Lyapunov stability of a biological model with piecewise constant argument

Many problems in engineering, physics, mechanics, chemistry, biology and similar areas are modeled mathematically by differential equations. A mathematical model is the mathematical expression of physical phenomena such as population growth, behaviour of a biological model, epidemics, the amount of a chemical substance in a chemical reaction and the change of this amount over time. The purpose of a mathematical model is to examine the process and enable future states to be predicted. For example, analysis of epidemic disease models provides to understand the current situation of the disease and obtain the necessary conditions to control it. Thus, the use of disease response measures and the management of medical resources are greatly affected. Therefore, mathematical modeling of epidemic diseases is of critical importance. If the rate of change of an unknown function in a system under consideration depends on the values in the past, such systems are modeled with the help of delay differential equations. Delay differential equations are frequently used in many fields such as biology, medicine, science, engineering and mechanics. Since even very small delay amounts can cause significant differences in the system, the use of delay differential equations in modeling the problems gives more realistic results. Differential equations with piecewise constant argument, which is the subject of this thesis, belong to the class of delay differential equations. Recently many studies have been carried out on the theory of differential equations with piecewise constant argument and on the applications of this theory. Hence, this type of differential equations is important for applications. In the present study, the issue model is an epidemic disease (SIR) model. In studies conducted in the literature, epidemic disease models in society have been created by SI, SIR, SIS, SIRS, SEIR, MSEIR epidemic models. Taking the disease incubation period of people who are in contact with an infected person into account, the epidemic disease in this study is modeled by differential equations with generalized piecewise constant argument, which is a delayed type differential equation. The stability of the solutions is investigated by applying the v second Lyapunov method to the handled epidemic model which includes a generalized piecewise constant argument.