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Lyapunov functions and global properties for SEIR and SEIS epidemic models. (English) Zbl 1055.92051

Introduction: We deal with global properties of classic SEIR and SEIS epidemic models. The phase space of the SEIR model is three-dimensional (the constant population size assumption allows to omit one equation), and the global analysis is non-trivial. So far the most successful approach to such a problem is the direct Lyapunov method. However, the method requires an auxiliary function with specific properties, a Lyapunov function, which is not easy to find, especially for multidimensional systems. In this work we construct explicit Lyapunov functions of the form \[ V=\sum a_i(x_i-x_i^*\ln x_i) \] (here \(a_i\) is a properly selected constant, \(x_i\) is the population of the \(i\)th compartment and \(x_i^*\) is the equilibrium level) for the SEIR and SEIS epidemic models. The functions are extensions to a three-dimensional case of the functions constructed earlier [A. Korobeinikov and G. C. Wake, Appl. Math. Lett. 15, 955–960 (2002; Zbl 1022.34044)] for SIR, SIRS and SIS models. The Lyapunov functions of this type are also proven to be useful for Lotka-Volterra predator-prey systems and it appears that they can be useful for more complex compartmental epidemic models as well.
Global stability of the SEIR and SEIS models has long been conjectured but only recently proven using a sophisticated technique – an extension of the Poincaré-Bendixson theorem for competitive systems in three-dimensional space. M. Y. Li and his co-authors [see Math. Biosci. 125, 155–164 (1995; Zbl 0821.92022); ibid. 160, 191–213 (1999; Zbl 0974.92029)] proved global stability for a variety of SEIR models, including a model with vertical transmission. The Lyapunov functions suggested in this paper make the proof of global stability of the SEIR and SEIS models almost elementary.

MSC:

92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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