Simulation of an epidemic model with nonlinear cross-diffusion

A spatially two-dimensional epidemic model is formulated by a reaction-diffusion system. The spatial pattern formation is driven by a cross-diffusion corresponding to a non-diagonal, uppertriangular diffusion matrix. Whereas the reaction terms describe the local dynamics of susceptible and infected species, the diffusion terms account for the spatial distribution dynamics. For both self-diffusion and cross-diffusion nonlinear constitutive assumptions are suggested. To simulate the pattern formation two finite volume formulations are proposed, which employ a conservative and a non-conservative discretization, respectively. Numerical examples illustrate the impact of the cross-diffusion on the pattern formation. have been proposed to study pattern formation induced by cross-diffusion (Ni 2004, Bendahmane et al. 2009b, Tian et al. 2010). In addition to a fundamental existence proof for general reactiondiffusion systems (Crandall et al. 1987), there are several approaches to analyze reaction-diffusion equations with one single ”cross-diffusion” that lead to a system with upper triangular diffusion matrix (Badraoui 2006, Daddiouaissa 2008). The structure of an upper triangular diffusion matrix has also been utilized in the existence analysis for systems of convection-diffusion equations with both Dirich-let and Neumann boundary conditions (see e.g. Frid & Shelukhin 2004, Frid & Shelukhin 2005, Berres et al. 2006). Besides numerous contributions to the development of numerical methods to solve reaction-diffusion equations in related contexts (Wong 2008, Phongthanapanich & Dechaumphai 2009), convergence proofs of associated finite volume schemes (Bendahmane & Sepúlveda 2009, Andreianov et al. 2011) and finite element formulations (Galiano et al. 2003, Barrett & Blowey 2004) have been provided. This contribution is a condensed version of Berres & Ruiz-Baier 2011. The goal is, on the one hand, to generate pattern formation in an epidemic model by a cross-diffusion term, and, on the other hand, to prevent blow-up by a nonlinear limitation of the cross-diffusion. These assumptions are designed to qualitatively reflect psychological behavior. The cross-diffusion term has the interpretation that the susceptible population moves away from increasing gradients of the