• Previous Article
    On the dynamics of two-consumers-one-resource competing systems with Beddington-DeAngelis functional response
  • DCDS-B Home
  • This Issue
  • Next Article
    Nonlinear dynamics of a mathematical model on action potential duration and calcium transient in paced cardiac cells
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 & 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. doi: 10.1137/S0036141099351693.

[3]

J. Crank, "Free and Moving Boundary Problems,", Clarendon Press, (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. 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. 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, ().

[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.

[8]

G. Röst and J. H. Wu, SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 5 (2008), 389. 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. doi: 10.1090/S0002-9939-00-05705-1.

[10]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (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. 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. 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, (1968).

[14]

Z. G. Lin, A free boundary problem for a predator-prey model,, Nonlinearity, 20 (2007), 1883. 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. 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. 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.

[18]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations,, Hiroshima Math. J., 17 (1987), 241.

[19]

L. I. Rubinstein, "The Stefan Problem,", American Mathematical Society, (1971).

show all references

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. doi: 10.1137/S0036141099351693.

[3]

J. Crank, "Free and Moving Boundary Problems,", Clarendon Press, (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. 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. 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, ().

[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.

[8]

G. Röst and J. H. Wu, SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 5 (2008), 389. 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. doi: 10.1090/S0002-9939-00-05705-1.

[10]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (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. 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. 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, (1968).

[14]

Z. G. Lin, A free boundary problem for a predator-prey model,, Nonlinearity, 20 (2007), 1883. 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. 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. 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.

[18]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations,, Hiroshima Math. J., 17 (1987), 241.

[19]

L. I. Rubinstein, "The Stefan Problem,", American Mathematical Society, (1971).

[1]

Gergely Röst, Jianhong Wu. SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2008, 5 (2) : 389-402. doi: 10.3934/mbe.2008.5.389

[2]

C. Connell McCluskey. Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2009, 6 (3) : 603-610. doi: 10.3934/mbe.2009.6.603

[3]

Zhenguo Bai. Threshold dynamics of a periodic SIR model with delay in an infected compartment. Mathematical Biosciences & Engineering, 2015, 12 (3) : 555-564. doi: 10.3934/mbe.2015.12.555

[4]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195

[5]

Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013

[6]

Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837

[7]

Yongzhi Xu. A free boundary problem model of ductal carcinoma in situ. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 337-348. doi: 10.3934/dcdsb.2004.4.337

[8]

Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks & Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583

[9]

Rui Peng, Xiao-Qiang Zhao. The diffusive logistic model with a free boundary and seasonal succession. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2007-2031. doi: 10.3934/dcds.2013.33.2007

[10]

Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039

[11]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic type chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 667-684. doi: 10.3934/krm.2015.8.667

[12]

Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045

[13]

Yihong Du, Zhigui Lin. The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3105-3132. doi: 10.3934/dcdsb.2014.19.3105

[14]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122

[15]

Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128

[16]

Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105

[17]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[18]

Yunfeng Liu, Zhiming Guo, Mohammad El Smaily, Lin Wang. A Leslie-Gower predator-prey model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2063-2084. doi: 10.3934/dcdss.2019133

[19]

Qun Liu, Daqing Jiang, Ningzhong Shi, Tasawar Hayat, Ahmed Alsaedi. Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2479-2500. doi: 10.3934/dcdsb.2017127

[20]

Zhisheng Shuai, P. van den Driessche. Impact of heterogeneity on the dynamics of an SEIR epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (2) : 393-411. doi: 10.3934/mbe.2012.9.393

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]