Exact multiplicity, bifurcation curves, and asymptotic profiles of endemic equilibria of a cross-diffusive epidemic model

Abstract

This study examines the global structure of endemic equilibrium (EE) solutions of a cross-diffusive epidemic model which incorporates the repulsive movement of the susceptible population away from the infected population. We show that the basic reproduction number $R_0$ alone cannot determine the existence of the EEs and the model may have multiple EEs when the repulsive movement rate χ is large. We prove that the set of EEs forms a simple and unbounded curve bifurcating from the curve of disease free equilibria at $R_0=1$ as $R_0$ varies from zero to infinity, where the bifurcation curve can be forward or backward. We find conditions under which a forward bifurcation curve is of S-shaped and show that a large χ tends to induce backward bifurcation curves. Results on the asymptotic profiles of the EEs are obtained as the repulsive movement rate is large or the random movement rates are small. Finally, we perform numerical simulations to illustrate the results.