Convergence results in \(SIR\) epidemic models with varying population sizes. (English) Zbl 0879.34054

Beretta, Edoardo; Takeuchi, Yasuhiro

Nonlinear Anal., Theory Methods Appl. 28, No. 12, 1909-1921 (1997).

The authors study an \(SIR\) epidemic model in which the force of infection involves a distributed delay and in which the population is not constant. The model exhibits both a disease-free equilibrium, \(E_0\), which exists for all parameter values, and one in which the infection is endemic, \(E_+\), which exists for certain parameter values. Their main result is that \(E_0\) is globally asymptotically stable when \(E_+\) doesn’t exist and that \(E_+\) is locally asymptotically stable when it exists. The proof involves Lyapunov functionals. They also calculate the radius of a ball contained in the domain of attraction of \(E_+\).

Reviewer: A.Hausrath (Boise)

MSC:

34D20	Stability of solutions to ordinary differential equations
92D30	Epidemiology

Keywords:

\(SIR\) epidemic model; time delay; stability of equilibria; Lyapunov functional

References:

[1]	Cooke, K. L., Stability analysis for a vector disease model, Rocky Mount. J. Math., 7, 253-263 (1979)
[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]	Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Rio de Janeiro · Zbl 0777.34002
[4]	Hethcote, H. W., Qualitative analyses of communicable disease models, Math. Biosci., 28, 335-356 (1976) · Zbl 0326.92017