×

zbMATH — the first resource for mathematics

Convergence results in \(SIR\) epidemic models with varying population sizes. (English) Zbl 0879.34054
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
PDF BibTeX XML Cite
Full Text: DOI
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 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
[5] BERETTA, E. and KUANG, Y., Convergence results in a well known delayed predator-prey system, preprint. · Zbl 0876.92021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.