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Traveling wave solutions in a reaction-diffusion epidemic model. (English) Zbl 1291.35418
Summary: We investigate the traveling wave solutions in a reaction-diffusion epidemic model. The existence of the wave solutions is derived through monotone iteration of a pair of classical upper and lower solutions. The traveling wave solutions are shown to be unique and strictly monotonic. Furthermore, we determine the critical minimal wave speed.

MSC:
35Q92PDEs in connection with biology and other natural sciences
92D30Epidemiology
35C07Traveling wave solutions of PDE
35K57Reaction-diffusion equations
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References:
[1] D. H. Sattinger, “Weighted norms for the stability of traveling waves,” Journal of Differential Equations, vol. 25, no. 1, pp. 130-144, 1977. · Zbl 0315.35010 · doi:10.1016/0022-0396(77)90185-1
[2] Y. Hosono and B. Ilyas, “Traveling waves for a simple diffusive epidemic model,” Mathematical Models & Methods in Applied Sciences, vol. 5, no. 7, pp. 935-966, 1995. · Zbl 0836.92023 · doi:10.1142/S0218202595000504
[3] J. Wu and X. Zou, “Traveling wave fronts of reaction-diffusion systems with delay,” Journal of Dynamics and Differential Equations, vol. 13, no. 3, pp. 651-687, 2001. · Zbl 0996.34053 · doi:10.1023/A:1016690424892
[4] P. Weng and X. Q. Zhao, “Spreading speed and traveling waves for a multi-type SIS epidemic model,” Journal of Differential Equations, vol. 229, no. 1, pp. 270-296, 2006. · Zbl 1126.35080 · doi:10.1016/j.jde.2006.01.020
[5] K. F. Zhang and X. Q. Zhao, “Spreading speed and travelling waves for a spatially discrete SIS epidemic model,” Nonlinearity, vol. 21, no. 1, pp. 97-112, 2008. · Zbl 1139.92023 · doi:10.1088/0951-7715/21/1/005
[6] 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, vol. 19, no. 6, pp. 1253-1273, 2006. · Zbl 1103.35049 · doi:10.1088/0951-7715/19/6/003
[7] S. Ma, “Traveling waves for non-local delayed diffusion equations via auxiliary equations,” Journal of Differential Equations, vol. 237, no. 2, pp. 259-277, 2007. · Zbl 1114.34061 · doi:10.1016/j.jde.2007.03.014 · eudml:194162
[8] R. Liu, V. R. S. K. Duvvuri, and J. Wu, “Spread pattern formation of H5N1-avian influenza and its implications for control strategies,” Mathematical Modelling of Natural Phenomena, vol. 3, no. 7, pp. 161-179, 2008. · doi:10.1051/mmnp:2008048
[9] K. Li and X. Li, “Travelling wave solutions in diffusive and competition-cooperation systems with delays,” IMA Journal of Applied Mathematics, vol. 74, no. 4, pp. 604-621, 2009. · Zbl 1185.35313 · doi:10.1093/imamat/hxp008
[10] Z. C. Wang, W. T. Li, and S. Ruan, “Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay,” Journal of Differential Equations, vol. 238, no. 1, pp. 153-200, 2007. · Zbl 1124.35089 · doi:10.1016/j.jde.2007.03.025
[11] Z. C. Wang and J. Wu, “Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission,” Proceedings of the Royal Society A, vol. 466, no. 2113, pp. 237-261, 2010. · Zbl 1195.35291 · doi:10.1098/rspa.2009.0377
[12] Z. C. Wang, J. Wu, and R. Liu, “Traveling waves of the spread of avian influenza,” Proceedings of the American Mathematical Society, vol. 140, no. 11, pp. 3931-3946, 2012. · Zbl 1275.35068 · doi:10.1090/S0002-9939-2012-11246-8
[13] A. Boumenir and V. M. Nguyen, “Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations,” Journal of Differential Equations, vol. 244, no. 7, pp. 1551-1570, 2008. · Zbl 1154.34031 · doi:10.1016/j.jde.2008.01.004
[14] X. Hou and A. W. Leung, “Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics,” Nonlinear Analysis: Real World Applications, vol. 9, no. 5, pp. 2196-2213, 2008. · Zbl 1156.35405 · doi:10.1016/j.nonrwa.2007.07.007
[15] X. Hou and W. Feng, “Traveling waves and their stability in a coupled reaction diffusion system,” Communications on Pure and Applied Analysis, vol. 10, no. 1, pp. 141-160, 2011. · Zbl 1229.35013 · doi:10.3934/cpaa.2011.10.141
[16] Q. Gan, R. Xu, and P. Yang, “Travelling waves of a delayed SIRS epidemic model with spatial diffusion,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 52-68, 2011. · Zbl 1202.35046 · doi:10.1016/j.nonrwa.2010.05.035
[17] Q. Ye, Z. Li, M. Wang, and Y. Wu, An Introduction to Reaction-Diffusion Equation, Science Press, Beijing, China, 2nd edition, 2011.
[18] H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599-653, 2000. · Zbl 0993.92033 · doi:10.1137/S0036144500371907
[19] Z. Ma, Y. Zhou, and J. Wu, Modeling and Dynamics of Infectious Diseases, vol. 11 of Series in Contemporary Applied Mathematics CAM, Higher Education Press, Beijing, China, 2009. · Zbl 1180.92081
[20] W. Wang, Y. Cai, M. Wu, K. Wang, and Z. Li, “Complex dynamics of a reaction-diffusion epidemic model,” Nonlinear Analysis: Real World Applications, vol. 13, no. 5, pp. 2240-2258, 2012. · Zbl 1327.92069 · doi:10.1016/j.nonrwa.2012.01.018
[21] Y. Cai, W. Liu, Y. Wang, and W. Wang, “Complex dynamics of a diffusive epidemic model with strong Allee effect,” Nonlinear Analysis: Real World Applications, vol. 14, no. 4, pp. 1907-1920, 2013. · Zbl 1274.92009 · doi:10.1016/j.nonrwa.2013.01.002
[22] F. Berezovsky, G. Karev, B. Song, and C. Castillo-Chavez, “A simple epidemic model with surprising dynamics,” Mathematical Biosciences and Engineering, vol. 2, no. 1, pp. 133-152, 2005. · Zbl 1061.92052
[23] W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics,” Proceedings of the Royal Society A, vol. 115, no. 772, pp. 700-721, 1927. · Zbl 53.0517.01 · doi:10.1098/rspa.1927.0118
[24] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Tata McGraw-Hill, New York, NY, USA, 1972. · Zbl 0064.33002