December  2011, 15(3): 867-892. doi: 10.3934/dcdsb.2011.15.867

Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model

1. 

School of Mathematics, South China Normal University, Guangzhou 510631, China

Received  January 2010 Revised  August 2010 Published  February 2011

In this article, the well-posedness of the initial value problem, the existence of traveling wavefronts and the asymptotic speed of propagation for a SIR epidemic model with stage structure and nonlocal response are studied. We further show that the minimum wave speed in fact coincides with the asymptotic speed of propagation.
Citation: Chufen Wu, Peixuan Weng. Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 867-892. doi: 10.3934/dcdsb.2011.15.867
References:
[1]

D. G. Aronson, The asymptotic speed of propagation of a simple epidemic,, in, (1977), 1.

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, in, 446 (1975), 5.

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5.

[4]

J. Al-Omari, S. A. Gourley, Monotone travelling fronts in an age-stuctured reaction-diffusion model for a single species,, J. Math. Biol., 45 (2002), 294. doi: 10.1007/s002850200159.

[5]

H. Berestycki, F. Hamel and L. Roques, Analisis of the periodically fragmented environment model: I-Species persistence,, J. Math. Biol., 51 (2005), 75. doi: 10.1007/s00285-004-0313-3.

[6]

H. Berestycki, F. Hamel and L. Roques, Analisis of the periodically fragmented environment model: II-Biological invasions and pulsating travelling fronts,, J. Math. Pures Appl., 84 (2005), 1101. doi: 10.1016/j.matpur.2004.10.006.

[7]

V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region,, Revue d'Epidemiologic et de Sant$\acutee$ Publique, 27 (1979), 121.

[8]

O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic,, J. Diff. Eqns., 33 (1979), 58. doi: 10.1016/0022-0396(79)90080-9.

[9]

J. Fang, J. Wei and X.-Q. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system,, J. Diff. Eqns., 245 (2008), 2749. doi: 10.1016/j.jde.2008.09.001.

[10]

J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications,, J. Diff. Eqns., 248 (2010), 2199. doi: 10.1016/j.jde.2010.01.009.

[11]

R. A. Fisher, The advance of advantageous genes,, Ann. Eugenics., 7 (1937), 355.

[12]

S. A. Gourley and Y. Kuang, Wavefront and global stability in a time-delayed population model with stage structure,, Proc. R. Soc. Lond. Ser. A., 459 (2003), 1563. doi: 10.1098/rspa.2002.1094.

[13]

S. A. Gourley and J. H. Wu, Delayed non-local diffusive sysrems in biological invasion and disease spread,, Nonlinear dynamics and evolution equations, 48 (2006), 137.

[14]

A. N. Kolmogorov, I. G. Petrowsky and N. S. Piscounov, Etude de l'equation de la diffusion avec croissance de la quantité de matiere et son application a un probleme biologique,, Univ. Bull. Moscow. Serie. Intern. Sec. A., 1 (1937), 1.

[15]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with application,, Commun. Pure Appl. Math., 60 (2007), 1.

[16]

X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems,, J. Diff. Eqns., 231 (2006), 57. doi: 10.1016/j.jde.2006.04.010.

[17]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory,, Math. Biosci., 93 (1989), 269. doi: 10.1016/0025-5564(89)90026-6.

[18]

R. Lui, Biological growth and spread modeled by systems of recursions. II. Biological theory,, Math. Biosci., 93 (1989), 297. doi: 10.1016/0025-5564(89)90027-8.

[19]

S. W. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Diff. Eqns., 171 (2001), 294. doi: 10.1006/jdeq.2000.3846.

[20]

J. Radcliffe and L. Rass, The asymptotic speed of propagation of the deterministic non-reducible n-type epidemic,, J. Math. Biol., 23 (1986), 341. doi: 10.1007/BF00275253.

[21]

L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics, Mathematical Surveys and Monographs,", AMS, (2003).

[22]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations,, Trans. Amer. Math. Soc., 302 (1987), 587.

[23]

H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equatons and asymptotic speeds for the spread of populatins,, J. Reine Angew. Math., 306 (1979), 94. doi: 10.1515/crll.1979.306.94.

[24]

H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread,, J. Math. Biol., 8 (1979), 173. doi: 10.1007/BF00279720.

[25]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion model,, J. Diff. Eqns., 195 (2003), 430. doi: 10.1016/S0022-0396(03)00175-X.

[26]

X. H. Tian and R. Xu, Stability analysis of a delayed SIR epidemic with stage structure and nonlinear incidence,, Discrete Dynamics in Nature and Society, (2009).

[27]

H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353. doi: 10.1137/0513028.

[28]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511. doi: 10.1007/s00285-002-0169-3.

[29]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183. doi: 10.1007/s002850200145.

[30]

P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction,, IMA J. Appl. Maths., 68 (2003), 409. doi: 10.1093/imamat/68.4.409.

[31]

P. X. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model,, J. Diff. Eqns., 229 (2006), 270. doi: 10.1016/j.jde.2006.01.020.

[32]

P. X. Weng and Z. T. Xu, Survey on progress for asymptotic speed of propagation and traveling wave solutions of some types of evolution equations(Chinese),, Advance in Mathematics, 39 (2009), 1.

[33]

C. F. Wu and P. X. Weng, Traveling wavefronts for a SIS epidemic model with stage structure,, Dynamic Systems and Applications, 19 (2010), 125.

[34]

X.-Q. Zhao and W. D. Wang, Fisher waves in an epidemic model,, Discrete Continuous Dynam. Systems - B, 4 (2004), 1117.

[35]

X.-Q. Zhao, Spatial dynamics of some evolution system in biology, "Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions,", World Scientific Publishing Co. Pte. Ltd. Singapore, (2009), 332.

[36]

X.-Q. Zhao and D. M. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model,, J. Dynam. Diff. Eqns., 18 (2006), 1001.

show all references

References:
[1]

D. G. Aronson, The asymptotic speed of propagation of a simple epidemic,, in, (1977), 1.

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, in, 446 (1975), 5.

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5.

[4]

J. Al-Omari, S. A. Gourley, Monotone travelling fronts in an age-stuctured reaction-diffusion model for a single species,, J. Math. Biol., 45 (2002), 294. doi: 10.1007/s002850200159.

[5]

H. Berestycki, F. Hamel and L. Roques, Analisis of the periodically fragmented environment model: I-Species persistence,, J. Math. Biol., 51 (2005), 75. doi: 10.1007/s00285-004-0313-3.

[6]

H. Berestycki, F. Hamel and L. Roques, Analisis of the periodically fragmented environment model: II-Biological invasions and pulsating travelling fronts,, J. Math. Pures Appl., 84 (2005), 1101. doi: 10.1016/j.matpur.2004.10.006.

[7]

V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region,, Revue d'Epidemiologic et de Sant$\acutee$ Publique, 27 (1979), 121.

[8]

O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic,, J. Diff. Eqns., 33 (1979), 58. doi: 10.1016/0022-0396(79)90080-9.

[9]

J. Fang, J. Wei and X.-Q. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system,, J. Diff. Eqns., 245 (2008), 2749. doi: 10.1016/j.jde.2008.09.001.

[10]

J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications,, J. Diff. Eqns., 248 (2010), 2199. doi: 10.1016/j.jde.2010.01.009.

[11]

R. A. Fisher, The advance of advantageous genes,, Ann. Eugenics., 7 (1937), 355.

[12]

S. A. Gourley and Y. Kuang, Wavefront and global stability in a time-delayed population model with stage structure,, Proc. R. Soc. Lond. Ser. A., 459 (2003), 1563. doi: 10.1098/rspa.2002.1094.

[13]

S. A. Gourley and J. H. Wu, Delayed non-local diffusive sysrems in biological invasion and disease spread,, Nonlinear dynamics and evolution equations, 48 (2006), 137.

[14]

A. N. Kolmogorov, I. G. Petrowsky and N. S. Piscounov, Etude de l'equation de la diffusion avec croissance de la quantité de matiere et son application a un probleme biologique,, Univ. Bull. Moscow. Serie. Intern. Sec. A., 1 (1937), 1.

[15]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with application,, Commun. Pure Appl. Math., 60 (2007), 1.

[16]

X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems,, J. Diff. Eqns., 231 (2006), 57. doi: 10.1016/j.jde.2006.04.010.

[17]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory,, Math. Biosci., 93 (1989), 269. doi: 10.1016/0025-5564(89)90026-6.

[18]

R. Lui, Biological growth and spread modeled by systems of recursions. II. Biological theory,, Math. Biosci., 93 (1989), 297. doi: 10.1016/0025-5564(89)90027-8.

[19]

S. W. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Diff. Eqns., 171 (2001), 294. doi: 10.1006/jdeq.2000.3846.

[20]

J. Radcliffe and L. Rass, The asymptotic speed of propagation of the deterministic non-reducible n-type epidemic,, J. Math. Biol., 23 (1986), 341. doi: 10.1007/BF00275253.

[21]

L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics, Mathematical Surveys and Monographs,", AMS, (2003).

[22]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations,, Trans. Amer. Math. Soc., 302 (1987), 587.

[23]

H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equatons and asymptotic speeds for the spread of populatins,, J. Reine Angew. Math., 306 (1979), 94. doi: 10.1515/crll.1979.306.94.

[24]

H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread,, J. Math. Biol., 8 (1979), 173. doi: 10.1007/BF00279720.

[25]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion model,, J. Diff. Eqns., 195 (2003), 430. doi: 10.1016/S0022-0396(03)00175-X.

[26]

X. H. Tian and R. Xu, Stability analysis of a delayed SIR epidemic with stage structure and nonlinear incidence,, Discrete Dynamics in Nature and Society, (2009).

[27]

H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353. doi: 10.1137/0513028.

[28]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511. doi: 10.1007/s00285-002-0169-3.

[29]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183. doi: 10.1007/s002850200145.

[30]

P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction,, IMA J. Appl. Maths., 68 (2003), 409. doi: 10.1093/imamat/68.4.409.

[31]

P. X. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model,, J. Diff. Eqns., 229 (2006), 270. doi: 10.1016/j.jde.2006.01.020.

[32]

P. X. Weng and Z. T. Xu, Survey on progress for asymptotic speed of propagation and traveling wave solutions of some types of evolution equations(Chinese),, Advance in Mathematics, 39 (2009), 1.

[33]

C. F. Wu and P. X. Weng, Traveling wavefronts for a SIS epidemic model with stage structure,, Dynamic Systems and Applications, 19 (2010), 125.

[34]

X.-Q. Zhao and W. D. Wang, Fisher waves in an epidemic model,, Discrete Continuous Dynam. Systems - B, 4 (2004), 1117.

[35]

X.-Q. Zhao, Spatial dynamics of some evolution system in biology, "Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions,", World Scientific Publishing Co. Pte. Ltd. Singapore, (2009), 332.

[36]

X.-Q. Zhao and D. M. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model,, J. Dynam. Diff. Eqns., 18 (2006), 1001.

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