Bulletin of Mathematical Biology

, Volume 72, Issue 5, pp 1192–1207

Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate


  • Gang Huang
    • Graduate School of Science and TechnologyShizuoka University
    • Graduate School of Science and TechnologyShizuoka University
  • Wanbiao Ma
    • Department of Mathematics and Mechanics, School of Applied ScienceUniversity of Science and Technology Beijing
  • Daijun Wei
    • Department of MathematicsHubei University for Nationalities
Original Article

DOI: 10.1007/s11538-009-9487-6

Cite this article as:
Huang, G., Takeuchi, Y., Ma, W. et al. Bull. Math. Biol. (2010) 72: 1192. doi:10.1007/s11538-009-9487-6


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 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.

Nonlinear incidence rateTime delayLyapunov functionalGlobal stability
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© Society for Mathematical Biology 2009