In recent years many delay epidemiological models have been proposed
to study at which stage of the epidemics the delays can destabilize
the disease free equilibrium, or the endemic equilibrium, giving
rise to stability switches. One of these models is the SEIR model
with constant latency time and infectious periods [2],
for which the authors have proved that the two delays are harmless
in inducing stability switches. However, it is left open the problem
of the global asymptotic stability of the endemic equilibrium
whenever it exists. Even the Lyapunov functions approach, recently
proposed by Huang and Takeuchi to study many delay epidemiological
models, fails to work on this model. In this paper, an age-infection
model is presented for the delay SEIR epidemic model, such that the
properties of global asymptotic stability of the equilibria of the
age-infection model imply the same properties for the original
delay-differential epidemic model. By introducing suitable Lyapunov
functions to study the global stability of the disease free
equilibrium
(when $\mathcal{R}_0\leq 1$) and of the endemic equilibria (whenever $
\mathcal{R}_0>1$) of the age-infection model, we can infer the
corresponding
global properties for the equilibria of the delay SEIR model in [2], thus proving that the endemic equilibrium in
[2] is globally asymptotically stable whenever it
exists.
Furthermore, we also present a review of the SIR, SEIR epidemic
models, with and without delays, appeared in literature, that can be
seen as particular cases of the approach presented in the paper.
Citation: Gang Huang, Edoardo Beretta, Yasuhiro Takeuchi. Global stability for epidemicmodel with constant latency and infectious periods[J]. Mathematical Biosciences and Engineering, 2012, 9(2): 297-312. doi: 10.3934/mbe.2012.9.297
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Abstract
In recent years many delay epidemiological models have been proposed
to study at which stage of the epidemics the delays can destabilize
the disease free equilibrium, or the endemic equilibrium, giving
rise to stability switches. One of these models is the SEIR model
with constant latency time and infectious periods [2],
for which the authors have proved that the two delays are harmless
in inducing stability switches. However, it is left open the problem
of the global asymptotic stability of the endemic equilibrium
whenever it exists. Even the Lyapunov functions approach, recently
proposed by Huang and Takeuchi to study many delay epidemiological
models, fails to work on this model. In this paper, an age-infection
model is presented for the delay SEIR epidemic model, such that the
properties of global asymptotic stability of the equilibria of the
age-infection model imply the same properties for the original
delay-differential epidemic model. By introducing suitable Lyapunov
functions to study the global stability of the disease free
equilibrium
(when $\mathcal{R}_0\leq 1$) and of the endemic equilibria (whenever $
\mathcal{R}_0>1$) of the age-infection model, we can infer the
corresponding
global properties for the equilibria of the delay SEIR model in [2], thus proving that the endemic equilibrium in
[2] is globally asymptotically stable whenever it
exists.
Furthermore, we also present a review of the SIR, SEIR epidemic
models, with and without delays, appeared in literature, that can be
seen as particular cases of the approach presented in the paper.