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

Jia-Bing Wang; Shao-Xia Qiao; Chufen Wu

doi:10.3934/dcdsb.2021152

Article Contents

2022, Volume 27, Issue 5: 2635-2660. Doi: 10.3934/dcdsb.2021152

This issue Previous Article Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions Next Article A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem

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

1.
School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China
2.
Center for Mathematical Sciences, China University of Geosciences, Wuhan, 430074, China
3.
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
4.
School of Mathematics and Big Data, Foshan University, Foshan 528000, China

^* Corresponding author: Chufen Wu
^* Corresponding author: Chufen Wu

Received: January 31, 2021

Revised: March 31, 2021

Early access: May 2021

Published: May 2022

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Abstract

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.

Keywords:

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

Citation:

Full Text(HTML)

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^* $

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

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

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

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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^* $

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

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

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

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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^* $

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

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

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

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

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