Global dynamics of a delayed SIRS epidemic model with a wide class of nonlinear incidence rates
详细信息 查看全文
- 作者：Yoichi Enatsu (1) yo1.gc-rw.docomo@akane.waseda.jp
Eleonora Messina (2) eleonora.messina@unina.it
Yukihiko Nakata (3) nakata@bcamath.org
Yoshiaki Muroya (4) ymuroya@waseda.jp
Elvira Russo (2) elvrusso@unina.it
Antonia Vecchio (5) a.vecchio@iac.cnr.it - 关键词：SIRS epidemic model – Nonlinear incidence rate – Global asymptotic stability – Lyapunov functional
- 刊名：Journal of Applied Mathematics and Computing
- 出版年：2012
- 出版时间：June 2012
- 年：2012
- 卷：39
- 期：1-2
- 页码：15-34
- 全文大小：691.3 KB
- 参考文献：1. Alexander, M.E., Moghadas, S.M.: Bifurcation analysis of an SIRS epidemic model with generalized incidence. SIAM J. Appl. Math. 65, 1794–1816 (2005) <Occurrence Type="DOI"><Handle>10.1137/040604947\end{DOI</Handle></Occurrence>
2. Beretta, E., Hara, T., Ma, W., Takeuchi, Y.: Global asymptotic stability of an SIR epidemic model with distributed time delay. Nonlinear Anal. 47, 4107–4115 (2001)
3. Capasso, V., Serio, G.: A generalization of the Kermack–Mckendrick deterministic epidemic model. Math. Biosci. 42, 43–61 (1978)
4. Cooke, K.L.: Stability analysis for a vector disease model. Rocky Mt. J. Math. 9, 31–42 (1979)
5. Enatsu, Y., Nakata, Y., Muroya, Y.: Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays. Discrete Contin. Dyn. Syst., Ser. B 15, 61–74 (2011)
6. Enatsu, Y., Nakata, Y., Muroya, Y.: Global stability of SIRS epidemic models with a class of nonlinear incidence rates and distributed delays, To appear
7. Huang, G., Takeuchi, Y., Ma, W., Wei, D.: Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate. Bull. Math. Biol. 72, 1192–1207 (2010)
8. Huang, G., Takeuchi, Y.: Global analysis on delay epidemiological dynamic models with nonlinear incidence. J. Math. Biol. 63, 125–139 (2011)
9. Jin, Y., Wang, W., Xiao, S.: An SIRS model with a nonlinear incidence rate. Chaos Solitons Fractals 34, 1482–1497 (2007)
10. Korobeinikov, A., Maini, P.K.: Nonlinear incidence and stability of infectious disease models. Math. Med. Biol. 22, 113–128 (2005)
11. Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 68, 615–626 (2006)
12. Korobeinikov, A.: Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol. 69, 1871–1886 (2007)
13. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, San Diego (1993)
14. McCluskey, C.C.: Complete global stability for an SIR epidemic model with delay-distributed or discrete. Nonlinear Anal., Real World Appl. 11, 55–59 (2010)
15. McCluskey, C.C.: Global stability for an SIR epidemic model with delay and nonlinear incidence. Nonlinear Anal., Real World Appl. 11, 3106–3109 (2010)
16. McCluskey, C.C.: Global stability of an SIR epidemic model with delay and general nonlinear incidence. Math. Biosci. Eng. 7, 837–850 (2010)
17. Mena-Lorcat, J., Hethcote, H.W.: Dynamic models of infectious diseases as regulators of population size. J. Math. Biol. 30, 693–716 (1992)
18. Muroya, Y., Enatsu, Y., Nakata, Y.: Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate. J. Math. Anal. Appl. 377, 1–14 (2011)
19. Muroya, Y., Enatsu, Y., Nakata, Y.: Monotone iterative techniques to SIRS epidemic models with nonlinear incidence rates and distributed delays. Nonlinear Anal., Real World Appl. 12, 1897–1910 (2011)
20. Nakata, Y., Enatsu, Y., Muroya, Y.: On the global stability of an SIRS epidemic model with distributed delays, To appear
21. Takeuchi, Y., Ma, W., Beretta, E.: Global asymptotic properties of a delay SIR epidemic model with finite incubation times. Nonlinear Anal. 42, 931–947 (2000)
22. Vargas-De-León, C., Gómez-Alcaraz, G.: Global stability conditions of delayed SIRS epidemiological model for vector diseases, Foro-Red-Mat: Revista Electrónica de Contenido Matemático 28 (2011)
23. Xu, R., Ma, Z.: Stability of a delayed SIRS epidemic model with a nonlinear incidence rate. Chaos Solitons Fractals 41, 2319–2325 (2009)
24. Zhou, X., Cui, J.: Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate. Nonlinear Dyn. 63, 639–653 (2011) - 作者单位：1. Department of Pure and Applied Mathematics, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan2. Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli “Federico II- Via Cintia, 80126 Napoli, Italy3. Basque Center for Applied Mathematics, Bizkaia Technology Park, Building 500, 48160 Derio, Spain4. Department of Mathematics, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan5. Ist. per Appl. del Calcolo “M. Picone-Sede di Napoli-CNR, Via P. Castellino, 111-80131 Napoli, Italy
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Computational Mathematics and Numerical Analysis
Applied Mathematics and Computational Methods of Engineering
Theory of Computation
Mathematics of Computing - 出版者：Springer Berlin / Heidelberg
- ISSN：1865-2085
文摘
In this paper, by constructing Lyapunov functionals, we consider the global dynamics of an SIRS epidemic model with a wide class of nonlinear incidence rates and distributed delays òh0 p(t)f(S(t),I(t-t)) dt\int^{h}_{0} p(\tau)f(S(t),I(t-\tau)) \mathrm{d}\tau under the condition that the total population converges to 1. By using a technical lemma which is derived from strong condition of strict monotonicity of functions f(S,I) and f(S,I)/I with respect to S≥0 and I>0, we extend the global stability result for an SIR epidemic model if R 0>1, where R 0 is the basic reproduction number. By using a limit system of the model, we also show that the disease-free equilibrium is globally asymptotically stable if R 0=1.