Fuzzy based Mathematical Model of Measles with Double-Dose Vaccination under Epidemiological Parameter Uncertainty
DOI:
https://doi.org/10.62054/ijdm/0303.15Abstract
This study develops a fuzzy-based mathematical model for the transmission dynamics of measles incorporating double-dose vaccination and treatment under epidemiological parameter uncertainty. The model extends a deterministic measles framework by representing selected parameters, including the transmission, vaccination, treatment and recovery rates, as triangular fuzzy numbers. Basic properties of the model are established, and the disease-free equilibrium and basic reproduction number are derived. The $r$-cut approach is employed to obtain interval-valued representations of the uncertain parameters and the fuzzy basic reproduction number. The results show that the uncertainty interval of the reproduction number decreases as the $r$-cut level increases and converges to the deterministic reproduction number at $r=1$. Intervention analysis further shows that a combined reduction in disease transmission with increased vaccination coverage and treatment can reduce both the deterministic reproduction number and the upper bound of the fuzzy reproduction number below unity. The proposed framework therefore provides a useful approach for assessing measles transmission and control when epidemiological parameters are imprecisely known.References
Berhe, H. W., Gebremeskel, A. A., Atsbaha, H. A., Kefela, Y. Y., Asgedom, A. A., Woldegerima, W. A., Osman, S., & Kabareh, L. (2024). Modelling and stability analysis of the dynamics of measles with application to Ethiopian data. Heliyon,
(13), e33594. https://doi.org/10.1016/j.heliyon.2024.e33594
Javidi, M., & Nyamoradi, N. (2017). A fuzzy SIR epidemic model with application to disease dynamics. Applied Mathematical Modelling, 43, 440452.
Khan, M. A., Ullah, S., & Farooq, M. (2019). Analysis of a fuzzy epidemic model with uncertain parameters. Journal of Intelligent & Fuzzy Systems, 37 (2), 23312342.
Kuddus, M. A., Mohiuddin, M., & Rahman, A. (2021). Mathematical analysis of a measles transmission dynamics model in Bangladesh with double dose vaccination. Scientic
Reports, 11, 16571. https://doi.org/10.1038/s41598-021-95913-8 Liu, S., Wang, A., Xue, Q., & Xie, N. (2025). Modelling the eect of vaccination on the transmission dynamics of measles in China. Advances in Continuous and Discrete Models, 2025, 155. https://doi.org/10.1186/s13662-025-03990-0
Mahmudov, E. N. (2011). Approximation and optimization of discrete and dierential inclusions. Elsevier. https://doi.org/10.1016/C2011-0-04277-1
Memon, Z., Qureshi, S., & Memon, B. R. (2020). Mathematical analysis for a new nonlinear measles epidemiological system using real incidence data from Pakistan. The European Physical Journal Plus, 135, 378. https://doi.org/10.1140/epjp/
s13360-020-00392-x
Moss, W. J. (2017). Measles. The Lancet, 390 (10111), 24902502. https://doi.org/ 10.1016/S0140-6736(17)31463-0
Patel, M. K., Goodson, J. L., Alexander, J. P., Jr., Kretsinger, K., Sodha, S. V., Steulet, C., Gacic-Dobo, M., Rota, P. A., McFarland, J., Menning, L., & Mulders, M. N. (2019). Progress toward regional measles elimination Worldwide, 20002018. MMWR Morbidity and Mortality Weekly Report, 68 (48), 11051111. https://doi.org/10.15585/mmwr.mm6848a1
van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180 (12), 2948. https://doi.org/10.1016/S0025-5564(02)00108
-6
World Health Organization. (2024). Measles. https://www.who.int/news-room/fact -sheets/detail/measles
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8 (3), 338353. https:// doi.org/10.1016/S0019-9958(65)90241-X
Zimmermann, H.-J. (2010). Fuzzy set theory and its applications (4th ed.). Springer.
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