September  2015, 20(7): 2039-2050. doi: 10.3934/dcdsb.2015.20.2039

The reaction-diffusion system for an SIR epidemic model with a free boundary

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150080, China

2. 

Natural Science Research Center, Harbin Institute of Technology, Harbin 150080

Received  September 2014 Revised  March 2015 Published  July 2015

The reaction-diffusion system for an $SIR$ epidemic model with a free boundary is studied. This model describes a transmission of diseases. The existence, uniqueness and estimates of the global solution are discussed first. Then some sufficient conditions for the disease vanishing are given. With the help of investigating the long time behavior of solution to the initial and boundary value problem in half space, the long time behavior of the susceptible population $S$ is obtained for the disease vanishing case.
Citation: Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039
References:
[1]

N. F. Britton, Essential Mathematical Biology, Springer Undergraduate Mathematics Series, Springer-Verlag London Ltd., London, 2003. doi: 10.1007/978-1-4471-0049-2.

[2]

V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomath., Springer Verlag, Berlin, 1993. doi: 10.1007/978-3-540-70514-7.

[3]

J. Crank, Free and Moving Boundary Problems, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1984.

[4]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089.

[5]

Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competition, Discrete Cont. Dyn. Syst. Ser. B, 19 (2014), 3105-3132. doi: 10.3934/dcdsb.2014.19.3105.

[6]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Diff. Equa., 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0.

[7]

Y. Kaneko, Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction-diffusion equations, Nonlinear Anal.: Real World Appl., 18 (2014), 121-140. doi: 10.1016/j.nonrwa.2014.01.008.

[8]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction diffusion equation appearing in ecology, Advan. Math. Sci. Appl., 21 (2011), 467-492.

[9]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London Series, A, 115 (1972), 700-721. Available from: http://rspa.royalsocietypublishing.org/content/115/772/700.

[10]

K. I. Kim, Z. G. Lin and Q. Y. Zhang, An SIR epidemic model with free boundary, Nonlinear Anal.: Real World Appl., 14 (2013), 1992-2001. doi: 10.1016/j.nonrwa.2013.02.003.

[11]

Z. G. Lin, Y. N. Zhao and P. Zhou, The infected frontier in an SEIR epidemic model with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2355-2376. doi: 10.3934/dcdsb.2013.18.2355.

[12]

J. D. Murray, Mathematical Biology. I. An Introduction, $3^{rd}$ edition, Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 2002.

[13]

L. I. Rubenstein, The Stefan Problem, Translations of Mathematical Monographs, Vol. 27, American Mathematical Society, Providence, R.I., 1971.

[14]

M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013.

[15]

M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266. doi: 10.1016/j.jde.2014.10.022.

[16]

M. X. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simulat., 23 (2015), 311-327. doi: 10.1016/j.cnsns.2014.11.016.

[17]

M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal.: Real World Appl., 24 (2015), 73-82. doi: 10.1016/j.nonrwa.2015.01.004.

[18]

M. X. Wang and J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Diff. Equat., 26 (2014), 655-672. doi: 10.1007/s10884-014-9363-4.

[19]

M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, preprint, arXiv:1312.7751.

[20]

J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal.: Real World Appl., 16 (2014), 250-263. doi: 10.1016/j.nonrwa.2013.10.003.

show all references

References:
[1]

N. F. Britton, Essential Mathematical Biology, Springer Undergraduate Mathematics Series, Springer-Verlag London Ltd., London, 2003. doi: 10.1007/978-1-4471-0049-2.

[2]

V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomath., Springer Verlag, Berlin, 1993. doi: 10.1007/978-3-540-70514-7.

[3]

J. Crank, Free and Moving Boundary Problems, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1984.

[4]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089.

[5]

Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competition, Discrete Cont. Dyn. Syst. Ser. B, 19 (2014), 3105-3132. doi: 10.3934/dcdsb.2014.19.3105.

[6]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Diff. Equa., 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0.

[7]

Y. Kaneko, Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction-diffusion equations, Nonlinear Anal.: Real World Appl., 18 (2014), 121-140. doi: 10.1016/j.nonrwa.2014.01.008.

[8]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction diffusion equation appearing in ecology, Advan. Math. Sci. Appl., 21 (2011), 467-492.

[9]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London Series, A, 115 (1972), 700-721. Available from: http://rspa.royalsocietypublishing.org/content/115/772/700.

[10]

K. I. Kim, Z. G. Lin and Q. Y. Zhang, An SIR epidemic model with free boundary, Nonlinear Anal.: Real World Appl., 14 (2013), 1992-2001. doi: 10.1016/j.nonrwa.2013.02.003.

[11]

Z. G. Lin, Y. N. Zhao and P. Zhou, The infected frontier in an SEIR epidemic model with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2355-2376. doi: 10.3934/dcdsb.2013.18.2355.

[12]

J. D. Murray, Mathematical Biology. I. An Introduction, $3^{rd}$ edition, Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 2002.

[13]

L. I. Rubenstein, The Stefan Problem, Translations of Mathematical Monographs, Vol. 27, American Mathematical Society, Providence, R.I., 1971.

[14]

M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013.

[15]

M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266. doi: 10.1016/j.jde.2014.10.022.

[16]

M. X. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simulat., 23 (2015), 311-327. doi: 10.1016/j.cnsns.2014.11.016.

[17]

M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal.: Real World Appl., 24 (2015), 73-82. doi: 10.1016/j.nonrwa.2015.01.004.

[18]

M. X. Wang and J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Diff. Equat., 26 (2014), 655-672. doi: 10.1007/s10884-014-9363-4.

[19]

M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, preprint, arXiv:1312.7751.

[20]

J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal.: Real World Appl., 16 (2014), 250-263. doi: 10.1016/j.nonrwa.2013.10.003.

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