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Identification of generic stable dynamical systems taking a nonlinear differential approach
A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment
School of Mathematics and Statistics, Lanzhou University, and Key Laboratory of Applied Mathematics and Complex Systems of Gansu province, Lanzhou, Gansu 730000, China |
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.
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.
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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. |
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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.
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[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.
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[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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