# 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 and Continuous Dynamical Systems, 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), 011912, 5 pp. [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), 519-534. doi: 10.1016/S0092-8240(03)00013-2. [3] M. S. Abual-Rub, Vaccination in a model of an epidemic, Int. J. Math. Math. Sci., 23 (2000), 425-429. doi: 10.1155/S0161171200002696. [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), 663-675. doi: 10.1017/S0308210500004054. [5] C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves, Math. Proc. Camb. Phil. Soc., 80 (1976), 315-330. doi: 10.1017/S0305004100052944. [6] M. S. Bartlett, Measles periodicity and community size (with discussion), J. Roy. Stat. Soc. A, 120 (1957), 48-70. doi: 10.2307/2342553. [7] F. van den Bosch, J. A. J. Metz and O. Diekmann, The velocity of spatial population expansion, J. Math. Biol., 28 (1990), 529-565. doi: 10.1007/BF00164162. [8] F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001. [9] K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Camb. Phil. Soc., 81 (1977), 431-433. doi: 10.1017/S0305004100053494. [10] V. Capasso and L. Maddalena, A nonlinear diffusion system modelling the spread of oro-faecal diseases, in "Nonlinear Phenomena in Mathematical Sciences" (ed. V. Lakshmikantham), Academic Press, New York, 1981. [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), 348-359. doi: 10.1086/341518. [12] J. Carr and A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5. [13] O. Diekmann, Thresholds and traveling waves for the geographical spread of an infection, J. Math. Biol., 6 (1978), 109-130. doi: 10.1007/BF02450783. [14] O. Diekmann and H. Kaper, On the bounded solutions of a nonliear convolution equation, Nonlinear Analysis, 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9. [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), 271-292. [16] A. Ducrot, Travelling wave solutions for a scalar age-structured equation, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 251-273. doi: 10.3934/dcdsb.2007.7.251. [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), 459-482. doi: 10.1017/S0308210507000455. [18] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32. doi: 10.1007/BF00276112. [19] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^4$, Trans. Amer. Math. Soc., 286 (1984), 557-594. [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), 839-845. doi: 10.2307/3212977. [21] J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3043-3058. doi: 10.3934/dcds.2012.32.3303. [22] R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. [23] W. E. Fitzgibbon, M. E. Parrott and G. F. Webb, Diffusion epidemic models with incubation and crisscross dynamics, Math. Biosci., 128 (1995), 131-155. doi: 10.1016/0025-5564(94)00070-G. [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), 52-68. doi: 10.1016/j.nonrwa.2010.05.035. [25] B. T. Grenfell, O. N. Bjornstad and J. Kappey, Travelling waves and spatial hierarchies in measles epidemics, Nature, 414 (2001), 716-723. doi: 10.1038/414716a. [26] Y. Hosono and B. Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model, Nonlinear World, 1 (1994), 277-290. [27] Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966. doi: 10.1142/S0218202595000504. [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), 243-256. doi: 10.1007/s10255-006-0300-0. [29] W. Huang, Traveling waves for a biological reaction-diffusion model, J. Dynam. Differential Equations, 16 (2004), 745-765. [30] A. Källén, Thresholds and travelling waves in an epidemic model for rabies, Nonlinear Anal., 8 (1984), 851-856. doi: 10.1016/0362-546X(84)90107-X. [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), 377-393. doi: 10.1016/S0022-5193(85)80276-9. [32] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. B, 115 (1927), 700-721. [33] D. G. Kendall, Discussion on Professor Bartlett's paper, J. Roy. Stat. Soc. A, 120 (1957), 64-67. [34] D. G. Kendall, Mathematical models of the spread of infection, in "Mathematics and Computer Science in Biology and Medicine," Medical Research Council, London, (1965), 213-225. [35] C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429. [36] M. N. Kuperman and H. S. Wio, Front propagation in epidemiological models with spatial dependence, Physica A, 272 (1999), 206-222. doi: 10.1016/S0378-4371(99)00284-8. [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), 1253-1273. doi: 10.1088/0951-7715/19/6/003. [38] X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77. [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), 1-40; Erratum: 61 (2008), 137-138, MR2361306. [40] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. [41] N. A. Maidana and H. M. Yang, Describing the geographic spread of dengue disease by traveling waves, Math. Biosci., 215 (2008), 64-77. doi: 10.1016/j.mbs.2008.05.008. [42] D. Mollison, Possible velocities for a simple epidemic, Adv. Appl. Prob., 4 (1972), 233-257. doi: 10.2307/1425997. [43] D. Mollison, Spatial contact models for ecological and epidemic spread, J. Roy. Stat. Soc. B, 39 (1977), 283-326. [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), 699-707. doi: 10.1080/00207720701596417. [45] J. D. Murray and W. L. Seward, On the spatial spread of rabies among foxes with immunity, J. Theor. Biol., 156 (1992), 327-348. doi: 10.1016/S0022-5193(05)80679-4. [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), 111-150. doi: 10.1098/rspb.1986.0078. [47] S. Pan, Traveling wave fronts in an epidemic model with nonlocal diffusion and time delay, Int. Journal of Math. Analysis, 2 (2008), 1083-1088. [48] L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics," Math. Surveys Monogr., 102, Amer. Math. Soc., Providence, RI, 2003. [49] E. Renshaw, Waveforms and velocities for models of spatial infection, J. Appl. Probab., 18 (1981), 715-720. doi: 10.2307/3213325. [50] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in "Mathematics for Life Science and Medicine" (eds. Y. Iwasa, K. Sato and Y. Takeuchi), Biol. Med. Phys. Biomed. Eng., Springer, Berlin, (2007), 97-122. [51] 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. [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), 991-1011. doi: 10.1017/S0308210500003590. [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), 165-183. doi: 10.1093/imammb/dqq016. [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), 28-47. doi: 10.1051/mmnp:2008069. [55] H. L. Smith and X.-Q. Zhao, Traveling waves in a bio-reactor model, Nonlinear Anal. Real World Appl., 5 (2004), 895-909. doi: 10.1016/j.nonrwa.2004.05.001. [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), 509-528. doi: 10.1016/j.bulm.2004.08.005. [57] H. 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. [58] H. Thieme and X.-Q. 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##### References:
 [1] G. Abramson and V. M. Kenkre, Spatiotemporal patterns in hantavirus infection, Phys. Rev. E, 66 (2002), 011912, 5 pp. [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), 519-534. doi: 10.1016/S0092-8240(03)00013-2. [3] M. S. Abual-Rub, Vaccination in a model of an epidemic, Int. J. Math. Math. Sci., 23 (2000), 425-429. doi: 10.1155/S0161171200002696. [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), 663-675. doi: 10.1017/S0308210500004054. [5] C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves, Math. Proc. Camb. Phil. Soc., 80 (1976), 315-330. doi: 10.1017/S0305004100052944. [6] M. S. Bartlett, Measles periodicity and community size (with discussion), J. Roy. Stat. Soc. A, 120 (1957), 48-70. doi: 10.2307/2342553. [7] F. van den Bosch, J. A. J. Metz and O. Diekmann, The velocity of spatial population expansion, J. Math. Biol., 28 (1990), 529-565. doi: 10.1007/BF00164162. [8] F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001. [9] K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Camb. Phil. Soc., 81 (1977), 431-433. doi: 10.1017/S0305004100053494. [10] V. Capasso and L. Maddalena, A nonlinear diffusion system modelling the spread of oro-faecal diseases, in "Nonlinear Phenomena in Mathematical Sciences" (ed. V. Lakshmikantham), Academic Press, New York, 1981. [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), 348-359. doi: 10.1086/341518. [12] J. Carr and A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5. [13] O. Diekmann, Thresholds and traveling waves for the geographical spread of an infection, J. Math. Biol., 6 (1978), 109-130. doi: 10.1007/BF02450783. [14] O. Diekmann and H. Kaper, On the bounded solutions of a nonliear convolution equation, Nonlinear Analysis, 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9. [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), 271-292. [16] A. Ducrot, Travelling wave solutions for a scalar age-structured equation, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 251-273. doi: 10.3934/dcdsb.2007.7.251. [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), 459-482. doi: 10.1017/S0308210507000455. [18] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32. doi: 10.1007/BF00276112. [19] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^4$, Trans. Amer. Math. Soc., 286 (1984), 557-594. [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), 839-845. doi: 10.2307/3212977. [21] J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3043-3058. doi: 10.3934/dcds.2012.32.3303. [22] R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. [23] W. E. Fitzgibbon, M. E. Parrott and G. F. Webb, Diffusion epidemic models with incubation and crisscross dynamics, Math. Biosci., 128 (1995), 131-155. doi: 10.1016/0025-5564(94)00070-G. [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), 52-68. doi: 10.1016/j.nonrwa.2010.05.035. [25] B. T. Grenfell, O. N. Bjornstad and J. Kappey, Travelling waves and spatial hierarchies in measles epidemics, Nature, 414 (2001), 716-723. doi: 10.1038/414716a. [26] Y. Hosono and B. Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model, Nonlinear World, 1 (1994), 277-290. [27] Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966. doi: 10.1142/S0218202595000504. [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), 243-256. doi: 10.1007/s10255-006-0300-0. [29] W. Huang, Traveling waves for a biological reaction-diffusion model, J. Dynam. Differential Equations, 16 (2004), 745-765. [30] A. Källén, Thresholds and travelling waves in an epidemic model for rabies, Nonlinear Anal., 8 (1984), 851-856. doi: 10.1016/0362-546X(84)90107-X. [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), 377-393. doi: 10.1016/S0022-5193(85)80276-9. [32] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. B, 115 (1927), 700-721. [33] D. G. Kendall, Discussion on Professor Bartlett's paper, J. Roy. Stat. Soc. A, 120 (1957), 64-67. [34] D. G. Kendall, Mathematical models of the spread of infection, in "Mathematics and Computer Science in Biology and Medicine," Medical Research Council, London, (1965), 213-225. [35] C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429. [36] M. N. Kuperman and H. S. Wio, Front propagation in epidemiological models with spatial dependence, Physica A, 272 (1999), 206-222. doi: 10.1016/S0378-4371(99)00284-8. [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), 1253-1273. doi: 10.1088/0951-7715/19/6/003. [38] X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77. [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), 1-40; Erratum: 61 (2008), 137-138, MR2361306. [40] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. [41] N. A. Maidana and H. M. Yang, Describing the geographic spread of dengue disease by traveling waves, Math. Biosci., 215 (2008), 64-77. doi: 10.1016/j.mbs.2008.05.008. [42] D. Mollison, Possible velocities for a simple epidemic, Adv. Appl. Prob., 4 (1972), 233-257. doi: 10.2307/1425997. [43] D. Mollison, Spatial contact models for ecological and epidemic spread, J. Roy. Stat. Soc. B, 39 (1977), 283-326. [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), 699-707. doi: 10.1080/00207720701596417. [45] J. D. Murray and W. L. Seward, On the spatial spread of rabies among foxes with immunity, J. Theor. Biol., 156 (1992), 327-348. doi: 10.1016/S0022-5193(05)80679-4. [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), 111-150. doi: 10.1098/rspb.1986.0078. [47] S. Pan, Traveling wave fronts in an epidemic model with nonlocal diffusion and time delay, Int. Journal of Math. Analysis, 2 (2008), 1083-1088. [48] L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics," Math. Surveys Monogr., 102, Amer. Math. Soc., Providence, RI, 2003. [49] E. Renshaw, Waveforms and velocities for models of spatial infection, J. Appl. Probab., 18 (1981), 715-720. doi: 10.2307/3213325. [50] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in "Mathematics for Life Science and Medicine" (eds. Y. Iwasa, K. Sato and Y. Takeuchi), Biol. Med. Phys. Biomed. Eng., Springer, Berlin, (2007), 97-122. [51] 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. [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), 991-1011. doi: 10.1017/S0308210500003590. [53] I. Sazonov and M. 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