models of Influenza: The Role of Cross-Immunity, Quarantine and Age-Structure 349 M. Nufio, C. Castillo-Chavez, Z. Feng and M. Martcheva 13.1 Introduction Basic Model Cross-Immunity and Quarantine Age-Structure Discussion and Future Work 362 References A Comparative Analysis of Models for West Nile Virus M.J. Wonham and M.A. Lewis 14.1 Introduction: Epidemiological Modeling Case Study: West Nile Virus Minimalist Model The Question Model Scope and Scale Model Formulation 370

7 xv Model Analysis Model Application Biological Assumptions 1: When does the Disease-Transmission Term Matter? Frequency Dependence Mass Action." Numerical Values of 7^ Biological Assumptions 2: When do Added Model Classes Matter? Model Parameterization, Validation, and Comparison Model Application #1: WN Control Model Application #2: Seasonal Mosquito Population Summary 384 References 386 Suggested Exercises and Projects Cholera Ebola Gonorrhea HIV/AIDS HrV in Cuba Human Papalonoma Virus Influenza Malaria Measles Poliomyelitis (Polio) Severe Acute Respiratory Syndrome (SARS) Smallpox Tuberculosis West Nile Virus Yellow Fever in Senegal Index 403