  This paper is concerned with two frequency-dependent SIS epidemic
reaction-diffusion models in heterogeneous environment, with a cross-diffusion
term modeling the effect that susceptible individuals tend to move away from
higher concentration of infected individuals. It is first shown that the
corresponding Neumann initial-boundary value problem in an $n$-dimensional
bounded smooth domain possesses a unique global classical solution which is
uniformly-in-time bounded regardless of the strength of the cross-diffusion and
the spatial dimension $n$. It is further shown that, even in the presence of
cross-diffusion, the models still admit threshold-type dynamics in terms of the
basic reproduction number $\mathcal R_0$; that is, the unique disease free
equilibrium 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, which
is globally stable in some special cases with weak chemotactic sensitivity. Our
results on the asymptotic profiles of endemic equilibrium illustrate that
restricting the motility of susceptible population may eliminate the infectious
disease entirely for the first model with constant total population but fails
for the second model with varying total population. In particular, this implies
that such cross-diffusion does not contribute to the elimination of the
infectious disease modelled by the second one.
