Numerical Investigation of a Diffusive SIR Model: Focus on Positivity Preservation

ABSTRACT In this paper, we consider a system of semilinear partial differential equations (PDEs) representing a spatially extended SIR epidemic model. A brief analytical investigation of the well‐posedness and positivity of the solutions is provided in the appendix, while the main focus is on the numerical treatment of the model. We examine the performance of several time‐stepping schemes, including the standard forward Euler, semi‐implicit Crank–Nicolson, their Mickens‐type nonstandard counterparts, and explicit exponential Runge–Kutta methods. Particular attention is given to the preservation of positivity in the numerical solutions, which is crucial for maintaining biological relevance. Two step‐size functions are employed, and the results are compared in terms of theoretical accuracy and computational runtime to identify the most efficient method for simulations. Our numerical results show that positivity is preserved by the standard and exponential methods only under certain restrictions on the time‐step size. In contrast, the corresponding nonstandard methods maintain positivity unconditionally, regardless of the time‐step size. These findings underscore the effectiveness of nonstandard schemes in modeling epidemic dynamics governed by reaction‐diffusion systems.