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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020217
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]

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[44]

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