Traveling waves in a diffusive epidemic model with criss‐cross mechanism

Abstract

In this paper, we propose a reaction‐diffusion system to describe the spread of infectious diseases within two population groups by self and criss‐cross infection mechanism. Firstly, based on the eigenvalues, we give two methods for the calculation of the critical wave speed c∗. Secondly, by constructing a pair of upper‐lower solutions and using the Schauder fixed‐point theorem, we prove that the system admits positive traveling wave solutions, which connect the initial disease‐free equilibrium false(u10,0,u30,0false) at t = −∞, but the traveling waves need not connect the final disease‐free equilibrium false(u1∗,0,u3∗,0false) at t = +∞. Hence, we study the asymptotic behaviors of the traveling wave solutions to show that the traveling wave solutions converge to false(u1∗,0,u3∗,0false) at t = +∞. Finally, by the two‐sided Laplace transform, we establish the nonexistence of traveling waves for the model. The approach in this paper provides an effective method to deal with the existence of traveling wave solutions for the nonmonotone reaction‐diffusion systems consisting of four equations.

Traveling waves for a cholera vaccination model with nonlocal dispersal

“…When c ≥ c ∗ , we derive the discussion into two cases: c > c ∗ and c = c ∗ , to both of which our model system admit a traveling wave solution, the asymptotic behaviors of the traveling wave solutions are studied to show that the traveling wave solutions converge to E ∗ at . One important feature of our paper is that we need to look for upper‐lower solutions for the reaction–diffusion systems with four equations, which is different from the ones in the literature 43,45,47,49 . Furthermore, when R 0 > 1 and c > c ∗ , the nonexistence of traveling wave solution of the system is also investigated.…”

“…In this section, we use the two‐sided Laplace transform 46,58,59t o establish the nonexistence of traveling wave solution for system (1–4) when R 0 > 1 and 0 < c < c ∗ . Firstly, by using the similar argument as those in other studies, 43,45,47 we only state the following Lemma 13 directly and skip the proof.…”

“…Wang et al 46 concerned with the existence and nonexistence of traveling wave solutions of a nonlocal dispersal SIR model with nonlocal delay, they concluded that the diffusion ability of the infectious can increase the wave speed and the incubation period can decrease the propagation while the nonlocality of interaction would speed up the spread of the disease. Xu and Guo 47 proposed a reaction–diffusion system consisting of four equations to describe the spread of infectious within two population groups by self and criss‐cross infection mechanism. They provided an effective method to explore the existence of traveling waves for the reaction–diffusion systems with four equations.…”

“…Zhang and Liu 45 studied an SVIR epidemic model with nonlocal dispersal and delay and investigated the existence and boundary asymptotic behavior of traveling wave solutions by using Schauder's fixed-point theorem and Lyapunov functional. Wang et al 46 concerned with the existence and nonexistence of traveling wave solutions of a nonlocal dispersal SIR model with nonlocal delay, they concluded that the diffusion ability of the infectious can increase the wave speed and the incubation period can decrease the propagation while the nonlocality of interaction would speed up the spread of the disease. Xu and Guo 47 proposed a reaction-diffusion system consisting of four equations to describe the spread of infectious within two population groups by self and criss-cross infection mechanism.…”

Analysis on critical waves for a diffusive epidemic model with saturating incidence rate and infinitely distributed delay

“…Traveling wave solutions are important in epidemiology since they can describe the epidemic invasion of uninfected regions. In the past decades, many works are done on the traveling wave solutions of these systems; we refer to previous studies [1][2][3][4][5][6][7] and the references quoted therein. Many epidemic models can be described by…”