J. Math. Biol. 49: 553–576 (2004) Digital Object Identifier (DOI): 10.1007/s00285-004-0267-5
Mathematical Biology
Pieter Trapman · Ronald Meester · Hans Heesterbeek
A branching model for the spread of infectious animal diseases in varying environments Received: 14 July 2003 / Revised version: 14 November 2003 / c Springer-Verlag 2004 Published online: 3 March 2004 – Abstract. This paper is concerned with a stochastic model, describing outbreaks of infectious diseases that have potentially great animal or human health consequences, and which can result in such severe economic losses that immediate sets of measures need to be taken to curb the spread. During an outbreak of such a disease, the environment that the infectious agent experiences is therefore changing due to the subsequent control measures taken. In our model, we introduce a general branching process in a changing (but not random) environment. With this branching process, we estimate the probability of extinction and the expected number of infected individuals for different control measures. We also use this branching process to calculate the generating function of the number of infected individuals at any given moment. The model and methods are designed using important infections of farmed animals, such as classical swine fever, foot-and-mouth disease and avian influenza as motivating examples, but have a wider application, for example to emerging human infections that lead to strict quarantine of cases and suspected cases (e.g. SARS) and contact and movement restrictions.
1. Introduction Recent outbreaks of infectious diseases of animals (e.g. classical swine fever (CSF), foot and mouth disease (FMD) and avian influenza (AI)) in Western Europe have had great impact on the economy, public life and animal health and welfare in the countries involved. During such an outbreak one would like to be able to compare the effectiveness of proposed control measures in, for example, their ability to reduce the expected final size and the expected duration of the outbreak. Typical for the strategies aimed at stopping outbreaks of important diseases of farm animals, is that infected herds are removed from the population by culling upon detection. A second characteristic is that due to increasing quantity and quality of the imposed control measures, the environment that the infectious agent experiences, is changing. By this we mean that consecutive measures can make, for example, contact opportunities between herds different in different phases of the outbreak, or can J. P. Trapman, J. A. P. Heesterbeek: Faculty of Veterinary Medicine, Utrecht University, Yalelaan 7, 3584 CL Utrecht, The Netherlands. e-mail:
[email protected] J. P. Trapman, R. W. J. Meester: Division of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands Key words or phrases: Stochastic epidemic – Branching process – Varying environments – Iterative method
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make the infectious period, or rate with which infectivity is produced, differ for farms infected at different times. In most cases, mathematical methods for computing outbreak characteristics such as expected final size and expected duration, assume a constant environment in that the control measures are not compounded in time and do not lead to changes in the rates that govern epidemic spread (see e.g. [2]). In this paper we aim to develop stochastic methods, based on branching processes, which allow us to compare the effectiveness of control strategies during such outbreaks in situations where the environment is varying because of changes in subsequent measures of control. Much work has already been done to describe the spread of classic