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June  2022, 27(6): 3053-3075. doi: 10.3934/dcdsb.2021173

Qualitative analysis of a diffusive SEIR epidemic model with linear external source and asymptomatic infection in heterogeneous environment

1. 

Institute of Geophysics and Geomatics, China University of Geosciences, School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

2. 

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

* Corresponding author: Shangjiang Guo

Received  February 2021 Revised  May 2021 Published  June 2022 Early access  July 2021

Fund Project: Research supported by the NSFC (Grant Nos. 11671123 & 12071446) and by the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (Grant No. CUGST2)

This paper is devoted to an SEIR epidemic model with variable recruitment and both exposed and infected populations having infectious in a spatially heterogeneous environment. The basic reproduction number is defined and the existence of endemic equilibrium is obtained, and the relationship between the basic reproduction number and diffusion coefficients is established. Then the global stability of the endemic equilibrium in a homogeneous environment is investigated. Finally, the asymptotic profiles of endemic equilibrium are discussed, when the diffusion rates of susceptible, exposed and infected individuals tend to zero or infinity. The theoretical results show that limiting the movement of exposed, infected and recovered individuals can eliminate the disease in low-risk sites, while the disease is still persistent in high-risk sites. Therefore, the presence of exposed individuals with infectious greatly increases the difficulty of disease prevention and control.

Citation: Xuan Tian, Shangjiang Guo, Zhisu Liu. Qualitative analysis of a diffusive SEIR epidemic model with linear external source and asymptomatic infection in heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3053-3075. doi: 10.3934/dcdsb.2021173
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[2]

L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.

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S. Chen and J. Shi, Asymptotic profiles of basic reproduction number for epidemic spreading in heterogeneous environment, SIAM J. Appl. Math., 80 (2020), 1247-1271.  doi: 10.1137/19M1289078.

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R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.

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R. CuiK. Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differential Equations, 263 (2017), 2343-2373.  doi: 10.1016/j.jde.2017.03.045.

[7]

K. Deng and Y. Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.  doi: 10.1017/S0308210515000864.

[8]

Y. DuR. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007.

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Z. Du and R. Peng, A priori $L^{\infty}$ estimates for solutions of a class of reaction-diffusion systems, J. Math. Biol., 72 (2016), 1429-1439.  doi: 10.1007/s00285-015-0914-z.

[10]

J. GaoS. Guo and W. Shen, Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2645-2676.  doi: 10.3934/dcdsb.2020199.

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J. GeK. I. KimZ. Lin and H. Zhu, An SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.

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J. GeL. Lin and L. Zhang, A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 2763-2776.  doi: 10.3934/dcdsb.2017134.

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S. Guo, Bifurcation in a reaction-diffusion model with nonlocal delay effect and nonlinear boundary condition, J. Differential Equations, 289 (2021), 236-278.  doi: 10.1016/j.jde.2021.04.021.

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S. Guo, S. Li and B. Sounvoravong, Oscillatory and stationary patterns in a diffusive model with delay effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2150035, 21 pp. doi: 10.1142/S0218127421500358.

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S. Guo and S. Li, On the stability of reaction-diffusion models with nonlocal delay effect and nonlinear boundary condition, Appl. Math. Lett., 103 (2020), 106197, 7 pp. doi: 10.1016/j.aml.2019.106197.

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[18]

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M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk (N. S.), 3 (1948), 3-95. 

[20]

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[21]

C. LeiJ. Xiong and X. Zhou, Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 81-98.  doi: 10.3934/dcdsb.2019173.

[22]

G. Li and Z. Jin, Global stability of an SEI epidemic model, Chaos Solitons Fractals, 21 (2004), 925-931. doi: 10.1016/j.chaos.2003.12.031.

[23]

G. Li and Z. Jin, Global stability of an SEI epidemic model with general contact rate, Chaos Solitons Fractals, 23 (2005), 997-1004.  doi: 10.1016/j.chaos.2004.06.012.

[24]

H. LiR. Peng and F. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913.  doi: 10.1016/j.jde.2016.09.044.

[25]

H. LiR. Peng and T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, European J. Appl. Math., 31 (2020), 26-56.  doi: 10.1017/S0956792518000463.

[26]

H. J. Li and S. Guo, Dynamics of a SIRC epidemiological model, Electron. J. Differential Equations, 2017 (2017), 1-18.

[27]

S. Li and S. Guo, Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2693-2719.  doi: 10.3934/dcdsb.2020201.

[28]

S. Li and S. Guo, Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5101-5134.  doi: 10.3934/dcdsb.2020335.

[29]

S. Li and S. Guo, Permanence and extinction of a stochastic prey-predator model with a general functional response, Math. Comput. Simulation, 187 (2021), 308-336.  doi: 10.1016/j.matcom.2021.02.025.

[30]

S. Li and S. Guo, Stability and Hopf bifurcation in a Hutchinson model, Appl. Math. Lett., 101 (2020), 106066, 7 pp. doi: 10.1016/j.aml.2019.106066.

[31]

C. Liu and S. Guo, Steady states of Lotka-Volterra competition models with nonlinear cross-diffusion, J. Differential Equations, 292 (2021), 247-286.  doi: 10.1016/j.jde.2021.05.014.

[32]

P. MagalG. F. Webb and Y. Wu, On the basic reproduction number of reaction-diffusion epidemic models, SIAM J. Appl. Math., 79 (2019), 284-304.  doi: 10.1137/18M1182243.

[33]

E. Mitidieri and G. Sweers, Weakly coupled elliptic systems and positivity, Math. Nachr., 173 (1995), 259-286.  doi: 10.1002/mana.19951730115.

[34]

J. D. Murray, Mathematical Biology. I. An Introduction, Interdisciplinary Applied Mathematics, Vol. 17, Springer-Verlag, New York, 2002. doi: 0-387-95223-3.

[35]

R. Peng, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model, J. Differential Equations, 247 (2009), 1096-1119.  doi: 10.1016/j.jde.2009.05.002.

[36]

R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.

[37]

R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.

[38]

H. Qiu, S. Guo and S. Li, Stability and bifurcation in a predator-prey system with prey-taxis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050022, 25 pp. doi: 10.1142/S0218127420500224.

[39]

P. SongY. Lou and Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differential Equations, 267 (2019), 5084-5114.  doi: 10.1016/j.jde.2019.05.022.

[40]

G. Sweers, Strong positivity in $C(\bar{\Omega})$ for elliptic systems, Math. Z., 209 (1992), 251-271.  doi: 10.1007/BF02570833.

[41]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.

[42]

Y. WangZ. Wang and C. Lei, Asymptotic profile of endemic equilibrium to a diffusive epidemic model with saturated incidence rate, Math. Biosci. Eng., 16 (2019), 3885-3913.  doi: 10.3934/mbe.2019192.

[43]

Y. Wang and S. Guo, Global existence and asymptotic behavior of a two-species competitive Keller-Segel system on $\mathbb{R}^N$, Nonlinear Anal. Real World Appl., 61 (2021), 103342, 41 pp. doi: 10.1016/j.nonrwa.2021.103342.

[44]

Y. Wang and S. Guo, Dynamics for a two-species competitive Keller-Segel chemotaxis system with a free boundary, J. Math. Anal. Appl., 502 (2021), 125259, 39 pp. doi: 10.1016/j.jmaa.2021.125259.

[45]

D. Wei and S. Guo, Hopf bifurcation of a diffusive SIS epidemic system with delay in heterogeneous environment, Applicable Analysis, (2021). doi: 10.1080/00036811.2021.1909724.

[46]

D. Wei and S. Guo, Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2599-2623.  doi: 10.3934/dcdsb.2020197.

[47]

Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differential Equations, 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.

[48]

S. Yan and S. Guo, Stability analysis of a stage-structure model with spatial heterogeneity, Math Meth Appl Sci., (2021). doi: 10.1002/mma.7464.

show all references

References:
[1]

L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM J. Appl. Math., 67 (2007), 1283-1309.  doi: 10.1137/060672522.

[2]

L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.

[4]

S. Chen and J. Shi, Asymptotic profiles of basic reproduction number for epidemic spreading in heterogeneous environment, SIAM J. Appl. Math., 80 (2020), 1247-1271.  doi: 10.1137/19M1289078.

[5]

R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.

[6]

R. CuiK. Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differential Equations, 263 (2017), 2343-2373.  doi: 10.1016/j.jde.2017.03.045.

[7]

K. Deng and Y. Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.  doi: 10.1017/S0308210515000864.

[8]

Y. DuR. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007.

[9]

Z. Du and R. Peng, A priori $L^{\infty}$ estimates for solutions of a class of reaction-diffusion systems, J. Math. Biol., 72 (2016), 1429-1439.  doi: 10.1007/s00285-015-0914-z.

[10]

J. GaoS. Guo and W. Shen, Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2645-2676.  doi: 10.3934/dcdsb.2020199.

[11]

J. GeK. I. KimZ. Lin and H. Zhu, An SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.

[12]

J. GeL. Lin and L. Zhang, A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 2763-2776.  doi: 10.3934/dcdsb.2017134.

[13]

S. Guo, Bifurcation in a reaction-diffusion model with nonlocal delay effect and nonlinear boundary condition, J. Differential Equations, 289 (2021), 236-278.  doi: 10.1016/j.jde.2021.04.021.

[14]

S. Guo, S. Li and B. Sounvoravong, Oscillatory and stationary patterns in a diffusive model with delay effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2150035, 21 pp. doi: 10.1142/S0218127421500358.

[15]

S. Guo and S. Li, On the stability of reaction-diffusion models with nonlocal delay effect and nonlinear boundary condition, Appl. Math. Lett., 103 (2020), 106197, 7 pp. doi: 10.1016/j.aml.2019.106197.

[16]

J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59.  doi: 10.1016/0022-247X(69)90175-9.

[17]

S.-B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwanese J. Math., 9 (2005), 151-173.  doi: 10.11650/twjm/1500407791.

[18]

W. O. Kermack and A. G. McKendrick, A contribution to mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721.  doi: 10.1007/BF02464424.

[19]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk (N. S.), 3 (1948), 3-95. 

[20]

K.-Y. Lam and Y. Lou, Asymptotic behavior of the principle eigenvalue for cooperative elliptic systems and applications, J. Dynam. Differential Equations, 28 (2016), 29-48.  doi: 10.1007/s10884-015-9504-4.

[21]

C. LeiJ. Xiong and X. Zhou, Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 81-98.  doi: 10.3934/dcdsb.2019173.

[22]

G. Li and Z. Jin, Global stability of an SEI epidemic model, Chaos Solitons Fractals, 21 (2004), 925-931. doi: 10.1016/j.chaos.2003.12.031.

[23]

G. Li and Z. Jin, Global stability of an SEI epidemic model with general contact rate, Chaos Solitons Fractals, 23 (2005), 997-1004.  doi: 10.1016/j.chaos.2004.06.012.

[24]

H. LiR. Peng and F. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913.  doi: 10.1016/j.jde.2016.09.044.

[25]

H. LiR. Peng and T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, European J. Appl. Math., 31 (2020), 26-56.  doi: 10.1017/S0956792518000463.

[26]

H. J. Li and S. Guo, Dynamics of a SIRC epidemiological model, Electron. J. Differential Equations, 2017 (2017), 1-18.

[27]

S. Li and S. Guo, Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2693-2719.  doi: 10.3934/dcdsb.2020201.

[28]

S. Li and S. Guo, Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5101-5134.  doi: 10.3934/dcdsb.2020335.

[29]

S. Li and S. Guo, Permanence and extinction of a stochastic prey-predator model with a general functional response, Math. Comput. Simulation, 187 (2021), 308-336.  doi: 10.1016/j.matcom.2021.02.025.

[30]

S. Li and S. Guo, Stability and Hopf bifurcation in a Hutchinson model, Appl. Math. Lett., 101 (2020), 106066, 7 pp. doi: 10.1016/j.aml.2019.106066.

[31]

C. Liu and S. Guo, Steady states of Lotka-Volterra competition models with nonlinear cross-diffusion, J. Differential Equations, 292 (2021), 247-286.  doi: 10.1016/j.jde.2021.05.014.

[32]

P. MagalG. F. Webb and Y. Wu, On the basic reproduction number of reaction-diffusion epidemic models, SIAM J. Appl. Math., 79 (2019), 284-304.  doi: 10.1137/18M1182243.

[33]

E. Mitidieri and G. Sweers, Weakly coupled elliptic systems and positivity, Math. Nachr., 173 (1995), 259-286.  doi: 10.1002/mana.19951730115.

[34]

J. D. Murray, Mathematical Biology. I. An Introduction, Interdisciplinary Applied Mathematics, Vol. 17, Springer-Verlag, New York, 2002. doi: 0-387-95223-3.

[35]

R. Peng, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model, J. Differential Equations, 247 (2009), 1096-1119.  doi: 10.1016/j.jde.2009.05.002.

[36]

R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.

[37]

R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.

[38]

H. Qiu, S. Guo and S. Li, Stability and bifurcation in a predator-prey system with prey-taxis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050022, 25 pp. doi: 10.1142/S0218127420500224.

[39]

P. SongY. Lou and Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differential Equations, 267 (2019), 5084-5114.  doi: 10.1016/j.jde.2019.05.022.

[40]

G. Sweers, Strong positivity in $C(\bar{\Omega})$ for elliptic systems, Math. Z., 209 (1992), 251-271.  doi: 10.1007/BF02570833.

[41]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.

[42]

Y. WangZ. Wang and C. Lei, Asymptotic profile of endemic equilibrium to a diffusive epidemic model with saturated incidence rate, Math. Biosci. Eng., 16 (2019), 3885-3913.  doi: 10.3934/mbe.2019192.

[43]

Y. Wang and S. Guo, Global existence and asymptotic behavior of a two-species competitive Keller-Segel system on $\mathbb{R}^N$, Nonlinear Anal. Real World Appl., 61 (2021), 103342, 41 pp. doi: 10.1016/j.nonrwa.2021.103342.

[44]

Y. Wang and S. Guo, Dynamics for a two-species competitive Keller-Segel chemotaxis system with a free boundary, J. Math. Anal. Appl., 502 (2021), 125259, 39 pp. doi: 10.1016/j.jmaa.2021.125259.

[45]

D. Wei and S. Guo, Hopf bifurcation of a diffusive SIS epidemic system with delay in heterogeneous environment, Applicable Analysis, (2021). doi: 10.1080/00036811.2021.1909724.

[46]

D. Wei and S. Guo, Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2599-2623.  doi: 10.3934/dcdsb.2020197.

[47]

Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differential Equations, 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.

[48]

S. Yan and S. Guo, Stability analysis of a stage-structure model with spatial heterogeneity, Math Meth Appl Sci., (2021). doi: 10.1002/mma.7464.

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