# American Institute of Mathematical Sciences

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

## 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

*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.

Citation: Ning Sun, Shaoyun Shi, Wenlei Li. Singular renormalization group approach to SIS problems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3577-3596. doi: 10.3934/dcdsb.2020073
##### References:
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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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##### 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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