American Institute of Mathematical Sciences

Advanced
December  2018, 23(10): 4557-4578. doi: 10.3934/dcdsb.2018176

A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment

Danhua Jiang , Zhi-Cheng Wang , and  Liang Zhang

School of Mathematics and Statistics, Lanzhou University, and Key Laboratory of Applied Mathematics and Complex Systems of Gansu province, Lanzhou, Gansu 730000, China

* Corresponding author: Zhi-Cheng Wang

Received  September 2017 Revised  February 2018 Published  June 2018

In this paper, we study the effects of diffusion and advection for an SIS epidemic reaction-diffusion-advection model in a spatially and temporally heterogeneous environment. We introduce the basic reproduction number $\mathcal{R}_{0}$ and establish the threshold-type results on the global dynamics in terms of $\mathcal{R}_{0}$. Some general qualitative properties of $\mathcal{R}_{0}$ are presented, then the paper is devoted to studying how the advection and diffusion of the infected individuals affect the reproduction number $\mathcal{R}_{0}$ for the special case that $γ(x,t)-β(x,t) = V(x,t)$ is monotone with respect to spatial variable $x$. Our results suggest that if $V_{x}(x,t)≥0,\not\equiv0$ and $V(x, t)$ changes sign about $x$, the advection is beneficial to eliminate the disease, whereas if $V_{x}(x,t)≤0,\not\equiv0$ and $V(x, t)$ changes sign about $x$, the advection is bad for the elimination of disease.

Keywords: SIS epidemic model, reaction-diffusion-advection, spatial heterogeneity, temporal periodic.
Mathematics Subject Classification: 35K57, 37N25, 92B05.
Citation: Danhua Jiang, Zhi-Cheng Wang, Liang Zhang. A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4557-4578. doi: 10.3934/dcdsb.2018176
References:
[1]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33 (1979), 201-225.  doi: 10.1016/0022-0396(79)90088-3.  Google Scholar

[2]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic disease patch model, SIAM J. Appl. Math., 67 (2007), 1283-1309.  doi: 10.1137/060672522.  Google Scholar

[3]

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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[7]

J. Ge, K. I. Kim, Z.-G. Lin and H.-P. Zhu, An 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

[8]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647.  Google Scholar

[10]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity,, Pitman Res. Notes Math., vol. 247, Longman Scientific & Technical, Harlow, 1991.  Google Scholar

[11]

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.  Google Scholar

[12]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar

[13]

H.-C. Li, R. 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

[14]

F. Lutscher, M. 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.  Google Scholar

[15]

F. Lutscher, E. McCauley and M. A. Lewis, Spatial patterns and coexistence mechanisms in systems with unidirectional flow, Theor. Popul. Biol., 71 (2007), 267-277.  doi: 10.1016/j.tpb.2006.11.006.  Google Scholar

[16]

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

[17]

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

[18]

R. Peng, Asymptotic profiles 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.  Google Scholar

[19]

R. Peng and S.-Q. 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

[20]

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

[21]

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

[22]

R. Peng and X.-Q. Zhao, Effects of diffusion and advection on the principal eigenvalue of a periodic parabolic problem with applications, Calc. Var. Partial Differential Equations, 54 (2015), 1611-1642.  doi: 10.1007/s00526-015-0838-x.  Google Scholar

[23]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[24]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., vol. 41, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[25]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential. Equations, 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.  Google Scholar

[26]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of Dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168.  doi: 10.1137/090775890.  Google Scholar

[27]

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

[28]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

show all references

References:
[1]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33 (1979), 201-225.  doi: 10.1016/0022-0396(79)90088-3.  Google Scholar

[2]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic disease patch model, SIAM J. Appl. Math., 67 (2007), 1283-1309.  doi: 10.1137/060672522.  Google Scholar

[3]

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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[7]

J. Ge, K. I. Kim, Z.-G. Lin and H.-P. Zhu, An 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

[8]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647.  Google Scholar

[10]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity,, Pitman Res. Notes Math., vol. 247, Longman Scientific & Technical, Harlow, 1991.  Google Scholar

[11]

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.  Google Scholar

[12]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar

[13]

H.-C. Li, R. 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

[14]

F. Lutscher, M. 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.  Google Scholar

[15]

F. Lutscher, E. McCauley and M. A. Lewis, Spatial patterns and coexistence mechanisms in systems with unidirectional flow, Theor. Popul. Biol., 71 (2007), 267-277.  doi: 10.1016/j.tpb.2006.11.006.  Google Scholar

[16]

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

[17]

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

[18]

R. Peng, Asymptotic profiles 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.  Google Scholar

[19]

R. Peng and S.-Q. 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

[20]

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

[21]

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

[22]

R. Peng and X.-Q. Zhao, Effects of diffusion and advection on the principal eigenvalue of a periodic parabolic problem with applications, Calc. Var. Partial Differential Equations, 54 (2015), 1611-1642.  doi: 10.1007/s00526-015-0838-x.  Google Scholar

[23]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[24]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., vol. 41, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[25]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential. Equations, 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.  Google Scholar

[26]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of Dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168.  doi: 10.1137/090775890.  Google Scholar

[27]

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

[28]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[1]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989
[2]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841
[3]

Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124
[4]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017
[5]

Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1
[6]

Wenzhang Huang, Maoan Han, Kaiyu Liu. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 51-66. doi: 10.3934/mbe.2010.7.51
[7]

Chris Cosner. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701
[8]

Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173
[9]

Mostafa Bendahmane, Kenneth H. Karlsen. Renormalized solutions of an anisotropic reaction-diffusion-advection system with $L^1$ data. Communications on Pure & Applied Analysis, 2006, 5 (4) : 733-762. doi: 10.3934/cpaa.2006.5.733
[10]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116
[11]

Linfeng Mei, Xiaoyan Zhang. On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 221-243. doi: 10.3934/dcdsb.2012.17.221
[12]

Bo Duan, Zhengce Zhang. A two-species weak competition system of reaction-diffusion-advection with double free boundaries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 801-829. doi: 10.3934/dcdsb.2018208
[13]

W. E. Fitzgibbon, M.E. Parrott, Glenn Webb. Diffusive epidemic models with spatial and age dependent heterogeneity. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 35-57. doi: 10.3934/dcds.1995.1.35
[14]

Yijun Lou, Xiao-Qiang Zhao. Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 169-186. doi: 10.3934/dcdsb.2009.12.169
[15]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[16]

Shuling Yan, Shangjiang Guo. Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1559-1579. doi: 10.3934/dcdsb.2018059
[17]

Zhisheng Shuai, P. van den Driessche. Impact of heterogeneity on the dynamics of an SEIR epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (2) : 393-411. doi: 10.3934/mbe.2012.9.393
[18]

Xueying Wang, Drew Posny, Jin Wang. A reaction-convection-diffusion model for cholera spatial dynamics. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2785-2809. doi: 10.3934/dcdsb.2016073
[19]

Benedetto Bozzini, Deborah Lacitignola, Ivonne Sgura. Morphological spatial patterns in a reaction diffusion model for metal growth. Mathematical Biosciences & Engineering, 2010, 7 (2) : 237-258. doi: 10.3934/mbe.2010.7.237
[20]

Zhen-Hui Bu, Zhi-Cheng Wang. Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media. Communications on Pure & Applied Analysis, 2016, 15 (1) : 139-160. doi: 10.3934/cpaa.2016.15.139

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (197)
  • HTML views (317)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences