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  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 & Continuous Dynamical Systems - B, 2018, 23 (10) : 4499-4517. doi: 10.3934/dcdsb.2018173
References:
[1]

L. J. S. AllenB. M. BolkerY. 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

[2]

K. J. BrownP. 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.  Google Scholar

[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.  Google Scholar

[4]

J. CuiX. 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.  Google Scholar

[5]

R. CuiK.-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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

W. DingW. 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.  Google Scholar

[11]

Y. DuR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. GeK. I. KimZ. 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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

H. W. HethcoteM. 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.  Google Scholar

[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.  Google Scholar

[21]

S.-Z. HsuF.-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.  Google Scholar

[22]

W. HuangM. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

H. LiR. 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.  Google Scholar

[29]

C. LinW.-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.  Google Scholar

[30]

W. M. LiuH. 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.  Google Scholar

[31]

W. M. LiuS. 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.  Google Scholar

[32]

J. LiuB. 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.  Google Scholar

[33]

M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epidemiology, J. Math. Biol., 35 (1996), 21-36.  doi: 10.1007/s002850050040.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

R. PengJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

X.-Q. Zhao, Dynamical Systems in Population Biology, $2^{nd}$ edition, Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

show all references

References:
[1]

L. J. S. AllenB. M. BolkerY. 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

[2]

K. J. BrownP. 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.  Google Scholar

[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.  Google Scholar

[4]

J. CuiX. 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.  Google Scholar

[5]

R. CuiK.-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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

W. DingW. 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.  Google Scholar

[11]

Y. DuR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. GeK. I. KimZ. 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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

H. W. HethcoteM. 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.  Google Scholar

[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.  Google Scholar

[21]

S.-Z. HsuF.-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.  Google Scholar

[22]

W. HuangM. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

H. LiR. 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.  Google Scholar

[29]

C. LinW.-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.  Google Scholar

[30]

W. M. LiuH. 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.  Google Scholar

[31]

W. M. LiuS. 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.  Google Scholar

[32]

J. LiuB. 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.  Google Scholar

[33]

M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epidemiology, J. Math. Biol., 35 (1996), 21-36.  doi: 10.1007/s002850050040.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

R. PengJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

X.-Q. Zhao, Dynamical Systems in Population Biology, $2^{nd}$ edition, Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

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