A numerical method for space‐fractional diffusion models with mass‐conserving boundary conditions

Abstract

The study of the spread of epidemics has gained significant attention in recent years, due to ongoing and recurring outbreaks of diseases such as COVID‐19, dengue, Ebola, and West Nile virus. In particular, modeling the spatial spread of these epidemics is crucial. This article explores the use of fractional diffusion as a means of describing non‐local infection spread. The Grünwald–Letnikov formulation of fractional diffusion is presented, along with several mass‐conserving boundary conditions, that is, we aim to design the boundary conditions in a mass‐conserving way, by not allowing gain or loss of the total population. The stationary points of the model for both sticky and reflecting boundary conditions are discussed, with numerical examples provided to illustrate the results. It is shown that reflecting boundary conditions are more reasonable, as the stationary point for sticky boundary conditions is infinite at the boundaries, while reflecting boundary conditions only have the trivial stationary point, given sufficiently fine discretization. The numerical results were applied to an SI model with fractional diffusion, highlighting the dependence of the system on the value of the fractional derivative. Results indicate that as the order of the derivative increases, the diffusivity also increases, accompanied by a slight increase in the average number of infected individuals. These models have the potential to provide valuable insights into the dynamics of disease spread and aid in the development of effective control strategies.