Markov SIRD Epidemic Model with Semi-Markov Sojourn-Time Analysis of COVID-19

Document Type : Original Article

Authors
Faihatuz Zuhairoh 1
M. Rais Ridwan 2
1 Department of Mathematics Education, STKIP YPUP Makassar, South Sulawesi, Indonesia
2 Study Program of Mathematics Education, STKIP YPUP Makassar, South Sulawesi, Indonesia
10.22034/jirss.2026.2073280.1143
Abstract
The SIRD (Susceptible–Infected–Recovered–Deceased) model is a standard framework for analyzing infectious disease dynamics. Classical continuous-time Markov formulations assume constant transition rates and memoryless (exponential) sojourn-times, which may oversimplify empirical epidemic processes. In this study, the Markov SIRD model is employed as a baseline, with transition parameters estimated analytically via maximum likelihood. To assess the validity of the exponential sojourn-time assumption, a semi-Markov framework is introduced exclusively for duration analysis of the infected state. Specifically, the sojourn-times associated with recovery (I --> R) and death (I -->D) transitions are modeled using exponential and Weibull distributions and compared using likelihood-based criteria. Using COVID-19 data from the Special Region of Yogyakarta, Indonesia, the results show that Weibull distributions provide a substantially better fit than the exponential assumption for both recovery and mortality durations. These findings indicate significant deviations from the memoryless assumption underlying the Markov model. This study does not construct a full dynamic semi-Markov epidemic simulator; instead, the semi-Markov framework is used to statistically characterize and evaluate the temporal structure of infected-state durations. The results highlight the importance of realistic sojourn-time modeling for understanding epidemic progression and for assessing the limitations of classical Markov-based epidemic models.
Keywords
COVID-19
maximum likelihood method
semi-Markov
SIRD model
sojourn-time
Subjects
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Journal of the Iranian Statistical Society
Volume 24, Issue 2
December 2025
Pages 91-106

  • Receive Date 01 October 2025
  • Revise Date 02 January 2026
  • Accept Date 27 January 2026