Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate
 Gang Huang,
 Yasuhiro Takeuchi,
 Wanbiao Ma,
 Daijun Wei
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In this paper, 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 Lyapunovâ€“LaSalle invariance principle, we prove the global stability of the endemic equilibrium and the diseasefree 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.
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 Title
 Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate
 Journal

Bulletin of Mathematical Biology
Volume 72, Issue 5 , pp 11921207
 Cover Date
 20100701
 DOI
 10.1007/s1153800994876
 Print ISSN
 00928240
 Online ISSN
 15229602
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Nonlinear incidence rate
 Time delay
 Lyapunov functional
 Global stability
 Industry Sectors
 Authors

 Gang Huang ^{(1)}
 Yasuhiro Takeuchi ^{(1)}
 Wanbiao Ma ^{(2)}
 Daijun Wei ^{(3)}
 Author Affiliations

 1. Graduate School of Science and Technology, Shizuoka University, Hamamatsu, 4328561, Japan
 2. Department of Mathematics and Mechanics, School of Applied Science, University of Science and Technology Beijing, Beijing, 100083, P.R. China
 3. Department of Mathematics, Hubei University for Nationalities, Enshi, 445000, P.R. China