A new analytical approach in studying some epidemic PDE model with singular chemotaxis sensitivity

We investigate a cross-diffusive susceptible-infected-susceptible (SIS) epidemic reaction-diffusion model, where susceptible individuals attempt to avoid areas with a relatively high density of infected populations. We aim to classify the structure of positive steady states for this SIS model. Unfortunately, the standard bifurcation approach is inadequate because there is no clear candidate for a disease-free equilibrium, from which positive solutions could bifurcate. Furthermore, defining the basic reproduction number for this model, which includes singular chemotaxis sensitivity, is even more challenging. To overcome these analytical difficulties, we transform the system into a scalar nonlinear elliptic equation with an integral constraint. Through intricate analysis of this reduced system, we establish the existence, uniqueness, and multiplicity of positive steady states, which helps us understand where the positive solutions could bifurcate from. Additionally, we establish various asymptotic behaviors of the positive steady states, e.g. when the repulsive-taxis coefficient is sufficiently large, the infected individuals tend to become spatially homogeneously distributed, which supports the numerical simulation findings in [19]. Lastly, our mathematical analysis even helps us identify a critical threshold quantity that shares similar important properties with the basic reproduction number for models without singular chemotaxis sensitivity.