Discrete-Time Stochastic Epidemic Models and Their Statistical Inference

Abstract

There is a wide range of models, both stochastic and deterministic, for the spread of an epidemic. Usually, when the population is constituted of a large number of individuals, a deterministic model is useful as a first approximation, and random variations can be neglected. As an alternative, a stochastic model could be more appropriate for describing the epidemic, but it is less tractable and its mathematical analysis is usually possible only when the population size is very small. However, most populations are not large enough to neglect the effect of statistical fluctuations, nor are they small enough to avoid cumbersome mathematical calculations in the stochastic model. In these cases, it uses to be convenient to take into account both types of models and their relationship. The interplay between ordinary differential equations and Markovian counting processes has been widely investigated in the literature. Major references on this subject can be found in [11, 20–22]. Concerning the deterministic epidemic models, those using ordinary differential equations in their formulation have received special attention and a great number of epidemics is modeled by means of Markovian counting processes. For example, some epidemic models known as SIR, SI, SIS, and others derived from these ones, use differential equations andMarkovian counting processes in their formulations. Furthermore, stochastic models based onMarkovian counting processes and differential equations are mainly used to carry out the statistical analysis of the model parameters. The Mathematical Theory of Infectious Diseases by Bailey [2] represents a classical reference containing a presentation and analysis of these models. However a more recent book by Andersson and Britton [1] entitled Stochastic Epidemic Models and their Statistical Analysis is a more appropriate reference according to the point of view of this chapter. The spread of these epidemics is developed in a closed population, which is divided into three individual compartments, i.e. susceptible, infective and removed cases; different types of transitions can occur among these three groups of individuals. These models include the stochastic and deterministic versions of the Kermack and McKendrick model [19] and the SIS epidemic model, among others. Moreover, a number of variations of these models has been widely studied. Modeling of epidemics by continuous-time Markov chains has a long history; thus, it seems pertinent to cite the works by [4–6, 18, 24, 28].