Journal Article

Stochastic effects in a model of nematode infection in ruminants


in Mathematical Medicine and Biology: A Journal of the IMA

Published on behalf of Institute of Mathematics and its Applications

Volume 15, issue 2, pages 97-116
Published in print June 1998 | ISSN: 1477-8599
Published online June 1998 | e-ISSN: 1477-8602 | DOI:
Stochastic effects in a model of nematode infection in ruminants

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  • Applied Mathematics
  • Biomathematics and Statistics


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We illustrate the importance of stochastic effects in population models of biological systems and demonstrate a number of analytic and simulation-based approaches that can usefully be applied to such models. In so doing, we compare the stochastic approach to the more usual deterministic one. The model studied represents the gastrointestinal infection of ruminants by nematodes when the hosts maintain a fixed density. The incorporation of a feedback mechanism, which accounts for the immune response of the infected animals, results in a highly nonlinear model; similar forms of nonlinearity are a feature of many plausible models in population biology. In the absence of an analytic solution to the full stochastic model we explore a number of approximations and compare them to simulations of the full stochastic process. We explore three modes of behaviour of the system. In the endemic regime the stochastic system fluctuates widely around the non-zero fixed points of the deterministic model. In the managed regime, where the system is subject to external periodic perturbation, stochastic effects are negligible. Finally, we find that in a regime in which the deterministic model predicts the long-term persistence of oscillations the stochastic model shows that extinction can occur. Of the approximation procedures we consider, the Normal approximation to the full stochastic process is the most generally applicable, and it is also the most accurate in the light of simulation results. Local linearization provides reasonably accurate prediction of the vanance-covariance structure, and a transfer function approach allows calculation of the time-lagged auto- and cross-correlations in the endemic regime. Linearization of the stochastic updates themselves results in poor prediction of the population variances.

Keywords: stochastic processes; population models; birth-death processes; Normal approximation; simulation; local linearization

Journal Article.  0 words. 

Subjects: Applied Mathematics ; Biomathematics and Statistics

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