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All Journal Limits: Journal of Mathematics and Its Applications
Hadi Sumarno
Institut Pertanian Bogor
Computer Science & IT Mathematics

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Kajian Analisis dan Simulasi Model Stokastik SIRS Pendekatan CTMC Putranto Utomo; Astri Wiliastri; Hadi Sumarno
Limits: Journal of Mathematics and Its Applications Vol. 23 No. 2 (2026): Limits: Journal of Mathematics and Its Applications Volume 23 Nomor 2 Edisi Ju
Publisher : Pusat Publikasi Ilmiah LPPM Institut Teknologi Sepuluh Nopember

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.12962/limits.v23i2.9115

Abstract

Infectious disease modelling has been widely developed through deterministic and stochastic approaches to describe transmission dynamics, disease persistence, and extinction. Various compartmental models, including SIS, SIRS, and SEIRS models and their modifications, have been proposed to represent different epidemiological mechanisms. This study investigates the transmission behaviour of infectious diseases using a stochastic Susceptible-Infected-Recovered-Susceptible (SIRS) model based on a Continuous-Time Markov Chain (CTMC) approach, with parameter values referring to tuberculosis data. The analysis shows that the effective reproduction number approaches the basic reproduction number as the ratio of initially susceptible individuals to the total population increases. When this ratio is close to one, the effective reproduction number approximates the basic reproduction number. The expected number of new infections is consistent with the basic reproduction number. The theoretical disease-free probability obtained from the branching-process formulation is 0.7075. Numerical simulations were conducted using 1,000 trajectories with a total population of 1,000 individuals. The simulation produced a disease-free probability of 0.7030, which is consistent with the theoretical result, with a relative error of 0.64%. The time to reach the disease-free state ranged from one day to 21 years, with an average of three years. For trajectories that remained endemic, the proportion of infected individuals ranged from 1.2% to 9.8%, with an average of 5.8%. These findings indicate that the stochastic CTMC approach provides a more detailed probabilistic interpretation of disease-free and endemic outcomes
Co-Authors Astri Wiliastri Putranto Utomo