Basic methods for modeling the invasion and spread of contagious diseases

Abstract

The evolution of disease requires a flrm understanding of hetero- geneity among pathogen strains and hosts with regard to the processes of transmission, movement, recovery, and pathobiology. In this and a companion chapter (Getz et al. this volume), we focus on the question of how to model the invasion and spread of diseases in heterogeneous environments, without making an explicit link to natural selection{the topic of other chapters in this volume. We begin in this chapter by providing an overview of current methods used to model epidemics in homogeneous populations, covering continuous and discrete time formulations in both deterministic and stochastic frameworks. In particular, we introduce Kermack and McKendricks SIR (susceptible, infected, removed) formulation for the case where the removed (R) disease class is parti- tioned into immune (V class) and dead (D class) individuals. We also focus on transmission, contrasting mass-action and frequency-dependent formulations and results. This is followed by a presentation of various extensions includ- ing the consideration of the latent period of infection, the staging of disease classes, and the addition of vital and demographic processes. We then discuss the relative merits of continuous versus discrete time formulations to model real systems, particularly in the context of stochastic analyses. The overview is completed with a presentation of basic branching process theory as a sto- chastic generation-based model for the invasion of disease into populations of inflnite size, with numerical extensions generalizing results to populations of flnite size. In framework of branching process theory, we explore the question of minor versus major stochastic epidemics and illuminate the relationship be- tween minor epidemics and a deterministic theory of disease invasion, as well as major epidemics and the deterministic theory of disease establishment. We conclude this chapter with a demonstration of how the basic ideas can be used to model containment policies associated with the outbreak of SARS in Asia in the early part of 2003.