Response of a deterministic epidemiological system to a stochastically varying environment

Fluctuations in the natural environment introduce variability into the biological systems that exist within them. In this paper, we develop a model for the influence of random fluctuations in the environment on a simple epidemiological system. The model describes the infection of a dynamic host population by an environmentally sensitive pathogen and is based on the infection of sugar beet plants by the endoparasitic slime-mold vector Polymyxa betae. The infection process is switched on only when the temperature is above a critical value. We discuss some of the problems inherent in modeling such a system and analyze the resulting model by using asymptotic techniques to generate closed-form solutions for the mean and variance of the net amount of new inoculum produced within a season. In this way, the variance of temperature profile can be linked with that of the inoculum produced in a season and hence the risk of disease. We also examine the connection between the model developed in this paper and discrete Markov-chain models for weather.