June  2021, 26(6): 2997-3022. doi: 10.3934/dcdsb.2020217

Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate

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

* Corresponding author: Renhao Cui

Received  December 2019 Revised  May 2020 Published  June 2021 Early access  July 2020

Fund Project: Renhao Cui is partially supported by National Natural Science Foundation of China (No.11571364), Natural Science Foundation of Heilongjiang Province (JJ2016ZR0019) and the Fundamental Research Funds for Heilongjiang Provincial Universities (2018-KYYWF-0996)

In this paper, we consider a reaction-diffusion SIS epidemic model with saturated incidence rate in advective heterogeneous environments. The existence of the endemic equilibrium (EE) is established when the basic reproduction number is greater than one. We further investigate the effects of diffusion, advection and saturation on asymptotic profiles of the endemic equilibrium. The individuals concentrate at the downstream end when the advection rate tends to infinity. As the the diffusion rate of the susceptible individuals tends to zero, a certain portion of the susceptible population concentrates at the downstream end, and the remaining portion of the susceptible population distributes in the habitat in a non-homogeneous way; on the other hand, the density of infected population is positive on the entire habitat. The density of the infected vanishes on the habitat for small diffusion rate of infected individuals or the large saturation. The results may provide some implications on disease control and prediction.

Citation: Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2997-3022. doi: 10.3934/dcdsb.2020217
References:
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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]

I. Averill, K.-Y. Lam and Y. Lou, The role of advection in a two-species competition model: A bifurcation approach, Mem. Amer. Math. Soc., 245 (2017), v+117 pp. doi: 10.1090/memo/1161.

[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.

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R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons Ltd., Chichester, 2003. doi: 10.1002/0470871296.

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V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.

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X. ChenK.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst., 32 (2012), 3841-3859.  doi: 10.3934/dcds.2012.32.3841.

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X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658.  doi: 10.1512/iumj.2008.57.3204.

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X. Chen and Y. Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operator and their applications, Indiana Univ. Math. J., 61 (2012), 45-80.  doi: 10.1512/iumj.2012.61.4518.

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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, submitted, (2019), 37 pp.

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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. 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.

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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.

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K.-Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics II, SIAM J. Math. Anal., 44 (2012), 1808-1830.  doi: 10.1137/100819758.

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K.-Y. LamY. Lou and F. Lutscher, The emergence of range limits in advective environments, SIAM J. Appl. Math., 76 (2016), 641-662.  doi: 10.1137/15M1027887.

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K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst., 28 (2010), 1051-1067.  doi: 10.3934/dcds.2010.28.1051.

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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.

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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.

[30]

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.

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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.

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Y. LouD. Xiao and P. Zhou, Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment, Discrete Contin. Dyn. Syst., 36 (2016), 953-969.  doi: 10.3934/dcds.2016.36.953.

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Y. LouX.-Q. Zhao and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. Math. Pures Appl., 121 (2019), 47-82.  doi: 10.1016/j.matpur.2018.06.010.

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Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 259 (2015), 141-171.  doi: 10.1016/j.jde.2015.02.004.

[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.

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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.

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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.

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R. Peng and Y. Wu, Global $L^\infty$-bounds and long-time behavior of a diffusive epidemic system in heterogeneous environment, preprint (arXiv: 1906.11699).

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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.

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R. PengG. Zhang and M. Zhou, Asymptotic behavior of the principal eigenvalue of a second order linear elliptic operator with small/large diffusion coefficient, SIAM J. Math. Anal., 51 (2019), 4724-4753.  doi: 10.1137/18M1217577.

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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.

[45]

R. Peng and M. Zhou, Effects of large degenerate advection and boundary conditions on the principal eigenvalue and its eigenfunction of a linear second order elliptic operator, Indiana Univ. Math. J., 67 (2018), 2523-2568.  doi: 10.1512/iumj.2018.67.7547.

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D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219-1237. 

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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.

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O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Q., 18 (2010), 439-469. 

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O. Vasilyeva and F. Lutscher, Competition of three species in an advective environment, Nonl. Anal. RWA, 13 (2012), 1730-1748.  doi: 10.1016/j.nonrwa.2011.12.004.

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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]

I. Averill, K.-Y. Lam and Y. Lou, The role of advection in a two-species competition model: A bifurcation approach, Mem. Amer. Math. Soc., 245 (2017), v+117 pp. doi: 10.1090/memo/1161.

[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.

[4]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons Ltd., Chichester, 2003. doi: 10.1002/0470871296.

[5]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.

[6]

X. ChenK.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst., 32 (2012), 3841-3859.  doi: 10.3934/dcds.2012.32.3841.

[7]

X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658.  doi: 10.1512/iumj.2008.57.3204.

[8]

X. Chen and Y. Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operator and their applications, Indiana Univ. Math. J., 61 (2012), 45-80.  doi: 10.1512/iumj.2012.61.4518.

[9]

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.

[10]

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, submitted, (2019), 37 pp.

[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]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infective Diseases: Model Building, Analysis and Interpretation, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2000.

[15]

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.

[16]

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.

[17]

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.

[18]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.

[19]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, AMS, Providence, RI, 1988.

[20]

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.

[21]

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), Art. 112, 28 pp. doi: 10.1007/s00526-017-1207-8.

[22]

K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differential Equations, 250 (2011), 161-181.  doi: 10.1016/j.jde.2010.08.028.

[23]

K.-Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics II, SIAM J. Math. Anal., 44 (2012), 1808-1830.  doi: 10.1137/100819758.

[24]

K.-Y. LamY. Lou and F. Lutscher, The emergence of range limits in advective environments, SIAM J. Appl. Math., 76 (2016), 641-662.  doi: 10.1137/15M1027887.

[25]

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

[26]

K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst., 28 (2010), 1051-1067.  doi: 10.3934/dcds.2010.28.1051.

[27]

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.

[28]

B. Li, H. Li and Y. 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.

[29]

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.

[30]

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.

[31]

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.

[32]

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.

[33]

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.

[34]

Y. LouD. Xiao and P. Zhou, Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment, Discrete Contin. Dyn. Syst., 36 (2016), 953-969.  doi: 10.3934/dcds.2016.36.953.

[35]

Y. LouX.-Q. Zhao and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. Math. Pures Appl., 121 (2019), 47-82.  doi: 10.1016/j.matpur.2018.06.010.

[36]

Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 259 (2015), 141-171.  doi: 10.1016/j.jde.2015.02.004.

[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. 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.

[41]

R. Peng and Y. Wu, Global $L^\infty$-bounds and long-time behavior of a diffusive epidemic system in heterogeneous environment, preprint (arXiv: 1906.11699).

[42]

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.

[43]

R. PengG. Zhang and M. Zhou, Asymptotic behavior of the principal eigenvalue of a second order linear elliptic operator with small/large diffusion coefficient, SIAM J. Math. Anal., 51 (2019), 4724-4753.  doi: 10.1137/18M1217577.

[44]

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.

[45]

R. Peng and M. Zhou, Effects of large degenerate advection and boundary conditions on the principal eigenvalue and its eigenfunction of a linear second order elliptic operator, Indiana Univ. Math. J., 67 (2018), 2523-2568.  doi: 10.1512/iumj.2018.67.7547.

[46]

X. Sun and R. Cui, Analysis on a diffusive SIS epidemic model with saturated incidence rate and linear source in a heterogenous environment, J. Math. Anal. Appl., 490 (2020), 124212. doi: 10.1016/j.jmaa.2020.124212.

[47]

D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219-1237. 

[48]

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.

[49]

O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Q., 18 (2010), 439-469. 

[50]

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