The preva‎lence of infectious diseases such as AIDS, cancer, and currently Covid 19 has affected human societies without a definitive cure. Predicting the disease process can control the spread of the disease to some extent. Scientists have found that mathematical models can predict this process in the long run. A mathematical model is a mathematical description of a real-world phenomenon. To better analyze an infectious disease, all the factors affecting in disease can be expressed as a mathematical model. Mathematical models are usually represented as a system of ordinary differential equations that the disease process can be understood by examining whose dynamics and the existence of bifurcations. Nowadays, dynamical system play a significant role in the prediction of infectious diseases. In general, the dynamics of infectious diseases is an important approach to investigating the transmission of infectious diseases, in which mathematical models are used a lot. Mathematical models are used to describe the mechanism of disease transmission and the dynamics of infectious agents. These models are built based on population dynamics, disease transmission behavior, characteristics of infectious agents, and their relationship with other social and physiological factors. They provide a good understanding of the spread of disease infection. One of the diseases whose models have been studied intracellularly is AIDS or Acquired Immune Deficiency Syndrome. It is a viral disease that is the fourth cause of death worldwide. The cause of this pandemic is HIV. The virus slowly destroys the immune system over several years. Thus, when the immune system is weakened, AIDS occurs. The virus is transmitted through sharp objects, breastfeeding, blood transfusions, etc. In this thesis, we examine the dynamics of advanced intracellular models of cancer and AIDS that play a critical role in predicting and spreading the disease. We will first review some basic concepts of the immune system and the virus. We will examine the stage of the virus entering the body until the onset of AIDS. Then, we will review various mathematical models from the beginning until now. In the following, we will express the new disease model with the available information about the disease and the development of existing models. In the extended models, many factors in the process of disease spread have been considered. Their dynamical analysis includes finding equilibrium points, stability of equilibrium points, basic reproduction rate, the presence of bifurcation in the system, and the effect of treatment on the disease as a parameter and control function using the principle of optimal control of Pontriagin and its simulation. With the dynamical analysis of the related mathematical model, it is possible to find out the process of spreading the disease.

رسول عاشقي

مريم نصيريان

حميدرضا ظهوري زنگنه، حميد شريفي، رسول كاظمي