| Abstract: |
| We consider two 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 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. This is a joint work with H. Li and R. Peng.
In 2-D, it is shown that, for any regular initial data, the system has a unique global -in-time smooth solution for arbitrary size of chi, and it is uniformly bounded in time in the case that the net growth rate of prey
are non-positive. In the latter case, we further study its long time dynamics, implying the cross-diffusion and the instability of certain semi-trivial constant equilibria are still unable to induce pattern formations. In particular, we find that the long time behaviors of the PDE may not always be determined by its corresponding ODE system. This seems to be a new phenomenon compared to existing long time dynamics. The technical condition of the convexity on the domain and the smallness condition on the prey-taxis sensitivity are removed. The (boundedness and) convergence are proved in n-D by the simple and transparent energy method rather than the dissipative dynamical system techniques and Lyapunov function techniques used in early related works. |
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