Global stability for epidemicmodel with constant latency and infectious periods - Gang Huang; Edoardo Beretta; Yasuhiro Takeuchi (Mathematical Biosciences and Engineering)
In recent years many delay epidemiological models have been proposedto study at which stage of the epidemics the delays can destabilizethe disease free equilibrium, or the endemic equilibrium, givingrise to stability switches. One of these models is the SEIR modelwith constant latency time and infectious periods [2],for which the authors have proved that the two delays are harmlessin inducing stability switches. However, it is left open the problemof the global asymptotic stability of the endemic equilibriumwhenever it exists. Even the Lyapunov functions approach, recentlyproposed by Huang and Takeuchi to study many delay epidemiologicalmodels, fails to work on this model. In this paper, an age-infectionmodel is presented for the delay SEIR epidemic model, such that theproperties of global asymptotic stability of the equilibria of theage-infection model imply the same properties for the originaldelay-differential epidemic model. By introducing suitable Lyapunovfunctions to study the global stability of the disease freeequilibrium(when $\mathcal{R}_0≤ 1$) and of the endemic equilibria (whenever $\mathcal{R}_0>1$) of the age-infection model, we can infer thecorrespondingglobal properties for the equilibria of the delay SEIR model in [2], thus proving that the endemic equilibrium in[2] is globally asymptotically stable whenever itexists. &nbsp Furthermore, we also present a review of the SIR, SEIR epidemicmodels, with and without delays, appeared in literature, that can beseen as particular cases of the approach presented in the paper.