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Dynamics of a waterborne pathogen model with spatial heterogeneity and general incidence rate. (English) Zbl 1431.35215
Summary: This paper is concerned with the dynamics of a waterborne pathogen model with spatial heterogeneity and general incidence rate. We first establish the well-posedness of this model. Then we clarify the relationship between the local basic reproduction number $$\tilde{\mathcal{R}}$$ and the basic reproduction number $$\mathcal{R}_0$$. It could be seen that $$\mathcal{R}_0$$ plays an important role in determining the global dynamics of this model. In fact, we show that the disease-free equilibrium is globally asymptotically stable when $$\mathcal{R}_0 < 1$$. If $$\mathcal{R}_0 = 1$$, then the disease-free equilibrium is globally asymptotically stable under some assumptions. In addition, the phenomena of uniform persistence occurs when $$\mathcal{R}_0 > 1$$. We also consider the local and global stability of endemic equilibrium when all the parameters of this model are constant. In the case $$\mathcal{R}_0 > 1$$, we further establish the existence of traveling wave solutions of this model. Moreover, we provide an example and numerical simulations to support our theoretical results. Our model extended some known results.
##### MSC:
 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92C60 Medical epidemiology 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs 35C07 Traveling wave solutions 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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