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

Haomin  Huang; Mingxin Wang

doi:10.3934/dcdsb.2015.20.2039

Article Contents

2015, Volume 20, Issue 7: 2039-2050. Doi: 10.3934/dcdsb.2015.20.2039

This issue Previous Article Protection zone in a modified Lotka-Volterra model Next Article Dynamics of stochastic fractional Boussinesq equations

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

Haomin Huang^1, and
Mingxin Wang^2,

1.
Department of Mathematics, Harbin Institute of Technology, Harbin 150080
2.
Natural Science Research Center, Harbin Institute of Technology, Harbin 150080

Received: August 31, 2014

Revised: February 28, 2015

Published: August 2015

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 35B35, 35K57, 35R35, 92D30.

Citation:

Related Papers

Cited by

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.