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Dynamics of non-autonomous fractional Ginzburg-Landau equations driven by colored noise
Singular renormalization group approach to SIS problems
1. | School of Mathematics, Jilin University, Changchun 130012, China |
2. | School of Mathematics & State Key Laboratory of Automotive, Simulation and Control, Jilin University, Changchun 130012, China |
In this paper, we consider the boundary value problems of a one-dimensional steady-state SIS epidemic reaction-diffusion-advection system in the following two cases: (ⅰ) the advection rate is relatively large comparing to the diffusion rates of infected and susceptible populations; (ⅱ) the diffusion rate of the susceptible population approaches zero. By introducing a singular parameter, the system can be viewed as a singularly perturbed problem. By the renormalization group method, we construct the first-order approximate solutions and obtain error estimates.
References:
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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. |
[2] |
D. Bl$\ddot{o}$mker, C. Gugg and S. Maier-Paape,
Stochastic Navier-Stokes equation and renormalization group theory, Phys. D, 173 (2002), 137-152.
doi: 10.1016/S0167-2789(02)00621-8. |
[3] |
D. Boyanovsky and H. J. D. Vega,
Dynamical renormalization group approach to relaxation in quantum field theory, Ann. Phys., 307 (2003), 335-371.
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Y. L. Cai and W. M. Wang,
Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 989-1013.
doi: 10.3934/dcdsb.2015.20.989. |
[5] |
L. Y. Chen, N. Goldenfeld and Y. Oono,
Renormalization group theory for global asymptotic analysis, Phys. Rev. Lett., 73 (1994), 1311-1315.
doi: 10.1103/PhysRevLett.73.1311. |
[6] |
H. Chiba,
$C^1$ Approximation of vector fields based on the renormalization group method, SIAM J. Appl. Dyn. Syst., 7 (2008), 895-932.
doi: 10.1137/070694892. |
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H. Chiba, Approximation of center manifolds on the renormalization group method, J. Math. Phys., 49 (2008), 102703, 11pp.
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R. H. 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. |
[9] |
R. H. 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. |
[10] |
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. |
[11] |
K. Deng and Y. X. Wu,
Dynamics of an susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.
doi: 10.1017/S0308210515000864. |
[12] |
R. E. L. DeVille, A. Harkin, M. Holzer, K. Josi$\acute{c}$ and T. J. Kaper,
Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations, Phys. D, 237 (2008), 1029-1052.
doi: 10.1016/j.physd.2007.12.009. |
[13] |
N. Glatt-Holtz and M. Ziane,
Singular perturbation systems with stochastic forcing and the renormalization group method, Discrete Contin. Dyn. Syst., 26 (2010), 1241-1268.
doi: 10.3934/dcds.2010.26.1241. |
[14] |
A. N. Gorban, I. V. Karlin and A. Y. Zinovyev,
Constructive methods of invariant manifolds for kinetic problems, Phys. Rep., 396 (2004), 197-403.
doi: 10.1016/j.physrep.2004.03.006. |
[15] |
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. |
[16] |
K. Y. Lam, Y. Lou and F. Lutscher,
Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), 188-212.
doi: 10.1080/17513758.2014.969336. |
[17] |
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. |
[18] |
W. L. Li and S. Y. Shi,
Singular perturbed renormalization group theory and its application to highly oscillatory problems, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1819-1833.
|
[19] |
Y. Lou and F. Lutscher,
Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.
doi: 10.1007/s00285-013-0730-2. |
[20] |
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. |
[21] |
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.
|
[22] |
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. |
[23] |
I. Moise and R. Temam,
Renormalization group method. Applications to Navier-Stokes equation, Discrete Contin. Dyn. Syst., 6 (2000), 191-200.
doi: 10.3934/dcds.2000.6.191. |
[24] |
I. Moise and M. Ziane,
Renormalization group method. Applications to partial differential equations, J. Dynam. Differential Equations, 13 (2001), 275-321.
doi: 10.1023/A:1016680007953. |
[25] |
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. |
[26] |
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. |
[27] |
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. |
[28] |
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. |
[29] |
R. Zhou, S. Y. Shi and W. L. Li,
Renormalization group approach to boundary layer problems, Commun. Nonlinear Sci. Numer. Simul., 71 (2019), 220-230.
doi: 10.1016/j.cnsns.2018.11.012. |
[30] |
M. Ziane,
On a certain renormalization group method, J. Math. Phys., 41 (2003), 3290-3299.
doi: 10.1063/1.533307. |
show all references
References:
[1] |
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. |
[2] |
D. Bl$\ddot{o}$mker, C. Gugg and S. Maier-Paape,
Stochastic Navier-Stokes equation and renormalization group theory, Phys. D, 173 (2002), 137-152.
doi: 10.1016/S0167-2789(02)00621-8. |
[3] |
D. Boyanovsky and H. J. D. Vega,
Dynamical renormalization group approach to relaxation in quantum field theory, Ann. Phys., 307 (2003), 335-371.
doi: 10.1016/S0003-4916(03)00115-5. |
[4] |
Y. L. Cai and W. M. Wang,
Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 989-1013.
doi: 10.3934/dcdsb.2015.20.989. |
[5] |
L. Y. Chen, N. Goldenfeld and Y. Oono,
Renormalization group theory for global asymptotic analysis, Phys. Rev. Lett., 73 (1994), 1311-1315.
doi: 10.1103/PhysRevLett.73.1311. |
[6] |
H. Chiba,
$C^1$ Approximation of vector fields based on the renormalization group method, SIAM J. Appl. Dyn. Syst., 7 (2008), 895-932.
doi: 10.1137/070694892. |
[7] |
H. Chiba, Approximation of center manifolds on the renormalization group method, J. Math. Phys., 49 (2008), 102703, 11pp.
doi: 10.1063/1.2996290. |
[8] |
R. H. 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. |
[9] |
R. H. 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. |
[10] |
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. |
[11] |
K. Deng and Y. X. Wu,
Dynamics of an susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.
doi: 10.1017/S0308210515000864. |
[12] |
R. E. L. DeVille, A. Harkin, M. Holzer, K. Josi$\acute{c}$ and T. J. Kaper,
Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations, Phys. D, 237 (2008), 1029-1052.
doi: 10.1016/j.physd.2007.12.009. |
[13] |
N. Glatt-Holtz and M. Ziane,
Singular perturbation systems with stochastic forcing and the renormalization group method, Discrete Contin. Dyn. Syst., 26 (2010), 1241-1268.
doi: 10.3934/dcds.2010.26.1241. |
[14] |
A. N. Gorban, I. V. Karlin and A. Y. Zinovyev,
Constructive methods of invariant manifolds for kinetic problems, Phys. Rep., 396 (2004), 197-403.
doi: 10.1016/j.physrep.2004.03.006. |
[15] |
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. |
[16] |
K. Y. Lam, Y. Lou and F. Lutscher,
Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), 188-212.
doi: 10.1080/17513758.2014.969336. |
[17] |
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. |
[18] |
W. L. Li and S. Y. Shi,
Singular perturbed renormalization group theory and its application to highly oscillatory problems, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1819-1833.
|
[19] |
Y. Lou and F. Lutscher,
Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.
doi: 10.1007/s00285-013-0730-2. |
[20] |
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. |
[21] |
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.
|
[22] |
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. |
[23] |
I. Moise and R. Temam,
Renormalization group method. Applications to Navier-Stokes equation, Discrete Contin. Dyn. Syst., 6 (2000), 191-200.
doi: 10.3934/dcds.2000.6.191. |
[24] |
I. Moise and M. Ziane,
Renormalization group method. Applications to partial differential equations, J. Dynam. Differential Equations, 13 (2001), 275-321.
doi: 10.1023/A:1016680007953. |
[25] |
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. |
[26] |
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. |
[27] |
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. |
[28] |
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. |
[29] |
R. Zhou, S. Y. Shi and W. L. Li,
Renormalization group approach to boundary layer problems, Commun. Nonlinear Sci. Numer. Simul., 71 (2019), 220-230.
doi: 10.1016/j.cnsns.2018.11.012. |
[30] |
M. Ziane,
On a certain renormalization group method, J. Math. Phys., 41 (2003), 3290-3299.
doi: 10.1063/1.533307. |
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