Abstract
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 Ω⊆Rn, we investigate the global stability of the model. More precisely, it is shown that, if the threshold value R0≤1, then the disease-free equilibrium E0 is globally asymptotically stable; if R0>1, then there exists a unique disease equilibrium E∗ which is globally asymptotically stable. Third, in the case of the unbounded spatial habitat Ω=Rn, we study the existence of traveling wave solutions of the model. Here we show that when the threshold value R0>1, then there exists c∗>0 such that there exist positive traveling wave solutions of the model connecting the two equilibria E0 and E∗ with speed c>c∗. And when R0>1, there is not such a traveling wave solution with speed c<c∗. 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 R0.
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