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Zbl 1169.92041
Korobeinikov, Andrei
Global properties of $SIR$ and $SEIR$ epidemic models with multiple parallel infectious stages. (English)
[J] Bull. Math. Biol. 71, No. 1, 75-83 (2009). ISSN 0092-8240; ISSN 1522-9602/e

Summary: We consider global properties of compartment $SIR$ and $SEIR$ models of infectious diseases, where there are several parallel infective stages. For instance, such a situation may arise if a fraction of the infected are detected and treated, while the rest of the infected remains undetected and untreated. We assume that the horizontal transmission is governed by the standard bilinear incidence rate. The direct Lyapunov method enables us to prove that the considered models are globally stable: There is always a globally asymptotically stable equilibrium state. Depending on the value of the basic reproduction number $R _{0}$, this state can be either endemic ($R _{0}>1$), or infection-free ($R _{0}\leq 1$).

MSC 2000:: *92D30 Epidemiology
34D23 Global stability
34D20 Lyapunov stability of ODE
37N25 Dynamical systems in biology

Keywords: infectious disease; mass-action; endemic equilibrium state; global stability; direct Lyapunov method; Lyapunov function

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Zentralblatt MATH Berlin [Germany]