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December  2021, 14(12): 4259-4292. doi: 10.3934/dcdss.2021131

Analysis on a diffusive SEI epidemic model with/without immigration of infected hosts

1. 

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, Jiangsu Province, China

2. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

3. 

School of Mathematics, Tianjin University, Tianjin 300350, China

* Corresponding author: Guanghui Zhang

Received  August 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

Fund Project: C. Lei is partially supported by NSF of China (No. 11671175, 11801232), the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Natural Science Foundation of the Jiangsu Province (No. BK20180999) and the Foundation of Jiangsu Normal University (No. 17XLR008). G. Zhang is partially supported by NSF of China (No. 11501225) and the Fundamental Research Funds for the Central Universities (No. 5003011008). Y. Zhang is partially supported by NSF of China (No. 11701415)

In this paper, we study a reaction-diffusion SEI epidemic model with/without immigration of infected hosts. Our results show that if there is no immigration for the infected (exposed) individuals, the model admits a threshold behaviour in terms of the basic reproduction number, and if the system includes the immigration, the disease always persists. In each case, we explore the global attractivity of the equilibrium via Lyapunov functions in the case of spatially homogeneous environment, and investigate the asymptotic behavior of the endemic equilibrium (when it exists) with respect to the small migration rate of the susceptible, exposed or infected population in the case of spatially heterogeneous environment. Our results suggest that the strategy of controlling the migration rate of population can not eradicate the disease, and the disease transmission risk will be underestimated if the immigration of infected hosts is ignored.

Citation: Chengxia Lei, Yi Shen, Guanghui Zhang, Yuxiang Zhang. Analysis on a diffusive SEI epidemic model with/without immigration of infected hosts. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4259-4292. doi: 10.3934/dcdss.2021131
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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.

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H. CuiP. E. Kloeden and W. Zhao, Strong $(L^2, L^\gamma \cap H^1_0)$-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension, Electron. Res. Arch., 28 (2020), 1357-1374.  doi: 10.3934/era.2020072.

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R. Cui, Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2997-3022.  doi: 10.3934/dcdsb.2020217.

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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.

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R. Cui, H. Li, R. Peng and M. Zhou, Concentration behavior of endemic equilibrium for a reaction-diffusion-advection SIS epidemic model with mass action infection mechanism, Calc. Var. Partial Differential Equations, 60 (2021), 38pp. doi: 10.1007/s00526-021-01992-w.

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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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K. Deng and Y. Wu, Asymptotic behavior of an SIR reaction-diffusion model with a linear source, Discrete Contin. Dyn. Syst. Ser. B, 24 (2018), 5945-5957. 

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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.

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M. HuangM. TangJ. Yu and B. Zheng, A stage structured model of delay differential equations for Aedes mosquito population suppression, Discrete Contin. Dyn. Syst., 40 (2020), 3467-3484.  doi: 10.3934/dcds.2020042.

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Y. Kabeya, Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere, Discrete Contin. Dyn. Syst., 40 (2020), 3529-3559.  doi: 10.3934/dcds.2020040.

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H. LiR. Peng and F.-B. Wang, Vary 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.

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H. LiR. Peng and Z.-A. Wang, On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: Analysis, simulations, and comparison with other mechanisms, SIAM J. Appl. Math., 78 (2018), 2129-2153.  doi: 10.1137/18M1167863.

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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.

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B. LiJ. Zhou and X. Zhou, Asymptotic profiles of endemic equilibrium of a diffusive SIS epidemic system with nonlinear incidence function in a heterogeneous environment, Proc. Amer. Math. Soc., 148 (2020), 4445-4453.  doi: 10.1090/proc/15117.

[26]

Q. LiuD. JiangT. Hayat and A. Alsaedi, Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching, Discrete Contin. Dyn. Syst., 39 (2019), 5683-5706.  doi: 10.3934/dcds.2019249.

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S. Pan, Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle, Electron. Res. Arch., 27 (2019), 89-99.  doi: 10.3934/era.2019011.

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

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[34]

R. PengJ. Shi and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488.  doi: 10.1088/0951-7715/21/7/006.

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R. Peng and Y. Wu, Global $L^\infty$-bounds and long-time behavior of a diffusive epidemic system in a heterogeneous environment, SIAM J. Math. Anal., 53 (2021), 2776-2810.  doi: 10.1137/19M1276030.

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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.

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R. Peng and X. 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.

[38]

L. Pu and Z. Lin, A diffusive SIS epidemic model in a heterogeneous and periodically evolving environment, Math. Biosci. Eng., 16 (2019), 3094-3110.  doi: 10.3934/mbe.2019153.

[39]

S. Rokn-e-vafa and H. T. Tehrani, Diffusive logistic equations with harvesting and heterogeneity under strong growth rate, Adv. Nonlinear Anal., 8 (2019), 455-467.  doi: 10.1515/anona-2016-0208.

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T. Saanouni, Global and non global solutions for a class of coupled parabolic systems, Adv. Nonlinear Anal., 9 (2020), 1383-1401.  doi: 10.1515/anona-2020-0073.

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R. P. Sigdel and C. C. McCluskey, Global stability for an SEI model of infectious disease with immigration, Appl. Math. Comput., 243 (2014), 684-689.  doi: 10.1016/j.amc.2014.06.020.

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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.

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X. Sun and R. Cui, Analysis on a diffusive SIS epidemic model with saturated incidence rate and linear source in a heterogeneous environment, J. Math. Anal. Appl., 490 (2020), 124212, 22pp. doi: 10.1016/j.jmaa.2020.124212.

[44]

J. Suo and B. Li, Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent incidence function in a heterogeneous environment, Math. Biosci. Eng., 17 (2020), 418-441.  doi: 10.3934/mbe.2020023.

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H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.

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Y. Tong and C. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment, Nonlinear Anal. Real World Appl., 41 (2018), 443-460.  doi: 10.1016/j.nonrwa.2017.11.002.

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J. Wang and R. Cui, Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates, Adv. Nonlinear Anal., 10 (2021), 922-951.  doi: 10.1515/anona-2020-0161.

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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.

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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 reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.

[2]

H. CuiP. E. Kloeden and W. Zhao, Strong $(L^2, L^\gamma \cap H^1_0)$-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension, Electron. Res. Arch., 28 (2020), 1357-1374.  doi: 10.3934/era.2020072.

[3]

R. Cui, Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2997-3022.  doi: 10.3934/dcdsb.2020217.

[4]

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.

[5]

R. Cui, H. Li, R. Peng and M. Zhou, Concentration behavior of endemic equilibrium for a reaction-diffusion-advection SIS epidemic model with mass action infection mechanism, Calc. Var. Partial Differential Equations, 60 (2021), 38pp. doi: 10.1007/s00526-021-01992-w.

[6]

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.

[7]

K. Deng and Y. Wu, Asymptotic behavior of an SIR reaction-diffusion model with a linear source, Discrete Contin. Dyn. Syst. Ser. B, 24 (2018), 5945-5957. 

[8]

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.

[9]

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.

[10]

S. Han and C. Lei, Global stability of equilibria of a diffusive SEIR epidemic model with nonlinear incidence, Appl. Math. Lett., 98 (2019), 114-120.  doi: 10.1016/j.aml.2019.05.045.

[11]

M. HuangM. TangJ. Yu and B. Zheng, A stage structured model of delay differential equations for Aedes mosquito population suppression, Discrete Contin. Dyn. Syst., 40 (2020), 3467-3484.  doi: 10.3934/dcds.2020042.

[12]

Y. Kabeya, Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere, Discrete Contin. Dyn. Syst., 40 (2020), 3529-3559.  doi: 10.3934/dcds.2020040.

[13]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Am. Math. Soc. Transl., 3 (1948), 3-95. 

[14]

K. Kuto, H. Matsuzawa and R. Peng, Concentration profile of the endemic equilibria of a reaction-diffusion-advection SIS epidemic model, Calc. Var. Partial Differential Equations, 56 (2017), 28pp. doi: 10.1007/s00526-017-1207-8.

[15]

C. LeiF. Li and J. Liu, Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4499-4517.  doi: 10.3934/dcdsb.2018173.

[16]

C. LeiZ. Lin and Q. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166.  doi: 10.1016/j.jde.2014.03.015.

[17]

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.

[18]

C. Lei and J. Zhou, Long-time dynamics of a reaction-diffusion system with negative feedback and inhibition, Appl. Math. Lett., 107 (2020), 106475, 8pp. doi: 10.1016/j.aml.2020.106475.

[19]

B. Li and Q. Bie, Long-time dynamics of an SIRS reaction-diffusion epidemic model, J. Math. Anal. Appl., 475 (2019), 1910-1926.  doi: 10.1016/j.jmaa.2019.03.062.

[20]

B. Li, H. Li and Y. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), 25pp. doi: 10.1007/s00033-017-0845-1.

[21]

H. Li and R. Peng, Dynamics and asymptotic profiles of endemic equilibrium for SIS epidemic patch models, J. Math. Biol., 79 (2019), 1279-1317.  doi: 10.1007/s00285-019-01395-8.

[22]

H. LiR. Peng and F.-B. Wang, Vary 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.

[23]

H. LiR. Peng and Z.-A. Wang, On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: Analysis, simulations, and comparison with other mechanisms, SIAM J. Appl. Math., 78 (2018), 2129-2153.  doi: 10.1137/18M1167863.

[24]

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.

[25]

B. LiJ. Zhou and X. Zhou, Asymptotic profiles of endemic equilibrium of a diffusive SIS epidemic system with nonlinear incidence function in a heterogeneous environment, Proc. Amer. Math. Soc., 148 (2020), 4445-4453.  doi: 10.1090/proc/15117.

[26]

Q. LiuD. JiangT. Hayat and A. Alsaedi, Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching, Discrete Contin. Dyn. Syst., 39 (2019), 5683-5706.  doi: 10.3934/dcds.2019249.

[27]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.

[28]

C. C. McCluskey, Lyapunov functions for disease models with immigration of infected hosts, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 4479-4491.  doi: 10.3934/dcdsb.2020296.

[29]

C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression, Math. Biosci. Eng., 3 (2006), 603-614.  doi: 10.3934/mbe.2006.3.603.

[30]

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

[31]

S. Pan, Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle, Electron. Res. Arch., 27 (2019), 89-99.  doi: 10.3934/era.2019011.

[32]

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

[33]

R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal., 71 (2009), 239-247.  doi: 10.1016/j.na.2008.10.043.

[34]

R. PengJ. Shi and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488.  doi: 10.1088/0951-7715/21/7/006.

[35]

R. Peng and Y. Wu, Global $L^\infty$-bounds and long-time behavior of a diffusive epidemic system in a heterogeneous environment, SIAM J. Math. Anal., 53 (2021), 2776-2810.  doi: 10.1137/19M1276030.

[36]

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.

[37]

R. Peng and X. 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.

[38]

L. Pu and Z. Lin, A diffusive SIS epidemic model in a heterogeneous and periodically evolving environment, Math. Biosci. Eng., 16 (2019), 3094-3110.  doi: 10.3934/mbe.2019153.

[39]

S. Rokn-e-vafa and H. T. Tehrani, Diffusive logistic equations with harvesting and heterogeneity under strong growth rate, Adv. Nonlinear Anal., 8 (2019), 455-467.  doi: 10.1515/anona-2016-0208.

[40]

T. Saanouni, Global and non global solutions for a class of coupled parabolic systems, Adv. Nonlinear Anal., 9 (2020), 1383-1401.  doi: 10.1515/anona-2020-0073.

[41]

R. P. Sigdel and C. C. McCluskey, Global stability for an SEI model of infectious disease with immigration, Appl. Math. Comput., 243 (2014), 684-689.  doi: 10.1016/j.amc.2014.06.020.

[42]

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.

[43]

X. Sun and R. Cui, Analysis on a diffusive SIS epidemic model with saturated incidence rate and linear source in a heterogeneous environment, J. Math. Anal. Appl., 490 (2020), 124212, 22pp. doi: 10.1016/j.jmaa.2020.124212.

[44]

J. Suo and B. Li, Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent incidence function in a heterogeneous environment, Math. Biosci. Eng., 17 (2020), 418-441.  doi: 10.3934/mbe.2020023.

[45]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.

[46]

Y. Tong and C. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment, Nonlinear Anal. Real World Appl., 41 (2018), 443-460.  doi: 10.1016/j.nonrwa.2017.11.002.

[47]

J. Wang and R. Cui, Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates, Adv. Nonlinear Anal., 10 (2021), 922-951.  doi: 10.1515/anona-2020-0161.

[48]

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