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September  2020, 25(9): 3577-3596. doi: 10.3934/dcdsb.2020073

Singular renormalization group approach to SIS problems

Ning Sun 1, , Shaoyun Shi 2, and  Wenlei Li 1,,

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

*Corresponding author: Wenlei Li (lwlei@jlu.edu.cn)

Received  August 2019 Revised  November 2019 Published  September 2020 Early access  April 2020

Fund Project: This work is supported by NSFC grant (No. 11771177, 11301210), China Automobile Industry Innovation and Development Joint Fund (No. U1664257), Program for Changbaishan Scholars of Jilin Province and Program for JLU Science, Technology Innovative Research Team (No. 2017TD-20), NSF grant (No. 20190201132JC) and ESF grant (No. JJKH20170776KJ) of Jilin, 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.

Keywords: Renormalization group method, SIS epidemic model, error estimate.
Mathematics Subject Classification: Primary: 34C60, 34D15, 34E10, 34E15, 34K26.
Citation: Ning Sun, Shaoyun Shi, Wenlei Li. Singular renormalization group approach to SIS problems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3577-3596. doi: 10.3934/dcdsb.2020073
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.  Google Scholar

[2]

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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.  Google Scholar

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[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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

M. Ziane, On a certain renormalization group method, J. Math. Phys., 41 (2003), 3290-3299.  doi: 10.1063/1.533307.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

H. Chiba, Approximation of center manifolds on the renormalization group method, J. Math. Phys., 49 (2008), 102703, 11pp. doi: 10.1063/1.2996290.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

M. Ziane, On a certain renormalization group method, J. Math. Phys., 41 (2003), 3290-3299.  doi: 10.1063/1.533307.  Google Scholar

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