On the stochastic engine of transmittable diseases in exponentially growing populations

Lindstrom, Torsten

The purpose of this paper is to analyze the interplay of deterministic and stochastic models for epidemic diseases. Deterministic models for epidemic diseases are prone to predict global stability. If the natural birth and death rates are assumed small in comparison to disease parameters like the contact rate and the recovery rate, then the globally stable endemic equilibrium corresponds to a tiny proportion of infected individuals. Asymptotic equilibrium levels corresponding to low numbers of individuals invalidate the deterministic results. Diffusion effects force frequency functions of the stochastic model to possess similar stability properties as the deterministic model. Particular simulations of the stochastic model are, however, oscillatory and predict oscillatory patterns. Smaller or isolated populations show longer periods, more violent oscillations, and larger probabilities of extinction. We prove that evolution maximizes the infectiousness of the disease as measured by the ability to increase the proportion of infected individuals. This hold provided the stochastic oscillations are moderate enough to remain in the vicinity of the deterministic equilibrium. We close our paper with a discussion of the herd-immunity concept and stress its close relation to vaccination-campaigns.