Traveling waves of a delayed diffusive SIR epidemic model

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May 2015, 14(3): 1001-1022. doi: 10.3934/cpaa.2015.14.1001

Traveling waves of a delayed diffusive SIR epidemic model

Yan Li ^1, , Wan-Tong Li ^2, and Guo Lin ^2,

1.	School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000
2.	School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China

Received September 2014 Revised January 2015 Published March 2015

Abstract
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This paper is concerned with the minimal wave speed of a delayed diffusive SIR epidemic model with Holling-II incidence rate and constant external supplies. By presenting the existence and nonexistence of traveling wave solutions for any positive wave speed, the minimal wave speed is established. In particular, the minimal wave speed decreases when the latency of infection increases. Biologically speaking, the longer the latency of infection in a vector is, the slower the disease spreads.

Keywords: constant external supplies., traveling waves, minimal wave speed, SIR epidemic model.

Mathematics Subject Classification: Primary: 35K57, 35R20; Secondary: 92D2.

Citation: Yan Li, Wan-Tong Li, Guo Lin. Traveling waves of a delayed diffusive SIR epidemic model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1001-1022. doi: 10.3934/cpaa.2015.14.1001

References:

[1]	S. Ai and R. Albashaireh, Traveling waves in spatial SIRS models,, \emph{J. Dyn. Diff. Equat.}, 26 (2014), 143. doi: 10.1007/s10884-014-9348-3. Google Scholar
[2]	M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states,, \emph{Appl. Math. Lett., 25 (2012), 2095. doi: 10.1016/j.aml.2012.05.006. Google Scholar
[3]	M. S. Bartlett, Deterministic and stochastic models for recurrent epidemics,, in\emph{Proc. 3rd Berkeley Symp. Mathematical Statistics and Probability}, (1956). Google Scholar
[4]	C. Briggs and H. Godfray, The dynamics of insect-pathogen interactions in stage-structured populations,, \emph{Amer. Nat., 145 (1995), 855. Google Scholar
[5]	K. Brown and J. Carr, Deterministic epidemic waves of critical velocity,, \emph{Math. Proc. Camb. Phil. Soc., 81 (1977), 431. doi: 10.1017/S0305004100053494. Google Scholar
[6]	V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model,, \emph{Math. Biosci., 42 (1978), 41. doi: 10.1016/0025-5564(78)90006-8. Google Scholar
[7]	Z. Chen and Z. Zhao, Harnack principle for weakly coupled elliptic systems,, \emph{J. Differential Equations, 139 (1997), 261. doi: 10.1006/jdeq.1997.3300. Google Scholar
[8]	K. Cooke, Stability analysis for a vector disease model,, \emph{Rocky Mountain J. Math., 9 (1979), 31. doi: 10.1216/RMJ-1979-9-1-31. Google Scholar
[9]	O. Diekmann, Thresholds and traveling waves for the geographical spread of infection,, \emph{J. Math. Biol., 6 (1978), 109. doi: 10.1007/BF02450783. Google Scholar
[10]	W. Ding, W. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model,, \emph{Discrete Contin. Dyn. Syst. B, 18 (2013), 1291. doi: 10.3934/dcdsb.2013.18.1291. Google Scholar
[11]	A. Ducrot, M. Langlais and P. Magal, Qualitative analysis and travelling wave solutions for the SI model with vertical transmission,, \emph{Commum. Pure Appl. Anal., 11 (2012), 97. doi: 10.3934/cpaa.2012.11.97. Google Scholar
[12]	A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion,, \emph{Proc. R. Soc. Edin. Ser. A Math., 139 (2009), 459. doi: 10.1017/S0308210507000455. Google Scholar
[13]	A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies,, \emph{Nonlinearity, 24 (2011), 2891. doi: 10.1088/0951-7715/24/10/012. Google Scholar
[14]	A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structure epidemic models,, \emph{Arch. Ration. Mech. Anal., 195 (2010), 311. doi: 10.1007/s00205-008-0203-8. Google Scholar
[15]	L. Evans, Partial Differential Equaitons, Second edition,, Graduate studies in mathematics, (2010). Google Scholar
[16]	J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications,, \emph{J. Differential Equations, 248 (2010), 2199. doi: 10.1016/j.jde.2010.01.009. Google Scholar
[17]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Vol. 224. Springer, (2001). Google Scholar
[18]	H. Hethcote, The mathematics of infectious diseases,, \emph{SIAM. Rev., 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar
[19]	W. Hirsch, H. Hamisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior,, \emph{Comm. Pure Appl. Math., 38 (1948), 221. doi: 10.1002/cpa.3160380607. Google Scholar
[20]	Y. Hosono and B. Ilyas, Travelling waves for a simple diffusive epidemic model,, \emph{Math. Model Meth. Appl. Sci., 5 (1995), 935. doi: 10.1142/S0218202595000504. Google Scholar
[21]	G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,, \emph{Bull. Math. Biol., 72 (2010), 1192. doi: 10.1007/s11538-009-9487-6. Google Scholar
[22]	W. T. Li, G. Lin, C. Ma and F. Y. Yang, Traveling waves of a nonlocal delayed SIR epidemic model without outbreak threshold,, \emph{Discrete Contin. Dyn. Syst. B, 19 (2014), 467. doi: 10.3934/dcdsb.2014.19.467. Google Scholar
[23]	G. Lin, Invasion traveling wave solution of a predator-prey system,, \emph{Nonlinear Anal.}, 96 (2014), 47. doi: 10.1016/j.na.2013.10.024. Google Scholar
[24]	G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays,, \emph{J. Dyn. Diff. Equat.}, 26 (2014), 583. doi: 10.1007/s10884-014-9355-4. Google Scholar
[25]	C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, \emph{Nonlinear Anal. Real Word Appl., 11 (2010), 3106. doi: 10.1016/j.nonrwa.2009.11.005. Google Scholar
[26]	J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications,, 3rd edn, (2003). Google Scholar
[27]	J. Roughgarden, Theory of Population Genetics and Evolutionary Ecology: An Introduction,, New York, (1979). Google Scholar
[28]	S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models,, in \emph{Mathematics for Life Science and Medicine} (eds. Y. Takeuchi, (2007), 97. Google Scholar
[29]	S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts,, in \emph{Spatial Ecology, (2009), 293. Google Scholar
[30]	H. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations,, \emph{SIAM J. Math. Anal., 31 (2000), 514. doi: 10.1137/S0036141098346785. Google Scholar
[31]	H. R. Thieme and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models,, \emph{J. Differential Equations, 195 (2003), 430. doi: 10.1016/S0022-0396(03)00175-X. Google Scholar
[32]	H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems,, \emph{J. Nonlinear Sci., 21 (2011), 747. doi: 10.1007/s00332-011-9099-9. Google Scholar
[33]	X. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation,, \emph{Discrete Contin. Dyn. Syst., 32 (2012), 3303. doi: 10.3934/dcds.2012.32.3303. Google Scholar
[34]	Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission,, \emph{Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237. doi: 10.1098/rspa.2009.0377. Google Scholar
[35]	Z. C. Wang, J. Wu and R. Liu, Traveling waves of avian influenza spread,, \emph{Proc. Amer. Math. Soc., 140 (2012), 3931. doi: 10.1090/S0002-9939-2012-11246-8. Google Scholar
[36]	M. Wu and P. Weng, Stability of a stage-structured diffusive SIR model with delays (Chinese),, \emph{J. South China Normal Univ. Natur. Sci. Ed., 45 (2013), 20. Google Scholar
[37]	S. L. Wu and P. Weng, Entire solutions for a multi-type SIS nonlocal epidemic model in R or Z,, \emph{J. Math. Anal. Appl., 394 (2012), 603. doi: 10.1016/j.jmaa.2012.05.009. Google Scholar
[38]	J. Yang, S. Liang and Y. Zhang, Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion,, \emph{PLoS One, 6 (2011). doi: 10.1371/journal.pone.0021128. Google Scholar
[39]	Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction-diffusion Equations,, Science Press, (2011). Google Scholar
[40]	E. Zeilder, Nonlinear Functional Analysis and Its Applications: I, Fixed-piont Theorems,, New York, (1986). Google Scholar

show all references

References:

[1]	S. Ai and R. Albashaireh, Traveling waves in spatial SIRS models,, \emph{J. Dyn. Diff. Equat.}, 26 (2014), 143. doi: 10.1007/s10884-014-9348-3. Google Scholar
[2]	M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states,, \emph{Appl. Math. Lett., 25 (2012), 2095. doi: 10.1016/j.aml.2012.05.006. Google Scholar
[3]	M. S. Bartlett, Deterministic and stochastic models for recurrent epidemics,, in\emph{Proc. 3rd Berkeley Symp. Mathematical Statistics and Probability}, (1956). Google Scholar
[4]	C. Briggs and H. Godfray, The dynamics of insect-pathogen interactions in stage-structured populations,, \emph{Amer. Nat., 145 (1995), 855. Google Scholar
[5]	K. Brown and J. Carr, Deterministic epidemic waves of critical velocity,, \emph{Math. Proc. Camb. Phil. Soc., 81 (1977), 431. doi: 10.1017/S0305004100053494. Google Scholar
[6]	V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model,, \emph{Math. Biosci., 42 (1978), 41. doi: 10.1016/0025-5564(78)90006-8. Google Scholar
[7]	Z. Chen and Z. Zhao, Harnack principle for weakly coupled elliptic systems,, \emph{J. Differential Equations, 139 (1997), 261. doi: 10.1006/jdeq.1997.3300. Google Scholar
[8]	K. Cooke, Stability analysis for a vector disease model,, \emph{Rocky Mountain J. Math., 9 (1979), 31. doi: 10.1216/RMJ-1979-9-1-31. Google Scholar
[9]	O. Diekmann, Thresholds and traveling waves for the geographical spread of infection,, \emph{J. Math. Biol., 6 (1978), 109. doi: 10.1007/BF02450783. Google Scholar
[10]	W. Ding, W. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model,, \emph{Discrete Contin. Dyn. Syst. B, 18 (2013), 1291. doi: 10.3934/dcdsb.2013.18.1291. Google Scholar
[11]	A. Ducrot, M. Langlais and P. Magal, Qualitative analysis and travelling wave solutions for the SI model with vertical transmission,, \emph{Commum. Pure Appl. Anal., 11 (2012), 97. doi: 10.3934/cpaa.2012.11.97. Google Scholar
[12]	A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion,, \emph{Proc. R. Soc. Edin. Ser. A Math., 139 (2009), 459. doi: 10.1017/S0308210507000455. Google Scholar
[13]	A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies,, \emph{Nonlinearity, 24 (2011), 2891. doi: 10.1088/0951-7715/24/10/012. Google Scholar
[14]	A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structure epidemic models,, \emph{Arch. Ration. Mech. Anal., 195 (2010), 311. doi: 10.1007/s00205-008-0203-8. Google Scholar
[15]	L. Evans, Partial Differential Equaitons, Second edition,, Graduate studies in mathematics, (2010). Google Scholar
[16]	J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications,, \emph{J. Differential Equations, 248 (2010), 2199. doi: 10.1016/j.jde.2010.01.009. Google Scholar
[17]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Vol. 224. Springer, (2001). Google Scholar
[18]	H. Hethcote, The mathematics of infectious diseases,, \emph{SIAM. Rev., 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar
[19]	W. Hirsch, H. Hamisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior,, \emph{Comm. Pure Appl. Math., 38 (1948), 221. doi: 10.1002/cpa.3160380607. Google Scholar
[20]	Y. Hosono and B. Ilyas, Travelling waves for a simple diffusive epidemic model,, \emph{Math. Model Meth. Appl. Sci., 5 (1995), 935. doi: 10.1142/S0218202595000504. Google Scholar
[21]	G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,, \emph{Bull. Math. Biol., 72 (2010), 1192. doi: 10.1007/s11538-009-9487-6. Google Scholar
[22]	W. T. Li, G. Lin, C. Ma and F. Y. Yang, Traveling waves of a nonlocal delayed SIR epidemic model without outbreak threshold,, \emph{Discrete Contin. Dyn. Syst. B, 19 (2014), 467. doi: 10.3934/dcdsb.2014.19.467. Google Scholar
[23]	G. Lin, Invasion traveling wave solution of a predator-prey system,, \emph{Nonlinear Anal.}, 96 (2014), 47. doi: 10.1016/j.na.2013.10.024. Google Scholar
[24]	G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays,, \emph{J. Dyn. Diff. Equat.}, 26 (2014), 583. doi: 10.1007/s10884-014-9355-4. Google Scholar
[25]	C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, \emph{Nonlinear Anal. Real Word Appl., 11 (2010), 3106. doi: 10.1016/j.nonrwa.2009.11.005. Google Scholar
[26]	J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications,, 3rd edn, (2003). Google Scholar
[27]	J. Roughgarden, Theory of Population Genetics and Evolutionary Ecology: An Introduction,, New York, (1979). Google Scholar
[28]	S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models,, in \emph{Mathematics for Life Science and Medicine} (eds. Y. Takeuchi, (2007), 97. Google Scholar
[29]	S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts,, in \emph{Spatial Ecology, (2009), 293. Google Scholar
[30]	H. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations,, \emph{SIAM J. Math. Anal., 31 (2000), 514. doi: 10.1137/S0036141098346785. Google Scholar
[31]	H. R. Thieme and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models,, \emph{J. Differential Equations, 195 (2003), 430. doi: 10.1016/S0022-0396(03)00175-X. Google Scholar
[32]	H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems,, \emph{J. Nonlinear Sci., 21 (2011), 747. doi: 10.1007/s00332-011-9099-9. Google Scholar
[33]	X. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation,, \emph{Discrete Contin. Dyn. Syst., 32 (2012), 3303. doi: 10.3934/dcds.2012.32.3303. Google Scholar
[34]	Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission,, \emph{Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237. doi: 10.1098/rspa.2009.0377. Google Scholar
[35]	Z. C. Wang, J. Wu and R. Liu, Traveling waves of avian influenza spread,, \emph{Proc. Amer. Math. Soc., 140 (2012), 3931. doi: 10.1090/S0002-9939-2012-11246-8. Google Scholar
[36]	M. Wu and P. Weng, Stability of a stage-structured diffusive SIR model with delays (Chinese),, \emph{J. South China Normal Univ. Natur. Sci. Ed., 45 (2013), 20. Google Scholar
[37]	S. L. Wu and P. Weng, Entire solutions for a multi-type SIS nonlocal epidemic model in R or Z,, \emph{J. Math. Anal. Appl., 394 (2012), 603. doi: 10.1016/j.jmaa.2012.05.009. Google Scholar
[38]	J. Yang, S. Liang and Y. Zhang, Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion,, \emph{PLoS One, 6 (2011). doi: 10.1371/journal.pone.0021128. Google Scholar
[39]	Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction-diffusion Equations,, Science Press, (2011). Google Scholar
[40]	E. Zeilder, Nonlinear Functional Analysis and Its Applications: I, Fixed-piont Theorems,, New York, (1986). Google Scholar

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