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 and 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 "Nonlinear Diffusion" (W. E. Fitzgibbon, H. F. Walker, eds.), Pitman, London, (1977), 1-23.

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" (J. A. Goldstein, ed.), Leture Notes in Mathematics, 446, Springer, Berlin, (1975), 5-49.

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. 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), 294-312. 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), 75-113. 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), 1101-1146. 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), 121-132.

[8]

O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Diff. Eqns., 33 (1979), 58-73. 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), 2749-2770. 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), 2199-2226. doi: 10.1016/j.jde.2010.01.009.

[11]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics., 7 (1937), 355-369.

[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), 1563-1579. 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, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 137-200.

[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), 1-26.

[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), 1-40. Erratum: 61 (2008), 137-138.

[16]

X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Diff. Eqns., 231 (2006), 57-77. 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), 269-295. 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), 297-312. 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), 294-314. 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), 341-359. doi: 10.1007/BF00275253.

[21]

L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics, Mathematical Surveys and Monographs," AMS, Rhode Island, USA, 2003.

[22]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.

[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), 94-121. 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), 173-187. 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), 430-470. 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), Art. ID 979217, 17pp.

[27]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. 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), 511-548. 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), 183-218. 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), 409-439. 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), 270-296. 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), 1-22.

[33]

C. F. Wu and P. X. Weng, Traveling wavefronts for a SIS epidemic model with stage structure, Dynamic Systems and Applications, 19 (2010), 125-146.

[34]

X.-Q. Zhao and W. D. Wang, Fisher waves in an epidemic model, Discrete Continuous Dynam. Systems - B, 4 (2004), 1117-1128.

[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), 332-363.

[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), 1001-1019; Erratum: 20 (2008), 277-279.

show all references

References:
[1]

D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, in "Nonlinear Diffusion" (W. E. Fitzgibbon, H. F. Walker, eds.), Pitman, London, (1977), 1-23.

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" (J. A. Goldstein, ed.), Leture Notes in Mathematics, 446, Springer, Berlin, (1975), 5-49.

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. 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), 294-312. 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), 75-113. 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), 1101-1146. 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), 121-132.

[8]

O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Diff. Eqns., 33 (1979), 58-73. 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), 2749-2770. 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), 2199-2226. doi: 10.1016/j.jde.2010.01.009.

[11]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics., 7 (1937), 355-369.

[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), 1563-1579. 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, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 137-200.

[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), 1-26.

[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), 1-40. Erratum: 61 (2008), 137-138.

[16]

X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Diff. Eqns., 231 (2006), 57-77. 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), 269-295. 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), 297-312. 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), 294-314. 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), 341-359. doi: 10.1007/BF00275253.

[21]

L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics, Mathematical Surveys and Monographs," AMS, Rhode Island, USA, 2003.

[22]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.

[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), 94-121. 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), 173-187. 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), 430-470. 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), Art. ID 979217, 17pp.

[27]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. 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), 511-548. 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), 183-218. 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), 409-439. 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), 270-296. 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), 1-22.

[33]

C. F. Wu and P. X. Weng, Traveling wavefronts for a SIS epidemic model with stage structure, Dynamic Systems and Applications, 19 (2010), 125-146.

[34]

X.-Q. Zhao and W. D. Wang, Fisher waves in an epidemic model, Discrete Continuous Dynam. Systems - B, 4 (2004), 1117-1128.

[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), 332-363.

[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), 1001-1019; Erratum: 20 (2008), 277-279.

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