Continuous time Markov chain (CTMC) and Itô stochastic differential equation (SDE) models are derived for a population with births, immigration and deaths (BID model). Differential equations are derived for the moments of the distribution for each stochastic model. Each moment differential equation depends on higher-order moments. Assumptions are made regarding higher-order moments to form a finite, solvable system. Conditions are given under which the CTMC and SDE BID models have the same moment solution or the same stationary solution. The close agreement between the CTMC and SDE models is illustrated in three numerical examples basedon normal or log-normal moment closure assumptions.

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For wildlife species living among patchy habitats, disease and the Allee effect (reduced per capita birth rates at low population densities) may together drive a patch's population to extinction, particularly if births are seasonal. Yet local extinction may not be indicative of global extinction, and a patch may become recolonized by migrating individuals. We introduce deterministic and stochastic susceptible, infectious, and immune epidemic models with vector species to study disease in a metapopulation with an Allee effect and seasonal birth and dispersal. We obtain conditions for the existence of a strong Allee effect and existence and stability of a disease-free positive periodic solution. These general models have application to many wildlife diseases. As a case study, we apply them to evaluate dynamics of the sylvatic plague in prairie dog colonies interconnected through dispersal. We further evaluate the effects of control of the vector population and control by immunization on plague eradication.

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Two stochastic models are derived for a susceptible–infectious–susceptible epidemic spreading through a metapopulation: a continuous time Markov chain (CTMC) model and an Itô stochastic differential equation (SDE) model. The stochastic models are numerically compared. Close agreement suggests that computationally intense CTMC simulations can be approximated by simpler SDE simulations. Differential equations for the moments of the SDE probability distribution are also derived, the steady states are solved numerically using a moment closure technique, and these results are compared to simulations. The moment closure technique only coarsely approximates simulation results. The effect of model parameters on stability of the disease-free equilibrium is also numerically investigated.