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Traveling waves in spatial SIRS models. (English) Zbl 1293.35069

Summary: We study traveling wavefront solutions for two reaction-diffusion systems, which are derived respectively as diffusion approximations to two nonlocal spatial SIRS models. These solutions characterize the propagating progress and speed of the spatial spread of underlying epidemic waves. For the first diffusion system, we find a lower bound for wave speeds and prove that the traveling waves exist for all speeds bigger than this bound. For the second diffusion system, we find the minimal wave speed and show that the traveling waves exist for all speeds bigger than or equal to the minimal speed. We further prove the uniqueness (up to translation) of these solutions for sufficiently large wave speeds. The existence of these solutions are proved by a shooting argument combining with LaSalle’s invariance principle, and their uniqueness by a geometric singular perturbation argument.

MSC:

35C07 Traveling wave solutions
35K57 Reaction-diffusion equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
92D30 Epidemiology
35B25 Singular perturbations in context of PDEs
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[1] Ai, S.: Traveling wave fronts for generalied Fisher equations with spatio-temporal delays. J. Differ. Equ. 232, 104-133 (2007) · Zbl 1113.34024 · doi:10.1016/j.jde.2006.08.015
[2] Anderson, R.M., May, R.M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, New York (1992)
[3] Aronson, D.G.: The asymptotic speed of propagation of a simple epidemic. Research Notes in Math, vol. 14, pp. 1-23. Pitman, London, (1977) · Zbl 0361.35011
[4] Bailey, N.T.J.: The Mathematical Theory of Infectious Diseases, 2nd edn. Macmillan, New York (1975) · Zbl 0334.92024
[5] Barbour, A.D.: The uniqueness of Atkinson and Reuters epidemic waves. Math. Proc. Camb. Philos. Soc. 82, 127-130 (1977) · Zbl 0358.92012 · doi:10.1017/S0305004100053755
[6] Brauer, F., Castillo-Chvez, C.: Mathematical Models in Population Biology and Epidemiology. Springer, New York (2001) · Zbl 0967.92015 · doi:10.1007/978-1-4757-3516-1
[7] Brown, K.J., Carr, J.: Deterministic epidemic waves of critical velocity. Math. Proc. Camb. Philos. Soc. 81, 431-433 (1977) · Zbl 0351.92022 · doi:10.1017/S0305004100053494
[8] Dunbar, S.R.: Traveling wave solutions of diffusive Lotka-Volterra equations. J. Math. Biol. 17, 11-32 (1983) · Zbl 0509.92024 · doi:10.1007/BF00276112
[9] Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53-98 (1979) · Zbl 0476.34034 · doi:10.1016/0022-0396(79)90152-9
[10] Fisher, R.A.: The wave of advance of advantageous gene. Ann. Eugen. 7, 353-369 (1937)
[11] Fife, P.: Mathematical Aspects of Reacting and Diffusing Systems. Springer, Berlin (1979) · Zbl 0403.92004 · doi:10.1007/978-3-642-93111-6
[12] Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42, 599-633 (2000) · Zbl 0993.92033 · doi:10.1137/S0036144500371907
[13] Hoppensteadt, F.: Mathematical Theories of Populations: Demographics, Genetics and Epidemics. SIAM, Philadelphia (1975) · Zbl 0304.92012 · doi:10.1137/1.9781611970487
[14] Huang, J., Lu, G., Ruan, S.: Existence of traveling wave solutions in a diffusive predator-prey model. J. Math. Biol. 46, 132-152 (2003) · Zbl 1018.92026 · doi:10.1007/s00285-002-0171-9
[15] Huang, W.: Traveling wave solutions for a class of predator-prey systems. J. Dyn. Differ. Equ. 24, 633-644 (2012) · Zbl 1365.35056 · doi:10.1007/s10884-012-9255-4
[16] Hutson, V., Martinez, S., Mischaikow, K., Vickers, G.T.: The evolution of dispersal. J. Math. Biol. 46, 483-517 (2003) · Zbl 1052.92042 · doi:10.1007/s00285-003-0210-1
[17] Jones, C.: Geometric singular perturbation theory. Lecture Notes in Mathematics, vol. 1069, pp. 44-118. Springer, Berlin (1995) · Zbl 0840.58040
[18] Kendall, D.G.: Mathematical Models of the Spread of Infection, Mathematics and Computer Science in Biology and Medicine. H.M. Stationary Off, London (1965)
[19] Kolmogorov, A.N., Petrovskii, I.G., Piskunov, N.S.: A study of the diffusion equation with increase in the quantity of matter, and its application to a biological problem. Bull. Mosc. Univ. Math. Ser. A 1, 1-25 (1937)
[20] Li, T., Li, Y., Hethcote, H.W.: Periodic traveling waves in SIRS endemic models. Math. Comput. Model. 49, 393-401 (2009) · Zbl 1165.35311 · doi:10.1016/j.mcm.2008.07.033
[21] Lin, X., Wu, C., Weng, P.: Traveling wave solutions for a predator-prey system with Sigmoidal response function. J. Dyn. Differ. Equ. 23, 903-921 (2011) · Zbl 1252.34048 · doi:10.1007/s10884-011-9220-7
[22] Lutscher, F.: Non-local dispersal and averaging in heterogeneous landscapes. Appl. Anal. 89, 1091-1108 (2010) · Zbl 1204.35039 · doi:10.1080/00036811003735816
[23] Ma, Z., Li, J.: Dynamical Modeling and Analysis of Epidemics. World Scientific, Singapore (2009) · Zbl 1170.92026 · doi:10.1142/9789812797506
[24] Medlock, J., Kot, M.: Spreading disease: integro-differential equations old and new. Math. Biosci. 184, 201-222 (2003) · Zbl 1036.92030 · doi:10.1016/S0025-5564(03)00041-5
[25] Mollison, D.: Possible velocities for a simple epidemic. Adv. Appl. Probab. 4, 233-257 (1972) · Zbl 0251.92012 · doi:10.2307/1425997
[26] Mora, A.: Spatial spread of infectious diseases. Master thesis, University of New Mexico (2010). http://hdl.handle.net/1928/10839. Accessed 20 Jan 2014 · Zbl 1113.34024
[27] Murray, J.D.: Mathematical Biology. Springer, Berlin (1989) · Zbl 0682.92001 · doi:10.1007/978-3-662-08539-4
[28] Rass, L., Radcliffe, J.: Spatial Deterministic Epidemics, Mathematical Surveys and Monographs. AMS, Providence (2001)
[29] Ruan, S.; Iwasa, Y. (ed.); Sato, K. (ed.); Takeuchi, Y. (ed.), Spatial-temporal dynamics in nonlocal epidemiological models, 99-122 (2007), New York
[30] Thieme, H.R.: Mathematics in Population Biology. Princeton University Press, Princeton (2003) · Zbl 1054.92042
[31] Volpert, A.I., Volpert, V.A., Volpert, V.A.: Traveling Wave Solutions of Parabolic Systems. American Mathematical Society, Providence (1994) · Zbl 0805.35143
[32] Volpert, V., Petrovskii, S.: Reaction-diffusion waves in biology. Phys. Life Rev. 6, 267-310 (2009) · doi:10.1016/j.plrev.2009.10.002
[33] Wu, J.: Spatial structure: partial differential equations models. Lecture Notes in Mathematics, vol. 1945, pp. 191-203. Springer, Berlin (2008) · Zbl 1206.92029
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