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November  2013, 18(9): 2355-2376. doi: 10.3934/dcdsb.2013.18.2355

## The infected frontier in an SEIR epidemic model with infinite delay

 1 School of Mathematical Science, Yangzhou University, Yangzhou 225002, China 2 Department of Mathematics, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240

Received  February 2013 Revised  June 2013 Published  September 2013

An SEIR epidemic model with infinite delay and the Neumann boundary condition is investigated, as well as the corresponding free boundary problem, in which the free boundary exactly describes the spreading frontier of the disease. For the problem in a fixed domain with null Neumann boundary condition, the transmission dynamics of the disease is determined by the basic reproduction number $R_0$. More specifically, whether the disease will die out or not depends on $R_0<1$ or $R_0>1$; while for the free boundary problem, we show that under certain conditions the disease will die out even $R_0>1$. Our results indicate that besides the basic reproduction number, the initial size of the infected domain and the diffusivity of the disease in a new region also produce a non-negligible influence to the disease transmission, and it seems more reasonable and acceptable.
Citation: Zhigui Lin, Yinan Zhao, Peng Zhou. The infected frontier in an SEIR epidemic model with infinite delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2355-2376. doi: 10.3934/dcdsb.2013.18.2355
##### References:
 [1] L. Caffarelli and S. Salsa, "A Geometric Approach to Free Boundary Problems," American Mathematical Society, 68 2005. [2] X. F. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693. [3] J. Crank, "Free and Moving Boundary Problems," Clarendon Press, Oxford 1984. [4] Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, J. Differential Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011. [5] 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. [6] Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, preprint, arXiv:1303.0454. [7] M. Fila and P. Souplet, Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem, Interfaces Free Bound., 3 (2001), 337-344. [8] G. Röst and J. H. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5 (2008), 389-402. doi: 10.3934/mbe.2008.5.389. [9] H. Ghidouche, P. Souplet and D. Tarzia, Decay of global solutions, stability and blowup for a reaction-diffusion problem with free boundary, Proc. Amer. Math. Soc., 129 (2001), 781-792. doi: 10.1090/S0002-9939-00-05705-1. [10] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, third ed., 840, Springer-Verlag, Berlin, New York 1981. [11] D. Hilhorst, M. Mimura and R. Schätzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285. doi: 10.1016/S1468-1218(02)00009-3. [12] K. I. Kim, Z. G. Lin and Z. Ling, Global existence and blowup of solutions to a free boundary problem for mutualistic model, Sci. China Math., 53 (2010), 2085-2095. doi: 10.1007/s11425-010-4007-6. [13] O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc, Providence, RI, 1968. [14] Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004. [15] Z. G. Lin and M. Pedersen, Stability in a diffusive food-chain model with Michaelis-Menten functional response, Nonlinear Anal., 57 (2004), 421-433. doi: 10.1016/j.na.2004.02.022. [16] M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186. doi: 10.1007/BF03167042. [17] M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498. [18] M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280. [19] L. I. Rubinstein, "The Stefan Problem," American Mathematical Society, Providence, RI 1971.

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##### References:
 [1] L. Caffarelli and S. Salsa, "A Geometric Approach to Free Boundary Problems," American Mathematical Society, 68 2005. [2] X. F. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693. [3] J. Crank, "Free and Moving Boundary Problems," Clarendon Press, Oxford 1984. [4] Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, J. Differential Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011. [5] 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. [6] Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, preprint, arXiv:1303.0454. [7] M. Fila and P. Souplet, Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem, Interfaces Free Bound., 3 (2001), 337-344. [8] G. Röst and J. H. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5 (2008), 389-402. doi: 10.3934/mbe.2008.5.389. [9] H. Ghidouche, P. Souplet and D. Tarzia, Decay of global solutions, stability and blowup for a reaction-diffusion problem with free boundary, Proc. Amer. Math. Soc., 129 (2001), 781-792. doi: 10.1090/S0002-9939-00-05705-1. [10] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, third ed., 840, Springer-Verlag, Berlin, New York 1981. [11] D. Hilhorst, M. Mimura and R. Schätzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285. doi: 10.1016/S1468-1218(02)00009-3. [12] K. I. Kim, Z. G. Lin and Z. Ling, Global existence and blowup of solutions to a free boundary problem for mutualistic model, Sci. China Math., 53 (2010), 2085-2095. doi: 10.1007/s11425-010-4007-6. [13] O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc, Providence, RI, 1968. [14] Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004. [15] Z. G. Lin and M. Pedersen, Stability in a diffusive food-chain model with Michaelis-Menten functional response, Nonlinear Anal., 57 (2004), 421-433. doi: 10.1016/j.na.2004.02.022. [16] M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186. doi: 10.1007/BF03167042. [17] M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498. [18] M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280. [19] L. I. Rubinstein, "The Stefan Problem," American Mathematical Society, Providence, RI 1971.
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