# American Institute of Mathematical Sciences

September  2012, 32(9): 3303-3324. doi: 10.3934/dcds.2012.32.3303

## Traveling waves of diffusive predator-prey systems: Disease outbreak propagation

 1 Mprime Centre for Disease Modelling, York Institute for Health Research, Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, M3J 1P3, Canada, Canada 2 Division of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069-7100, United States

Received  January 2012 Revised  March 2012 Published  April 2012

We study the traveling waves of reaction-diffusion equations for a diffusive SIR model. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed. Our proof is based on Schauder fixed point theorem and Laplace transform.
Citation: Xiang-Sheng Wang, Haiyan Wang, Jianhong Wu. Traveling waves of diffusive predator-prey systems: Disease outbreak propagation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3303-3324. doi: 10.3934/dcds.2012.32.3303
##### References:
 [1] G. Abramson and V. M. Kenkre, Spatiotemporal patterns in hantavirus infection,, Phys. Rev. E, 66 (2002). Google Scholar [2] G. Abramson, V. M. Kenkre, T. L. Yates and R. R. Parmenter, Traveling waves of infection in the hantavirus epidemics,, Bull. Math. Biol., 65 (2003), 519. doi: 10.1016/S0092-8240(03)00013-2. Google Scholar [3] M. S. Abual-Rub, Vaccination in a model of an epidemic,, Int. J. Math. Math. Sci., 23 (2000), 425. doi: 10.1155/S0161171200002696. Google Scholar [4] S. Ai and W. Huang, Traveling waves for a reaction-diffusion system in population dynamics and epidemiology,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 663. doi: 10.1017/S0308210500004054. Google Scholar [5] C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves,, Math. Proc. Camb. Phil. Soc., 80 (1976), 315. doi: 10.1017/S0305004100052944. Google Scholar [6] M. S. Bartlett, Measles periodicity and community size (with discussion),, J. Roy. Stat. Soc. A, 120 (1957), 48. doi: 10.2307/2342553. Google Scholar [7] F. van den Bosch, J. A. J. Metz and O. Diekmann, The velocity of spatial population expansion,, J. Math. Biol., 28 (1990), 529. doi: 10.1007/BF00164162. Google Scholar [8] F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology,", Texts in Applied Mathematics, 40 (2001). Google Scholar [9] K. Brown and J. Carr, Deterministic epidemic waves of critical velocity,, Math. Proc. Camb. Phil. Soc., 81 (1977), 431. doi: 10.1017/S0305004100053494. Google Scholar [10] V. Capasso and L. Maddalena, A nonlinear diffusion system modelling the spread of oro-faecal diseases,, in, (1981). Google Scholar [11] T. Caraco, S. Glavanakov, G. Chen, J. E. Flaherty, T. K. Ohsumi and B. K. Szymanski, Stage-structured infection transmission and a spatial epidemic: A model for lyme disease,, Am. Nat., 160 (2002), 348. doi: 10.1086/341518. Google Scholar [12] J. Carr and A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar [13] O. Diekmann, Thresholds and traveling waves for the geographical spread of an infection,, J. Math. Biol., 6 (1978), 109. doi: 10.1007/BF02450783. Google Scholar [14] O. Diekmann and H. Kaper, On the bounded solutions of a nonliear convolution equation,, Nonlinear Analysis, 2 (1978), 721. doi: 10.1016/0362-546X(78)90015-9. Google Scholar [15] S. Djebali, Traveling front solutions for a diffusive epidemic model with external sources,, Annales de la Faculté des Sciences de Toulouse Sér. 6, 10 (2001), 271. Google Scholar [16] A. Ducrot, Travelling wave solutions for a scalar age-structured equation,, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 251. doi: 10.3934/dcdsb.2007.7.251. Google Scholar [17] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion,, Proc. R. Soc. Edin. Sect. A, 139 (2009), 459. doi: 10.1017/S0308210507000455. Google Scholar [18] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations,, J. Math. Biol., 17 (1983), 11. doi: 10.1007/BF00276112. Google Scholar [19] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^4$,, Trans. Amer. Math. Soc., 286 (1984), 557. Google Scholar [20] M. J. Faddy and I. H. Slorach, Bounds on the velocity of spread of infection for a spatially connected epidemic process,, J. Appl. Probab., 17 (1980), 839. doi: 10.2307/3212977. Google Scholar [21] J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems,, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3043. doi: 10.3934/dcds.2012.32.3303. Google Scholar [22] R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar [23] W. E. Fitzgibbon, M. E. Parrott and G. F. Webb, Diffusion epidemic models with incubation and crisscross dynamics,, Math. Biosci., 128 (1995), 131. doi: 10.1016/0025-5564(94)00070-G. Google Scholar [24] Q. Gan, R. Xu and P. Yang, Travelling waves of a delayed SIRS epidemic model with spatial diffusion,, Nonlinear Anal. Real World Appl., 12 (2011), 52. doi: 10.1016/j.nonrwa.2010.05.035. Google Scholar [25] B. T. Grenfell, O. N. Bjornstad and J. Kappey, Travelling waves and spatial hierarchies in measles epidemics,, Nature, 414 (2001), 716. doi: 10.1038/414716a. Google Scholar [26] Y. Hosono and B. Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model,, Nonlinear World, 1 (1994), 277. Google Scholar [27] Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model,, Math. Models Methods Appl. Sci., 5 (1995), 935. doi: 10.1142/S0218202595000504. Google Scholar [28] J.-H. Huang and X.-F. Zou, Travelling wave solutions in delayed reaction diffusion systems with partial monotonicity,, Acta Mathematicae Applicatae Sinica Engl. Ser., 22 (2006), 243. doi: 10.1007/s10255-006-0300-0. Google Scholar [29] W. Huang, Traveling waves for a biological reaction-diffusion model,, J. Dynam. Differential Equations, 16 (2004), 745. Google Scholar [30] A. Källén, Thresholds and travelling waves in an epidemic model for rabies,, Nonlinear Anal., 8 (1984), 851. doi: 10.1016/0362-546X(84)90107-X. Google Scholar [31] A. Källén, P. Arcuri and J. D. Murray, A simple model for the spatial spread and control of rabies,, J. Theor. Biol., 116 (1985), 377. doi: 10.1016/S0022-5193(85)80276-9. Google Scholar [32] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. R. Soc. Lond. B, 115 (1927), 700. Google Scholar [33] D. G. Kendall, Discussion on Professor Bartlett's paper,, J. Roy. Stat. Soc. A, 120 (1957), 64. Google Scholar [34] D. G. Kendall, Mathematical models of the spread of infection,, in, (1965), 213. Google Scholar [35] C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death,, Bull. Math. Biol., 42 (1980), 397. Google Scholar [36] M. N. Kuperman and H. S. Wio, Front propagation in epidemiological models with spatial dependence,, Physica A, 272 (1999), 206. doi: 10.1016/S0378-4371(99)00284-8. Google Scholar [37] W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems,, Nonlinearity, 19 (2006), 1253. doi: 10.1088/0951-7715/19/6/003. Google Scholar [38] X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems,, J. Differential Equations, 231 (2006), 57. Google Scholar [39] 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. Google Scholar [40] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294. Google Scholar [41] N. A. Maidana and H. M. Yang, Describing the geographic spread of dengue disease by traveling waves,, Math. Biosci., 215 (2008), 64. doi: 10.1016/j.mbs.2008.05.008. Google Scholar [42] D. Mollison, Possible velocities for a simple epidemic,, Adv. Appl. Prob., 4 (1972), 233. doi: 10.2307/1425997. Google Scholar [43] D. Mollison, Spatial contact models for ecological and epidemic spread,, J. Roy. Stat. Soc. B, 39 (1977), 283. Google Scholar [44] B. Mukhopadhyay and R. Bhattacharyya, Existence of epidemic waves in a disease transmission model with two-habitat population,, Internat. J. Systems Sci., 38 (2007), 699. doi: 10.1080/00207720701596417. Google Scholar [45] J. D. Murray and W. L. Seward, On the spatial spread of rabies among foxes with immunity,, J. Theor. Biol., 156 (1992), 327. doi: 10.1016/S0022-5193(05)80679-4. Google Scholar [46] J. D. Murray, E. A. Stanley and D. L. Brown, On the spatial spread of rabies among foxes,, Proc. R. Soc. Lond. B, 229 (1986), 111. doi: 10.1098/rspb.1986.0078. Google Scholar [47] S. Pan, Traveling wave fronts in an epidemic model with nonlocal diffusion and time delay,, Int. Journal of Math. Analysis, 2 (2008), 1083. Google Scholar [48] L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics,", Math. Surveys Monogr., 102 (2003). Google Scholar [49] E. Renshaw, Waveforms and velocities for models of spatial infection,, J. Appl. Probab., 18 (1981), 715. doi: 10.2307/3213325. Google Scholar [50] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models,, in, (2007), 97. Google Scholar [51] S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts,, in, (2009), 293. Google Scholar [52] S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model,, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991. doi: 10.1017/S0308210500003590. Google Scholar [53] I. Sazonov and M. Kelbert, Travelling waves in a network of SIR epidemic nodes with an approximation of weak coupling,, Math. Med. Biol., 28 (2011), 165. doi: 10.1093/imammb/dqq016. Google Scholar [54] I. Sazonov, M. Kelbert and M. B. Gravenor, The speed of epidemic waves in a one-dimensional lattice of SIR models,, Mathematical Modelling of Natural Phenomena, 3 (2008), 28. doi: 10.1051/mmnp:2008069. Google Scholar [55] H. L. Smith and X.-Q. Zhao, Traveling waves in a bio-reactor model,, Nonlinear Anal. Real World Appl., 5 (2004), 895. doi: 10.1016/j.nonrwa.2004.05.001. Google Scholar [56] L. T. Takahashi, N. A. Maidana, W. C. Ferreira, Jr., P. Pulino and H. M. Yang, Mathematical models for the Aedes aegypti dispersal dynamics: Travelling waves by wing and wind,, Bull. Math. Biol., 67 (2005), 509. doi: 10.1016/j.bulm.2004.08.005. Google Scholar [57] H. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread,, J. Math. Biol., 8 (1979), 173. doi: 10.1007/BF00279720. Google Scholar [58] H. 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. Google Scholar [59] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [60] H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations,, J. Differential Equations, 247 (2009), 887. Google Scholar [61] X.-S. Wang, J. Wu and Y. Yang, Richards model revisited: Validation by and application to infection dynamics,, submitted., (). Google Scholar [62] Z.-C. Wang and J. Wu, Traveling 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. doi: 10.1098/rspa.2009.0377. Google Scholar [63] Z.-C. Wang and J. Wu, Traveling waves in a bio-reactor model with stage-structure,, J. Math. Anal. Appl., 385 (2012), 683. doi: 10.1016/j.jmaa.2011.06.084. Google Scholar [64] P. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model,, J. Differential Equations, 229 (2006), 270. Google Scholar [65] S.-L. Wu and S.-Y. Liu, Existence and uniqueness of traveling waves for non-monotone integral equations with application,, J. Math. Anal. Appl., 365 (2010), 729. doi: 10.1016/j.jmaa.2009.11.028. Google Scholar [66] J. Wylie, H. Huang and R. M. Miura, Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves,, Discrete Contin. Dyn. Syst., 23 (2009), 561. doi: 10.3934/dcds.2009.23.561. Google Scholar [67] D. Xu and X.-Q. Zhao, Bistable waves in an epidemic model,, J. Dynamics and Differential Equations, 16 (2004), 679. Google Scholar [68] D. Xu and X.-Q. Zhao, Asymptotic speed of spread and traveling waves for a nonlocal epidemic model,, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 1043. doi: 10.3934/dcdsb.2005.5.1043. Google Scholar [69] 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). doi: 10.1371/journal.pone.0021128. Google Scholar [70] Y.-R. Yang, W.-T. Li and S.-L. Wu, Exponential stability of traveling fronts in a diffusion epidemic system with delay,, Nonlinear Anal. Real World Appl., 12 (2011), 1223. doi: 10.1016/j.nonrwa.2010.09.017. Google Scholar [71] F. Zhang and X.-Q. Zhao, Spreading speed and travelling waves for a spatially discrete SIS epidemic model,, Nonlinearity, 21 (2008), 97. doi: 10.1088/0951-7715/21/1/005. Google Scholar [72] X.-Q. Zhao and W. Wang, Fisher waves in an epidemic model,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117. doi: 10.3934/dcdsb.2004.4.1117. Google Scholar

show all references

##### References:
 [1] G. Abramson and V. M. Kenkre, Spatiotemporal patterns in hantavirus infection,, Phys. Rev. E, 66 (2002). Google Scholar [2] G. Abramson, V. M. Kenkre, T. L. Yates and R. R. Parmenter, Traveling waves of infection in the hantavirus epidemics,, Bull. Math. Biol., 65 (2003), 519. doi: 10.1016/S0092-8240(03)00013-2. Google Scholar [3] M. S. Abual-Rub, Vaccination in a model of an epidemic,, Int. J. Math. Math. Sci., 23 (2000), 425. doi: 10.1155/S0161171200002696. Google Scholar [4] S. Ai and W. Huang, Traveling waves for a reaction-diffusion system in population dynamics and epidemiology,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 663. doi: 10.1017/S0308210500004054. Google Scholar [5] C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves,, Math. Proc. Camb. Phil. Soc., 80 (1976), 315. doi: 10.1017/S0305004100052944. Google Scholar [6] M. S. Bartlett, Measles periodicity and community size (with discussion),, J. Roy. Stat. Soc. A, 120 (1957), 48. doi: 10.2307/2342553. Google Scholar [7] F. van den Bosch, J. A. J. Metz and O. Diekmann, The velocity of spatial population expansion,, J. Math. Biol., 28 (1990), 529. doi: 10.1007/BF00164162. Google Scholar [8] F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology,", Texts in Applied Mathematics, 40 (2001). Google Scholar [9] K. Brown and J. Carr, Deterministic epidemic waves of critical velocity,, Math. Proc. Camb. Phil. Soc., 81 (1977), 431. doi: 10.1017/S0305004100053494. Google Scholar [10] V. Capasso and L. Maddalena, A nonlinear diffusion system modelling the spread of oro-faecal diseases,, in, (1981). Google Scholar [11] T. Caraco, S. Glavanakov, G. Chen, J. E. Flaherty, T. K. Ohsumi and B. K. Szymanski, Stage-structured infection transmission and a spatial epidemic: A model for lyme disease,, Am. Nat., 160 (2002), 348. doi: 10.1086/341518. Google Scholar [12] J. Carr and A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar [13] O. Diekmann, Thresholds and traveling waves for the geographical spread of an infection,, J. Math. Biol., 6 (1978), 109. doi: 10.1007/BF02450783. Google Scholar [14] O. Diekmann and H. Kaper, On the bounded solutions of a nonliear convolution equation,, Nonlinear Analysis, 2 (1978), 721. doi: 10.1016/0362-546X(78)90015-9. Google Scholar [15] S. Djebali, Traveling front solutions for a diffusive epidemic model with external sources,, Annales de la Faculté des Sciences de Toulouse Sér. 6, 10 (2001), 271. Google Scholar [16] A. Ducrot, Travelling wave solutions for a scalar age-structured equation,, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 251. doi: 10.3934/dcdsb.2007.7.251. Google Scholar [17] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion,, Proc. R. Soc. Edin. Sect. A, 139 (2009), 459. doi: 10.1017/S0308210507000455. Google Scholar [18] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations,, J. Math. Biol., 17 (1983), 11. doi: 10.1007/BF00276112. Google Scholar [19] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^4$,, Trans. Amer. Math. Soc., 286 (1984), 557. Google Scholar [20] M. J. Faddy and I. H. Slorach, Bounds on the velocity of spread of infection for a spatially connected epidemic process,, J. Appl. Probab., 17 (1980), 839. doi: 10.2307/3212977. Google Scholar [21] J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems,, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3043. doi: 10.3934/dcds.2012.32.3303. Google Scholar [22] R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar [23] W. E. Fitzgibbon, M. E. Parrott and G. F. Webb, Diffusion epidemic models with incubation and crisscross dynamics,, Math. Biosci., 128 (1995), 131. doi: 10.1016/0025-5564(94)00070-G. Google Scholar [24] Q. Gan, R. Xu and P. Yang, Travelling waves of a delayed SIRS epidemic model with spatial diffusion,, Nonlinear Anal. Real World Appl., 12 (2011), 52. doi: 10.1016/j.nonrwa.2010.05.035. Google Scholar [25] B. T. Grenfell, O. N. Bjornstad and J. Kappey, Travelling waves and spatial hierarchies in measles epidemics,, Nature, 414 (2001), 716. doi: 10.1038/414716a. Google Scholar [26] Y. Hosono and B. Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model,, Nonlinear World, 1 (1994), 277. Google Scholar [27] Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model,, Math. Models Methods Appl. Sci., 5 (1995), 935. doi: 10.1142/S0218202595000504. Google Scholar [28] J.-H. Huang and X.-F. Zou, Travelling wave solutions in delayed reaction diffusion systems with partial monotonicity,, Acta Mathematicae Applicatae Sinica Engl. Ser., 22 (2006), 243. doi: 10.1007/s10255-006-0300-0. Google Scholar [29] W. Huang, Traveling waves for a biological reaction-diffusion model,, J. Dynam. Differential Equations, 16 (2004), 745. Google Scholar [30] A. Källén, Thresholds and travelling waves in an epidemic model for rabies,, Nonlinear Anal., 8 (1984), 851. doi: 10.1016/0362-546X(84)90107-X. Google Scholar [31] A. Källén, P. Arcuri and J. D. Murray, A simple model for the spatial spread and control of rabies,, J. Theor. Biol., 116 (1985), 377. doi: 10.1016/S0022-5193(85)80276-9. Google Scholar [32] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. R. Soc. Lond. B, 115 (1927), 700. Google Scholar [33] D. G. Kendall, Discussion on Professor Bartlett's paper,, J. Roy. Stat. Soc. A, 120 (1957), 64. Google Scholar [34] D. G. Kendall, Mathematical models of the spread of infection,, in, (1965), 213. Google Scholar [35] C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death,, Bull. Math. Biol., 42 (1980), 397. Google Scholar [36] M. N. Kuperman and H. S. Wio, Front propagation in epidemiological models with spatial dependence,, Physica A, 272 (1999), 206. doi: 10.1016/S0378-4371(99)00284-8. Google Scholar [37] W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems,, Nonlinearity, 19 (2006), 1253. doi: 10.1088/0951-7715/19/6/003. Google Scholar [38] X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems,, J. Differential Equations, 231 (2006), 57. Google Scholar [39] 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. Google Scholar [40] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294. Google Scholar [41] N. A. Maidana and H. M. Yang, Describing the geographic spread of dengue disease by traveling waves,, Math. Biosci., 215 (2008), 64. doi: 10.1016/j.mbs.2008.05.008. Google Scholar [42] D. Mollison, Possible velocities for a simple epidemic,, Adv. Appl. Prob., 4 (1972), 233. doi: 10.2307/1425997. Google Scholar [43] D. Mollison, Spatial contact models for ecological and epidemic spread,, J. Roy. Stat. Soc. B, 39 (1977), 283. Google Scholar [44] B. Mukhopadhyay and R. Bhattacharyya, Existence of epidemic waves in a disease transmission model with two-habitat population,, Internat. J. Systems Sci., 38 (2007), 699. doi: 10.1080/00207720701596417. Google Scholar [45] J. D. Murray and W. L. Seward, On the spatial spread of rabies among foxes with immunity,, J. Theor. Biol., 156 (1992), 327. doi: 10.1016/S0022-5193(05)80679-4. Google Scholar [46] J. D. Murray, E. A. Stanley and D. L. Brown, On the spatial spread of rabies among foxes,, Proc. R. Soc. Lond. B, 229 (1986), 111. doi: 10.1098/rspb.1986.0078. Google Scholar [47] S. Pan, Traveling wave fronts in an epidemic model with nonlocal diffusion and time delay,, Int. Journal of Math. Analysis, 2 (2008), 1083. Google Scholar [48] L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics,", Math. Surveys Monogr., 102 (2003). Google Scholar [49] E. Renshaw, Waveforms and velocities for models of spatial infection,, J. Appl. Probab., 18 (1981), 715. doi: 10.2307/3213325. Google Scholar [50] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models,, in, (2007), 97. Google Scholar [51] S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts,, in, (2009), 293. Google Scholar [52] S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model,, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991. doi: 10.1017/S0308210500003590. Google Scholar [53] I. Sazonov and M. Kelbert, Travelling waves in a network of SIR epidemic nodes with an approximation of weak coupling,, Math. Med. Biol., 28 (2011), 165. doi: 10.1093/imammb/dqq016. Google Scholar [54] I. Sazonov, M. Kelbert and M. B. Gravenor, The speed of epidemic waves in a one-dimensional lattice of SIR models,, Mathematical Modelling of Natural Phenomena, 3 (2008), 28. doi: 10.1051/mmnp:2008069. Google Scholar [55] H. L. Smith and X.-Q. Zhao, Traveling waves in a bio-reactor model,, Nonlinear Anal. Real World Appl., 5 (2004), 895. doi: 10.1016/j.nonrwa.2004.05.001. Google Scholar [56] L. T. Takahashi, N. A. Maidana, W. C. Ferreira, Jr., P. Pulino and H. M. Yang, Mathematical models for the Aedes aegypti dispersal dynamics: Travelling waves by wing and wind,, Bull. Math. Biol., 67 (2005), 509. doi: 10.1016/j.bulm.2004.08.005. Google Scholar [57] H. 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