Article Contents
Article Contents

Traveling wave solutions for a diffusive sis epidemic model

• In this paper, we study the traveling wave solutions of an SIS reaction-diffusion epidemic model. The techniques of qualitative analysis has been developed which enable us to show the existence of traveling wave solutions connecting the disease-free equilibrium point and an endemic equilibrium point. In addition, we also find the precise value of the minimum speed that is significant to study the spreading speed of the population towards to the endemic steady state.
Mathematics Subject Classification: 35C07, 92B05.

 Citation:

•  [1] L. J. S. Allen, B. M. Boller, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. A, 21 (2008), 1-20.doi: 10.3934/dcds.2008.21.1. [2] I. Beardmore and R. Beardmore, The global structure of a spatial model of infections disease, Proc. Roy. Soc. Lond. A., 459 (2003), 1427-1448.doi: 10.1098/rspa.2002.1080. [3] O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130,doi: 10.1007/BF02450783. [4] O. Diekmann, Run for your life, A note on the aymptotic speed of propagation of an epidimic, J. Diff. Equations, 33 (1979), 58-73.doi: 10.1016/0022-0396(79)90080-9. [5] S. R. Dunbar, Traveling wave solutions of Diffusive Lotka-Volterra Equations: A heteroclinic connection in $\mathbbR^4$, Trans. Amer. Math. Society, 286 (1984), 557-594.doi: 10.2307/1999810. [6] P. C. Fife, "Mathematical Aspects of Reacting and Diffusing systems," Lecture Notes in Biomath, 28, Springer-Verlag, New York, 1979. [7] W. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model of the spread of Feline Leukemia Virus through a highly heterogeneous spatial domain, SIAM J. Math. Anal., 33 (2001), 570-588.doi: 10.1137/S0036141000371757. [8] W. Fitzgibbon, M. Langlais and J. J. Morgan, A reaction-diffusion system modelling direct and indirect transmission of diseases, Discrete and Continuous Dynamical Systems, Series B, 4 (2004), 893-910.doi: 10.3934/dcdsb.2004.4.893. [9] W. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on non coincident domains, Structured Population Models in Biology and Epidemiology Lecture Notes in Mathematics, 1936 (2008), 115-164. [10] R. A. Gardner, Review on traveling wave solutions of parabolic systems by A. I. Volpert, V. A. Volpert, Bull. Aner. Math. Soc., 32 (1995), 446-452.doi: 10.1090/S0273-0979-1995-00607-5. [11] W. Huang, Traveling wave solutions for a class of predator-prey systems, Journal of Dynamics and Differential Equations, 24 (2012), 633-644.doi: 10.1007/s10884-012-9255-4. [12] W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-dissusion Epidemic Model for Disease transmission, Mathematical Biosciences and Engineering, 7 (2010), 51-66.doi: 10.3934/mbe.2010.7.51. [13] J. Keener and J. Sneyd, "Mathematical Physiology," Springer-Verlag, New York, Inc., 1998. [14] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics-I, Original Research Article Bulletin of Mathematical Biology, 53 (1991), 33-55. [15] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics-II. the problem of endemicity, Original Research Article Bulletin of Mathematical Biology, 53 (1991), 57-87. [16] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics-III. Further studies of the problem of endemicity, Original Research Article Bulletin of Mathematical Biology, 53 (1991), 89-118. [17] M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.doi: 10.1007/s002850200144. [18] B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci, 196 (2005), 82-98.doi: 10.1016/j.mbs.2005.03.008. [19] X. Liang and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure and Appl. Math., 60 (2007), 1-40.doi: 10.1002/cpa.20154. [20] R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Non. Analysis, 71 (2009), 239-247.doi: 10.1016/j.na.2008.10.043. [21] 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), e21128.doi: 10.1371/journal.pone.0021128.