Continuous-time Hybrid Markov/semi-Markov Model with Sojourn Time Approach in the Spread of Infectious Diseases

The semi-Markov model is another name for the continuous-time Markov model. This model exponentially disperses the amount of time spent on each sojourn. This work aims to develop a model of infectious disease transmission using a continuous-time hybrid Markov/semi-Markov model. The model will combine assumptions from the Markov and semi-Markov models. The sojourn time in the semi-Markov model may have an exponential or a Weibull distribution, depending on the circumstances. A hybrid Markov/semi-Markov model can consist of both types of sojourn time distributions. In general, there are two parameterizations in the semi-Markov model: the transition intensity function and the sojourn time distribution, each of which has a different probabilistic and inferential perspective. This paper uses data on COVID-19 cases in DIY, Indonesia, from March 15, 2020, to July 31, 2020. This research uses four states: susceptible, infected, recovered, and deceased, where the sojourn time in the susceptible state is in the Weibull distribution while the sojourn time in other states has an Exponential distribution. This is because, at the beginning of the spread of COVID-19, few cases were found, so the distribution of sojourn time in each state tends to be constant except for the sojourn time in susceptible states. Under the semi-Markov model, the Weibull distribution leads to a dynamic probability with a higher degree of decline and a slight difference. In the final section, a comparison is made of the Markov, semi-Markov, and hybrid Markov/semi-Markov models. The hybrid Markov/semi-Markov model shows the best results with the smallest AIC value. Next, make a prediction equation for the SIRD model assuming a hybrid Markov/semi-Markov, which gives a MAPE < 20%. This means that the model’s ability to predict COVID-19 cases is good.