Article Contents
Article Contents

Wave phenomena in a compartmental epidemic model with nonlocal dispersal and relapse

• * Corresponding author: Chufen Wu
• This paper is concerned with the wave phenomena in a compartmental epidemic model with nonlocal dispersal and relapse. We first show the well-posedness of solutions for such a problem. Then, in terms of the basic reproduction number and the wave speed, we establish a threshold result which reveals the existence and non-existence of the strong traveling waves accounting for phase transitions between the disease-free equilibrium and the endemic steady state. Further, we clarify and characterize the minimal wave speed of traveling waves. Finally, numerical simulations and discussions are also given to illustrate the analytical results. Our result indicates that the relapse can encourage the spread of the disease.

Mathematics Subject Classification: Primary: 45K05, 35R20; Secondary: 92D30, 35C07.

 Citation:

• Figure 1.  The profile of $(S,I,R)$ connecting $(\frac{7}{3},0,0)$ and $(\frac{11}{6},\frac{3}{22},\frac{1}{11})$ for $c>c^*$

Figure 2.  The profile of $S$ connecting $\frac{7}{3}$ and $\frac{11}{6}$ for $c>c^*$ at time steps $t = 5,12,20$

Figure 3.  The profile of $I$ connecting $0$ and $\frac{3}{22}$ for $c>c^*$ at time steps $t = 5,12,20$

Figure 4.  The profile of $R$ connecting $0$ and $\frac{1}{11}$ for $c>c^*$ at time steps $t = 5,12,20$

Figure 5.  The profile of $(S,I,R)$ connecting $(\frac{7}{3},0,0)$ and $(\frac{11}{6},\frac{3}{22},\frac{1}{11})$ for $c = c^*$

Figure 6.  The profile of $S$ connecting $\frac{7}{3}$ and $\frac{11}{6}$ for $c = c^*$ at time steps $t = 5,12,20$

Figure 7.  The profile of $I$ connecting $0$ and $\frac{3}{22}$ for $c = c^*$ at time steps $t = 5,12,20$

Figure 8.  The profile of $R$ connecting $0$ and $\frac{1}{11}$ for $c = c^*$ at time steps $t = 5,12,20$

Figure 9.  The profile of $(S,I,R)$ connecting $(5,0,0)$ and $(\frac{24}{5},\frac{1}{24},\frac{5}{48})$ for $c<c^*$

Figure 10.  The profile of $S$ connecting $5$ and $\frac{24}{5}$ for $c<c^*$ at time steps $t = 5,12,20$

Figure 11.  The profile of $I$ connecting $0$ and $\frac{1}{24}$ for $c<c^*$ at time steps $t = 5,12,20$

Figure 12.  The profile of $R$ connecting $0$ and $\frac{5}{48}$ for $c<c^*$ at time steps $t = 5,12,20$

Figure 13.  The minimal wave speed $c^*$ with respect to the relapse rate $\delta$

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