| Abstract: |
| We study the structure of positive steady states in a cross-diffusive susceptible-infected-susceptible (SIS) epidemic reaction-diffusion model, in which susceptible individuals tend to avoid regions with relatively high densities of infected individuals. Standard bifurcation methods are not applicable in this setting, as there is no evident disease-free equilibrium from which positive solutions can bifurcate. In addition, defining the basic reproduction number becomes significantly more difficult due to the presence of singular chemotaxis sensitivity in the model. To address these issues, we develop a new analytical framework. Using this approach, we establish results on the existence, uniqueness, and multiplicity of positive steady states. Furthermore, we characterize the asymptotic profiles of these steady states in the regime of large chemotaxis sensitivity coefficients. |
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