# American Institute of Mathematical Sciences

August  2021, 26(8): 4045-4057. doi: 10.3934/dcdsb.2020273

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

 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  August 2021 Early access  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.

Citation: Yachun Tong, Inkyung Ahn, Zhigui Lin. Effect of diffusion in a spatial SIS epidemic model with spontaneous infection. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4045-4057. doi: 10.3934/dcdsb.2020273
##### References:
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Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157. [23] L. Nirenberg, Topics in Nonlinear Functional Analysis, Providence, RI: American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/006. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [7] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Ser. Math. Comput. Biol., 2003. doi: 10.1002/0470871296. [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. [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. [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. [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. [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. [13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001. [14] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [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. [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. [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. [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. [19] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008. [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. [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. [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. [23] L. Nirenberg, Topics in Nonlinear Functional Analysis, Providence, RI: American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/006. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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