Stability and traveling waves of a vaccination model with nonlinear incidence

In this paper, we propose a spatial vaccination model with nonlinear incidence. First, we consider the well-posedness of solutions of the model. Second, in the case of the bounded spatial habitat Omega subset of R-n, we investigate the global stability of the model. More precisely, it is shown that, if the threshold value R-0 <= 1, then the disease-free equilibrium E-0 is globally asymptotically stable; if R-0 > 1, then there exists a unique disease equilibrium E* which is globally asymptotically stable. Third, in the case of the unbounded spatial habitat Omega = R-n, we study the existence of traveling wave solutions of the model. Here we show that when the threshold value R-0 > 1, then there exists c* > 0 such that there exist positive traveling wave solutions of the model connecting the two equilibria E-0 and E* with speed c < c*. And when , there is not such a traveling wave solution with speed . Numerical simulations are performed to illustrate our analytic results. Our results indicate that the global dynamics of the model are completely determined by the threshold value R-0. (C) 2017 Elsevier Ltd. All rights reserved.