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Complete global stability for an SIR epidemic model with delay - distributed or discrete. (English) Zbl 1185.37209
In modelling the transmission of an infectious disease, a common model structure involves dividing the population into susceptible, infectious and recovered individuals. If the immunity that is obtained upon recovery is permanent, then one gets an SIR model. In this paper the author considers SIR models with mass action incidence and constant recruitment. In Section 2 an SIR model with distributed delay is given. In Section 3, some results from the literature relating to earlier work on this model are given. Section 4 contains a proof of the global asymptotic stability of the endemic equilibrium for $${\mathfrak R}_0> 1$$. In Section 5, an SIR model with discrete delay is presented and the endemic equilibrium is shown to be globally asymptotically stable for $${\mathfrak R}_0> 1$$.

##### MSC:
 37N40 Dynamical systems in optimization and economics 92D30 Epidemiology
##### Keywords:
delay; distributed delay; global stability; Lyapunov functional
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##### References:
 [1] Beretta, E.; Hara, T.; Ma, W.; Takeuchi, Y., Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear anal., 47, 4107-4115, (2001) · Zbl 1042.34585 [2] Beretta, E.; Takeuchi, Y., Global stability of an SIR epidemic model with time delays, J. math. biol., 33, 250-260, (1995) · Zbl 0811.92019 [3] Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A.J., On the definition and the computation of the basic reproduction $$\mathcal{R}_0$$ in models for infectious diseases in heterogeneous populations, J. math. biol., 28, 365-382, (1990) · Zbl 0726.92018 [4] Guo, H.; Li, M.Y., Global dynamics of a staged progression model for infectious diseases, Math. biosci. eng., 3, 3, 513-525, (2006) · Zbl 1092.92040 [5] Hale, J.; Verduyn Lunel, S., Introduction to functional differential equations, (1993), Springer-Verlag · Zbl 0787.34002 [6] Hethcote, H.W., Qualitative analyses of communicable disease models, Math. biosci., 28, 335-356, (1976) · Zbl 0326.92017 [7] Ma, W.; Song, M.; Takeuchi, Y., Global stability of an SIR epidemic model with time delay, Appl. math. lett., 17, 1141-1145, (2004) · Zbl 1071.34082 [8] Ma, W.; Takeuchi, Y.; Hara, T.; Beretta, E., Permanence of an SIR epidemic model with distributed time delays, Tohoku math. J., 54, 581-591, (2002) · Zbl 1014.92033 [9] C.C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay (submitted for publication) · Zbl 1190.34108 [10] Takeuchi, Y.; Ma, W.; Beretta, E., Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear anal., 42, 931-947, (2000) · Zbl 0967.34070
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