Koptleuova, DaianaKashkynbayev, ArdakTourassis, Vassilios D.2019-08-292019-08-292019-05-06http://nur.nu.edu.kz/handle/123456789/4193Submitted to the Department of Mathematics on May 6, 2019, in partial fulfillment of the requirements for the degree of Master of Science in Applied MathematicsThis thesis consider three type of epidemiological models: SIR, SIS and SIRS with nonlinear incidence rate and piecewise constant delay of generalized type. In this paper the total population size is varied with time elapse. We study the global asymptotic stability of the disease-free and endemic equilibrium states of models by constructing suitable Lyapunov functions and Lyapunov–LaSalle technique. The main contribution of this master thesis is to develop more realistic compartmental models by extending the literature of models with piecewise constant delay. The theoretical findings are illustrated through numerical simulations.enAttribution-NonCommercial-ShareAlike 3.0 United StatesResearch Subject Categories::MATHEMATICS::Applied mathematicsSIRSISSIRSepidemiological modelglobal asymptotic stabilityLyapunov functionsLyapunov–LaSalle techniqueendemic equilibrium states of modelsGlobal stability analysis for a tick-borne modelMaster's thesis