Mathematical Epidemiology

Epidemics of many infectious diseases occur periodically. Why?

We describe and analyze compartmental models for disease transmission. We begin with models for epidemics, showing how to calculate the basic reproduction number and the final size of the epidemic. We also study models with multiple compartments, including treatment or isolation of infectives. We then consider models including births and deaths in which there may be an endemic equilibrium and study the asymptotic stability of equilibria. We conclude by studying age of infection models which give a unifying framework for more complicated compartmental models.

A brief introduction to the formulation of various types of stochastic epidemic models is presented based on the well-known deterministic SIS and SIR epidemic models. Three different types of stochastic model formulations are discussed: discrete time Markov chain, continuous time Markov chain and stochastic differential equations. Properties unique to the stochastic models are presented: probability of disease extinction, probability of disease outbreak, quasistationary probability distribution, final size distribution, and expected duration of an epidemic. The chapter ends with a discussion of two stochastic formulations that cannot be directly related to the SIS and SIR epidemic models. They are discrete time Markov chain formulations applied in the study of epidemics within households (chain binomial models) and in the prediction of the initial spread of an epidemic (branching processes).

We use a stochastic branching process to describe the beginning of a disease outbreak. Unlike compartmental models, if the basic reproduction number is greater than one there may be a minor outbreak or a major epidemic with a probability depending on the nature of the contact network. We use a network approach to determine the distribution of outbreak and epidemic sizes.

The basic compartmental models for disease transmission are extended to include three separate biological features. The first such feature is vertical transmission of disease, for which two ordinary differential equation models (SIR and SEIR) are formulated and analyzed. In particular, vertical transmission is shown to increase the basic reproduction number. Immigration of infective individuals is considered as a second feature, and the resulting model has a unique endemic equilibrium (with no disease-free state). An illustration is provided that includes screening and isolating infectives to reduce the spread of disease. A constant period of temporary immunity is introduced in an SIRS model as the third feature. This results in an integrodifferential equation for the fraction of infectives. Analysis shows that, for some parameter values, Hopf bifurcation can give rise to periodic solutions.

The basic reproduction number, R ₀ is a measure of the potential for disease spread in a population. Mathematically, R ₀ is a threshold for stability of a disease-free equilibrium and is related to the peak and final size of an epidemic. The purpose of these notes is to give a precise definition and algorithm for obtaining R ₀ for a general compartmental ordinary differential equation model of disease transmission. Several examples of calculating R ₀ are included, and the epidemiological interpretation of this threshold parameter is connected to the local and global stability of a disease-free equilibrium.

Discrete spatial heterogenity is introduced into disease transmission models, resulting in large systems of ordinary differential equations. Such metapopulation models describe disease spread on a number of spatial patches. In the first model considered, there is no explicit movement of individuals; rather infectives can pass the disease to susceptibles in other patches. The second type of model explicitly includes rates of travel between patches and also takes account of the resident patch as well as the current patch of individuals. A formula for and useful bounds on the basic reproduction number of the system are determined. Brief descriptions of application of this type of metapopulation model are given to investigate the spread of bovine tuberculosis and the effect of quarantine on the spread of influenza.

This chapter introduces some basic concepts and techniques in modeling spatial spread of diseases involving hosts moving randomly during certain stages of the disease progression. First we derive some reaction diffusion models using the conservation law and Fick's law of diffusion. We then discuss the usefulness of these models in describing disease spread rates and evaluating the effectiveness of some spatially relevant disease control strategies. We illustrate the general theory via two case studies, one about the spread of rabies in continental Europe during the period 1945–1985 and another about spread rates of West Nile virus in North America.

The occurrence of a major outbreak, the shape of the epidemic curves, as well as the final sizes of outbreaks, are realizations of some stochastic events with some probability distributions. These distributions are manifested through some stochastic mechanisms. This chapter divides a typical outbreak in a closed population into two phases, the initial phase and beyond the initial phase. For the initial phase, this chapter addresses several aspects: the invasion (i.e. the risk of a large outbreak); quantities associated with a small outbreak; and characteristics of a large outbreak. In a large outbreak beyond the initial phase, the focus is on its final size. After a review of distribution theories and stochastic processes, this chapter separately addresses each of these issues by asking questions such as: Are the latent period and/or the infectious period distributions playing any role? What is the role of the contact process for this issue? Is the basic reproduction number R ₀ sufficient to address this issue? How many stochastic mechanisms may manifest observations that may resemble a power-law distribution, and how much detail is really needed to address this specific issue? etc. This chapter uses distribution theory and stochastic processes to capture the agent–host–environment interface during an outbreak of an infectious disease. With different phases of an outbreak and special issues in mind, modellers need to choose which detailed aspects of the distributions and the stochastic mechanisms need to be included, and which detailed aspects need to be ignored. With these discussions, this chapter provides some syntheses for the concepts and models discussed in some proceeding chapters, as well as some food for thought for following chapters on case studies.

Infectious childhood diseases such as measles are characterized by recurrent outbreaks. Mathematicians have long used models in an effort to better understand and predict these recurrent outbreak patterns. This paper summarizes and comments upon those efforts, providing a historical outline of childhood disease models that have been developed since the start of the twentieth century. This paper also discusses the influence of data analysis techniques, such as spectral analysis, on the understanding and modelling of childhood disease dynamics.

We describe and analyze compartmental models for influenza, including pre-epidemic vaccination and antiviral treatment. The analysis is based on the final size relation for compartmental epidemic models. We consider models of increasing complexity and compare their predictions using parameter values appropriate to the 1957 pandemic.

This chapter compiles some of the results on influenza dynamics that involve a single strain, as well as two competing strains. The emphasis is on the role of cross-immunity, quarantine and age-structure as mechanisms capable of supporting recurrent influenza epidemic outbreaks. Quarantine or age-structure alone can support oscillations while cross-immunity enhances the likelihood of strain coexistence and impacts the length of the period. It is the hope that the perspective provided here will instigate others to use mathematical models in the study of disease transmission and its evolution, particularly in a setting that involves highly variable pathogens.

This chapter describes the steps needed to formulate, analyze and apply epidemiological models to vector-borne diseases. Our models focus on West Nile (WN) virus, an emerging infectious disease in North America, first identified in Africa. We begin by introducing a minimalist model for WN dynamics to illustrate the processes of model formulation, analysis, and application. We then revisit the question of model formulation to examine how two major biological assumptions affect the model structure and therefore its predictions. Next, we briefly compare these different model structures in an introductory exercise of model parameterization, validation, and comparison. Finally, we address model applications in more detail with two examples of how the model output can usefully be connected to public health applications.