Epidemic Reaction-Diffusion System With Cross-Diffusion: Modeling and Numerical Solution

Abstract

Reaction-diffusion systems with cross-diffusion are analyzed here for modeling the population dynamics of epidemic systems. In this paper specific attention is devoted to the numerical analysis and simulation of such systems to show that, far from possible pathologies, the qualitative behaviour of the systems may well interpret the dynamics of real systems.

Spatial behavioural responses to the spread of an infectious disease can suppress Turing and Turing–Hopf patterning of the disease

“…where D i,j > 0 and D (0) i,j > 0, first introduced in mathematical epidemiology by Capasso and coworkers [52] are termed in the literature non linear cross-diffusion fluxes.…”

A reaction-diffusion system with cross-diffusion modeling the spread of an epidemic disease

“…This means that the mobility in each subclass is influenced by the spatial gradient of others subclasses (see e.g. [8,9,21]). A prototype of a nonlinear system that governs this S-I-R model with cross-diffusion is, (d 1 + k 1 u 1 + δ 1,2 u 2 + δ 1,3 u 3 )∇u 1 + δ 1,2 u 1 ∇u 2 + δ 1,3 u 1 ∇u 3 · η = 0, δ 2,1 u 2 ∇u 1 + (d 2 + δ 2,1 u 1 + k 2 u 2 + δ 2,3 u 3 )∇u 2 + δ 2,3 u 2 ∇u 3 · η = 0,…”

“…Other classes of epidemic cross-diffusion systems have been the object of researches; the asymptotic behavior of solutions as well as the numerical analysis and simulations of such systems are studied by Capasso et al [8,9].…”