Stochastic Models for Nematode Infection in Ruminants
Eric Renshaw
Dept. of Statistics and Modelling Science, Univ. of Strathclyde,
Glasgow, UK
Abstract
We examine the importance of stochastic effects in a model for
the gastro-intestinal infection of ruminants by nematodes where hosts
maintain a fixed density and exhibit acquired immunity. Such a
situation might arise in a commercial agricultural setting where
fluctuations in the host density are negligible. The infection is
directly transmitted to the hosts in grazing, and acquired immunity
impedes the processes of parasite establishment, development and
reproduction. The incorporation of this feed-back mechanism results
in a highly non-linear model, and renders the full stochastic model
intractable to theoretical solution, a feature that is characteristic
of many plausible models in population biology. We therefore explore
a variety of analytic approximations, and compare them to simulations
of the full process. This system is of sufficient complexity to
demonstrate the general applicability and practical utility of these
procedures.
The model is a natural stochastic formulation of a deterministic system proposed by Roberts and Grenfell (1991), and is particularly suitable for our purposes since it captures the essence of more complicated formulations of the processes of parasite demography and herd immunity found in the literature. Here we consider two specific host-replacement regimes, namely the endemic and the managed. In practice birth and death rates are unlikely to remain constant, and so we explore the effect of allowing the model parameters to vary: sinusoidally; through additive white and pink noise; and, as stochastic processes in their own right. Further realism is incorporated by considering the effects of micro-climatic fluctuations on the contact rate. The resultant weather-driven model of Helminth infection in ewes and lambs shows increased variances of population fluctuations in the endemic regime whilst marked stochasticity is introduced into the managed regime.
