# American Institute of Mathematical Sciences

March  2014, 19(2): 467-484. doi: 10.3934/dcdsb.2014.19.467

## Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold

 1 School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000 2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China, China

Received  April 2013 Revised  October 2013 Published  February 2014

This paper is concerned with the traveling wave solutions of a diffusive SIR system with nonlocal delay. We obtain the existence and nonexistence of traveling wave solutions, which formulate the propagation of disease without outbreak threshold. Moreover, it is proved that at any fixed moment, the faster the disease spreads, the more the infected individuals, and the larger the recovery/remove ratio is, the less the infected individuals.
Citation: Wan-Tong Li, Guo Lin, Cong Ma, Fei-Ying Yang. Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 467-484. doi: 10.3934/dcdsb.2014.19.467
##### References:
 [1] R. M. Anderson and R. M. May, Population Biology of Infectious Diseases, Springer-Verlag, Berlin, 1982. doi: 10.1007/978-3-642-68635-1. [2] R. M. Anderson, R. M. May and B. Anderson, Infectious Diseases of Humans: Dynamics and Control, Oxford university press, Oxford, 1992. [3] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics (eds. J.A. Goldstein), Lecture Notes in Mathematics, Vol. 446, Springer, Berlin, (1975), pp. 5-49. [4] N. F. Britton, Aggregation and the competitive exclusion principle, J. Theor. Biol., 136 (1989), 57-66. doi: 10.1016/S0022-5193(89)80189-4. [5] N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099. [6] J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5. [7] M. C. M. De Jong, O. Diekmann and H. Heesterbeek, How does transmission of infection depend on population size, Epidemic models: their structure and relation to data, 5 (1995), 84-94. [8] O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 69 (1978), 109-130. doi: 10.1007/BF02450783. [9] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33 (1979), 58-73. doi: 10.1016/0022-0396(79)90080-9. [10] 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. [11] A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Rational Mech. Anal., 195 (2010), 311-331. doi: 10.1007/s00205-008-0203-8. [12] T. Faria, W. Huang and J. Wu, Traveling waves for delayed reaction- diffusion equations with global response, Proc. R. Soc. Lond., 462A (2006), 229-261. doi: 10.1098/rspa.2005.1554. [13] T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376. doi: 10.1016/j.jde.2006.05.006. [14] S. A. Gourley and S. Ruan, Convergence and traveling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822. doi: 10.1137/S003614100139991. [15] S. A. Gourley, J. W.-H. So and J. Wu, Nonlocality of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics, J. Math. Sci., 124 (2004), 5119-5153. doi: 10.1023/B:JOTH.0000047249.39572.6d. [16] S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear dynamics and evolution equations (eds. H. Brunner, X.Q. Zhao and X. Zou), Fields Inst. Commun., 48, AMS, Providence, RI, (2006), pp. 137-200,. [17] 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. [18] J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction-diffusion systems without monotonicity, Discrete Cont. Dyn. Sys., 9 (2003), 925-936. doi: 10.3934/dcds.2003.9.925. [19] J. Huang and X. Zou, Traveling wave solutions in delayed reaction diffusion systems with partial monotonicity, Acta Math. Appl. Sinica, 22 (2006), 243-256. doi: 10.1007/s10255-006-0300-0. [20] W. Huang, Traveling waves for a biological reaction-diffusion model, J. Dynam. Differential Equations, 16 (2004), 745-765. doi: 10.1007/s10884-004-6115-x. [21] B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems, Nonlinearity, 24 (2011), 1759-1776. doi: 10.1088/0951-7715/24/6/004. [22] J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079. doi: 10.1007/s11538-009-9457-z. [23] W. T. Li, G. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003. [24] W. T. Li and F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equ. Appl., in press. [25] X. Liang and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. [26] G. Lin, W. T. Li and M. Ma, Travelling wave solutions in delayed reaction-diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414. doi: 10.3934/dcdsb.2010.13.393. [27] G. Lin, W. T. Li and S. Ruan, Monostable wavefronts in cooperative Lotka-Volterra systems with nonlocal delays, Discrete Contin. Dyn. Syst., 31 (2011), 1-23. doi: 10.3934/dcds.2011.31.1. [28] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846. [29] S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277. doi: 10.1016/j.jde.2007.03.014. [30] S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J. Dynam. Diff. Eqns., 19 (2007), 391-436. doi: 10.1007/s10884-006-9065-7. [31] M. Mei, C. K. Lin, C. T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: (I) Local nonlinearity, J. Differential Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026. [32] M. Mei, C. K. Lin, C. T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: (II) Nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529. doi: 10.1016/j.jde.2008.12.020. [33] M. Mei, C. Ou and X. Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790. doi: 10.1137/090776342. [34] C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations, J. Differential Equations, 235 (2007), 219-261. doi: 10.1016/j.jde.2006.12.010. [35] 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. [36] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. doi: 10.2307/2000859. [37] I. Sazonov, M. Kelbert, M.B. Gravenor, The speed of epidemic waves in a one-dimensional lattice of SIR models, Math. Model. Nat. Phenom, 3 (2008), 28-47. doi: 10.1051/mmnp:2008069. [38] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, RI, 1995. [39] H. R. Thieme, A model for the spatial spread of an epidemic, J. Math. Biol., 4 (1977), 337-351. doi: 10.1007/BF00275082. [40] 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. [41] Y. Tian and P. Weng, Spreading speed and wavefronts for parabolic functional differential equations with spatio-temporal delays, Nonlinear Anal. TMA, 71 (2009), 3374-3388. doi: 10.1016/j.na.2009.01.237. [42] X. S. Wang, H. Y. 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. [43] X. S. Wang, J. Wu and Y. Yang, Richards model revisited: Validation by and application to infection dynamics, J. Theoretical Biology, 313 (2012), 12-19. doi: 10.1016/j.jtbi.2012.07.024. [44] Z. Wang, W. T. Li and S. Ruan, Traveling wave fronts of reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010. [45] Z. Wang, W. T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction-advection-diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025. [46] Z. Wang, W. T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Diff. Eqns., 20 (2008), 573-607. doi: 10.1007/s10884-008-9103-8. [47] Z. C. Wang, W. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084. doi: 10.1090/S0002-9947-08-04694-1. [48] 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. [49] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Diff. Eqns., 13 (2001), 651-687. doi: 10.1023/A:1016690424892. [50] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, Science Press, Beijing, 1990. [51] X. Q. Zhao, Spatial dynamics of some evolution systems in biology, In Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions (ed. by Y. Du, H. Ishii and W.Y. Lin), pp.332-363, World Scientific, Singapore, 2009. doi: 10.1142/9789812834744_0015.

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##### References:
 [1] R. M. Anderson and R. M. May, Population Biology of Infectious Diseases, Springer-Verlag, Berlin, 1982. doi: 10.1007/978-3-642-68635-1. [2] R. M. Anderson, R. M. May and B. Anderson, Infectious Diseases of Humans: Dynamics and Control, Oxford university press, Oxford, 1992. [3] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics (eds. J.A. Goldstein), Lecture Notes in Mathematics, Vol. 446, Springer, Berlin, (1975), pp. 5-49. [4] N. F. Britton, Aggregation and the competitive exclusion principle, J. Theor. Biol., 136 (1989), 57-66. doi: 10.1016/S0022-5193(89)80189-4. [5] N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099. [6] J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5. [7] M. C. M. De Jong, O. Diekmann and H. Heesterbeek, How does transmission of infection depend on population size, Epidemic models: their structure and relation to data, 5 (1995), 84-94. [8] O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 69 (1978), 109-130. doi: 10.1007/BF02450783. [9] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33 (1979), 58-73. doi: 10.1016/0022-0396(79)90080-9. [10] 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. [11] A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Rational Mech. Anal., 195 (2010), 311-331. doi: 10.1007/s00205-008-0203-8. [12] T. Faria, W. Huang and J. Wu, Traveling waves for delayed reaction- diffusion equations with global response, Proc. R. Soc. Lond., 462A (2006), 229-261. doi: 10.1098/rspa.2005.1554. [13] T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376. doi: 10.1016/j.jde.2006.05.006. [14] S. A. Gourley and S. Ruan, Convergence and traveling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822. doi: 10.1137/S003614100139991. [15] S. A. Gourley, J. W.-H. So and J. Wu, Nonlocality of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics, J. Math. Sci., 124 (2004), 5119-5153. doi: 10.1023/B:JOTH.0000047249.39572.6d. [16] S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear dynamics and evolution equations (eds. H. Brunner, X.Q. Zhao and X. Zou), Fields Inst. Commun., 48, AMS, Providence, RI, (2006), pp. 137-200,. [17] 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. [18] J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction-diffusion systems without monotonicity, Discrete Cont. Dyn. Sys., 9 (2003), 925-936. doi: 10.3934/dcds.2003.9.925. [19] J. Huang and X. Zou, Traveling wave solutions in delayed reaction diffusion systems with partial monotonicity, Acta Math. Appl. Sinica, 22 (2006), 243-256. doi: 10.1007/s10255-006-0300-0. [20] W. Huang, Traveling waves for a biological reaction-diffusion model, J. Dynam. Differential Equations, 16 (2004), 745-765. doi: 10.1007/s10884-004-6115-x. [21] B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems, Nonlinearity, 24 (2011), 1759-1776. doi: 10.1088/0951-7715/24/6/004. [22] J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079. doi: 10.1007/s11538-009-9457-z. [23] W. T. Li, G. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003. [24] W. T. Li and F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equ. Appl., in press. [25] X. Liang and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. [26] G. Lin, W. T. Li and M. Ma, Travelling wave solutions in delayed reaction-diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414. doi: 10.3934/dcdsb.2010.13.393. [27] G. Lin, W. T. Li and S. Ruan, Monostable wavefronts in cooperative Lotka-Volterra systems with nonlocal delays, Discrete Contin. Dyn. Syst., 31 (2011), 1-23. doi: 10.3934/dcds.2011.31.1. [28] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846. [29] S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277. doi: 10.1016/j.jde.2007.03.014. [30] S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J. Dynam. Diff. Eqns., 19 (2007), 391-436. doi: 10.1007/s10884-006-9065-7. [31] M. Mei, C. K. Lin, C. T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: (I) Local nonlinearity, J. Differential Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026. [32] M. Mei, C. K. Lin, C. T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: (II) Nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529. doi: 10.1016/j.jde.2008.12.020. [33] M. Mei, C. Ou and X. Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790. doi: 10.1137/090776342. [34] C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations, J. Differential Equations, 235 (2007), 219-261. doi: 10.1016/j.jde.2006.12.010. [35] 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. [36] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. doi: 10.2307/2000859. [37] I. Sazonov, M. Kelbert, M.B. Gravenor, The speed of epidemic waves in a one-dimensional lattice of SIR models, Math. Model. Nat. Phenom, 3 (2008), 28-47. doi: 10.1051/mmnp:2008069. [38] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, RI, 1995. [39] H. R. Thieme, A model for the spatial spread of an epidemic, J. Math. Biol., 4 (1977), 337-351. doi: 10.1007/BF00275082. [40] 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. [41] Y. Tian and P. Weng, Spreading speed and wavefronts for parabolic functional differential equations with spatio-temporal delays, Nonlinear Anal. TMA, 71 (2009), 3374-3388. doi: 10.1016/j.na.2009.01.237. [42] X. S. Wang, H. Y. 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. [43] X. S. Wang, J. Wu and Y. Yang, Richards model revisited: Validation by and application to infection dynamics, J. Theoretical Biology, 313 (2012), 12-19. doi: 10.1016/j.jtbi.2012.07.024. [44] Z. Wang, W. T. Li and S. Ruan, Traveling wave fronts of reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010. [45] Z. Wang, W. T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction-advection-diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025. [46] Z. Wang, W. T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Diff. Eqns., 20 (2008), 573-607. doi: 10.1007/s10884-008-9103-8. [47] Z. C. Wang, W. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084. doi: 10.1090/S0002-9947-08-04694-1. [48] 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. [49] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Diff. Eqns., 13 (2001), 651-687. doi: 10.1023/A:1016690424892. [50] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, Science Press, Beijing, 1990. [51] X. Q. Zhao, Spatial dynamics of some evolution systems in biology, In Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions (ed. by Y. Du, H. Ishii and W.Y. Lin), pp.332-363, World Scientific, Singapore, 2009. doi: 10.1142/9789812834744_0015.
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