Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models. (English) Zbl 1022.34044

The aim of this note is to prove the global stability of the endemic equilibrium states for classical SIR, SIRS and SIS epidemiological models by using the direct Lyapunov metod. Although this result is known in the related literature, the obtention of a Lyapunov function has some interesting advantages like the possibility to compare the rates of convergence towards the equilibrium between the different models.


34D23 Global stability of solutions to ordinary differential equations
92D30 Epidemiology
34C60 Qualitative investigation and simulation of ordinary differential equation models
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[1] Lyapunov, A.M., ()
[2] Hethcote, H.W.; Levin, S.A., Periodicity in epidemiological models, (), 193-211
[3] Hethcote, H.W.; Stech, H.W.; van den Driessche, P., Periodicity and stability in epidemiological models: A survey, (), 65-85
[4] Mena-Lorca, J.; Hethcote, H.W., Dynamics models of infectious diseases as regulator of population sizes, J. math. biol., 30, 693-716, (1992) · Zbl 0748.92012
[5] Busenberg, S.; Cooke, K., ()
[6] Busenberg, S.N.; van den Driessche, P., A method for proving the non-existence of limit cycles, J. math. anal. appl., 172, 463-479, (1993) · Zbl 0779.34026
[7] Anderson, R.M.; May, R.M., ()
[8] Bailey, N.T.J., ()
[9] Barbashin, E.A., ()
[10] La Salle, J.; Lefschetz, S., ()
[11] Goh, B.-S., ()
[12] Takeuchi, Y., ()
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