Modeling and analysis of the spread of some epidemics with Markov chains

The aim of this thesis is to analyze the spread dynamics of infectious diseases through mathematical modeling techniques. In particular, SIS (Susceptible–Infected–Susceptible) type epidemic models, which are of critical importance in public health, are examined from both deterministic and stochastic perspectives. By utilizing discrete-time stochastic models, the random nature of epidemic processes is taken into account, and the temporal behavior of the system is evaluated under various parameter settings. In this study, based on fundamental probability theory and Markov chains, the SIS model is reconstructed as a discrete-time Markov chain. The transition probabilities are defined parametrically, and the system behavior is observed through numerical simulations. The outcomes of the stochastic model are compared with their deterministic counterparts, demonstrating that stochastic models provide a more realistic representation of epidemic progression. The results reveal that the randomness in transmission, along with birth and transition probabilities, has a significant impact on epidemic dynamics. In this regard, the study shows that investigating SIS models through stochastic structures contributes to a more accurate analysis of the spread of infectious diseases.