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August  2021, 29(3): 2325-2358. doi: 10.3934/era.2020118

Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

* Corresponding author: Hai-Feng Huo

Received  August 2020 Revised  October 2020 Published  August 2021 Early access  November 2020

Fund Project: This work is supported by the National Natural Science Foundation of China (11861044 and 11661050), and the HongLiu first-class disciplines Development Program of Lanzhou University of Technology

A reaction-diffusion SEIR model, including the self-protection for susceptible individuals, treatments for infectious individuals and constant recruitment, is introduced. The existence of traveling wave solution, which is determined by the basic reproduction number $ R_0 $ and wave speed $ c, $ is firstly proved as $ R_0>1 $ and $ c\geq c^* $ via the Schauder fixed point theorem, where $ c^* $ is minimal wave speed. Asymptotic behavior of traveling wave solution at infinity is also proved by applying the Lyapunov functional. Furthermore, when $ R_0\leq1 $ or $ R_0>1 $ with $ c\in(0,\ c^*), $ we derive the non-existence of traveling wave solution with utilizing two-sides Laplace transform. We take advantage of numerical simulations to indicate the existence of traveling wave, and show that self-protection and treatment can reduce the spread speed at last.

Citation: Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, 2021, 29 (3) : 2325-2358. doi: 10.3934/era.2020118
References:
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H.-F. HuoP. Yang and H. Xiang, Stability and bifurcation for an SEIS epidemic model with the impact of media, Phys. A, 490 (2018), 702-720.  doi: 10.1016/j.physa.2017.08.139.  Google Scholar

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H.-F. HuoP. Yang and H. Xiang, Dynamics for an SIRS epidemic model with infection age and relapse on a scale-free network, J. Franklin Inst., 356 (2019), 7411-7443.  doi: 10.1016/j.jfranklin.2019.03.034.  Google Scholar

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J. LiY. Yang and Y. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Anal. Real World Appl., 12 (2011), 2163-2173.  doi: 10.1016/j.nonrwa.2010.12.030.  Google Scholar

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S. Riley, Large-scale spatial-transmission models of infectious disease, Science, 316 (2007), 1298-1301.  doi: 10.1126/science.1134695.  Google Scholar

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P. Song and Y. Xiao, Global Hopf bifurcation of a delayed equation describing the lag effect of media impact on the spread of infectious disease, J. Math. Biol., 76 (2018), 1249-1267.  doi: 10.1007/s00285-017-1173-y.  Google Scholar

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P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

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J.-B. Wang and C. Wu, Forced waves and gap formations for a Lotka–Volterra competition model with nonlocal dispersal and shifting habitats, Nonlinear Analysis: Real World Applications, 58 (2021), 103208, 19 pp. doi: 10.1016/j.nonrwa.2020.103208.  Google Scholar

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T. Zhang and W. Wang, Existence of traveling wave solutions for influenza model with treatment, J. Math. Anal. Appl., 419 (2014), 469-495.  doi: 10.1016/j.jmaa.2014.04.068.  Google Scholar

[30]

T. ZhangW. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusion–reaction system, J. Differential Equations, 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.  Google Scholar

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L. Zhao and Z.-C. Wang, Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages, IMA J. Appl. Math., 81 (2016), 795-823.  doi: 10.1093/imamat/hxw033.  Google Scholar

[32]

L. ZhaoZ.-C. Wang and S. Ruan, Traveling wave solutions in a two-group epidemic model with latent period, Nonlinearity, 30 (2017), 1287-1325.  doi: 10.1088/1361-6544/aa59ae.  Google Scholar

[33]

L. ZhaoZ.-C. Wang and S. Ruan, Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol., 77 (2018), 1871-1915.  doi: 10.1007/s00285-018-1227-9.  Google Scholar

show all references

References:
[1]

S. Ai and R. Albashaireh, Traveling waves in spatial SIRS models, J. Dynam. Differential Equations, 26 (2014), 143-164.  doi: 10.1007/s10884-014-9348-3.  Google Scholar

[2]

A. ArapostathisM. K. Ghosh and S. I. Marcus, Harnack's inequality for cooperative weakly coupled elliptic systems: Harnack's inequality, Comm. Partial Differential Equations, 24 (1999), 1555-1571.  doi: 10.1080/03605309908821475.  Google Scholar

[3]

A. DucrotP. Magal and S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331.  doi: 10.1007/s00205-008-0203-8.  Google Scholar

[4]

J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.  doi: 10.1137/140953939.  Google Scholar

[5]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 2008. Google Scholar

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015. Google Scholar

[7]

K. P. Hadeler, Hyperbolic travelling fronts, Proc. Edinburgh Math. Soc., 31 (1988), 89-97.  doi: 10.1017/S001309150000660X.  Google Scholar

[8]

K. P. Hadeler, Travelling fronts for correlated random walks, Canad. Appl. Math. Quart., 2 (1994), 27-43.   Google Scholar

[9]

H.-F. HuoP. Yang and H. Xiang, Stability and bifurcation for an SEIS epidemic model with the impact of media, Phys. A, 490 (2018), 702-720.  doi: 10.1016/j.physa.2017.08.139.  Google Scholar

[10]

H.-F. HuoP. Yang and H. Xiang, Dynamics for an SIRS epidemic model with infection age and relapse on a scale-free network, J. Franklin Inst., 356 (2019), 7411-7443.  doi: 10.1016/j.jfranklin.2019.03.034.  Google Scholar

[11]

J. S. Jia, X. Lu, Y. Yuan, G. Xu, J. Jia and N. A. Christakis, Population flow drives spatio-temporal distribution of COVID-19 in China, Nature, 1–5. Google Scholar

[12]

S.-L. Jing, H.-F. Huo and H. Xiang, Modeling the effects of meteorological factors and unreported cases on seasonal influenza outbreaks in Gansu province, China, Bull. Math. Biol., 82 (2020), Paper No. 73, 36 pp. doi: 10.1007/s11538-020-00747-6.  Google Scholar

[13]

S. LaiN. W. RuktanonchaiL. ZhouO. ProsperW. LuoJ. R. FloydA. WesolowskiM. SantillanaC. ZhangX. DuH. Yu and A. J. Tatem, Effect of non-pharmaceutical interventions to contain COVID-19 in China, Nature, 585 (2020), 410-413.   Google Scholar

[14]

Y. LiW.-T. Li and F.-Y. Yang, Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072.  Google Scholar

[15]

J. LiY. Yang and Y. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Anal. Real World Appl., 12 (2011), 2163-2173.  doi: 10.1016/j.nonrwa.2010.12.030.  Google Scholar

[16]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[17]

B. F. Maier and D. Brockmann, Effective containment explains subexponential growth in recent confirmed COVID-19 cases in China, Science, 368 (2020), 742-746.  doi: 10.1126/science.abb4557.  Google Scholar

[18]

J. D. Murray, Mathematical Biology, Springer, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[19]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, 2012. Google Scholar

[20]

L. Rass and J. Radcliffe, Spatial Deterministic Epidemics, American Mathematical Society, 2003. doi: 10.1090/surv/102.  Google Scholar

[21]

S. Riley, Large-scale spatial-transmission models of infectious disease, Science, 316 (2007), 1298-1301.  doi: 10.1126/science.1134695.  Google Scholar

[22]

P. Song and Y. Xiao, Global Hopf bifurcation of a delayed equation describing the lag effect of media impact on the spread of infectious disease, J. Math. Biol., 76 (2018), 1249-1267.  doi: 10.1007/s00285-017-1173-y.  Google Scholar

[23]

P. Song and Y. Xiao, Analysis of an epidemic system with two response delays in media impact function, Bull. Math. Biol., 81 (2019), 1582-1612.  doi: 10.1007/s11538-019-00586-0.  Google Scholar

[24]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[25]

J.-B. Wang and C. Wu, Forced waves and gap formations for a Lotka–Volterra competition model with nonlocal dispersal and shifting habitats, Nonlinear Analysis: Real World Applications, 58 (2021), 103208, 19 pp. doi: 10.1016/j.nonrwa.2020.103208.  Google Scholar

[26]

Z.-C. WangJ. Wu and R. Liu, Traveling waves of the spread of avian influenza, Proc. Amer. Math. Soc., 140 (2012), 3931-3946.  doi: 10.1090/S0002-9939-2012-11246-8.  Google Scholar

[27]

F.-Y. YangY. LiW.-T. Li and Z.-C. Wang, Traveling waves in a nonlocal anisotropic dispersal Kermack-Mckendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969.  Google Scholar

[28]

T. Zhang, Minimal wave speed for a class of non-cooperative reaction–diffusion systems of three equations, J. Differential Equations, 262 (2017), 4724-4770.  doi: 10.1016/j.jde.2016.12.017.  Google Scholar

[29]

T. Zhang and W. Wang, Existence of traveling wave solutions for influenza model with treatment, J. Math. Anal. Appl., 419 (2014), 469-495.  doi: 10.1016/j.jmaa.2014.04.068.  Google Scholar

[30]

T. ZhangW. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusion–reaction system, J. Differential Equations, 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.  Google Scholar

[31]

L. Zhao and Z.-C. Wang, Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages, IMA J. Appl. Math., 81 (2016), 795-823.  doi: 10.1093/imamat/hxw033.  Google Scholar

[32]

L. ZhaoZ.-C. Wang and S. Ruan, Traveling wave solutions in a two-group epidemic model with latent period, Nonlinearity, 30 (2017), 1287-1325.  doi: 10.1088/1361-6544/aa59ae.  Google Scholar

[33]

L. ZhaoZ.-C. Wang and S. Ruan, Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol., 77 (2018), 1871-1915.  doi: 10.1007/s00285-018-1227-9.  Google Scholar

Figure 1.  The numerical simulations of existence for traveling wave solution of system (2)
Figure 2.  Cross section curve of traveling wave solution for system (2) as $ t = 200. $
Figure 3.  Show the effects of self-protection $ \sigma $ and treatment $ \theta $ on minimal spread speed $ c^*, $ where $ \sigma $ and $ \theta $ are taken from $ 0.05 $ to $ 1. $
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