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This paper is concerned with two frequency-dependent SIS epidemicreaction-diffusion models in heterogeneous environment, with a cross-diffusionterm modeling the effect that susceptible individuals tend to move away fromhigher concentration of infected individuals. It is first shown that thecorresponding Neumann initial-boundary value problem in an $n$-dimensionalbounded smooth domain possesses a unique global classical solution which isuniformly-in-time bounded regardless of the strength of the cross-diffusion andthe spatial dimension $n$. It is further shown that, even in the presence ofcross-diffusion, the models still admit threshold-type dynamics in terms of thebasic reproduction number $\mathcal R_0$; that is, the unique disease freeequilibrium is globally stable if $\mathcal R_0<1$, while if $\mathcal R_0>1$,the disease is uniformly persistent and there is an endemic equilibrium, whichis globally stable in some special cases with weak chemotactic sensitivity. Ourresults on the asymptotic profiles of endemic equilibrium illustrate thatrestricting the motility of susceptible population may eliminate the infectiousdisease entirely for the first model with constant total population but failsfor the second model with varying total population. In particular, this impliesthat such cross-diffusion does not contribute to the elimination of theinfectious disease modelled by the second one.

Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion