doi: 10.3934/dcdsb.2021173
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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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021173
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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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