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Traveling wave solutions for a class of discrete diffusive SIR epidemic model. (English) Zbl 1462.35130

Summary: This paper is concerned with the conditions of existence and nonexistence of traveling wave solutions (TWS) for a class of discrete diffusive epidemic model. We find that the existence of TWS is determined by the so-called basic reproduction number and the critical wave speed: When the basic reproduction number \(\mathfrak{R}_0>1\), there exists a critical wave speed \(c^*>0\), such that for each \(c\ge c^*\) the system admits a nontrivial TWS and for \(c<c^*\) there exists no nontrivial TWS for the system. In addition, the boundary asymptotic behavior of TWS is obtained by constructing a suitable Lyapunov functional and employing Lebesgue dominated convergence theorem. Finally, we apply our results to two discrete diffusive epidemic models to verify the existence and nonexistence of TWS.

MSC:

35C07 Traveling wave solutions
35K45 Initial value problems for second-order parabolic systems
35K57 Reaction-diffusion equations
39A12 Discrete version of topics in analysis
92D30 Epidemiology
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References:

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