الفهرس 
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Abstract
Nonlinear systems are exposed to stochastic and environmental effects. Therefore, it is advisable to study these systems in a probabilistic (stochastic) sense as well as in the deterministic sense. Moreover, it should be noted the time delay occurred in these systems, and ignoring this time delay may lead to false conclusions. Consequently, noise and delay in the nonlinear system must be taken into consideration when modeling the real phenomena. In this thesis, the stability of nonlinear deterministic systems, stochastic systems, and delayed stochastic systems is investigated. The thesis consists of four chapters, and it is structured as follows. Chapter 1 makes the thesis as much as possible a self-contained piece of research. It includes most of the main definitions, concepts, and theoretical results. Based on these basic concepts, we formulate the stability theory of the nonlinear systems. Three basic mathematical models in the deterministic sense are presented. Epidemics modeled by the SVIR mathematical model with continuous vaccination strategy, the dynamics of the COVID-19 epidemic within the human body, and the dynamics of the Nicholson’s blowflies model in ecology. Chapter 2 introduces a new technique using Lyapunov functional for investigating the mean-square stability of the zero solution of the famous Nicholson’s blowflies equation. Sufficient conditions of the possible extinction of these species are obtained under some stochastic perturbations. The impact of the delay in time and the noise is shown. Chapter 3 studies the stochastic differential equation in the general form. Stochastic stability and stochastic global exponential stability of the equilibrium states are investigated. Dynamics of the stochastic HIV/AIDS epidemic is considered. Extinction of the disease is studied via the global exponential stability of the disease-free equilibrium of the stochastic HIV/AIDS model. Stability regions of the disease free equilibrium ad some numerical simulation are presented. Chapter 4 focuses on the general stochastic functional differential equation of the retarded type. Stability theory is presented with some examples. Global exponential mean-square stability of the Nicholson’s blowflies equation is investigated as well as the stability of the stochastic delayed Black-Sholes market model.