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Protection zone in a modified Lotka-Volterra model
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 |
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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