# American Institute of Mathematical Sciences

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

## Traveling waves of a delayed diffusive SIR epidemic model

 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

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.
Citation: Yan Li, Wan-Tong Li, Guo Lin. Traveling waves of a delayed diffusive SIR epidemic model. Communications on Pure and 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, J. Dyn. Diff. Equat., 26 (2014), 143-164. doi: 10.1007/s10884-014-9348-3. [2] M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099. doi: 10.1016/j.aml.2012.05.006. [3] M. S. Bartlett, Deterministic and stochastic models for recurrent epidemics, inProc. 3rd Berkeley Symp. Mathematical Statistics and Probability, Vol. 4. No. 81. Berkeley: University of California Press, 1956. [4] C. Briggs and H. Godfray, The dynamics of insect-pathogen interactions in stage-structured populations, Amer. Nat., 145 (1995), 855-887. [5] K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Camb. Phil. Soc., 81 (1977), 431-433. doi: 10.1017/S0305004100053494. [6] V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 41-61. doi: 10.1016/0025-5564(78)90006-8. [7] Z. Chen and Z. Zhao, Harnack principle for weakly coupled elliptic systems, J. Differential Equations, 139 (1997), 261-282. doi: 10.1006/jdeq.1997.3300. [8] K. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42. doi: 10.1216/RMJ-1979-9-1-31. [9] O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130. doi: 10.1007/BF02450783. [10] W. Ding, W. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model, Discrete Contin. Dyn. Syst. B, 18 (2013), 1291-1304. doi: 10.3934/dcdsb.2013.18.1291. [11] A. Ducrot, M. Langlais and P. Magal, Qualitative analysis and travelling wave solutions for the SI model with vertical transmission, Commum. Pure Appl. Anal., 11 (2012), 97-113. doi: 10.3934/cpaa.2012.11.97. [12] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. R. Soc. Edin. Ser. A Math., 139 (2009), 459-482. doi: 10.1017/S0308210507000455. [13] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911. doi: 10.1088/0951-7715/24/10/012. [14] A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structure epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331. doi: 10.1007/s00205-008-0203-8. [15] L. Evans, Partial Differential Equaitons, Second edition, Graduate studies in mathematics, American Mathematical Society, 2010. [16] J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009. [17] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Vol. 224. Springer, 2001. [18] H. Hethcote, The mathematics of infectious diseases, SIAM. Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [19] W. Hirsch, H. Hamisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1948), 221-246. doi: 10.1002/cpa.3160380607. [20] Y. Hosono and B. Ilyas, Travelling waves for a simple diffusive epidemic model, Math. Model Meth. Appl. Sci., 5 (1995), 935-966. doi: 10.1142/S0218202595000504. [21] G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207. doi: 10.1007/s11538-009-9487-6. [22] W. T. Li, G. Lin, C. Ma and F. Y. Yang, Traveling waves of a nonlocal delayed SIR epidemic model without outbreak threshold, Discrete Contin. Dyn. Syst. B, 19 (2014), 467-484. doi: 10.3934/dcdsb.2014.19.467. [23] G. Lin, Invasion traveling wave solution of a predator-prey system, Nonlinear Anal., 96 (2014), 47-58. doi: 10.1016/j.na.2013.10.024. [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, J. Dyn. Diff. Equat., 26 (2014), 583-605. doi: 10.1007/s10884-014-9355-4. [25] C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. Real Word Appl., 11 (2010), 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005. [26] J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, 3rd edn, Springer-Verlag, New York, 2003. [27] J. Roughgarden, Theory of Population Genetics and Evolutionary Ecology: An Introduction, New York, Macmillan, 1979. [28] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in Mathematics for Life Science and Medicine (eds. Y. Takeuchi, K. Sato and Y. Iwasa), Springer-Verlag, New York, (2007), pp. 97-122. [29] S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in Spatial Ecology, Chapman & Hall/CRC, Boca Raton, FL, (2009), 293-316. [30] H. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785. [31] H. R. Thieme and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [32] H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), 747-783. doi: 10.1007/s00332-011-9099-9. [33] X. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst., 32 (2012), 3303-3324. doi: 10.3934/dcds.2012.32.3303. [34] Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261. doi: 10.1098/rspa.2009.0377. [35] Z. C. Wang, J. Wu and R. Liu, Traveling waves of avian influenza spread, Proc. Amer. Math. Soc., 140 (2012), 3931-3946. doi: 10.1090/S0002-9939-2012-11246-8. [36] M. Wu and P. Weng, Stability of a stage-structured diffusive SIR model with delays (Chinese), J. South China Normal Univ. Natur. Sci. Ed., 45 (2013), 20-23. [37] S. L. Wu and P. Weng, Entire solutions for a multi-type SIS nonlocal epidemic model in R or Z, J. Math. Anal. Appl., 394 (2012), 603-615. doi: 10.1016/j.jmaa.2012.05.009. [38] J. Yang, S. Liang and Y. Zhang, Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion, PLoS One, 6 (2011), e21128. doi: 10.1371/journal.pone.0021128. [39] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction-diffusion Equations, Science Press, Beijing, 2011. [40] E. Zeilder, Nonlinear Functional Analysis and Its Applications: I, Fixed-piont Theorems, New York, Springer-Verlag, 1986.

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##### References:
 [1] S. Ai and R. Albashaireh, Traveling waves in spatial SIRS models, J. Dyn. Diff. Equat., 26 (2014), 143-164. doi: 10.1007/s10884-014-9348-3. [2] M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099. doi: 10.1016/j.aml.2012.05.006. [3] M. S. Bartlett, Deterministic and stochastic models for recurrent epidemics, inProc. 3rd Berkeley Symp. Mathematical Statistics and Probability, Vol. 4. No. 81. Berkeley: University of California Press, 1956. [4] C. Briggs and H. Godfray, The dynamics of insect-pathogen interactions in stage-structured populations, Amer. Nat., 145 (1995), 855-887. [5] K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Camb. Phil. Soc., 81 (1977), 431-433. doi: 10.1017/S0305004100053494. [6] V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 41-61. doi: 10.1016/0025-5564(78)90006-8. [7] Z. Chen and Z. Zhao, Harnack principle for weakly coupled elliptic systems, J. Differential Equations, 139 (1997), 261-282. doi: 10.1006/jdeq.1997.3300. [8] K. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42. doi: 10.1216/RMJ-1979-9-1-31. [9] O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130. doi: 10.1007/BF02450783. [10] W. Ding, W. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model, Discrete Contin. Dyn. Syst. B, 18 (2013), 1291-1304. doi: 10.3934/dcdsb.2013.18.1291. [11] A. Ducrot, M. Langlais and P. Magal, Qualitative analysis and travelling wave solutions for the SI model with vertical transmission, Commum. Pure Appl. Anal., 11 (2012), 97-113. doi: 10.3934/cpaa.2012.11.97. [12] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. R. Soc. Edin. Ser. A Math., 139 (2009), 459-482. doi: 10.1017/S0308210507000455. [13] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911. doi: 10.1088/0951-7715/24/10/012. [14] A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structure epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331. doi: 10.1007/s00205-008-0203-8. [15] L. Evans, Partial Differential Equaitons, Second edition, Graduate studies in mathematics, American Mathematical Society, 2010. [16] J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009. [17] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Vol. 224. Springer, 2001. [18] H. Hethcote, The mathematics of infectious diseases, SIAM. Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [19] W. Hirsch, H. Hamisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1948), 221-246. doi: 10.1002/cpa.3160380607. [20] Y. Hosono and B. Ilyas, Travelling waves for a simple diffusive epidemic model, Math. Model Meth. Appl. Sci., 5 (1995), 935-966. doi: 10.1142/S0218202595000504. [21] G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207. doi: 10.1007/s11538-009-9487-6. [22] W. T. Li, G. Lin, C. Ma and F. Y. Yang, Traveling waves of a nonlocal delayed SIR epidemic model without outbreak threshold, Discrete Contin. Dyn. Syst. B, 19 (2014), 467-484. doi: 10.3934/dcdsb.2014.19.467. [23] G. Lin, Invasion traveling wave solution of a predator-prey system, Nonlinear Anal., 96 (2014), 47-58. doi: 10.1016/j.na.2013.10.024. [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, J. Dyn. Diff. Equat., 26 (2014), 583-605. doi: 10.1007/s10884-014-9355-4. [25] C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. Real Word Appl., 11 (2010), 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005. [26] J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, 3rd edn, Springer-Verlag, New York, 2003. [27] J. Roughgarden, Theory of Population Genetics and Evolutionary Ecology: An Introduction, New York, Macmillan, 1979. [28] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in Mathematics for Life Science and Medicine (eds. Y. Takeuchi, K. Sato and Y. Iwasa), Springer-Verlag, New York, (2007), pp. 97-122. [29] S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in Spatial Ecology, Chapman & Hall/CRC, Boca Raton, FL, (2009), 293-316. [30] H. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785. [31] H. R. Thieme and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [32] H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), 747-783. doi: 10.1007/s00332-011-9099-9. [33] X. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst., 32 (2012), 3303-3324. doi: 10.3934/dcds.2012.32.3303. [34] Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261. doi: 10.1098/rspa.2009.0377. [35] Z. C. Wang, J. Wu and R. Liu, Traveling waves of avian influenza spread, Proc. Amer. Math. Soc., 140 (2012), 3931-3946. doi: 10.1090/S0002-9939-2012-11246-8. [36] M. Wu and P. Weng, Stability of a stage-structured diffusive SIR model with delays (Chinese), J. South China Normal Univ. Natur. Sci. Ed., 45 (2013), 20-23. [37] S. L. Wu and P. Weng, Entire solutions for a multi-type SIS nonlocal epidemic model in R or Z, J. Math. Anal. Appl., 394 (2012), 603-615. doi: 10.1016/j.jmaa.2012.05.009. [38] J. Yang, S. Liang and Y. Zhang, Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion, PLoS One, 6 (2011), e21128. doi: 10.1371/journal.pone.0021128. [39] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction-diffusion Equations, Science Press, Beijing, 2011. [40] E. Zeilder, Nonlinear Functional Analysis and Its Applications: I, Fixed-piont Theorems, New York, Springer-Verlag, 1986.
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