# American Institute of Mathematical Sciences

December  2018, 23(10): 4499-4517. doi: 10.3934/dcdsb.2018173

## Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment

 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, China

Received  August 2017 Revised  January 2018 Published  December 2018 Early access  June 2018

Fund Project: The research was partially supported by NSF of China (No. 11671175), the Priority Academic Program Development of Jiangsu Higher Education Institutions.

In this paper, we deal with a diffusive SIR epidemic model with nonlinear incidence of the form $I^pS^q$ for $0<p≤1$ in a heterogeneous environment. We establish the boundedness and uniform persistence of solutions to the system, and the global stability of the constant endemic equilibrium in the case of homogeneous environment. When the spatial environment is heterogeneous, we determine the asymptotic profile of endemic equilibrium if the diffusion rate of the susceptible or infected population is small. Our theoretical analysis shows that, different from the studies of [1,28,38,44] for the SIS models, restricting the movement of the susceptible or infected population can not lead to the extinction of infectious disease for such an SIR system.

Citation: Chengxia Lei, Fujun Li, Jiang Liu. Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4499-4517. doi: 10.3934/dcdsb.2018173
##### 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] 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. [3] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley and Sons Ltd., Chichester, UK, 2003. doi: 10.1002/0470871296. [4] J. Cui, X. Tao and H. Zhu, An SIS infection model incorporating media coverage, Rocky Mount. J. Math., 38 (2008), 1323-1334.  doi: 10.1216/RMJ-2008-38-5-1323. [5] 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. [6] 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. [7] K. Deng and Y. Wu, Dynamics of an SIS epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.  doi: 10.1017/S0308210515000864. [8] W. R. Derrick and P. van den Driessche, A disease transmission model in a nonconstant population, J. Math. Biol., 31 (1993), 495-512.  doi: 10.1007/BF00173889. [9] W. R. Derrick and P. van den Driessche, Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 299-309.  doi: 10.3934/dcdsb.2003.3.299. [10] W. Ding, W. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1291-1304.  doi: 10.3934/dcdsb.2013.18.1291. [11] 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. [12] Z. Du and R. Peng, A priori $L^∞$ 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. Gao and S. Ruan, An SIS patch model with variable transmission coefficients, Math. Biosci., 232 (2011), 110-115.  doi: 10.1016/j.mbs.2011.05.001. [14] 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. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, 2001. [16] P. Glendinning and L. P. Perry, Melnikov analysis of chaos in a simple epidemiological model, J. Math. Biol., 35 (1997), 359-373.  doi: 10.1007/s002850050056. [17] H. W. Hethcote, Epidemiology models with variable population size, Mathematical understanding of infectious disease dynamics, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., World Sci. Publ., Hackensack, NJ, 16 (2009), 63-89.  doi: 10.1142/9789812834836_0002. [18] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [19] H. W. Hethcote, M. A. Lewis and P. van den Driessche, An epidemiological model with delay and a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49-64.  doi: 10.1007/BF00276080. [20] H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287.  doi: 10.1007/BF00160539. [21] S.-Z. Hsu, F.-B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013), 265-297.  doi: 10.1016/j.jde.2013.04.006. [22] W. Huang, M. Han and K. 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. [23] H. Jiang and W.-M. Ni, A priori estimates of stationary solutions of an activator-inhibitor system, Indiana Univ. Math. J., 56 (2007), 681-732.  doi: 10.1512/iumj.2007.56.2982. [24] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700-721. [25] A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.  doi: 10.3934/mbe.2004.1.57. [26] A. Korobeinikov and G. C. Wake, A Lyapunov function and global stability for SIR, SEIR and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960.  doi: 10.1016/S0893-9659(02)00069-1. [27] K. Kuto, H. Matsuzawa and R. Peng, Concentration profile of the endemic equilibria of a reaction-diffusion-advection SIS epidemic model, Calc. Var. Partial Differential Equations, 56 (2017), Art. 112, 28 pp. doi: 10.1007/s00526-017-1207-8. [28] H. Li, R. Peng and F.-B. Wang, Vary 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. [29] C. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotais systems, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7. [30] W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162. [31] W. M. Liu, S. A. Levin and Y. Isawa, Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956. [32] J. Liu, B. Peng and T. Zhang, Effect of discretization on dynamical behavior of SEIR and SIR models with nonlinear incidence, Appl. Math. Lett., 39 (2015), 60-66.  doi: 10.1016/j.aml.2014.08.012. [33] M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epidemiology, J. Math. Biol., 35 (1996), 21-36.  doi: 10.1007/s002850050040. [34] D. Le, Dissipativity and global attractors for a class of quasilinear parabolic systems, Comm. Partial Differential Equations, 22 (1997), 413-433.  doi: 10.1080/03605309708821269. [35] 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. [36] 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. [37] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173. [38] 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. [39] 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. [40] R. Peng, J. Shi and M. 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. [41] 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. [42] 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. [43] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2. [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] X.-Q. Zhao, Dynamical Systems in Population Biology, $2^{nd}$ edition, Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.

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 reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1. [2] 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. [3] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley and Sons Ltd., Chichester, UK, 2003. doi: 10.1002/0470871296. [4] J. Cui, X. Tao and H. Zhu, An SIS infection model incorporating media coverage, Rocky Mount. J. Math., 38 (2008), 1323-1334.  doi: 10.1216/RMJ-2008-38-5-1323. [5] 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. [6] 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. [7] K. Deng and Y. Wu, Dynamics of an SIS epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.  doi: 10.1017/S0308210515000864. [8] W. R. Derrick and P. van den Driessche, A disease transmission model in a nonconstant population, J. Math. Biol., 31 (1993), 495-512.  doi: 10.1007/BF00173889. [9] W. R. Derrick and P. van den Driessche, Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 299-309.  doi: 10.3934/dcdsb.2003.3.299. [10] W. Ding, W. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1291-1304.  doi: 10.3934/dcdsb.2013.18.1291. [11] 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. [12] Z. Du and R. Peng, A priori $L^∞$ 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. Gao and S. Ruan, An SIS patch model with variable transmission coefficients, Math. Biosci., 232 (2011), 110-115.  doi: 10.1016/j.mbs.2011.05.001. [14] 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. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, 2001. [16] P. Glendinning and L. P. Perry, Melnikov analysis of chaos in a simple epidemiological model, J. Math. Biol., 35 (1997), 359-373.  doi: 10.1007/s002850050056. [17] H. W. Hethcote, Epidemiology models with variable population size, Mathematical understanding of infectious disease dynamics, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., World Sci. Publ., Hackensack, NJ, 16 (2009), 63-89.  doi: 10.1142/9789812834836_0002. [18] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [19] H. W. Hethcote, M. A. Lewis and P. van den Driessche, An epidemiological model with delay and a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49-64.  doi: 10.1007/BF00276080. [20] H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287.  doi: 10.1007/BF00160539. [21] S.-Z. Hsu, F.-B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013), 265-297.  doi: 10.1016/j.jde.2013.04.006. [22] W. Huang, M. Han and K. 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. [23] H. Jiang and W.-M. Ni, A priori estimates of stationary solutions of an activator-inhibitor system, Indiana Univ. Math. J., 56 (2007), 681-732.  doi: 10.1512/iumj.2007.56.2982. [24] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700-721. [25] A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.  doi: 10.3934/mbe.2004.1.57. [26] A. Korobeinikov and G. C. Wake, A Lyapunov function and global stability for SIR, SEIR and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960.  doi: 10.1016/S0893-9659(02)00069-1. [27] K. Kuto, H. Matsuzawa and R. Peng, Concentration profile of the endemic equilibria of a reaction-diffusion-advection SIS epidemic model, Calc. Var. Partial Differential Equations, 56 (2017), Art. 112, 28 pp. doi: 10.1007/s00526-017-1207-8. [28] H. Li, R. Peng and F.-B. Wang, Vary 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. [29] C. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotais systems, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7. [30] W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162. [31] W. M. Liu, S. A. Levin and Y. Isawa, Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956. [32] J. Liu, B. Peng and T. Zhang, Effect of discretization on dynamical behavior of SEIR and SIR models with nonlinear incidence, Appl. Math. Lett., 39 (2015), 60-66.  doi: 10.1016/j.aml.2014.08.012. [33] M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epidemiology, J. Math. Biol., 35 (1996), 21-36.  doi: 10.1007/s002850050040. [34] D. Le, Dissipativity and global attractors for a class of quasilinear parabolic systems, Comm. Partial Differential Equations, 22 (1997), 413-433.  doi: 10.1080/03605309708821269. [35] 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. [36] 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. [37] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173. [38] 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. [39] 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. [40] R. Peng, J. Shi and M. 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. [41] 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. [42] 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. [43] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2. [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] X.-Q. Zhao, Dynamical Systems in Population Biology, $2^{nd}$ edition, Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.
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