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SEIR epidemic model with delay

Published online by Cambridge University Press:  17 February 2009

Ping Yan  and
Shengqiang Liu
Ping Yan
Affiliation:
Department of Mathematics, University of Helsinki, FIN-00014, Finland; e-mail: [email protected].
Shengqiang Liu
Affiliation:
Department of Mathematics, Xiamen University, Xiamen 361005, P. R., China; e-mail: [email protected].
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Abstract

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A disease transmission model of SEIR type with exponential demographic structure is formulated, with a natural death rate constant and an excess death rate constant for infective individuals. The latent period is assumed to be constant, and the force of the infection is assumed to be of the standard form, namely, proportional to I(t)/N(t) where N(t) is the total (variable) population size and I(t) is the size of the infective population. The infected individuals are assumed not to be able to give birth and when an individual is removed fromthe I-class, it recovers, acquiring permanent immunity with probability f (0 ≤ f ≤ 1) and dies from the disease with probability 1 − f. The global attractiveness of the disease-free equilibrium, existence of the endemic equilibrium as well as the permanence criteria are investigated. Further, it is shown that for the special case of the model with zero latent period, R0 > 1 leads to the global stability of the endemic equilibrium, which completely answers the conjecture proposed by Diekmann and Heesterbeek.

Keywords

SEIR modeldelayconjecturepermanenceextinctionglobal stability.
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 1 , July 2006 , pp. 119 - 134
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Alexander, M. E. and Moghadas, S. M., “Periodicity in an epidemic model with a generalized non-linear incidence”, Math. Biosc. 189 (2004) 75–96.Google Scholar
[2]Beretta, E., Hara, T., Ma, W. B. and Takeuchi, Y., “Global asymptotic stability of an SIR epidemic model with distributed time delay”, Nonlinear Anal. 47 (2001) 4107–4115.Google Scholar
[3]Beretta, E. and Takeuchi, Y., “Global stability of an sir epidemic model with time delays”, J. Math. Biol. 33 (1995) 250–260.CrossRefGoogle ScholarPubMed
[4]Brauer, F., “Models for the spread of universally fatal diseases”, J. Math. Biol. 28 (1990) 451–462.CrossRefGoogle ScholarPubMed
[5]Brauer, F., “Models for the spread of universally fatal diseases, II”, in Differential Equation Models in Biology, Epidemiology and Ecology (eds. Busenberg, S. and Martelli, M.), Lecture Notes in Biomath. 92, (Springer, New York, 1991) 57–69.CrossRefGoogle Scholar
[6]Busenberg, S. and Cooke, K. L., Vertically transmitted diseases, Biomathematics 23 (Springer-Verlag, Berlin, 1993).Google Scholar
[7]Busenberg, S., Cooke, K. L. and Pozio, A., “Analysis of a model of a vertically transmitted disease”, Math. Biol. 17 (1983) 305–329.Google Scholar
[8]Cooke, K. and Driessche, P., “Analysis of an SEIRS epidemic model with two delays”, J. Math.Biol. 35 (1996) 240–260.CrossRefGoogle ScholarPubMed
[9]Diekmann, O. and Heesterbeek, J. A. P., Mathematical Epidemiology of Infectious Diseases (John Wiley & Sons, Chichester, 2000).Google Scholar
[10]Freedman, H. I. and Moson, P., “Persistence definitions and their connections”, Proc. Amer. Math. Soc. 109 (1990) 1025–1033.Google Scholar
[11]Gao, L. Q. and Herbert, H. W., “Disease transmission models with density-dependent demographics”, J. Math. Biol. 30 (1992) 717–731.CrossRefGoogle ScholarPubMed
[12]Gourley, S. A. and Kuang, Y., “A stage structured predator-prey model and its dependence on through-stage delay and death rate”, J. Math. Biol. 49 (2004) 188–200.CrossRefGoogle ScholarPubMed
[13]Hale, J. K. and Waltman, P., “Persistence in infinite-dimensional systems”, SIAM J. Math. Anal. 20 (1989) 388–395.Google Scholar
[14]Hethcote, H. W., “Qualitative analyses of communicable disease models”, Math. Biosci. 28 (1976) 335–356.Google Scholar
[15]Hethcote, H. W., “Three basic epidemiological models”, in Applied Mathematical Ecology (eds. Gross, L., Hallam, T. G. and Levin, S. A.), (Springer, Berlin, 1989) 119–144.CrossRefGoogle Scholar
[16]Hethcote, H. W., “The mathematics of infectious diseases”, SIAM Rev. 42 (2000) 599–653.CrossRefGoogle Scholar
[17]Hethcote, H. W. and Driessche, P., “An SIS epidemic model with variable population size and a delay”, J. Math. Biol. 34 (1995) 177–194.Google Scholar
[18]Hethcote, H. W. and Driessche, P., “Two SIS epidemiologic models with delays”, J. Math. Biol. 40 (2000) 3–26.Google Scholar
[19]Hethcote, H. W., Stech, H. W. and Driessche, P. V. D., “Nonlinear oscilllations in epidemic models”, SIAM J. Appl. Math. 40 (1981) 1–9.Google Scholar
[20]Kermack, W. O. and McKendrick, A. G., “Contributions to the mathematical theory of epidemics”, Proc. Roy. Soc. A 115 (1927) 700–721.Google Scholar
[21]Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (Academic Press, Boston, 1993).Google Scholar
[22]Liu, S., Chen, L., Luo, G. and Jiang, Y., “Asymptotic behavior of competitive Lotka-Volterra system with stage structure”, J. Math. Anal. Appl. 271 (2002) 124–138.Google Scholar
[23]Liu, S. Q. and Beretta, E., “A stage-structured predator-prey model of Beddington-De Angelis type”, SIAM J. Appl. Math. 66 (2006) 1101–1129.Google Scholar
[24]Liu, S. Q. and Liu, Z. J., “Permanence of general stage-structured consumer-resource models”, J. Comp. Appl. Math, (in press).Google Scholar
[25]Ma, W. B., Takeuchi, Y., Hara, T. and Beretta, E., “Permanence of an SIR epidemic model with distributed time delays”, Tohoku Math. J. 54 (2002) 581–591.Google Scholar
[26]Mena-Lorca, J. and Hethcote, H. W., “Dynamic models of infectious disease as regulators of population size”, J. Math. Biol. 30 (1992) 693–716.CrossRefGoogle Scholar
[27]Thieme, H. R., “Persistence under relaxed point-dissipativity (with application to an endemic model)”, SIAM J. Math. Anal. 24 (1993) 407–35.CrossRefGoogle Scholar
[28]Thieme, H. R., “Uniform persistence and permanence for non-autonomous semiflows in population biology”. Math. Biosc. 166 (2000) 173–201.Google Scholar
[29]Wang, W. D., “Global behavior of an SEIRS epidemic model with time delays”, Appl. Math. Let. 15 (2002) 423–428.Google Scholar
[30]Wang, W. D. and Ma, Z. E., “Global dynamics of an epidemic model with time delay”, Nonlinear Analysis: Real World Applications 3 (2002) 365–373.Google Scholar
[31]Xiao, Y. N. and Chen, L. S., “Modeling and analysis of a predator-prey model with disease in the prey”, Math. Biosc. 171 (2001) 59–82.CrossRefGoogle ScholarPubMed