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Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate. (English) Zbl 1197.92040
Summary: Based on SIR and SEIR epidemic models with a general nonlinear incidence rate, we incorporate time delays into the ordinary differential equation models. In particular, we consider two delay differential equation models in which delays are caused (i) by the latency of the infection in a vector, and (ii) by the latent period in an infected host. By constructing suitable Lyapunov functionals and using the Lyapunov-LaSalle invariance principle, we prove the global stability of the endemic equilibrium and the disease-free equilibrium for time delays of any length in each model. Our results show that the global properties of equilibria also only depend on the basic reproductive number and that the latent period in a vector does not affect the stability, but the latent period in an infected host plays a positive role to control disease development.

92D30 Epidemiology
34K20 Stability theory of functional-differential equations
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[1] Beretta, E., Hara, T., Ma, W., Takeuchi, Y., 2001. Global asymptotic stability of an SIR epidemic model with distributed time delay. Nonlinear Anal. 47, 4107–4115. · Zbl 1042.34585
[2] Beretta, E., Takeuchi, Y., 1995. Global stability of an SIR model with time delays. J. Math. Biol. 33, 250–260. · Zbl 0811.92019
[3] Cooke, K.L., van den Driessche, P., 1996. Analysis of an SEIRS epidemic model with two delays. J. Math. Biol. 35, 240–260. · Zbl 0865.92019
[4] Cooke, K.L., van den Driessche, P., Zou, X., 1999. Interaction of maturation delay and nolinear birth in population and epidemic models. J. Math. Biol. 39, 332–352. · Zbl 0945.92016
[5] Derrick, W.R., van den Driessche, P., 2003. Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population. Discrete Continuous Dyn. Syst., Ser. B 3, 299–309. · Zbl 1126.34337
[6] Kermack, W.O., McKendrick, A.G., 1927. A contribution to the mathematical theory of epidemics. Proc. R. Soc. A115, 700–721. · JFM 53.0517.01
[7] Korobeinikov, A., Maini, P.K., 2005. Nonlinear incidence and stability of infectious disease models. Math. Med. Biol. 22, 113–128. · Zbl 1076.92048
[8] Korobeinikov, A., 2006. Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 68, 615–626. · Zbl 1334.92410
[9] Korobeinikov, A., 2007. Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol. 69, 1871–1886. · Zbl 1298.92101
[10] Korobeinikov, A., 2009a. Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence, and nonlinear incidence rate. Math. Med. Biol. 26, 225–239. · Zbl 1171.92034
[11] Korobeinikov, A., 2009b. Stability of ecosystem: Global properties of a general prey-predator model. Math. Medic. Biol. (in print). http://imammb.oxfordjournals.org/cgi/reprint/dqp009 . · Zbl 1178.92053
[12] Kuang, Y., 1993. Delay Differential Equations with Applications in Population Dynamics. Academic Press, San Diego. · Zbl 0777.34002
[13] Kyrychko, Y.N., Blyuss, K.B., 2005. Global properties of a delay SIR model with temporary immunity and nonlinear incidence rate. Nonlinear Anal. 6, 495–507. · Zbl 1144.34374
[14] Liu, W., Hethcote, H.W., Levin, S.A., 1987. Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25, 359–380. · Zbl 0621.92014
[15] Ma, W., Song, M., 2004. Global stability for an SIR epidemic model with time delay. Appl. Math. Lett. 17, 1141–1145. · Zbl 1071.34082
[16] McCluskey, C.C., 2009a. Complete global stability for an SIR epidemic model with delay-distributed or discrete. Nonlinear Anal. doi: 10.10.16/j.nonrwa.2008.10.014 .
[17] McCluskey, C.C., 2009b. Global stability for an SIER epidemiological model with varying infectivity and infinite delay. Math. Biosci. Eng. 6, 603–610. · Zbl 1190.34108
[18] Meng, X., Chen, L., Wu, B., 2009. A delay SIR epidemic model with pulse vaccination and incubation times. Nonlinear Anal. doi: 10.1016/j.nonrwa.2008.10.041 . · Zbl 1184.92044
[19] Rost, G., Wu, J., 2008. SEIR epidemiological model with varying infectivity and infinite delay. Math. Biosci. Eng. 5, 389–402. · Zbl 1165.34421
[20] Smith, H.L., 1983. Subharmonic bifurcation in an SIR epidemic model. J. Math. Biol. 17, 163–177. · Zbl 0578.92023
[21] van den Driessche, P., Watmough, J., 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48. · Zbl 1015.92036
[22] Xu, R., Ma, Z., 2009. Global stability of a SIR epidemic model with nonlinear incidence rate and time delay. Nonlinear Anal. 10, 3175–3189. · Zbl 1183.34131
[23] Zhao, Z., Chen, L., Song, X., 2008. Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate. Math. Comput. Simul. 79, 500–510. · Zbl 1151.92030
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