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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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