doi: 10.3934/cpaa.2021120
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Spatial dynamics of a diffusive SIRI model with distinct dispersal rates and heterogeneous environment

1. 

School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan 232001, Anhui, China

2. 

School of Computer Engineering and Applied Mathematics, Changsha University, Changsha 410022, Hunan, China

3. 

School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, Hunan, China

* Corresponding author

Received  January 2021 Revised  June 2021 Early access July 2021

Fund Project: This work was jointly supported by the Major Program of University Natural Science Research Fund of Anhui Province(KJ2020ZD32), Anhui Provincial Natural Science Foundation (1808085QA01), China Postdoctoral Science Foundation (2018M640579), Anhui Provincial Postdoctoral Science Foundation (2019B329), and National Natural Science Foundation of China (11701007, 11771059, 1197107)

In this paper, we are concerned with the dynamics of a diffusive SIRI epidemic model with heterogeneous parameters and distinct dispersal rates for the susceptible and infected individuals. We first establish the basic properties of solutions to the model, and then identify the basic reproduction number $ \mathscr{R}_{0} $ which serves as a threshold parameter that predicts whether epidemics will persist or become globally extinct. Moreover, we study the asymptotic profiles of the positive steady state as the dispersal rate of the susceptible or infected individuals approaches zero. Our analytical results reveal that the epidemics can be extinct by limiting the movement of the susceptible individuals, and the infected individuals concentrate on certain points in some circumstances when limiting their mobility.

Citation: Lian Duan, Lihong Huang, Chuangxia Huang. Spatial dynamics of a diffusive SIRI model with distinct dispersal rates and heterogeneous environment. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021120
References:
[1]

L. AllenB. 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

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[3]

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B. BerrhaziM. FatiniT. Caraballo and R. Pettersson, A stochastic SIRI epidemic model with Lévy noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2415-2431.  doi: 10.3934/dcdsb.2018057.  Google Scholar

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Y. CaiX. LianZ. Peng and W. Wang, Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment, Nonlinear Anal. Real World Appl., 46 (2019), 178-194.  doi: 10.1016/j.nonrwa.2018.09.006.  Google Scholar

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T. CaraballoM. FatiniR. Pettersson and R. Taki, A stochastic SIRI epidemic model with relapse and media coverage, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3483-3501.  doi: 10.3934/dcdsb.2018250.  Google Scholar

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Y. ChenJ. Li and S. Zou, Global dynamics of an epidemic model with relapse and nonlinear incidence, Math. Methods Appl. Sci., 42 (2019), 1283-1291.  doi: 10.1002/mma.5439.  Google Scholar

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L. Duan and Z. Xu, A note on the dynamics analysis of a diffusive cholera epidemic model with nonlinear incidence rate, Appl. Math. Lett., 106 (2020), 106356. doi: 10.1016/j.aml.2020.106356.  Google Scholar

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M. Fatini, M. Khalifi, R. Gerlach, A. Laaribi and R. Taki, Stationary distribution and threshold dynamics of a stochastic SIRS model with a general incidence, Phys. A, 534 (2019), 120696. doi: 10.1016/j.physa.2019.03.061.  Google Scholar

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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.  Google Scholar

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Z. GuoF. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. Math. Biol., 65 (2012), 1387-1410.  doi: 10.1007/s00285-011-0500-y.  Google Scholar

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H. Hethcote and W. Van, Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs, Math. Biosci., 84 (1987), 85-118.  doi: 10.1016/0025-5564(87)90044-7.  Google Scholar

[16]

H. HuX. YuanL. Huang and C. Huang, Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks, Math. Biosci. Eng., 16 (2019), 5729-5749.  doi: 10.3934/mbe.2019286.  Google Scholar

[17]

H. Hu and X. Zou, Traveling waves of a diffusive SIR epidemic model with general nonlinear incidence and infinitely distributed latency but without demography, Nonlinear Anal. Real World Appl., 58 (2021), 103224. doi: 10.1016/j.nonrwa.2020.103224.  Google Scholar

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P. Magal and X. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[20]

R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[21]

H. Moreira and W. Yuquan, Classroom Note: Global Stability in an S$\to$I$\to$R$\to$I Model, SIAM Rev., 39 (1997), 496-502.  doi: 10.1137/S0036144595295879.  Google Scholar

[22]

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[23]

M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[24]

H. Smith, Monotone Dynamic Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[25]

H. Smith and X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[26]

H. Thieme, Spectral bound and reproduction number for intinite-dimensional population structure and time heterogeneity, SIAMJ. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.  Google Scholar

[27]

D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990), 136-139.  doi: 10.1137/1032003.  Google Scholar

[28]

J. Wang and J. Wang, Analysis of a reaction-diffusion cholera model with distinct dispersal rates in the human population, J. Dyn. Differ. Equ., 33 (2020), 549-575.  doi: 10.1007/s10884-019-09820-8.  Google Scholar

[29]

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

[30]

Y. Wu and X. Zou, Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates, J. Differ. Equ., 264 (2018), 4989-5024.  doi: 10.1016/j.jde.2017.12.027.  Google Scholar

[31]

Y. YangJ. Zhou and C. Hsu, Threshold dynamics of a diffusive SIRI model with nonlinear incidence rate, J. Math. Anal. Appl., 478 (2019), 874-896.  doi: 10.1016/j.jmaa.2019.05.059.  Google Scholar

[32]

Y. YangL. ZouT. Zhang and Y. Xu, Dynamical analysis of a diffusive SIRS model with general incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2433-2451.  doi: 10.3934/dcdsb.2020017.  Google Scholar

[33]

W. Zhang and X. Meng, Stochastic analysis of a novel nonautonomous periodic SIRI epidemic system with random disturbances, Phys. A, 492 (2018), 1290-1301.  doi: 10.1016/j.physa.2017.11.057.  Google Scholar

[34]

X. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

show all references

References:
[1]

L. AllenB. 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

[2] R. Anderson and R. May, Infectious Diesases of Humans: Dynamics and Control, Oxford University Press, 1992.   Google Scholar
[3]

R. Anderson and R. May, Population biology of infectious diseases, part I, Nature, 280 (1979), 361-367.   Google Scholar

[4]

B. BerrhaziM. FatiniT. Caraballo and R. Pettersson, A stochastic SIRI epidemic model with Lévy noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2415-2431.  doi: 10.3934/dcdsb.2018057.  Google Scholar

[5]

J. BenedettiL. Corey and R. Ashley, Recurrence rates in genital herpes after symptomatic first-episode infection, Ann. Int. Med., 121 (1994), 847-854.   Google Scholar

[6]

Y. CaiX. LianZ. Peng and W. Wang, Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment, Nonlinear Anal. Real World Appl., 46 (2019), 178-194.  doi: 10.1016/j.nonrwa.2018.09.006.  Google Scholar

[7]

R. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley & Sons, Ltd., 2004. doi: 10.1002/0470871296.  Google Scholar

[8]

T. CaraballoM. FatiniR. Pettersson and R. Taki, A stochastic SIRI epidemic model with relapse and media coverage, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3483-3501.  doi: 10.3934/dcdsb.2018250.  Google Scholar

[9]

Y. ChenJ. Li and S. Zou, Global dynamics of an epidemic model with relapse and nonlinear incidence, Math. Methods Appl. Sci., 42 (2019), 1283-1291.  doi: 10.1002/mma.5439.  Google Scholar

[10]

L. Duan and Z. Xu, A note on the dynamics analysis of a diffusive cholera epidemic model with nonlinear incidence rate, Appl. Math. Lett., 106 (2020), 106356. doi: 10.1016/j.aml.2020.106356.  Google Scholar

[11]

M. Fatini, M. Khalifi, R. Gerlach, A. Laaribi and R. Taki, Stationary distribution and threshold dynamics of a stochastic SIRS model with a general incidence, Phys. A, 534 (2019), 120696. doi: 10.1016/j.physa.2019.03.061.  Google Scholar

[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.  Google Scholar

[13]

Z. GuoF. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. Math. Biol., 65 (2012), 1387-1410.  doi: 10.1007/s00285-011-0500-y.  Google Scholar

[14]

J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. doi: 10.1090/surv/025.  Google Scholar

[15]

H. Hethcote and W. Van, Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs, Math. Biosci., 84 (1987), 85-118.  doi: 10.1016/0025-5564(87)90044-7.  Google Scholar

[16]

H. HuX. YuanL. Huang and C. Huang, Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks, Math. Biosci. Eng., 16 (2019), 5729-5749.  doi: 10.3934/mbe.2019286.  Google Scholar

[17]

H. Hu and X. Zou, Traveling waves of a diffusive SIR epidemic model with general nonlinear incidence and infinitely distributed latency but without demography, Nonlinear Anal. Real World Appl., 58 (2021), 103224. doi: 10.1016/j.nonrwa.2020.103224.  Google Scholar

[18] Z. MaY. Zhou and J. Wu, Modeling and Dynamics of Infectious Diseases, Higher Education Press, Beijing, 2009.  doi: 10.1142/7223.  Google Scholar
[19]

P. Magal and X. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[20]

R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[21]

H. Moreira and W. Yuquan, Classroom Note: Global Stability in an S$\to$I$\to$R$\to$I Model, SIAM Rev., 39 (1997), 496-502.  doi: 10.1137/S0036144595295879.  Google Scholar

[22]

L. OlsenG. Truty and W. Schaffer, Oscillations and chaos in epidemics: A nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark, Theoret. Population Biol., 33 (1988), 344-370.  doi: 10.1016/0040-5809(88)90019-6.  Google Scholar

[23]

M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[24]

H. Smith, Monotone Dynamic Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[25]

H. Smith and X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[26]

H. Thieme, Spectral bound and reproduction number for intinite-dimensional population structure and time heterogeneity, SIAMJ. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.  Google Scholar

[27]

D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990), 136-139.  doi: 10.1137/1032003.  Google Scholar

[28]

J. Wang and J. Wang, Analysis of a reaction-diffusion cholera model with distinct dispersal rates in the human population, J. Dyn. Differ. Equ., 33 (2020), 549-575.  doi: 10.1007/s10884-019-09820-8.  Google Scholar

[29]

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

[30]

Y. Wu and X. Zou, Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates, J. Differ. Equ., 264 (2018), 4989-5024.  doi: 10.1016/j.jde.2017.12.027.  Google Scholar

[31]

Y. YangJ. Zhou and C. Hsu, Threshold dynamics of a diffusive SIRI model with nonlinear incidence rate, J. Math. Anal. Appl., 478 (2019), 874-896.  doi: 10.1016/j.jmaa.2019.05.059.  Google Scholar

[32]

Y. YangL. ZouT. Zhang and Y. Xu, Dynamical analysis of a diffusive SIRS model with general incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2433-2451.  doi: 10.3934/dcdsb.2020017.  Google Scholar

[33]

W. Zhang and X. Meng, Stochastic analysis of a novel nonautonomous periodic SIRI epidemic system with random disturbances, Phys. A, 492 (2018), 1290-1301.  doi: 10.1016/j.physa.2017.11.057.  Google Scholar

[34]

X. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

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