Leonid Shaikhet

Stability of epidemic model with time delays influenced by stochastic perturbations

Many processes in automatic regulation, physics, mechanics, biology, economy, ecology etc. can be modelled by hereditary equations (see, e.g. [1–6]). 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. In [7–11], the procedure is 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.

Some peculiarities of the general method of Lyapunov functionals construction

The general method of Lyapunov functionals construction for stability investigation of stochastic hereditary systems which was proposed and developed before is considered. Some features of this method for difference systems which allow one to use the method more effectively are discussed.

Necessary and Sufficient Conditions of Asymptotic Mean Square Stability for Stochastic Linear Difference Equations

Abstract Many processes in automatic regulation, physics, mechanics, biology, economy, ecology, etc. can be modelled by hereditary systems (see, e.g., [1–4]). One of the main problems for the theory of such systems and their applications is connected with stability (see, e.g., [2–4]). Many stability results were obtained by the construction of appropriate Lyapunov functionals. At present, the method is proposed which allows us, in some sense, to formalize the procedure of the corresponding Lyapunov functionals construction [5–10]. In this work, by virtue of the proposed procedure, the necessary and sufficient conditions of asymptotic mean square stability for stochastic linear difference equations are obtained.

Lyapunov functionals and stability of stochastic difference equations

Lyapunov-type Theorems and Procedure for Lyapunov Functional Construction.- Illustrative Example.- Linear Equations with Stationary Coefficients.- Linear Equations with Nonstationary Coefficients.- Some Peculiarities of the Method.- Systems of Linear Equations with Varying Delays.- Nonlinear Systems.- Volterra Equations of the Second Type.- Difference Equations with Continuous Time.- Difference Equations as Difference Analogues of Differential Equations.

Lyapunov functionals and stability of stochastic functional differential equations

Short Introduction to Stability Theory of Deterministic Functional Differential Equations -- Stability of Linear Scalar Equations -- Stability of Linear Systems of Two Equations -- Stability of Systems with Nonlinearities -- Matrix Riccati Equations in Stability of Linear Stochastic Differential Equations with Delays -- Stochastic Systems with Markovian Switching -- Stabilization of the Controlled Inverted Pendulum by Control with Delay -- Stability of Equilibrium Points of Nicholson’s Blowflies Equation with Stochastic Perturbations -- Stability of Positive Equilibrium Point of Nonlinear System of Type of Predator-Prey with Aftereffect and Stochastic Perturbations -- Stability of SIR Epidemic Model Equilibrium Points -- Stability of Some Social Mathematical Models with Delay by Stochastic Perturbations.

About Lyapunov functionals construction for difference equations with continuous time

Abstract Stability investigation of hereditary systems is connected often with construction of Lyapunov functionals. One general method of Lyapunov functionals construction was proposed and developed in [1–9] both for differential equations with aftereffect and for difference equations with discrete time. Here, some modification of Lyapunov-type stability theorem is proposed, which allows one to use this method for difference equations with continuous time also.

Stability of the Positive Point of Equilibrium of Nicholson's Blowflies Equation with Stochastic Perturbations: Numerical Analysis

An optical brancher branches an input optical signal into two. An optical detector converts one optical signal branched by the optical brancher into an electrical signal. A first controller generates a control electrical signal having a waveform obtained by inverting the envelope of the electrical signal. Based on the control electrical signal, an optical signal generator produces a dummy optical signal having a waveform lambdd and an amplitude alpha/2. The other signal branched by the optical brancher is delayed by a delay unit for a predetermined time, and then multiplexed by an optical multiplexer with the dummy optical signal from the optical signal generator. An optical amplifier amplifies amultiplexed optical signal. An optical filter separates an optical signal of a wavelength lambd1 from the amplified optical signal. Thus, optical signal amplification can be carried out without optical surges.

Some New Aspects of Lyapunov-Type Theorems for Stochastic Differential Equations of Neutral Type

Some new Lyapunov-type theorems for stochastic differential equations of neutral type are proved. It is shown that these theorems simplify an application of Kolmanovskii and Shaikhets general method of Lyapunov functionals construction for stability investigation of different mathematical models.

About stability of nonlinear stochastic difference equations

Abstract Using the method of Lyapunov functionals construction, it is shown that investigation of stability in probability of nonlinear stochastic difference equation with order of nonlinearity more than one can be reduced to the investigation of asymptotic mean square stability of the linear part of this equation.

Lyapunov functionals construction for stochastic difference second-kind Volterra equations with continuous time

The general method of Lyapunov functionals construction which was developed during the last decade for stability investigation of stochastic differential equations with aftereffect and stochastic difference equations is considered. It is shown that after some modification of the basic Lyapunov-type theorem, this method can be successfully used also for stochastic difference Volterra equations with continuous time usable in mathematical models. The theoretical results are illustrated by numerical calculations.