Much of the literature and intuition associated with mathematical epidemiology is driven by deterministic models, which are a reasonable assumption when the population size is large. Stochastic models, especially individual based models, are however considered vital when dealing with small population sizes, especially at times of invasion or extinction. The overwhelming majority of these models (both deterministic and stochastic) assume that the underlying parameters are fixed (or follow a regular seasonal pattern). Here, we consider an analytic framework for dealing with randomly varying parameters through the use of stochastic differential equations - thereby capturing the action of external noisy processes such as weather. In particular, we focus on when the transmission rate, {beta}, varies as the solution to a Cox-Ingersoll-Ross Model, such that {beta} is gamma distributed with autocorrelation. We consider the impact of this parameter variation on a stochastic version of the Susceptible-Infected-Recovered model, and for this 'double-stochastic' model show through simulation and analytical results that exogenous noise increases the impact of stochasticity, potentially leading to more early extinctions, wider variations in the number of cases at equilibrium, but that early growth rate can be faster or slower depending on the precise parameters.
Higgins, D. R., Keeling, M. J., Dyson, L.
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