Numerical Analysis of BIMs for Stochastic SIR and SIS Models with Variable Contact Diffusion Rates

Qualitative properties of adequate numerical methods for stochastic differential equations to simulate the stochastic counterparts of some known models with variable contact diffusion rates in mathematical epidemiology are investigated, especially SIS and SIR models with non-constant total population sizes on bounded domains. We show the convergence of numerical approximations based on Balanced Implicit Methods (BIMs) by proving several properties; including positivity, invariance, stability, mean and mean square consistency, local uniform boundedness, mean square Hölder continuity and mean square contractivity. We also present some simulation results for those models with realistic parameters.