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June  2022, 27(6): 3077-3100. doi: 10.3934/dcdsb.2021174

Concentration phenomenon of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with spontaneous infection

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

* Corresponding author: Chengxia Lei

Received  February 2021 Revised  May 2021 Published  June 2022 Early access  July 2021

Fund Project: The research was partially supported by NSF of China (No. 11971454, 11801232, 11671175), the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Natural Science Foundation of the Jiangsu Province(No. BK20180999), the Foundation of Jiangsu Normal University (No. 17XLR008)

In this paper, we investigate the effect of spontaneous infection and advection for a susceptible-infected-susceptible epidemic reaction-diffusion-advection model in a heterogeneous environment. The existence of the endemic equilibrium is proved, and the asymptotic behaviors of the endemic equilibrium in three cases (large advection; small diffusion of the susceptible population; small diffusion of the infected population) are established. Our results suggest that the advection can cause the concentration of the susceptible and infected populations at the downstream, and the spontaneous infection can enhance the persistence of infectious disease in the entire habitat.

Citation: Chengxia Lei, Xinhui Zhou. Concentration phenomenon of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with spontaneous infection. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3077-3100. doi: 10.3934/dcdsb.2021174
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[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.

[2]

F. Altarelli, A. Braunstein, L. Dall'Asta, J. R. Wakeling and R. Zecchina, Containing epidemic outbreaks by message-passing techniques, Phys. Rev. X, 4 (2014), 021024.

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H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Jpn., 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.

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Y. Cai and W. Wang, Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 989-1013.  doi: 10.3934/dcdsb.2015.20.989.

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R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Ser. Math. Comput. Biol., John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.

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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.

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R. Cui, H. Li, R. Peng and M. Zhou, Concentration behavior of endemic equilibrium for a reaction-diffusion-advection SIS epidemic model with mass action infection mechanism, preprint, (2019).

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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.

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K. A. DahmenD. R. Nelson and N. M. Shnerb, Life and death near a windy oasis, J. Math. Biol., 41 (2000), 1-23.  doi: 10.1007/s002850000025.

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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.

[11]

Z. 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.

[12]

L. Dung, Dissipativity and global attractors for a class of quasilinear parabolic systems, Comm. Partial Differential Equations, 22 (1997), 413-433.  doi: 10.1080/03605309708821269.

[13]

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.

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, Berlin, 2010.

[15]

S. Han and C. Lei, Global stability of equilibria of a diffusive SEIR epidemic model with nonlinear incidence, Appl. Math. Lett., 98 (2019), 114-120.  doi: 10.1016/j.aml.2019.05.045.

[16]

A. HillD. G. RandM. A. Nowak and N. A. Christakis, Emotions as infectious diseases in a large social network: The SISa model, Proc. R. Soc. B, 277 (2010), 3827-3835. 

[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. 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.

[19] M. J. Keeling and P. Rohani, Modeling Infectious Disease in Humans and Animals, Princeton University Press, Princeton, NJ, 2016. 
[20]

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), Paper No. 112, 28 pp. doi: 10.1007/s00526-017-1207-8.

[21]

K.-Y. LamY. Lou and F. Lutscher, Evolution of dispersal in closed advective environment, J. Biol. Dyn., 9 (2015), 188-212.  doi: 10.1080/17513758.2014.969336.

[22]

C. LeiF. Li and J. Liu, Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4499-4517.  doi: 10.3934/dcdsb.2018173.

[23]

B. Li and Q. Bie, Long-time dynamics of an SIRS reaction-diffusion epidemic model, J. Math. Anal. Appl., 475 (2019), 1910-1926.  doi: 10.1016/j.jmaa.2019.03.062.

[24]

H. LiR. 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.

[25]

H. LiR. 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.

[26]

H. LiR. Peng and T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, European J. Appl. Math., 31 (2020), 26-56.  doi: 10.1017/S0956792518000463.

[27]

Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2.

[28]

F. LutscherM. A. Lewis and E. McCauley, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160.  doi: 10.1007/s11538-006-9100-1.

[29]

F. LutscherE. McCauley and M. A. Lewis, Spatial patterns and coexistence mechanisms in systems with unidirectional flow, Theor. Popul. Biol., 71 (2007), 267-277. 

[30]

F. LutscherE. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772.  doi: 10.1137/050636152.

[31]

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.

[32]

M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.

[33]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.

[34]

M. T. O'Regan and J. M. Drake, Theory of early warning signals of disease emergence and leading indicators of elimination, Theor. Econ., 6 (2013), 333-357. 

[35]

R. Peng, Asymptotic profile 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.

[36]

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.

[37]

R. Peng and Y. Wu, Global $L^\infty$-bounds and long-time behavior of a diffusive epidemic system in heterogeneous environment, SIAM J. Math. Anal., 53 (2021), 2776-2810.  doi: 10.1137/19M1276030.

[38]

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.

[39]

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.

[40]

H. ShiZ. Duan and G. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144. 

[41]

P. SongY. Lou and Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differential Equations, 267 (2019), 5084-5114.  doi: 10.1016/j.jde.2019.05.022.

[42]

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.

[43]

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.

[44]

M. YangG. Chen and X. Fu, A modified SIS model with an infective medium on complex networks and its global stability, Physica A, 390 (2011), 2408-2413. 

[45]

X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Canad. Appl. Math. Quart., 3 (1995), 473-495. 

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.

[2]

F. Altarelli, A. Braunstein, L. Dall'Asta, J. R. Wakeling and R. Zecchina, Containing epidemic outbreaks by message-passing techniques, Phys. Rev. X, 4 (2014), 021024.

[3]

H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Jpn., 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.

[4]

Y. Cai and W. Wang, Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 989-1013.  doi: 10.3934/dcdsb.2015.20.989.

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Ser. Math. Comput. Biol., John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.

[6]

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.

[7]

R. Cui, H. Li, R. Peng and M. Zhou, Concentration behavior of endemic equilibrium for a reaction-diffusion-advection SIS epidemic model with mass action infection mechanism, preprint, (2019).

[8]

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.

[9]

K. A. DahmenD. R. Nelson and N. M. Shnerb, Life and death near a windy oasis, J. Math. Biol., 41 (2000), 1-23.  doi: 10.1007/s002850000025.

[10]

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.

[11]

Z. 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.

[12]

L. Dung, Dissipativity and global attractors for a class of quasilinear parabolic systems, Comm. Partial Differential Equations, 22 (1997), 413-433.  doi: 10.1080/03605309708821269.

[13]

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.

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, Berlin, 2010.

[15]

S. Han and C. Lei, Global stability of equilibria of a diffusive SEIR epidemic model with nonlinear incidence, Appl. Math. Lett., 98 (2019), 114-120.  doi: 10.1016/j.aml.2019.05.045.

[16]

A. HillD. G. RandM. A. Nowak and N. A. Christakis, Emotions as infectious diseases in a large social network: The SISa model, Proc. R. Soc. B, 277 (2010), 3827-3835. 

[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. 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.

[19] M. J. Keeling and P. Rohani, Modeling Infectious Disease in Humans and Animals, Princeton University Press, Princeton, NJ, 2016. 
[20]

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), Paper No. 112, 28 pp. doi: 10.1007/s00526-017-1207-8.

[21]

K.-Y. LamY. Lou and F. Lutscher, Evolution of dispersal in closed advective environment, J. Biol. Dyn., 9 (2015), 188-212.  doi: 10.1080/17513758.2014.969336.

[22]

C. LeiF. Li and J. Liu, Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4499-4517.  doi: 10.3934/dcdsb.2018173.

[23]

B. Li and Q. Bie, Long-time dynamics of an SIRS reaction-diffusion epidemic model, J. Math. Anal. Appl., 475 (2019), 1910-1926.  doi: 10.1016/j.jmaa.2019.03.062.

[24]

H. LiR. 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.

[25]

H. LiR. 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.

[26]

H. LiR. Peng and T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, European J. Appl. Math., 31 (2020), 26-56.  doi: 10.1017/S0956792518000463.

[27]

Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2.

[28]

F. LutscherM. A. Lewis and E. McCauley, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160.  doi: 10.1007/s11538-006-9100-1.

[29]

F. LutscherE. McCauley and M. A. Lewis, Spatial patterns and coexistence mechanisms in systems with unidirectional flow, Theor. Popul. Biol., 71 (2007), 267-277. 

[30]

F. LutscherE. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772.  doi: 10.1137/050636152.

[31]

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.

[32]

M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.

[33]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.

[34]

M. T. O'Regan and J. M. Drake, Theory of early warning signals of disease emergence and leading indicators of elimination, Theor. Econ., 6 (2013), 333-357. 

[35]

R. Peng, Asymptotic profile 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.

[36]

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.

[37]

R. Peng and Y. Wu, Global $L^\infty$-bounds and long-time behavior of a diffusive epidemic system in heterogeneous environment, SIAM J. Math. Anal., 53 (2021), 2776-2810.  doi: 10.1137/19M1276030.

[38]

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.

[39]

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.

[40]

H. ShiZ. Duan and G. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144. 

[41]

P. SongY. Lou and Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differential Equations, 267 (2019), 5084-5114.  doi: 10.1016/j.jde.2019.05.022.

[42]

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.

[43]

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.

[44]

M. YangG. Chen and X. Fu, A modified SIS model with an infective medium on complex networks and its global stability, Physica A, 390 (2011), 2408-2413. 

[45]

X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Canad. Appl. Math. Quart., 3 (1995), 473-495. 

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