Mathematical epidemiology

Mathematical Epidemiology

Part I. Introduction and General Framework

1. A Light Introduction to Modelling Recurrent Epidemics

1.1. Introduction

1.2. Plague

1.3. Measles

1.4. The SIR Model

1.5. Solving the Basic SIR Equations

1.6. SIR with Vital Dynamics

1.7. Demographic Stochasticity

1.8. Seasonal Forcing

1.9. Slow Changes in Susceptible Recruitment

1.10. Not the Whole Story

1.11. Take Home Message

References

2. Compartmental Models in Epidemiology

2.1. Introduction

2.1.1. Simple Epidemic Models

2.1.2. The Kermack-McKendrick Model

2.1.3. Kermack-McKendrick Models with General Contact Rates

2.1.4. Exposed Periods

2.1.5. Treatment Models

2.1.6. An Epidemic Management (Quarantine-Isolation) Model

2.1.7. Stochastic Models for Disease Outbreaks

2.2. Models with Demographic Effects

2.2.1. The SIR Model

2.2.2. The SIS Model

2.3. Some Applications

2.3.1. Herd Immunity

2.3.2. Age at Infection

2.3.3. The Interepidemic Period

2.3.4. "Epidemic" Approach to the Endemic Equilibrium

2.3.5. Disease as Population Control

2.4. Age of Infection Models

2.4.1. The Basic SI* R Model

2.4.2. Equilibria

2.4.3. The Characteristic Equation

2.4.4. The Endemic Equilibrium

2.4.5. An SI* S Model

2.4.6. An Age of Infection Epidemic Model

References

3. An Introduction to Stochastic Epidemic Models

3.1. Introduction

3.2. Review of Deterministic SIS and SIR Epidemic Models

3.3. Formulation of DTMC Epidemic Models

3.3.1. SIS Epidemic Model

3.3.2. Numerical Example

3.3.3. SIR Epidemic Model

3.3.4. Numerical Example

3.4. Formulation of CTMC Epidemic Models

3.4.1. SIS Epidemic Model

3.4.2. Numerical Example

3.4.3. SIR Epidemic Model

3.5. Formulation of SDE Epidemic Models

3.5.1. SIS Epidemic Model

3.5.2. Numerical Example

3.5.3. SIR Epidemic Model

3.5.4. Numerical Example

3.6. Properties of Stochastic SIS and SIR Epidemic Models

3.6.1. Probability of an Outbreak

3.6.2. Quasistationary Probability Distribution

3.6.3. Final Size of an Epidemic

3.6.4. Expected Duration of an Epidemic

3.7. Epidemic Models with Variable Population Size

3.7.1. Numerical Example

3.8. Other Types of DTMC Epidemic Models

3.8.1. Chain Binomial Epidemic Models

3.8.2. Epidemic Branching Processes

3.9. MatLab Programs

References

Part II. Advanced Modeling and Heterogeneities

4. An Introduction to Networks in Epidemic Modeling

4.1. Introduction

4.2. The Probability of a Disease Outbreak

4.3. Transmissibility

4.4. The Distribution of Disease Outbreak and Epidemic Sizes

4.5. Some Examples of Contact Networks

4.6. Conclusions

References

5. Deterministic Compartmental Models: Extensions of Basic Models

5.1. Introduction

5.2. Vertical Transmission

5.2.1. Kermack-McKendrick SIR Model

5.2.2. SEIR Model

5.3. Immigration of Infectives

5.4. General Temporary Immunity

References

6. Further Notes on the Basic Reproduction Number

6.1. Introduction

6.2. Compartmental Disease Transmission Models

6.3. The Basic Reproduction Number

6.4. Examples

6.4.1. The SEIR Model

6.4.2. A Variation on the Basic SEIR Model

6.4.3. A Simple Treatment Model

6.4.4. A Vaccination Model

6.4.5. A Vector-Host Model

6.4.6. A Model with Two Strains

6.5. R[subscript o] and the Local Stability of the Disease-Free Equilibrium

6.6. R[subscript o] and Global Stability of the Disease-Free Equilibrium

References

7. Spatial Structure: Patch Models

7.1. Introduction

7.2. Spatial Heterogeneity

7.3. Geographic Spread

7.4. Effect of Quarantine on Spread of 1918-1919 Influenza in Central Canada

7.5. Tuberculosis in Possums

7.6. Concluding Remarks

References

8. Spatial Structure: Partial Differential Equations Models

8.1. Introduction

8.2. Model Derivation

8.3. Case Study I: Spatial Spread of Rabies in Continental Europe

8.4. Case Study II: Spread Rates of West Nile Virus

8.5. Remarks

References

9. Continuous-Time Age-Structured Models in Population Dynamics and Epidemiology

9.1. Why Age-Structured Models?

9.2. Modeling Populations with Age Structure

9.2.1. Solutions along Characteristic Lines

9.2.2. Equilibria and the Characteristic Equation

9.3. Age-Structured Integral Equations Models

9.3.1. The Renewal Equation

9.4. Age-Structured Epidemic Models

9.5. A Simple Age-Structured AIDS Model

9.5.1. The Reproduction Number

9.5.2. Pair-Formation in Age-Structured Epidemic Models

9.5.3. The Semigroup Method

9.6. Modeling with Discrete Age Groups

9.6.1. Examples

References

10. Distribution Theory, Stochastic Processes and Infectious Disease Modelling

10.1. Introduction

10.2. A Review of Some Probability Theory and Stochastic Processes

10.2.1. Non-negative Random Variables and Their Distributions

10.2.2. Some Important Discrete Random Variables Representing Count Numbers

10.2.3. Continuous Random Variables Representing Time-to-Event Durations

10.2.4. Mixture of Distributions

10.2.5. Stochastic Processes

10.2.6. Random Graph and Random Graph Process

10.3. Formulating the Infectious Contact Process

10.3.1. The Expressions for R[subscript 0] and the Distribution of N such that R[subscript 0] = E[N]

10.3.2. Competing Risks, Independence and Homogeneity in the Transmission of Infectious Diseases

10.4. Some Models Under Stationary Increment Infectious Contact Process {K(x)}

10.4.1. Classification of some Epidemics Where N Arises from the Mixed Poisson Processes

10.4.2. Tail Properties for N

10.5. The Invasion and Growth During the Initial Phase of an Outbreak

10.5.1. Invasion and the Epidemic Threshold

10.5.2. The Risk of a Large Outbreak and Quantities Associated with a Small Outbreak

10.5.3. Behaviour of a Large Outbreak in its Initial Phase: The Intrinsic Growth

10.5.4. Summary for the Initial Phase of an Outbreak

10.6. Beyond the Initial Phase: The Final Size of Large Outbreaks

10.6.1. Generality of the Mean Final Size

10.6.2. Some Cautionary Remarks

10.7. When the Infectious Contact Process may not Have Stationary Increment

10.7.1. The Linear Pure Birth Processes and the Yule Process

10.7.2. Parallels to the Preferential Attachment Model in Random Graph Theory

10.7.3. Distributions for N when {K(x)} Arises as a Linear Pure Birth Process

References

Part III. Case Studies

11. The Role of Mathematical Models in Explaining Recurrent Outbreaks of Infectious Childhood Diseases

11.1. Introduction

11.2. The SIR Model with Demographics

11.3. Historical Development of Compartmental Models

11.3.1. Early Models

11.3.2. Stochasticity

11.3.3. Seasonality

11.3.4. Age Structure

11.3.5. Alternative Assumptions About Incidence Terms

11.3.6. Distribution of Latent and Infectious Period

11.3.7. Seasonality Versus Nonseasonality

11.3.8. Chaos

11.3.9. Transitions Between Outbreak Patterns

11.4. Spectral Analysis of Incidence Time Series

11.4.1. Power Spectra

11.4.2. Wavelet Power Spectra

11.5. Conclusions

References

12. Modeling Influenza: Pandemics and Seasonal Epidemics

12.1. Introduction

12.2. A Basic Influenza Model

12.3. Vaccination

12.4. Antiviral Treatment

12.5. A More Detailed Model

12.6. A Model with Heterogeneous Mixing

12.7. A Numerical Example

12.8. Extensions and Other Types of Models

References

13. Mathematical Models of Influenza: The Role of Cross-Immunity, Quarantine and Age-Structure

13.1. Introduction

13.2. Basic Model

13.3. Cross-Immunity and Quarantine

13.4. Age-Structure

13.5. Discussion and Future Work

References

14. A Comparative Analysis of Models for West Nile Virus

14.1. Introduction: Epidemiological Modeling

14.2. Case Study: West Nile Virus

14.3. Minimalist Model

14.3.1. The Question

14.3.2. Model Scope and Scale

14.3.3. Model Formulation

14.3.4. Model Analysis

14.3.5. Model Application

14.4. Biological Assumptions 1: When does the Disease-Transmission Term Matter?

14.4.1. Frequency Dependence

14.4.2. Mass Action

14.4.3. Numerical Values of R[subscript 0]

14.5. Biological Assumptions 2: When do Added Model Classes Matter?

14.6. Model Parameterization, Validation, and Comparison

14.7. Model Application #1: WN Control

14.8. Model Application #2: Seasonal Mosquito Population

14.9. Summary

References

Suggested Exercises and Projects

1. Cholera

2. Ebola

3. Gonorrhea

4. HIV/AIDS

5. HIV in Cuba

6. Human Papalonoma Virus

7. Influenza

8. Malaria

9. Measles

10. Poliomyelitis (Polio)

11. Severe Acute Respiratory Syndrome (SARS)

12. Smallpox

13. Tuberculosis

14. West Nile Virus

15. Yellow Fever in Senegal 2002

Index