On a Cross-Diffusive SIS Epidemic Model with Singular Sensitivity

Abstract

This work introduces a repulsive chemotaxis susceptible-infected-susceptible (SIS) epidemic model with logarithmic sensitivity and with mass-action transmission mechanism, in which the logarithmic sensitivity assumes that the chemotactic migration of susceptible populations is suppressed by large density of infected individuals while the biased movement is strongly sensitive to a density variation of small infected population. Under suitable regular assumption on the initial data, we firstly assert the global existence and boundedness of smooth solutions to the corresponding no-flux initial boundary value problem in the spatially one-dimensional setting. Second, we investigate the effect of strong chemotaxis sensitivity on the dynamics of solutions through extensive numerical simulations. Our numerical study suggests that although this chemotaxis model includes an unbounded force of infection, the blow-up of solutions cannot occur in two dimensions, which remains to be analytically verified. Moreover, the numerical studies on the asymptotic profiles of the endemic equilibrium indicate that the susceptible populations move to low-risk domains whereas infected individuals become spatially homogeneous when the repulsive-taxis coefficient is large. Furthermore, simulations performed in the one- and two-dimensional cases find that rich patterns, like periodic peaks, structured holes, dots and round circles, may arise at intermediate times, but eventually are smoothed out, and that clusters of infection can emerge in a heterogeneous environment. Additionally, our numerical simulations suggest that the susceptible population with larger chemosensitivity, tends to respond better to the infected population, revealing the effect of strong chemotaxis sensitivity coefficient on the dynamics of the disease.

Keywords

SIS model / chemotaxis / global existence / endemic equilibria / asymptotic profiles