The Influence of Basic Reproduction Number on Pattern Formations in a Spatial Epidemic Model with The Susceptible Cross-diffusion

Abstract

In this paper, we considered a spatial epidemic model with cross-diffusion of the susceptible. The cross-diffusion of the susceptible represents the tendency of susceptible to move away from the infected. We focused on the influence of basic reproduction numbers on the pattern formations of the model through Turing analysis. The Turing space in the parameter domain was given. From numerical simulation results, the model exhibited the following patterns, i.e., the holes, holes–stripes, stripes, spots–stripes, and spots. The holes indicated the occurrence of an outbreak in a region, whereas the spots indicated otherwise.  If the moderate outbreak occurred in a region, the susceptible cross-diffusion term could change the patterns which indicated that it might prevent the disease dispersion to wider areas although the basic reproduction number was greater than its threshold. However, once the outbreak occurred spatially, the action of the susceptible to move away from the infected could not prevent the outbreak anymore. The results showed that the basic reproduction number has a big influence on the pattern formations when the values of the susceptible cross-diffusion coefficient were small.

Keywords: Spatial epidemic model, cross-diffusion of the susceptible, basic reproduction number, pattern formation, Turing bifurcation.

DOI: 10.7176/MTM/11-4-02

Publication date:September 30^th 2021