We consider stochastic models of interacting populations, especially models for the progress of a recurrent epidemic in large communities. Diseases such as measles, and solutions of simple forms of model equations exhibit periodic behaviour. Solutions are therefore developed as perturbations of harmonic solutions, using the diffusion approximation to discrete processes.It is shown how measles, which has a natural period between one and two years, may be forced to oscillate with period two years due to parametric resonance, if the seasonal variations in reaction rates are large enough. It is shown how the same effect may lead to an apparent period of one year for sufficiently large communities. For chickenpox, which also has a natural period between one and two years, ordinary resonance is appropriate so that the forced oscillations always have major period one year.Estimates are obtained for the resulting effects on the critical population size required to support such periodic recurrent epidemics, and on the expected time to extinction. We also derive the stationary distribution of the amplitude of epidemic outbreaks.Models including latent periods in the development of the disease and spatial distribution are considered using similar methods, and the effects on critical community size and time to extinction are deduced.
展开▼
