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doi: 10.3934/dcdsb.2020273

Effect of diffusion in a spatial SIS epidemic model with spontaneous infection

Yachun Tong 1, , Inkyung Ahn 2, and  Zhigui Lin 3,,

1. 

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
2. 

Department of Mathematics, Korea University, 2511, Sejong-ro, Sejong 339-700, South Korea
3. 

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

* Corresponding author: Email: zglin68@hotmail.com (Z. Lin)

Received  February 2020 Revised  July 2020 Published  September 2020

This paper is concerned with an SIS epidemic reaction-diffusion model with mass-action incidence incorporating spontaneous infection in a spatially heterogeneous environment. The main goal of this article is to study the influence of spontaneous infection on the endemic equilibrium (EE) of the model. To achieve this, first the existence of EE is investigated. Furthermore, we discuss the asymptotic behavior of endemic equilibrium if the migration rate of the susceptible or infected population is sufficiently small. Compared to the case without spontaneous infection, our theoretical results show that spontaneous infection can enhance persistence of infectious disease.

Keywords: SIS epidemic model, Spontaneous infection, Endemic equilibria, Persistence.
Mathematics Subject Classification: Primary:35K57;35B40;Secondary:92D25.
Citation: Yachun Tong, Inkyung Ahn, Zhigui Lin. Effect of diffusion in a spatial SIS epidemic model with spontaneous infection. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020273
References:
[1]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic disease patch model, SIAM J. Appl. Math., 76 (2007), 1283-1309.  doi: 10.1137/060672522.  Google Scholar

[2]

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R. H. Cui, K. 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

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R. H. 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

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K. Deng and Y. X. 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

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H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

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H. W. Hethcote, Epidemiology models with variable population size, Mathematical Understanding of Infectious Disease Dynamics, 16 (2009), 63-89.  doi: 10.1142/9789812834836_0002.  Google Scholar

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A. Hill, D. G. Rand, M. A. Nowak and N. A. Christakis, Emotions as infectious diseases in a large social network: The SISa model, Proceedings of the Royal Society B, 277 (2010), 3827-3835.  doi: 10.1098/rspb.2010.1217.  Google Scholar

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H. C. Li, R. Peng and F. B. 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

[21]

C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[22]

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.  Google Scholar

[23]

L. Nirenberg, Topics in Nonlinear Functional Analysis, Providence, RI: American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/006.  Google Scholar

[24]

S. O'Regan and J. Drake, Theory of early warning signals of disease emergenceand leading indicators of elimination, Theoretical Ecology, 6 (2013), 333-357.  doi: 10.1007/s12080-013-0185-5.  Google Scholar

[25]

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

[26]

R. Peng and S. Q. 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.  Google Scholar

[27]

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

[28]

R. Peng and F. Q. 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

[29]

H. J. Shi, Z. S. Duan and G. R. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144.  doi: 10.1016/j.physa.2007.11.048.  Google Scholar

[30]

Y. C. Tong and C. X. 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.  Google Scholar

[31]

X. W. Wen, J. P. Ji and B. Li, Asymptotic profiles of the endemic equilibrium to a diffusive SIS epidemic model with mass action infection mechanism, J. Math. Anal. Appl., 458 (2018), 715-729.  doi: 10.1016/j.jmaa.2017.08.016.  Google Scholar

[32]

Y. X. Wu and X. F. 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

[33]

M. Yang, G. R. Chen and X. C. Fu, A modified SIS model with an infective medium on complex networks and its global stability, Physica A, 390 (2011), 2408-2413.  doi: 10.1016/j.physa.2011.02.007.  Google Scholar

show all references

References:
[1]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic disease patch model, SIAM J. Appl. Math., 76 (2007), 1283-1309.  doi: 10.1137/060672522.  Google Scholar

[2]

L. J. S. Allen, B. M. Bolker, Y. 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]

F. Altarelli, A. Braunstein, L. Dall'Asta, J. R. Wakeling and R. Zecchina, Containing epidemic outbreaks by message-passing techniques, Physical Review X, 4 (2014), 021024. doi: 10.1103/PhysRevX.4.021024.  Google Scholar

[4]

R. M. Anderson and R. M. May, Population biology of infectious diseases: Part â…, Discrete Contin. Dyn. Syst. Ser. A, 280 (1979), 361-367.  doi: 10.1038/280361a0.  Google Scholar

[5]

H. Amman, Invariant sets and existence theorems for semilinear parabolic and elliptic systems, J.Math. Anal.Appl., 65 (1978), 432-467.  doi: 10.1016/0022-247X(78)90192-0.  Google Scholar

[6]

H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.  Google Scholar

[7]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Ser. Math. Comput. Biol., 2003. doi: 10.1002/0470871296.  Google Scholar

[8]

R. H. Cui, K. 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

[9]

R. H. 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

[10]

K. Deng and Y. X. 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

[11]

Y. H. Du, R. Peng and M. X. 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

[12]

Z. J. 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

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001.  Google Scholar

[14]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[15]

H. W. Hethcote, Epidemiology models with variable population size, Mathematical Understanding of Infectious Disease Dynamics, 16 (2009), 63-89.  doi: 10.1142/9789812834836_0002.  Google Scholar

[16]

A. Hill, D. G. Rand, M. A. Nowak and N. A. Christakis, Emotions as infectious diseases in a large social network: The SISa model, Proceedings of the Royal Society B, 277 (2010), 3827-3835.  doi: 10.1098/rspb.2010.1217.  Google Scholar

[17]

A. Hill, D. G. Rand, M. A. Nowak and N. A. Christakis, Infectious disease modeling of social contagion in networks, Plos Comput. Biol., 6 (2010), e1000968, 15 pp. doi: 10.1371/journal.pcbi.1000968.  Google Scholar

[18]

W. Z. Huang, M. A. Han and K. Y. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Biosci. Eng., 7 (2010), 51-66.  doi: 10.3934/mbe.2010.7.51.  Google Scholar

[19] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008.   Google Scholar
[20]

H. C. Li, R. Peng and F. B. 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

[21]

C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[22]

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.  Google Scholar

[23]

L. Nirenberg, Topics in Nonlinear Functional Analysis, Providence, RI: American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/006.  Google Scholar

[24]

S. O'Regan and J. Drake, Theory of early warning signals of disease emergenceand leading indicators of elimination, Theoretical Ecology, 6 (2013), 333-357.  doi: 10.1007/s12080-013-0185-5.  Google Scholar

[25]

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

[26]

R. Peng and S. Q. 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.  Google Scholar

[27]

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

[28]

R. Peng and F. Q. 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

[29]

H. J. Shi, Z. S. Duan and G. R. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144.  doi: 10.1016/j.physa.2007.11.048.  Google Scholar

[30]

Y. C. Tong and C. X. 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.  Google Scholar

[31]

X. W. Wen, J. P. Ji and B. Li, Asymptotic profiles of the endemic equilibrium to a diffusive SIS epidemic model with mass action infection mechanism, J. Math. Anal. Appl., 458 (2018), 715-729.  doi: 10.1016/j.jmaa.2017.08.016.  Google Scholar

[32]

Y. X. Wu and X. F. 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

[33]

M. Yang, G. R. Chen and X. C. Fu, A modified SIS model with an infective medium on complex networks and its global stability, Physica A, 390 (2011), 2408-2413.  doi: 10.1016/j.physa.2011.02.007.  Google Scholar

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