Browse > Article
http://dx.doi.org/10.4134/CKMS.2007.22.4.597

A GENERAL UNIQUENESS RESULT OF AN ENDEMIC STATE FOR AN EPIDEMIC MODEL WITH EXTERNAL FORCE OF INFECTION  

Cha, Young-Joon (Department of Applied Mathematics Sejong University)
Publication Information
Communications of the Korean Mathematical Society / v.22, no.4, 2007 , pp. 597-608 More about this Journal
Abstract
We present a general uniqueness result of an endemic state for an S-I-R model with external force of infection. We reduce the problem of finding non-trivial steady state solutions to that of finding zeros of a real function of one variable so that we can easily prove the uniqueness of an endemic state. We introduce an assumption which was usually used to show stability of a non-trivial steady state. It turns out that such an assumption adopted from a stability analysis is crucial for proving the uniqueness as well, and the assumption holds for almost all cases in our model.
Keywords
endemic; S-I-R; uniqueness; inter-cohort; external force;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
Times Cited By SCOPUS : 0
연도 인용수 순위
1 M. Iannelli, F. A. Milner, and A. Pugliese, Analytical and Numerical Results for The Age-Structured S-I-S Model with Mixed Inter-Intracohort Transmission, SIAM J. Math. Anal. 23 (1992), 662-688   DOI
2 H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. BioI. 28 (1990), 411-434   DOI
3 A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc. 44 (1926), 98-130   DOI
4 H. von Foerster, The Kinetics of Cellular Proliferation, Grune and Stratton, New York, 1959
5 K. Dietz and D. Schenzle, Proportionate mixing models for age-dependent infection transmission, J. Math. BioI. 22 (1985), 117-120   DOI
6 El Doma, Analysis of nonlinear integro-differential equations arising in age dependent epidemic models, Nonl. Anal. T. M. A. 11 (1987), 913-937   DOI   ScienceOn
7 S. Busenberg, K. Cooke, and M. Iannelli, Endemic thresholds and stability in a class of age-structured epidemics, SIAM J. Appl. Math. 48 (1988), 1379-1395   DOI   ScienceOn
8 S. Busenberg, M. Iannelli, and H. Thieme, Dynamics of an age-structured epidemic model, Dynamical Systems, S.-T. Liao, T.-R. Ding, and Y.-Q. Ye (Eds.), Nankai Series in Pure, Applied Mathematics, and Theoretical Physics, vol. 4, World Scientific, 1993
9 Y. Cha, Existence and uniqueness of endemic states for an epidemic model with external force of infection, Commun. Korean Math. Soc. 17 (2002), no. 1, 175-187   과학기술학회마을   DOI   ScienceOn
10 Y. Cha, Local stability of endemic states for an epidemic model with external force of infection, Commun. Korean Math. Soc. 18 (2003), no. 1, 133-149   과학기술학회마을   DOI   ScienceOn
11 Y. Cha and F. A. Milner, A uniqueness result of an endemic state for an S-I-R epidemic model, submitted to a journal
12 Y. Cha, M. Iannelli, and F. A. Milner, Existence and uniqueness of endemic states for the age structured S-I-R model, Math. Biosc. 150 (1998), 177-190   DOI   ScienceOn
13 Y. Cha, Stability change of an epidemic model, Dynamic Systems and Applications 9 (2000), 361-376
14 H. Thieme, Stability change for the endemic equilibrium in age-structured models for the spread of S-I-R type infectious diseases, Differential Equations Models in Biology, Epidemiology and Ecology, Lectures Notes in Biomathematics, vol. 92, Springer Verlag, 1991, 139-158
15 V. Andreassen, Instability in an SIR-model with age-dependent susceptibility, Mathematical Population Dynamics: Analysis of Heterogeneity, vol. 1, Theory of Epidemics, Proc. 3rd International Conference on Mathematical Population Dynamics, Pau, France, 1992, Wuerz Publishing Ltd., Winnipeg, Canada, 1995, 3-14