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

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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	P. Hess, Periodic-parabolic Boundary Value Problems and Positivity,, Pitman Res. Notes Math., vol. 247, Longman Scientific & Technical, Harlow, 1991.
[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.
[12]	T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag New York, Inc., New York, 1966.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	P. Hess, Periodic-parabolic Boundary Value Problems and Positivity,, Pitman Res. Notes Math., vol. 247, Longman Scientific & Technical, Harlow, 1991.
[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.
[12]	T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag New York, Inc., New York, 1966.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

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