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May  2022, 27(5): 2635-2660. doi: 10.3934/dcdsb.2021152

## 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

Received  February 2021 Revised  April 2021 Published  May 2022 Early access  May 2021

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.

Citation: Jia-Bing Wang, Shao-Xia Qiao, Chufen Wu. Wave phenomena in a compartmental epidemic model with nonlocal dispersal and relapse. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2635-2660. doi: 10.3934/dcdsb.2021152
##### References:
 [1] Z. Bai and S. Zhang, Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1370-1381.  doi: 10.1016/j.cnsns.2014.07.005. [2] S. Blower, Modelling the genital herpes epidemic, Herpes 11, 3 (2004), 138A–146A. [3] S. M. Blower, T. C. Porco and G. Darby, Predicting and preventing the emergence of antiviral drug resistance in HSV-2, Nat. Med., 4 (1998), 673-678.  doi: 10.1038/nm0698-673. [4] X. Chen and J. S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0. [5] J. Coville and L. Dupaigne, On a non-local eqution arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A., 137 (2007), 727-755.  doi: 10.1017/S0308210504000721. [6] H. Cox, Tuberculosis recurrence and mortality after successful treatment: Impact of drug resistance, PLoS Med., 3 (2006), 1836-1843.  doi: 10.1371/journal.pmed.0030384. [7] O. Diekman, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 69 (1978), 109–130. doi: 10.1007/BF02450783. [8] A. Ducrot and P. Magal, Traveling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911.  doi: 10.1088/0951-7715/24/10/012. [9] A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Rational Mech. Anal., 195 (2010), 311-331.  doi: 10.1007/s00205-008-0203-8. [10] S. C. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20-37.  doi: 10.1016/j.jmaa.2015.09.069. [11] P. Guo, X. S. Yang and Z. C. Yang, Dynamical behaviors of an SIRI epidemic model with nonlinear incidence and latent period, Adv. Difference Equ., 2014 (2014), 164-181.  doi: 10.1186/1687-1847-2014-164. [12] P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse, Appl. Math. Comput., 219 (2013), 8496-8507.  doi: 10.1016/j.amc.2013.02.044. [13] S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137-200.  doi: 10.1007/s00285-006-0050-x. [14] G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.  doi: 10.1007/s11538-009-9487-6. [15] W. Huang and C. Wu, Non-monotone waves of a stage-structured SLIRM epidemic model with latent period, Proc. Roy. Soc. Edinburgh Sect. A. doi: 10.1017/prm.2020.65. [16] V. Hutson, S. Martinez, K. Mischailow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1. [17] C. Y. Kao, Y. Lou and W. Shen, Random diseprsal vs nonlocal dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551. [18] M. Kermack and A. Mckendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc. A, 115 (1927), 700-721. [19] T. Kuniya and J. Wang, Lyapunov functions and global stability for a spatially diffusive SIR epidemic model, Appl. Anal., 96 (2017), 1935-1960.  doi: 10.1080/00036811.2016.1199796. [20] W. T. Li, J. B. Wang and X.-Q. Zhao, Spatial dynamics of a nonlocal dispersal population model in a shifting environment, J. Nonlinear Sci., 28 (2018), 1189-1219.  doi: 10.1007/s00332-018-9445-2. [21] W. T. Li and F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equations Appl., 26 (2014), 243-273.  doi: 10.1216/JIE-2014-26-2-243. [22] Y. Li, W. T. Li and F. Y. Yang, Traveling waves for nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072. [23] J. Martins, A. Pinto and N. Stollenwerkc, A scaling analysis in the SIRI epidemiological model, J. Biol. Dyn., 3 (2009), 479-496.  doi: 10.1080/17513750802601058. [24] H. N. Moreira and Y. Wang, Global stability in an $S\rightarrow I\rightarrow R\rightarrow I$ model, SIAM Rev., 39 (1997), 496-502.  doi: 10.1137/S0036144595295879. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. doi: 10.1007/978-1-4612-5561-1. [26] D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990), 136-139.  doi: 10.1137/1032003. [27] P. van den Driessche and X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103.  doi: 10.1016/j.mbs.2006.09.017. [28] C. Vargas-De-León, On the global stability of infectious diseases models with relapse, Abstr. Appl., 9 (2013), 50-61. [29] J. B. Wang, W. T. Li and F. Y. Yang, Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission, Commun. Nonlinear Sci. Numer. Simulat., 27 (2015), 136-152.  doi: 10.1016/j.cnsns.2015.03.005. [30] J. B. Wang and C. Wu, Forced waves and gap formations for a Lotka-Volterra competition model with nonlocal dispersal and shifting habitats, Nonlinear Anal. Real World Appl., 58 (2021), 103208. doi: 10.1016/j.nonrwa.2020.103208. [31] X. Wang, H. Wang and J. Wu, Travelling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst., 32 (2012), 3303-3324.  doi: 10.3934/dcds.2012.32.3303. [32] Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-Mckendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A, 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377. [33] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, CRC Press, 1985. [34] P. Weng and X. Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020. [35] D. V. Widder, Laplace Transform, Princeton University Press, Princeton, NJ, 1941. [36] P. Wildy, H. J. Field and A. A. Nash, Classical herpes latency revisited, Virus Persistence Symposium, 33 (1982), 133-167. [37] C. Wu, Y. Yang, Q. Zhao, Y. Tian and Z. Xu, Epidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal, Appl. Math. Comput., 313 (2017), 122-143.  doi: 10.1016/j.amc.2017.05.068. [38] C. Wu, Y. Wang and X. Zou, Spatial-temporal dynamics of a Lotka-Volterra competition model with nonlocal dispersal under shifting environment, J. Differential Equations, 267 (2019), 4890-4921.  doi: 10.1016/j.jde.2019.05.019. [39] C. C. Wu, Existence of traveling waves with the critical speed for a discrete diffusive epidemic model, J. Differential Equations, 262 (2017), 272-282.  doi: 10.1016/j.jde.2016.09.022. [40] F. Y. Yang, Y. Li, W. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal Kermack-Mckendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B., 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969. [41] F. Y. Yang and W. T. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131-1146.  doi: 10.1016/j.jmaa.2017.10.016. [42] G. B. Zhang, W. T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Model., 49 (2009), 1021-1029.  doi: 10.1016/j.mcm.2008.09.007. [43] C. C. Zhu, W. T. Li and F. Y. Yang, Traveling waves in a nonlocal dispersal SIRH model with relapse, Comput. Math. Appl., 73 (2017), 1707-1723.  doi: 10.1016/j.camwa.2017.02.014.

show all references

##### References:
 [1] Z. Bai and S. Zhang, Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1370-1381.  doi: 10.1016/j.cnsns.2014.07.005. [2] S. Blower, Modelling the genital herpes epidemic, Herpes 11, 3 (2004), 138A–146A. [3] S. M. Blower, T. C. Porco and G. Darby, Predicting and preventing the emergence of antiviral drug resistance in HSV-2, Nat. Med., 4 (1998), 673-678.  doi: 10.1038/nm0698-673. [4] X. Chen and J. S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0. [5] J. Coville and L. Dupaigne, On a non-local eqution arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A., 137 (2007), 727-755.  doi: 10.1017/S0308210504000721. [6] H. Cox, Tuberculosis recurrence and mortality after successful treatment: Impact of drug resistance, PLoS Med., 3 (2006), 1836-1843.  doi: 10.1371/journal.pmed.0030384. [7] O. Diekman, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 69 (1978), 109–130. doi: 10.1007/BF02450783. [8] A. Ducrot and P. Magal, Traveling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911.  doi: 10.1088/0951-7715/24/10/012. [9] A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Rational Mech. Anal., 195 (2010), 311-331.  doi: 10.1007/s00205-008-0203-8. [10] S. C. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20-37.  doi: 10.1016/j.jmaa.2015.09.069. [11] P. Guo, X. S. Yang and Z. C. Yang, Dynamical behaviors of an SIRI epidemic model with nonlinear incidence and latent period, Adv. Difference Equ., 2014 (2014), 164-181.  doi: 10.1186/1687-1847-2014-164. [12] P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse, Appl. Math. Comput., 219 (2013), 8496-8507.  doi: 10.1016/j.amc.2013.02.044. [13] S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137-200.  doi: 10.1007/s00285-006-0050-x. [14] G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.  doi: 10.1007/s11538-009-9487-6. [15] W. Huang and C. Wu, Non-monotone waves of a stage-structured SLIRM epidemic model with latent period, Proc. Roy. Soc. Edinburgh Sect. A. doi: 10.1017/prm.2020.65. [16] V. Hutson, S. Martinez, K. Mischailow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1. [17] C. Y. Kao, Y. Lou and W. Shen, Random diseprsal vs nonlocal dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551. [18] M. Kermack and A. Mckendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc. A, 115 (1927), 700-721. [19] T. Kuniya and J. Wang, Lyapunov functions and global stability for a spatially diffusive SIR epidemic model, Appl. Anal., 96 (2017), 1935-1960.  doi: 10.1080/00036811.2016.1199796. [20] W. T. Li, J. B. Wang and X.-Q. Zhao, Spatial dynamics of a nonlocal dispersal population model in a shifting environment, J. Nonlinear Sci., 28 (2018), 1189-1219.  doi: 10.1007/s00332-018-9445-2. [21] W. T. Li and F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equations Appl., 26 (2014), 243-273.  doi: 10.1216/JIE-2014-26-2-243. [22] Y. Li, W. T. Li and F. Y. Yang, Traveling waves for nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072. [23] J. Martins, A. Pinto and N. Stollenwerkc, A scaling analysis in the SIRI epidemiological model, J. Biol. Dyn., 3 (2009), 479-496.  doi: 10.1080/17513750802601058. [24] H. N. Moreira and Y. Wang, Global stability in an $S\rightarrow I\rightarrow R\rightarrow I$ model, SIAM Rev., 39 (1997), 496-502.  doi: 10.1137/S0036144595295879. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. doi: 10.1007/978-1-4612-5561-1. [26] D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990), 136-139.  doi: 10.1137/1032003. [27] P. van den Driessche and X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103.  doi: 10.1016/j.mbs.2006.09.017. [28] C. Vargas-De-León, On the global stability of infectious diseases models with relapse, Abstr. Appl., 9 (2013), 50-61. [29] J. B. Wang, W. T. Li and F. Y. Yang, Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission, Commun. Nonlinear Sci. Numer. Simulat., 27 (2015), 136-152.  doi: 10.1016/j.cnsns.2015.03.005. [30] J. B. Wang and C. Wu, Forced waves and gap formations for a Lotka-Volterra competition model with nonlocal dispersal and shifting habitats, Nonlinear Anal. Real World Appl., 58 (2021), 103208. doi: 10.1016/j.nonrwa.2020.103208. [31] X. Wang, H. Wang and J. Wu, Travelling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst., 32 (2012), 3303-3324.  doi: 10.3934/dcds.2012.32.3303. [32] Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-Mckendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A, 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377. [33] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, CRC Press, 1985. [34] P. Weng and X. Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020. [35] D. V. Widder, Laplace Transform, Princeton University Press, Princeton, NJ, 1941. [36] P. Wildy, H. J. Field and A. A. Nash, Classical herpes latency revisited, Virus Persistence Symposium, 33 (1982), 133-167. [37] C. Wu, Y. Yang, Q. Zhao, Y. Tian and Z. Xu, Epidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal, Appl. Math. Comput., 313 (2017), 122-143.  doi: 10.1016/j.amc.2017.05.068. [38] C. Wu, Y. Wang and X. Zou, Spatial-temporal dynamics of a Lotka-Volterra competition model with nonlocal dispersal under shifting environment, J. Differential Equations, 267 (2019), 4890-4921.  doi: 10.1016/j.jde.2019.05.019. [39] C. C. Wu, Existence of traveling waves with the critical speed for a discrete diffusive epidemic model, J. Differential Equations, 262 (2017), 272-282.  doi: 10.1016/j.jde.2016.09.022. [40] F. Y. Yang, Y. Li, W. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal Kermack-Mckendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B., 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969. [41] F. Y. Yang and W. T. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131-1146.  doi: 10.1016/j.jmaa.2017.10.016. [42] G. B. Zhang, W. T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Model., 49 (2009), 1021-1029.  doi: 10.1016/j.mcm.2008.09.007. [43] C. C. Zhu, W. T. Li and F. Y. Yang, Traveling waves in a nonlocal dispersal SIRH model with relapse, Comput. Math. Appl., 73 (2017), 1707-1723.  doi: 10.1016/j.camwa.2017.02.014.
The profile of $(S,I,R)$ connecting $(\frac{7}{3},0,0)$ and $(\frac{11}{6},\frac{3}{22},\frac{1}{11})$ for $c>c^*$
The profile of $S$ connecting $\frac{7}{3}$ and $\frac{11}{6}$ for $c>c^*$ at time steps $t = 5,12,20$
The profile of $I$ connecting $0$ and $\frac{3}{22}$ for $c>c^*$ at time steps $t = 5,12,20$
The profile of $R$ connecting $0$ and $\frac{1}{11}$ for $c>c^*$ at time steps $t = 5,12,20$
The profile of $(S,I,R)$ connecting $(\frac{7}{3},0,0)$ and $(\frac{11}{6},\frac{3}{22},\frac{1}{11})$ for $c = c^*$
The profile of $S$ connecting $\frac{7}{3}$ and $\frac{11}{6}$ for $c = c^*$ at time steps $t = 5,12,20$
The profile of $I$ connecting $0$ and $\frac{3}{22}$ for $c = c^*$ at time steps $t = 5,12,20$
The profile of $R$ connecting $0$ and $\frac{1}{11}$ for $c = c^*$ at time steps $t = 5,12,20$
The profile of $(S,I,R)$ connecting $(5,0,0)$ and $(\frac{24}{5},\frac{1}{24},\frac{5}{48})$ for $c<c^*$
The profile of $S$ connecting $5$ and $\frac{24}{5}$ for $c<c^*$ at time steps $t = 5,12,20$
The profile of $I$ connecting $0$ and $\frac{1}{24}$ for $c<c^*$ at time steps $t = 5,12,20$
The profile of $R$ connecting $0$ and $\frac{5}{48}$ for $c<c^*$ at time steps $t = 5,12,20$
The minimal wave speed $c^*$ with respect to the relapse rate $\delta$
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