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July  2013, 18(5): 1291-1304. doi: 10.3934/dcdsb.2013.18.1291

Traveling wave solutions for a diffusive sis epidemic model

Wei Ding 1, , Wenzhang Huang 1, and  Siroj Kansakar 2,

1. 

Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China
2. 

Department of Mathematical Science, University of Alabama in Huntsville, Huntsville, Alabama 35899, United States

Received  August 2012 Revised  February 2013 Published  March 2013

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.
Keywords: minimum wave speed., traveling wave solution, SIS epidemic model.
Mathematics Subject Classification: 35C07, 92B0.
Citation: Wei Ding, Wenzhang Huang, Siroj Kansakar. Traveling wave solutions for a diffusive sis epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1291-1304. doi: 10.3934/dcdsb.2013.18.1291
References:
[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.  Google Scholar

[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.  Google Scholar

[3]

O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130, doi: 10.1007/BF02450783.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing systems," Lecture Notes in Biomath, 28, Springer-Verlag, New York, 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. Keener and J. Sneyd, "Mathematical Physiology," Springer-Verlag, New York, Inc., 1998.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3]

O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130, doi: 10.1007/BF02450783.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing systems," Lecture Notes in Biomath, 28, Springer-Verlag, New York, 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. Keener and J. Sneyd, "Mathematical Physiology," Springer-Verlag, New York, Inc., 1998.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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