The Effects of the Susceptible and Infected Cross-Diffusion Terms on Pattern Formations in an SI Model

In this paper, we discuss the pattern dynamics of an SI epidemic model caused by spatial dependency, which is represented by self- and cross-diffusion terms. Cross-diffusion of the susceptible represents a tendency of the susceptible to stay away from the infected. Meanwhile, cross-diffusion of the infected represents their movement to the location with a high density of the susceptible. This study focuses on the presence of the effects of cross-diffusion terms on the Turing instability. This study applies Turing analysis to yield the Turing space and Turing patterns corresponding to the model by involving the infection rate as the bifurcation parameter. The results show that the presence of cross-diffusion terms narrows the Turing space depending on the magnitude of the cross-diffusion coefficients itself. Dynamical behaviors of the model are then investigated through a series of numerical simulations that successfully perform five types of patterns, i.e., spots, spots–stripes, stripes, stripes–holes, and holes. Those patterns give a description of the spread of an infectious disease. The holes denote an outbreak situation in a region, whereas the non-outbreak situation is emphasized by the spots pattern. Further, the decreasing of the ratio of recruitment and death rates indicates that the increasing of the infection rate triggers an outbreak. The present study confirms that cross-diffusion terms have a significant role in infectious disease transmission, spatially.