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2015 | Buch

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An Introduction to Mathematical Epidemiology

verfasst von: Maia Martcheva

Verlag: Springer US

Buchreihe : Texts in Applied Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The book is a comprehensive, self-contained introduction to the mathematical modeling and analysis of infectious diseases. It includes model building, fitting to data, local and global analysis techniques. Various types of deterministic dynamical models are considered: ordinary differential equation models, delay-differential equation models, difference equation models, age-structured PDE models and diffusion models. It includes various techniques for the computation of the basic reproduction number as well as approaches to the epidemiological interpretation of the reproduction number. MATLAB code is included to facilitate the data fitting and the simulation with age-structured models.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

This chapter introduces the subject of epidemiology and includes a classification of infectious diseases based on their route of transmission. A number of definitions of key epidemiological concepts are given. The chapter introduces a brief history of the study and modeling of infectious diseases, highlighting the seminal contributions of Sir Ronald Ross. Finally, the chapter gives a description of the modeling process in biology and a list of the types of models used today in mathematical biology.

Maia Martcheva

Chapter 2. Introduction to Epidemic Modeling

Abstract

This chapter is the introductory chapter to epidemiological modeling. The chapter starts with the derivation of the Kermack–McKendrick SIR ODE epidemic model. Furthermore, it studies the basic mathematical properties of the model. The model is then fitted to data on influenza at an English boarding school, and its parameters are estimated from the data. In addition, a simple SIS model is introduced and reduced to a single-equation epidemic model. General techniques for studying the dynamics of single-equation ODE models are described and applied to the single-equation version of the SIS model. Furthermore, the SIS model is extended to an SIS model with saturating treatment. The analysis of the model with saturating treatment leads to the introduction of the concepts of multiple equilibria and bistability.

Maia Martcheva

Chapter 3. The SIR Model with Demography: General Properties of Planar Systems

Abstract

This chapter introduces a number of simple demographic models and fits them to population data. A demographic model is then integrated with the SIR model, which results in an SIR model with demography. The model is reduced to a 2 × 2 system and nondimensionalized. General tools for analysis of planar systems are presented and applied to the SIR model. The basic reproduction number is defined, and its mathematical and epidemiological significance is highlighted. Methods for establishing global stability of equilibria of planar systems are covered and applied to the SIR model. The concepts of Hopf bifurcation and periodic cycles are introduced and applied to the SIR model with more general incidence. Although much of the material presented in this chapter is basic for ODE books, its application to epidemic models that are characterized by multiple unknown parameters is nontrivial.

Maia Martcheva

Chapter 4. Vector-Borne Diseases

Abstract

This chapter introduces and studies vector-borne diseases. The chapter lists a number of vector-borne diseases with their prevalences. A simple two-species model of a vector-borne disease is introduced and studied mathematically. Delay-differential equations are introduced, and the simple vector-borne disease model is recast as a single delay-differential equation model. The simple model is studied both analytically and numerically, and it is shown to exhibit Hopf bifurcation and chaos. A couple of more complex ODE or DDE models of vector-borne diseases are studied.

Maia Martcheva

Chapter 5. Techniques for Computing ℛ 0 $$\mathcal{R}_{0}$$

Abstract

This chapter begins with introducing progressively more complex and realistic ODE models. It introduces the basic SEIR model, a model with an asymptomatic stage, a model with a carrier stage, a model with quarantine/isolation, a vaccination model, a tuberculosis model with treatment, and models with host and pathogen heterogeneities. The current state-of-the-art tools for the computation of the reproduction number in complex models, including the next-generation approach, are introduced and illustrated on examples.

Maia Martcheva

Chapter 6. Fitting Models to Data

Abstract

This chapter is focused on fitting models to data. Least square fit is explained and least square error is defined. Fitting models to data is illustrated on examples and MATLAB code for the fitting is given. This chapter also gives a point-by-point list of steps that should be followed when fitting is performed. The concepts of model selection and the Akaike Information Criterion are introduced and illustrated on examples. Computing elasticities and sensitivities is explained.

Maia Martcheva

Chapter 7. Analysis of Complex ODE Epidemic Models: Global Stability

Abstract

This chapter is concerned with mathematical analysis of complex ODE models and their global stability analysis. The chapter introduces Lyapunov function and LaSalle Theorem as tools and illustrates their use on the SEIR model. Hopf bifurcation theorem in higher dimensions is included and its use is illustrated on the SIR model with isolation. The concept of backward bifurcation is introduced and illustrated on an SEI model with standard incidence. Castillo-Chavez and Song Theorem for detecting backward bifurcation in higher dimensions is introduced and illustrated on the SEI example.

Maia Martcheva

Chapter 8. Multistrain Disease Dynamics

Abstract

This chapter treats multistrain disease dynamics. It discusses the concept of competitive exclusion and illustrates it on a specific example. It lists possible mechanisms that cause coexistence in multistrain models. It introduces the invasion reproduction numbers as a tool to determine stability of equilibria in multistrain models and illustrates their use on an example. It extends the next-generation approach to the computation of invasion reproduction numbers and illustrates the application to a two-strain model with isolation.

Maia Martcheva

Chapter 9. Control Strategies

Abstract

This chapter studies control strategies. Control strategies are listed and explained. It focuses on modeling vaccination in single-strain and multistrain diseases. Different modes of introducing vaccination in models are shown. Imperfect vaccination as a mechanism leading to backward bifurcation and strain replacement is explained. Strain replacement with perfect vaccination is demonstrated. Quarantine and isolation are discussed and included in a model. Introduction to optimal control theory is incorporated and the theory is illustrated on a specific example with optimal control treatment. Matlab code for computing the optimal control is included.

Maia Martcheva

Chapter 10. Ecological Context of Epidemiology

Abstract

This chapter is focused on ecoepidemiology. It introduces and studies a number of models related to infectious diseases in animal populations. Animals are typically subject to ecological interactions. The chapter first introduces SI and SIR models of species subject to a generalist predator and studies the impact of selective and indiscriminate predation. The classical Lotka–Volterra predator–prey and competition models are reviewed together with their basic mathematical properties. Furthermore, the chapter includes and discusses a Lotka–Volterra predator–prey model with disease in prey and a Lotka–Volterra competition model with disease in one of the species. Hopf bifurcation and chaos are found in some of the ecoepidemiological models.

Maia Martcheva

Chapter 11. Zoonotic Disease, Avian Influenza, and Nonautonomous Models

Abstract

This chapter treats avain influenza (AI) as an example of zoonotic diseases. Simple AI model, involving poultry and humans, is introduced and fitted to H5N1 human case data. Tools from Chap. 6 are employed to evaluate multiple AI control strategies. The chapter argues that more detailed H5N1 data suggest that there is seasonality in the transmission of AI. Non-autonomous models as a modeling tool to seasonality are introduced. Tools for the analysis of non-autonomous models, such as the Poincare map, are discussed and their use is illustrated on a simple non-autonomous SI model. Simulations suggest that the simple non-autonomous model with disease-induced mortality is capable of exhibiting chaotic behavior.

Maia Martcheva

Chapter 12. Age-Structured Epidemic Models

Abstract

This chapter introduces chronological age-structured models. The linear McKendrick age-structured model is derived and studied. Population reproduction number and growth rate are derived in the context of age-structured models. Probability of survival is explained and fitted to data. Separable solutions are studied. The chapter also introduces and studies an SIS age-structured model. Intracohort, intercohort, and mixed incidence are discussed. The reproduction number in the age-structured case is derived. Numerical methods for age-structured models are included.

Maia Martcheva

Chapter 13. Class-Age Structured Epidemic Models

Abstract

This chapter introduces and studies class-age structured models. These include models structured by age-since-infection, age-since-recovery, etc. The chapter first derives and studies an SIR model with age-since-infection and mass action incidence. A reproduction number is derived, and its threshold properties are explained. A time-since-recovery model is introduced. Destabilization of this model is studied, and oscillatory solutions are presented. Numerical methods for class-age structured models are introduced. Matlab code for simulations is given.

Maia Martcheva

Chapter 14. Immuno-Epidemiological Modeling

Abstract

This chapter introduces and studies multiscale nested immuno-epidemiological models. The chapter begins with brief introduction to within-host dynamics and within-host modeling. Then the chapter introduces nested immuno-epidemiological models. The first model studied is a nested SI model of HIV. Using data, the chapter argues that the transmission rate is not a linear function of the viral load. The epidemiological reproduction number and prevalence of HIV are derived and studied in terms of the within-host viral load. In addition, a nested immuno-epidemiological model with immune response in the within-host model is introduced and studied.

Maia Martcheva

Chapter 15. Spatial Heterogeneity in Epidemiological Models

Abstract

This chapter focuses on spatial heterogeneity of epidemic models. In the first part of this chapter, metapopulation epidemic modeling is discussed. Models with Lagrangian and Eulerian movement patterns are introduced and studied. In the second part of this chapter, spatial heterogeneity is modeled with diffusion. Single species model with diffusion is derived and combined with an SI epidemic model. Dirichlet, Neumann, and Mixed boundary conditions are reviewed. The SI epidemic model with diffusion is reduced to a single equation model with diffusion and studied. Equilibria and their stability are discussed. Traveling wave solutions are introduced and illustrated with the SI model. Turing instability is also introduced and illustrated on an SI model with distinct diffusion rates.

Maia Martcheva

Chapter 16. Discrete Epidemic Models

Abstract

This chapter introduces and studies discrete epidemic modeling. The chapter begins with single species discrete models of population growth. Tools for the analysis of single equation discrete models are introduced and applied to the population models. The concepts of 2-cycle, 4-cycle, and period doubling are introduced and illustrated. The chapter includes the mathematical tools for studying higher dimensional models, such as Jury conditions. It then applied these tools to study a discrete SIS and a discrete SEIS epidemic models. A generalization of the next-generation approach to discrete models is given and applied to the computation of the reproduction number of the SEIS and a two-patch SIS models. The chapter concludes with the introduction and analysis of a discrete SARS epidemic model.

Maia Martcheva

Backmatter

Titel: An Introduction to Mathematical Epidemiology
verfasst von: Maia Martcheva
Verlag: Springer US
Electronic ISBN: 978-1-4899-7612-3
Print ISBN: 978-1-4899-7611-6
DOI: https://doi.org/10.1007/978-1-4899-7612-3