Method Of Lyapunov Functionals Construction Research Articles

Stability of equilibria of exponential type system of three differential equations under stochastic perturbations

A system of three differential equations with exponential nonlinearity is considered. It is shown that the considered system has both the zero and nonzero (positive or negative) equilibria. It is supposed that the system is exposed to stochastic perturbations that are directly proportional to the deviation of a system state from an equilibrium. Via the general method of Lyapunov functionals construction and the method of linear matrix inequalities sufficient conditions of stability in probability for an each equilibrium are obtained. Numerical simulations and figures are presented to demonstrate the obtained results. The proposed investigation method can be applied for many other types of nonlinear systems.

About Stability of Nonlinear Stochastic Differential Equations with State-Dependent Delay

A nonlinear stage-structured population model with a state-dependent delay under stochastic perturbations is investigated. Delay-independent and delay-dependent conditions of stability in probability for two equilibria of the considered system are obtained via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The model under consideration is not the aim of the work and was chosen only to demonstrate the proposed research method, which can be used for the study of other types of nonlinear systems with a state-dependent delay.

Stability Analysis of Delayed Tumor-Antigen-ActivatedImmune Response in Combined BCG and IL-2Immunotherapy of Bladder Cancer

We use a system biology approach to translate the interaction of Bacillus Calmette-Gurin (BCG) + interleukin 2 (IL-2) for the treatment of bladder cancer into a mathematical model. The main goal of this research is to predict the outcome of BCG + IL-2 treatment combinations. We examined whether the delay effect caused by the proliferation of tumor antigen-specific effector cells after the immune system destroys BCG-infected urothelium cells after BCG and IL-2 immunotherapy influences success in bladder cancer treatment. To do this, we introduce a system of differential equations where the variables are the main participants in the immune response after BCG installations to fight cancer: the number of tumor cells, BCG cells, immune cells, and cytokines involved in the tumor-immune response. The relevant parameters describing the dynamics of the system are taken from a variety of biological, clinical literature and estimated using the mathematical models. We examine the local stability analysis of non-negative equilibrium states of the model. In theory, treatment could improve system stability, and we analyze the stability of all equilibria using the method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). Our results prove that the period for the proliferation of tumor antigen-specific effector cells does not influence to the success of the non-responsive patients after an intensified combined BCG + IL-2 treatment.

Improving Stability Conditions for Equilibria of SIR Epidemic Model with Delay under Stochastic Perturbations

So called SIR epidemic model with distributed delay and stochastic perturbations is considered. It is shown, that the known sufficient conditions of stability in probability of the equilibria of this model, formulated immediately in the terms of the system parameters, can be improved by virtue of the method of Lyapunov functionals construction and the method of Linear Matrix Inequalities (LMIs). It is also shown, that stability can be investigated immediately via numerical simulation of a solution of the considered model.

Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps

Delay differential equation is considered under stochastic perturbations of the type of white noise and Poisson's jumps. It is shown that if stochastic perturbations fade on the infinity quickly enough then sufficient conditions for asymptotic stability of the zero solution of the deterministic differential equation with delay provide also asymptotic mean square stability of the zero solution of the stochastic differential equation. Stability conditions are obtained via the general method of Lyapunov functionals construction and the method of Linear Matrix Inequalities (LMIs). Investigation of the situation when stochastic perturbations do not fade on the infinity or fade not enough quickly is proposed as an unsolved problem.

Stability of the neoclassical growth model under perturbations of the type of Poisson’s jumps: Analytical and numerical analysis

The well known delay differential neoclassical growth model is considered under stochastic perturbations of the type of Poisson’s jumps. Stability conditions for positive and the zero equilibria of this model are obtained via the Gikhman and Skorokhod theory of stochastic differential equations with the Poisson measure and the general method of Lyapunov functionals construction. The obtained analytical results are illustrated by detail numerical simulations of solutions of the considered stochastic differential equation and construction of appropriate stability regions.

Stability Analysis of Delayed Immune Response BCG Infection in Bladder Cancer Treatment Model by Stochastic Perturbations.

We present a revised mathematical model of the immune response to Bacillus Calmette-Guérin (BCG) treatment of bladder cancer, optimized according to biological and clinical data accumulated during the last decade. The improved model accounts for cytotoxic T lymphocyte differentiation as an integral element of the delayed immune response, as well as the logistic growth terms for cancer cell proliferation. Three equilibria are demonstrated for the proposed model, which is assumed to be influenced by white noise stochastic perturbations that are directly proportional to the system state deviation from an equilibrium. Stability conditions for all equilibria are analyzed using the Kolmanovskii-Shaikhet general method of Lyapunov functionals construction.

Analysing Social Epidemics by Delayed Stochastic Models

We investigate the dynamics of a delayed stochastic mathematical model to understand the evolution of the alcohol consumption in Spain. Sufficient condition for stability in probability of the equilibrium point of the dynamic model with aftereffect and stochastic perturbations is obtained via Kolmanovskii and Shaikhet general method of Lyapunov functionals construction. We conclude that alcohol consumption in Spain will be constant (with stability) in time with around 36.47% of nonconsumers, 62.94% of nonrisk consumers, and 0.59% of risk consumers. This approach allows us to emphasize the possibilities of the dynamical models in order to study human behaviour.

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 Shaikhet's general method of Lyapunov functionals construction for stability investigation of different mathematical models.

Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations

It is supposed that the fractional difference equation , has an equilibrium point and is exposed to additive stochastic perturbations type of that are directly proportional to the deviation of the system state from the equilibrium point . It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted.

Stability in Probability of Nonlinear Stochastic Volterra Difference Equations with Continuous Variable

The general method of Lyapunov functionals construction, that was proposed by Kolmanovskii and Shaikhet and successfully used already for functional-differential equations, difference equations with discrete time, difference equations with continuous time, and is used here to investigate the stability in probability of nonlinear stochastic Volterra difference equations with continuous time. It is shown that the investigation of the stability in probability of nonlinear stochastic difference equation with order of nonlinearity more than one can be reduced to investigation of the asymptotic mean square stability of the linear part of this equation.

Method of Lyapunov functionals construction in stability of delay evolution equations

The investigation of stability for hereditary systems is often related to the construction of Lyapunov functionals. The general method of Lyapunov functionals construction which was proposed by V. Kolmanovskii and L. Shaikhet and successfully used already for functional differential equations, for difference equations with discrete time, for difference equations with continuous time, is used here to investigate the stability of delay evolution equations, in particular, partial differential equations.

Mean Square Summability of Solution of Stochastic Difference Second-Kind Volterra Equation with Small Nonlinearity

Stochastic difference second-kind Volterra equation with continuous time and small nonlinearity is considered. Via the general method of Lyapunov functionals construction, sufficient conditions for uniform mean square summability of solution of the considered equation are obtained.

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

Known Nicholson's blowflies equation (which is one of the most important models in ecology) with stochastic perturbations is considered. Stability of the positive (nontrivial) point of equilibrium of this equation and also a capability of its discrete analogue to preserve stability properties of the original differential equation are studied. For this purpose, the considered equation is centered around the positive equilibrium and linearized. Asymptotic mean square stability of the linear part of the considered equation is used to verify stability in probability of nonlinear origin equation. From known previous results connected with B. Kolmanovskii and L. Shaikhet, general method of Lyapunov functionals construction, necessary and sufficient condition of stability in the mean square sense in the continuous case and necessary and sufficient conditions for the discrete case are deduced. Stability conditions for the discrete analogue allow to determinate an admissible step of discretization for numerical simulation of solution trajectories. The trajectories of stable and unstable solutions of considered equations are simulated numerically in the deterministic and the stochastic cases for different values of the parameters and of the initial data. Numerous graphical illustrations of stability regions and solution trajectories are plotted.

GENERAL METHOD OF LYAPUNOV FUNCTIONALS CONSTRUCTION IN STABILITY INVESTIGATIONS OF NONLINEAR STOCHASTIC DIFFERENCE EQUATIONS WITH CONTINUOUS TIME

The general method of Lyapunov functionals construction has been developed during the last decade for stability investigations of stochastic differential equations with after-effect and stochastic difference equations. After some modification of the basic Lyapunov type theorem this method was successfully used also for difference Volterra equations with continuous time. The latter often appear as useful mathematical models. Here this method is used for a stability investigation of some nonlinear stochastic difference equation with continuous time.

About Lyapunov functionals construction for difference equations with continuous time

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.

Construction of Lyapunov functionals for stochastic difference equations with continuous time

One general method of Lyapunov functionals construction which was used earlier both for stochastic differential equations with aftereffect and for stochastic difference equations with discrete time here is applied for stochastic difference equations with continuous time.

Application of the general method of Lyapunov functionals construction for difference Volterra equations

Lyapunov functionals are used usually for stability investigation of systems with aftereffect [1–3]. The general method of Lyapunov functionals construction which was proposed and developed in [4–19] is used here for stochastic second type Volterra difference equations. It is shown that using this method, there is a possibility to construct for a given equation, a sequence of extending stability regions.

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.

About one application of the general method of Lyapunov functionals construction

AbstractSome peculiarities of the general method of Lyapunov functionals construction proposed and developed by the authors during last decade for stability investigation of stochastic hereditary systems are considered. It is shown, in particular, that using this method one can construct a sequence of extended stability regions for stochastic difference Volterra equations. Copyright © 2003 John Wiley & Sons, Ltd.