Abstract
In this paper, we prove the existence of a critical traveling wave solution for a delayed diffusive SIR epidemic model with saturated incidence. Moreover, we establish the nonexistence of traveling wave solutions with nonpositive wave speed for this model. Our results solve some open problems left in the recent paper (Z. Xu in Nonlinear Anal. 111:66–81, 2014).
Highlights
1 Introduction In the past few decades, more research has focused on spatial propagation of communicable diseases in mathematical epidemiology and more reaction-diffusion SIR models have been proposed to describe the transmission of communicable diseases [1, 4, 6,7,8,9, 13, 19,20,21,22,23,24,25,26, 29,30,31]
In Theorem 2.1, we proved the existence of the traveling wave solutions, we obtained a lot of nice properties of the traveling wave solutions for (1.1)
3.3 Properties of the critical traveling wave solutions we focus on some properties of the critical traveling wave solution of (2.1), that is, the proof of the four properties in Theorem 2.1
Summary
In the past few decades, more research has focused on spatial propagation of communicable diseases in mathematical epidemiology and more reaction-diffusion SIR models have been proposed to describe the transmission of communicable diseases [1, 4, 6,7,8,9, 13, 19,20,21,22,23,24,25,26, 29,30,31]. He showed that if R0 < 1 and c ≥ 0 or R0 > 1 and c ∈ (0, c∗), the subsystem of (1.1) has no nontrivial and nonnegative traveling wave solutions Observing his results in [26], one can find that there exist some open problems listed as follows:. There has been some work on the existence of critical traveling wave solutions for diffusive epidemic systems [2, 7, 14, 19, 23, 25, 27, 28, 30]. Theorem 2.1 If R0 > 1 and c = c∗, system (1.1) admits a critical traveling wave solution (S(ξ ), I(ξ ), R(ξ )) satisfying (2.2).
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