# American Institute of Mathematical Sciences

May  2020, 25(5): 1999-2019. doi: 10.3934/dcdsb.2020013

## Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment

 School of Mathematical Science, Heilongjiang University, Harbin 150080, China

Received  April 2019 Revised  August 2019 Published  December 2019

In this paper, we investigate a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment. We first prove that the solution of the model is bounded when the susceptible and infected individuals have same or distinct dispersal rates. The global stability of the constant endemic equilibrium is proved by constructing suitable Lyapunov functionals when all parameters are positive constants. We employ the topological degree argument to show the existence of positive steady state. Most importantly, we have also investigated the asymptotic profiles of the positive steady state as the dispersal rate of susceptible or infected individuals tends to zero or infinity. Our result reveals that a linear source and spontaneous infection can significantly enhance disease persistence no matter what dispersal rate of the susceptible or infected population is small or large, which leads to the situation that when total population number allows to vary, disease becomes more difficult to control.

Citation: Siyao Zhu, Jinliang Wang. Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1999-2019. doi: 10.3934/dcdsb.2020013
##### 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 reaction-diffusion model, Disc. Cont. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar [2] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0.  Google Scholar [3] H. Brézis 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 [4] 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 [5] 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 [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar [7] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2006), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar [8] A. L. 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 [9] L. Dung, Dissipativity and global attractors for a class of quasilinear parabolic systems, Commun. Partial Differ. Equ., 22 (1997), 413-433.  doi: 10.1080/03605309708821269.  Google Scholar [10] 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 [11] B. Li, H. C. Li and Y. C. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), Art. 96, 25 pp. doi: 10.1007/s00033-017-0845-1.  Google Scholar [12] G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary $p$ in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406.  doi: 10.1137/S003614100343651X.  Google Scholar [13] Y. J. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Bio., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.  Google Scholar [14] L. Nirenberg, Topic in Nonlinear Functional Analysis, Courant Lecture Notes in Mathematics, 6. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/006.  Google Scholar [15] 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 [16] 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.  Google Scholar [17] 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, Physica D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.  Google Scholar [18] R. Peng, J. P. Shi and M. X. 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.  Google Scholar [19] Y. C. Tong and C. X. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment, Nonlinear Analysis: RWA, 41 (2018), 443-460.  doi: 10.1016/j.nonrwa.2017.11.002.  Google Scholar [20] H. F. Weinberger, Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat., 8 (1975), 295-310.   Google Scholar [21] 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

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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 reaction-diffusion model, Disc. Cont. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar [2] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0.  Google Scholar [3] H. Brézis 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 [4] 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 [5] 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 [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar [7] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2006), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar [8] A. L. 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 [9] L. Dung, Dissipativity and global attractors for a class of quasilinear parabolic systems, Commun. Partial Differ. Equ., 22 (1997), 413-433.  doi: 10.1080/03605309708821269.  Google Scholar [10] 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 [11] B. Li, H. C. Li and Y. C. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), Art. 96, 25 pp. doi: 10.1007/s00033-017-0845-1.  Google Scholar [12] G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary $p$ in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406.  doi: 10.1137/S003614100343651X.  Google Scholar [13] Y. J. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Bio., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.  Google Scholar [14] L. Nirenberg, Topic in Nonlinear Functional Analysis, Courant Lecture Notes in Mathematics, 6. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/006.  Google Scholar [15] 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 [16] 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.  Google Scholar [17] 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, Physica D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.  Google Scholar [18] R. Peng, J. P. Shi and M. X. 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.  Google Scholar [19] Y. C. Tong and C. X. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment, Nonlinear Analysis: RWA, 41 (2018), 443-460.  doi: 10.1016/j.nonrwa.2017.11.002.  Google Scholar [20] H. F. Weinberger, Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat., 8 (1975), 295-310.   Google Scholar [21] 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
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