# American Institute of Mathematical Sciences

December  2021, 14(12): 4503-4520. doi: 10.3934/dcdss.2021120

## Existence and asymptotic profiles of the steady state for a diffusive epidemic model with saturated incidence and spontaneous infection mechanism

 Y.Y. Tseng Functional Analysis Research Center and School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang 150025, China

* Corresponding author

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

Fund Project: Partially supported by National Natural Science Foundation of China (No. 12171125), Natural Science Foundation of Heilongjiang Province (No. LH2020A012), Heilongjiang Postdoctoral Science Foundation (No. LBH-Q17086) and the Fundamental Research Funds for Heilongjiang Provincial Universities (No. 2018-KYYWF-0996)

In this paper, we are concerned with a reaction-diffusion SIS epidemic model with saturated incidence rate, linear source and spontaneous infection mechanism. We derive the uniform bounds of parabolic system and obtain the global asymptotic stability of the constant steady state in a homogeneous environment. Moreover, the existence of the positive steady state is established. We mainly analyze the effects of diffusion, saturation and spontaneous infection on the asymptotic profiles of the steady state. These results show that the linear source and spontaneous infection can enhance the persistence of an infectious disease. Our mathematical approach is based on topological degree theory, singular perturbation technique, the comparison principles for elliptic equations and various elliptic estimates.

Citation: Xueying Sun, Renhao Cui. Existence and asymptotic profiles of the steady state for a diffusive epidemic model with saturated incidence and spontaneous infection mechanism. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4503-4520. doi: 10.3934/dcdss.2021120
##### References:
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Gardner, A semilinear parabolic system arising in the theory of superconductivity, J. Differential Equations, 40 (1981), 232-252.  doi: 10.1016/0022-0396(81)90020-6. [8] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, , Wiley Series in Mathematical and Computational Biology, vol. 17, Wiley, Chichester, UK, 2003. doi: 10.1002/0470871296. [9] V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8. [10] R. 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. [11] 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. [12] K. Deng, Asymptotic behavior of an SIR reaction-diffusion model with a linear source, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5945-5957.  doi: 10.3934/dcdsb.2019114. [13] K. Deng and Y. 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. [14] Y. Du, R. 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. [15] J. Ge, K. I. Kim, Z. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035. [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, , Springer, New York, 2001. [17] H. W. 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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. [27] H. Li, R. 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. [28] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7. [29] Q. Liu, D. Jiang, T. 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. [30] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010. [31] 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. [32] L. Ma and D. Tang, Evolution of dispersal in advective homogeneous environments, Discrete Contin. Dyn. Syst., 40 (2020), 5815-5830.  doi: 10.3934/dcds.2020247. [33] L. Nirenberg, Topic in Nonlinear Functional Analysis, , Providence, RI: American Mathe- Matical Society. doi: 10.1090/cln/006. [34] 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. [35] 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. [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.-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. [38] 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, 22 pp. doi: 10.1016/j.jmaa.2020.124212. [39] 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. [40] 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. [41] X. Wang, P. Kloeden and M. Yang, Asymptotic behaviour of a neural field lattice model with delays, Electron. Res. Arch., 28 (2020), 1037-1048.  doi: 10.3934/era.2020056. [42] Y. Wang, Z. Wang and C. Lei, Asymptotic profile of endemic equilibrium to a diffusive epidemic model with saturated incidence rate, Math. Biosci. Eng., 16 (2019), 3885-3913.  doi: 10.3934/mbe.2019192. [43] S.-L. Wu and C.-H. Hsu, Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity, Adv. Nonlinear Anal., 9 (2020), 923-957.  doi: 10.1515/anona-2020-0033. [44] Y. Wu and X. 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. [45] Y. Yang, Y.-R. Yang and X.-J. Jiao, Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence, Electron. Res. Arch., 28 (2020), 1-13.  doi: 10.3934/era.2020001. [46] J. Zhang and R. Cui, Asymptotic profiles of the endemic equilibrium of a diffusive SIS epidemic system with saturated incidence rate and spontaneous infection, Math. Methods Appl. Sci., 44 (2021), 517-532.  doi: 10.1002/mma.6754. [47] B. Zheng and J. Yu, Existence and uniqueness of periodic orbits in a discrete model on Wolbachia infection frequency, Adv. Nonlinear Anal., 11 (2022), 212-224.  doi: 10.1515/anona-2020-0194.

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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, Discrete Contin. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1. [2] R. M. Anderson and R. M. May, Population biology of infectious diseases, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0. [3] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control,, Oxford University Press, Oxford, 1991. [4] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, , Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9. [5] F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology, , Springer, Berlin, 2008. doi: 10.1007/978-3-540-78911-6. [6] 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. [7] K. J. Brown, P. C. Dunne and R. A. Gardner, A semilinear parabolic system arising in the theory of superconductivity, J. Differential Equations, 40 (1981), 232-252.  doi: 10.1016/0022-0396(81)90020-6. [8] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, , Wiley Series in Mathematical and Computational Biology, vol. 17, Wiley, Chichester, UK, 2003. doi: 10.1002/0470871296. [9] V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8. [10] R. 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. [11] 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. [12] K. Deng, Asymptotic behavior of an SIR reaction-diffusion model with a linear source, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5945-5957.  doi: 10.3934/dcdsb.2019114. [13] K. Deng and Y. 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. [14] Y. Du, R. 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. [15] J. Ge, K. I. Kim, Z. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035. [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, , Springer, New York, 2001. [17] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [18] A. L. Hill, D. G. Rand, M. A. Nowak and N. A. Christakis, Emotions as infectious diseases in a large social network: The SIS model, Proc. R. Soc. B, 277 (2010), 3827-3835.  doi: 10.1098/rspb.2010.1217. [19] 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. [20] C. Huang and Y. Tan, Global behavior of a reaction-diffusion model with time delay and Dirichlet condition, J. Differential Equations, 271 (2021), 186-215.  doi: 10.1016/j.jde.2020.08.008. [21] L. Huang, H. Ma, J. Wang and C. Huang, Global dynamics of a Filippov plant disease model with an economic threshold of infected-susceptible ratio, J. Appl. Anal. Comput., 10 (2020), 2263-2277.  doi: 10.11948/20190409. [22] M. Huang, M. Tang, J. 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. [23] H.-F. Huo, S.-K. Hu and X. Hong, Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment, Electron. Res. Arch., 29 (2021), 2325-2358.  doi: 10.3934/era.2020118. [24] K. Kuto, H. Matsuzawa and R. Peng, Concentration profile of endemic equilibrium of a reaction-diffusion-advection SIS epidemic model,, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 112, 28 pp. doi: 10.1007/s00526-017-1207-8. [25] C. Lei, J. 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. [26] H. 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. [27] H. Li, R. 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. [28] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7. [29] Q. Liu, D. Jiang, T. 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. [30] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010. [31] 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. [32] L. Ma and D. Tang, Evolution of dispersal in advective homogeneous environments, Discrete Contin. Dyn. Syst., 40 (2020), 5815-5830.  doi: 10.3934/dcds.2020247. [33] L. Nirenberg, Topic in Nonlinear Functional Analysis, , Providence, RI: American Mathe- Matical Society. doi: 10.1090/cln/006. [34] 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. [35] 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. [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.-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. [38] 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, 22 pp. doi: 10.1016/j.jmaa.2020.124212. [39] 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. [40] 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. [41] X. Wang, P. Kloeden and M. Yang, Asymptotic behaviour of a neural field lattice model with delays, Electron. Res. Arch., 28 (2020), 1037-1048.  doi: 10.3934/era.2020056. [42] Y. Wang, Z. Wang and C. Lei, Asymptotic profile of endemic equilibrium to a diffusive epidemic model with saturated incidence rate, Math. Biosci. Eng., 16 (2019), 3885-3913.  doi: 10.3934/mbe.2019192. [43] S.-L. Wu and C.-H. Hsu, Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity, Adv. Nonlinear Anal., 9 (2020), 923-957.  doi: 10.1515/anona-2020-0033. [44] Y. Wu and X. 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. [45] Y. Yang, Y.-R. Yang and X.-J. Jiao, Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence, Electron. Res. Arch., 28 (2020), 1-13.  doi: 10.3934/era.2020001. [46] J. Zhang and R. Cui, Asymptotic profiles of the endemic equilibrium of a diffusive SIS epidemic system with saturated incidence rate and spontaneous infection, Math. Methods Appl. Sci., 44 (2021), 517-532.  doi: 10.1002/mma.6754. [47] B. Zheng and J. Yu, Existence and uniqueness of periodic orbits in a discrete model on Wolbachia infection frequency, Adv. Nonlinear Anal., 11 (2022), 212-224.  doi: 10.1515/anona-2020-0194.
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