On small-data solution of the chemotaxis-SIS epidemic system with bilinear incidence rate

This paper deals with the initial-boundary value problem for chemotaxis susceptible-infected-susceptible epidemic system with bilinear incidence rate {u(t) = d(1) Delta u + chi del center dot (u del v) - beta(x)uv + gamma(x)v, x is an element of Omega, t > 0 v(t) = d(2) Delta v + beta(x)uv - gamma(x)v, x is an element of Omega, t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Omega subset of R-n (n >= 1), where d(1) > 0, d(2) > 0, chi is an element of R, 0 < beta is an element of C-1(<(Omega)over bar>). It is proved that if (u(x, 0), v(x, 0)) in L-infinity (Omega) x L-1(Omega) is suitably small and parallel to v(x, 0)parallel to(L infinity(Omega)) + parallel to del v(x, 0)parallel to(Lq(Omega)) <= for each K > 0 and q > n, then the problem possesses a unique global classical solution which is bounded in Omega x (0, infinity). Moreover, we prove that the solution exponentially stabilizes to the disease-free equilibrium (N/vertical bar Omega vertical bar, 0) with N = integral(Omega)(u(x, 0) + v(x, 0)) in L-infinity(Omega) as t -> infinity.