Article Contents
Article Contents

# Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model

• In this paper, we consider a Kermack-McKendrick epidemic model with nonlocal dispersal. We find that the existence and nonexistence of traveling wave solutions are determined by the reproduction number. To prove the existence of nontrivial traveling wave solutions, we construct an invariant cone in a bounded domain with initial functions being defined on, and apply Schauder's fixed point theorem as well as limiting argument. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Moreover, the nonexistence of traveling wave solutions is obtained by Laplace transform if the speed is less than the critical velocity.
Mathematics Subject Classification: 35K57, 37C65, 92D30.

 Citation:

•  [1] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, "Nonlocal Diffusion Problems," Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010. [2] P. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.doi: 10.1007/s002050050037. [3] P. Bates and G. Zhao, Existence, uniquenss and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.doi: 10.1016/j.jmaa.2006.09.007. [4] 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. [5] X. Chen, Existence, uniqueness and asymptotic stability of travelling waves in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. [6] J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.doi: 10.1016/j.jde.2007.11.002. [7] J. Coville, J. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.doi: 10.1137/060676854. [8] J. Coville and L. Dupaigne, On a nonlocal reaction diffusion equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755.doi: 10.1017/S0308210504000721. [9] P. De Mottoni, E. Orlandi and A. Tesei, Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection, Nonlinear Anal., 3 (1979), 663-675.doi: 10.1016/0362-546X(79)90095-6. [10] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.doi: 10.1017/S0308210507000455. [11] 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. [12] P. Fife, An integrodifferential analog of semilinear parabolic PDEs, in "Partial Differential Equations and Applications" in "Lect. Notes" Pure Appl. Math., 177, Dekker, New York, (1996), 137-145. [13] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in "Trends in Nonlinear Analysis," Springer, Berlin, (2003), 153-191. [14] 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. [15] J. García-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal., 8 (2009), 2037-2053.doi: 10.3934/cpaa.2009.8.2037. [16] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math., 17 (2006), 221-232.doi: 10.1017/S0956792506006462. [17] V. Hutson, S. Martinez, K. Mischailow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.doi: 10.1007/s00285-003-0210-1. [18] V. Hutson, W. Shen and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain J. Math., 38 (2008), 1147-1175.doi: 10.1216/RMJ-2008-38-4-1147. [19] 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. [20] C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.doi: 10.3934/dcds.2010.26.551. [21] C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072.doi: 10.3934/dcdsb.2012.17.2047. [22] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London Ser. B, 115 (1927), 700-721.doi: 10.1098/rspa.1927.0118. [23] W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinerity, 19 (2006), 1253-1273.doi: 10.1088/0951-7715/19/6/003. [24] W.-T. Li, Y.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real Word Appl., 11 (2010), 2302-2313.doi: 10.1016/j.nonrwa.2009.07.005. [25] G. Lin, W.-T. Li and M.-J. Ma, Traveling 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. [26] J. D. Murray, "Mathematical Biology, II, Spatial Models and Biomedical Applications," Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. [27] S. Ma, Travelling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.doi: 10.1006/jdeq.2000.3846. [28] J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.doi: 10.1016/S0025-5564(03)00041-5. [29] S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity, J. Math. Anal. Appl., 346 (2008), 415-424.doi: 10.1016/j.jmaa.2008.05.057. [30] S. Pan, W.-T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.doi: 10.1007/s00033-007-7005-y. [31] S. Pan, W.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158.doi: 10.1016/j.na.2009.12.008. [32] K. Schumacher, Travelling-front solutions for integro-differential equations.I., J. Reine Angew. Math., 316 (1980), 54-70.doi: 10.1515/crll.1980.316.54. [33] W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 15 (2010), 747-795.doi: 10.1016/j.jde.2010.04.012. [34] Y.-J. Sun, W.-T. Li and Z.-C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581.doi: 10.1016/j.jde.2011.04.020. [35] Y.-J. Sun, W.-T. Li and Z.-C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Anal., 74 (2011), 814-826.doi: 10.1016/j.na.2010.09.032. [36] X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations, 183 (2002), 434-461.doi: 10.1006/jdeq.2001.4129. [37] Z.-C. 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. [38] 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. [39] 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, 466 (2010), 237-261.doi: 10.1098/rspa.2009.0377. [40] Z.-C. Wang and J. Wu, Traveling waves in a bio-reactor model with stage-structure, J. Math. Anal. Appl., 385 (2012), 683-692.doi: 10.1016/j.jmaa.2011.06.084. [41] D. V. Widder, "The Laplace Transform," Princeton University Press, Princeton, NJ, 1941. [42] H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953.doi: 10.2977/prims/1260476648. [43] H. Yagisita, Existence of traveling waves for a nonlocal monostable equation: an abstract approach, Publ. Res. Inst. Math. Sci., 45 (2009), 955-979.doi: 10.2977/prims/1260476649. [44] L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neural networks, J. Differential Equations, 197 (2004), 162-196.doi: 10.1016/S0022-0396(03)00170-0. [45] G.-B. Zhang, W.-T. Li and Z.-C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124.doi: 10.1016/j.jde.2012.01.014.