## Global properties of $$SIR$$ and $$SEIR$$ epidemic models with multiple parallel infectious stages.(English)Zbl 1169.92041

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:

 92D30 Epidemiology 34D23 Global stability of solutions to ordinary differential equations 34D20 Stability of solutions to ordinary differential equations 37N25 Dynamical systems in biology
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### References:

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