A Simulation of the Discrete Time Markov ChainSusceptible Vaccinated Exposed Infected Recovered (DTMC SVEIR) Model with Relapse Cases in theSpread of Tuberculosis*
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Abstract
Tuberculosis (TB) is a contagious disease that pri-marily affects the lungs and spreads through the air whenan infected person coughs, sneezes, or spits. TB remains as asignificant global health issue, with millions of cases and deathsannually. There are mathematical epidemiological models thathave been developed, including SIR, SEIR, and SVEIR forexplaining the spread of disease pattern. The SVEIR modelexplains the impact of vaccination on disease transmission such asTB. In TB cases, the recovered individuals could be relapsed andbe infected again. This study discussed a Discrete Time MarkovChain (DTMC) SVEIR model and applied model simulation toTB disease. The population is divided into five groups such assusceptible, vaccinated, exposed, infected, and recovered. Thechanges of group’s state follow Markov process in discretetime observation. The model is extension from DTMC SEIRwith relapse by adding Vaccinated compartment. The modelis expressed as transition probability that explain the changesof state in each group. The simulation results indicate that thespread of the disease is significantly influenced by parametersvalue such as contact rate, vaccination rate, vaccine effectiveness,relapse rate, and recovery rate. If contact rate higher thanthe recovery rate then the number of infected individuals willincrease. The other hand, if vaccination works effectively thenthe number of susceptible individuals decrease. Lower relapserates lead to higher recovery numbers, whereas higher relapserates accelerate reinfection
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References
[1] P. Widyaningsih, A. A. Nugroho, D. R. S. Saputro, and Sutanto, “Tuber culosis transmission with relapse in Indonesia: Susceptible vaccinated infected recovered model,” J. Phys. Conf. Ser., vol. 1217, no. 1, 2019, doi: 10.1088/1742-6596/1217/1/012071.
[2] World Health Organization, “Tuberculosis.” [Online]. Available: https://www.who.int/news-room/fact-sheets/detail/tuberculosis
[3] World Health Organization, 2024 Global tuberculosis report. 2024.
[4] Ministry of Health of the Republic of Indonesia, “Tuberculosis Control Program Report 2022,” pp. 1–147, 2023, [Online]. Available: https://tbindonesia.or.id/pustaka tbc/laporan-tahunan-program-tbc-2021/
[5] N. Stojkovic, M. K. Paneva, A. S. Ilievska, and C. M. Bande, “SEEIR+DModel of Tuberculosis,” Balk. J. Appl. Math. Informatics, vol. VI, pp.115–124, 2023, doi: https://doi.org/10.46763/bjami.
[6] W. O. Kermack and A. G. McKendrick, “A Contribution to the Mathe matical Theory of Epidemics,” Proc. R. Soc. London Ser. A, vol. 772, no. 115, pp. 700–721, 1927, doi: https://doi.org/10.1098/rspa.1927.0118.
[7] Y. F. Valentino, Respatiwulan, and I. Slamet, “Simulation of the Discrete Time Markov Chain Susceptible Infected Recovered (DTMC SIR) Epidemic Model for COVID-19 Transmission in Central Java,” in ICHES: International Conference on Humanity Education and Society, 2024.
[8] J. D. Murray, Mathematical Biology I: An Introduction. New York: Springer Verlag, 2002.
[9] M. Martcheva, An Introduction to Mathematical Epidemiology. New York: Springer, 2015.
[10] M. J. Keeling and L. Danon, “Mathematical modelling of infectious diseases,” Br. Med. Bull., vol. 92, no. 1, pp. 33–42, 2009, doi:10.1093/bmb/ldp038.
[11] H. W. Hethcote, “The Mathematics of Infectious Diseases,” Soc. Ind. Appl. Math., vol. 42, no. 4, pp. 599–653, 2000, doi:
10.1137/S0036144500371907.
[12] L. J. S. Allen, “An Introduction to Stochastic Epidemic Models,” in Mathematical Epidemiology, Berlin, Heiderlberg: Springer, 2008. doi: 10.1007/978-3-540-78911-6.
[13] D. Wackerly, W. Mendenhall, and R. L. Scheaffer, Mathematical Statis tics with Applications. Thomson Brooks/Cole, 2008.
[14] R. W. West and J. R. Thompson, “Models for the Simple Epidemic,”Math. Biosci., vol. 141, no. 1, pp. 29–39, 1997, doi: 10.1016/s00255564(96)00169-1.
[15] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases. Chichester, UK: John Wiley & Sons, 2000.
[16] L. J. S. Allen, “A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis,” Infect. Dis. Model., vol. 2, no. 2, pp. 128–142, 2017, doi: 10.1016/j.idm.2017.03.001.
[17] E. Z. Nirmala, Respatiwulan, and Y. Susanti, “Simulasi Model DTMC SEIR pada Penyakit Tuberkulosis dengan Kasus Kambuh di Indonesia,” in Prosiding Seminar Nasional Pendidikan Matematika VI (Sandika VI), 2025, pp. 67–80. doi: https://doi.org/10.31941/prosandika.v6i1.2365.
[18] D. J. Daley and D. Gani, Epidemic modelling: An Introduction. Cambridge University Press, UK, 2001.
[19] C. Martin, N. Aguilo, and J. Gonzalo-Asensio, “Vaccination against tuberculosis,” Enferm. Infecc. Microbiol. Clin., vol. 36, no. 10, pp. 648–656, 2018, doi: 10.1016/j.actpha.2022.07.043.
[20] I. Abubakar et al., Systematic review and meta-analysis of the current evidence on the duration of protection by bacil
lus Calmette–Gu´erin vaccination against tuberculosis. Southampton (UK): NIHR Journals Library, 2013. [Online]. Available:
https://www.ncbi.nlm.nih.gov/books/NBK260958/
[21] I. N¨asell, “Stochastic models of some endemic infections,” Math. Biosci., vol. 179, no. 1, pp. 1–19, 2002, doi: doi.org/10.1016/S00255564(02)00098-6.
[22] T. Britton, “Stochastic epidemic models: A survey,” Math. Biosci., vol. 225, no. 1, pp. 24–35, 2010, doi: 10.1016/j.mbs.2010.01.006.
[23] F. Ball, T. Britton, C. Lar´edo, E. Pardoux, D. Sirl, and V. C. Tran, Stochastic Epidemic Models with Inference. Berlin: Springer, 2019.
[24] F. Brauer, C. Castillo-chavez, and Z. Feng, Mathematical Models in Epidemiology. New York: Springer, 2019. doi: doi:10.1007/978-1-49399828-9.
[25] C. I. Siettos and L. Russo, “Mathematical modeling of infectious disease dynamics,” Virulence, vol. 4, no. 4, pp. 295–306, 2013, doi: 10.4161/viru.24041.
[26] S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, 2nd ed. New York: Academic Press, 1975.
[27] O. Ige, “Markov Chain Epidemic Models and Parameter Estimation,”Marshall University, 2020.
[28] R. Durrett, Probability: Theory and Examples, 4th ed. Philadelphia: Duxbury Press, 2010.
[29] N. C. Grassly and C. Fraser, “Mathematical models of infectious disease transmission,” Nat Rev Microbiol, vol. 6, pp. 477–487, 2008, doi: 10.1038/nrmicro1845.
[30] W. Duan, Z. Fan, P. Zhang, G. Guo, and X. Qiu, “Mathematical and computational approaches to epidemic modeling: a comprehen