The SIRD Epidemic Model with Markov and Semi-Markov Assumptions

[Objective] This study aims to complement the shortcomings of the Markov model in modeling the spread of infectious diseases by adding the assumption of sojourn time in each state, called the semi-Markov model. The semi-Markov process does not have the limitations of the Markov model, where reality shows that the sojourn time distribution function in a continuous-time semi-Markov process can take any form. [Materials and Methods] There are two ways to define a multi-state model based on semi-Markov assumptions: the sojourn time approach and the transition intensity function approach. This study uses the sojourn time method to determine the appropriate distribution for each state transition. Sojourn time distribution comprises the Exponential distribution and the Weibull distribution. The model estimate employs maximum likelihood methods based on the sojourn time approach. [Results] The semi-Markov assumption is applied to the SIRD epidemic model, which consists of four states: susceptible, infected, recovered, and deceased, using COVID-19 data from the Special Region of Yogyakarta, Indonesia. Comparing the AIC values of the two models revealed that the semi-Markov model produced superior results than the Markov model. Under the semi-Markov model, the Weibull distribution makes dynamic probabilities with a faster rate of decline and smaller variance. [Conclusion] The SIRD epidemic model with a semi-Markov assumption can overcome the weaknesses of the Markov assumption. So to model the spread of infectious diseases, it is better to pay attention to the sojourn time in one state before transitioning to another