Stability and qualitative behavior in discrete and continuoustime mathematical models for the epidemiology of hepatitis B and monkey pox

Infectious diseases are global problems that pose serious threats to human health and can significantly reduce the quality of life. Humanity has faced numerous epidemics, such as plague, tuberculosis, and pneumonia, throughout history, and suffered heavy losses. However, the constant evolution of pathogens leads to the reemergence of past diseases with new variants or the emergence of entirely new diseases. Today, many infectious diseases, such as HIV, hepatitis B, COVID-19, monkeypox, and cholera, remain significant public health problems. In this context, mathematical modeling offers a powerful and systematic approach to understanding the spread of infectious diseases, predicting their future course, and developing effective control strategies. Mathematical models play a critical role in explaining disease transmission mechanisms, identifying critical parameters, assessing the impact of potential interventions, and providing scientifically based guidance to decision-makers. This thesis examines the transmission and spread dynamics of hepatitis B and monkeypox within the framework of mathematical modeling. While hepatitis B is a significant public health ABSTRACT (continued) problem that can cause serious problems and maintains its global prevalence, the rapid increase in monkeypox cases in recent years necessitates detailed investigations of this disease. To this end, mathematical models were developed for both diseases, ensuring the biological and mathematical consistency of the models and applying stability analysis, bifurcation analysis, sensitivity analysis, and optimal control methods. Numerous simulation studies were conducted to support the analysis results. The obtained results contribute to the identification of effective prevention and intervention strategies and provide important insights into the future course of the diseases. This study aims to provide public health experts and decision-makers with a scientifically based perspective in the fight against infectious diseases, thus making meaningful contributions to society.