Stability of epidemic model with time delays influenced by stochastic perturbations. (English) Zbl 1017.92504

Summary: Many processes in automatic regulation, physics, mechanics, biology, economy, ecology etc. can be modelled by hereditary equations. One of the main problems for the theory of stochastic hereditary equations and their applications is connected with stability. Many stability results were obtained by the construction of appropriate Lyapunov functionals. Earlier the procedure was proposed, allowing, in some sense, to formalize the algorithm of the corresponding Lyapunov functionals construction for stochastic functional differential equations, for stochastic difference equations. In this paper, stability conditions are obtained by using this procedure for the mathematical model of the spread of infections diseases with delays influenced by stochastic perturbations.


92D25 Population dynamics (general)
34K20 Stability theory of functional-differential equations
34K50 Stochastic functional-differential equations
92D30 Epidemiology
Full Text: DOI


[1] V.B. Kolmanovskii, V.R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986 · Zbl 0593.34070
[2] V.B. Kolmanovskii, A.D. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Boston, 1992 · Zbl 0917.34001
[3] V.B. Kolmanovskii, L.E. Shaikhet, Control of Systems with Aftereffect, Translations of mathematical monographs, 157, American Mathematical Society, Providence, RI, 1996
[4] Beretta, E.; Capasso, V.; Rinaldi, F.: Global stability results for a generalized Lotka–Volterra system with distributed delays. J. math. Biol. 26, 661-688 (1988) · Zbl 0716.92020
[5] Beretta, E.; Takeuchi, Y.: Qualitative properties of chemostat equations with time delays: boundedness local and global asymptotic stability. Diff. eqs. And dynamical systems 2, 19-40 (1994) · Zbl 0868.45002
[6] Beretta, E.; Fasano, A.; Hosono, Yu.; Kolmanovskii, V. B.: Stability analysis of the phytoplankton vertical steady states in a laboratory test tube. Math. methods in the applied sciences 17, 551-575 (1994) · Zbl 0807.92018
[7] Kolmanovskii, V. B.; Shaikhet, L. E.: Stability of stochastic hereditary systems. Avtomatika i telemekhanika (in russian) 7, 66-85 (1993)
[8] Kolmanovskii, V. B.: On stability of some hereditary systems. Avtomatika i telemekhanika (in russian) 11, 45-59 (1993)
[9] Kolmanovskii, V. B.; Shaikhet, L. E.: On one method of Lyapunov functional construction for stochastic hereditary systems. Differentsialniye uravneniya (in russian) 11, 1909-1920 (1993) · Zbl 1029.93057
[10] Kolmanovskii, V. B.; Shaikhet, L. E.: New results in stability theory for stochastic functional differential equations (SFDEs) and their applications, dynamical systems and applications. Dynamic publishers inc., N.Y. 1, 167-171 (1994) · Zbl 0811.34062
[11] Kolmanovskii, V. B.; Shaikhet, L. E.: General method of Lyapunov functionals construction for stability investigations of stochastic difference equations, dynamical systems and applications. World scientific series in applicable analysis, Singapore 4, 397-439 (1995)
[12] Cooke, K. L.: Stability analysis for a vector disease model. Rocky mount. J. math. 7, 253-263 (1979)
[13] Beretta, E.; Takeuchi, Y.: Global stability of an SIR epidemic model with time delays. J. math. Biol. 33, 250-260 (1995) · Zbl 0811.92019
[14] I.I. Gihman, A.V. Skorokhod, The Theory of Stochastic Processes, I,II,III, Springer, Berlin, 1974, 1975, 1979 · Zbl 0291.60019
[15] Shaikhet, L. E.: Stability in probability of nonlinear stochastic hereditary systems. Dynamic systems and applications 4, No. 2, 199-204 (1995) · Zbl 0831.60075
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.