Document Type : Research Article
Author
Department of Mathematics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran
Abstract
This paper presents a chickenpox-inspired mathematical modeling framework for analyzing varicella-like disease transmission dynamics, using a modified compartmental model that incorporates vaccination, isolation-induced contact reduction, treatment efficacy, and public awareness interventions. The objective is to develop an optimal-control framework and compare scenario-based intervention policies aimed at decreasing infection rates while considering economic and public-health constraints. For scenario comparison, the numerical simulations use fixed representative control levels; they do not numerically solve for time-dependent optimal control trajectories. The research applies Pontryagin’s Maximum Principle to formulate a theoretical optimal control framework, including the most critical intervention measures such as vaccination, isolation-induced contact reduction, treatment efficacy, and public awareness. Sensitivity analysis is utilized to examine the impact of model parameters on disease transmission, focusing on the basic reproduction number ℜ0. Numerical computations under various control regimes reveal that an appropriate combination of vaccination, isolation-induced contact reduction, treatment improvement, and public awareness campaigns can effectively reduce infection rates and provide theoretical insights into community-level intervention planning. The findings are highly beneficial for policymakers by supporting the evaluation of epidemic intervention scenarios and resource allocation decisions in public health administration.
Highlights
- A varicella-like SVEIR compartmental model with four integrated intervention controls is proposed.
- The disease-free equilibrium and basic reproduction number R0 are rigorously derived via the next-generation matrix.
- Sensitivity analysis identifies the transmission rate and recruitment rate as the most influential parameters.
- Pontryagin's Maximum Principle yields analytical characterizations of four optimal control functions.
- Scenario-based simulations show that a combined vaccination, isolation, treatment, and awareness strategy achieves the strongest infection reduction.
Keywords
- Chickenpox (Varicella)
- Optimal control
- Vaccination
- Isolation-induced contact reduction
- Sensitivity analysis
Main Subjects
[1] Abiola, I. O., Oyewole, A. S., Yusuf, T. T. (2024). “Mathematical modelling of Lassafever transmission dynamics with optimal control of selected control measures”. Modeling Earth Systems and Environment, 10(6), 7443–7458. https://doi.org/10.1007/ s40808-024-02168-z
[2] Ahmad, W., Butt, A. I. K., Rafiq, M., Asif, Z., Ismaeel, T., Ahmad, N. (2024). “Modeling, analyzing and simulating the measles transmission dynamics through efficient computational optimal control technique”. The European Physical Journal Plus, 139(7), 1–30. https://doi.org/10.1140/epjp/s13360-024-05368-9
[3] Akbari, R., Navaei, L., Shahriari, M. (2024). “Dynamical behaviour of fractional order SEIR mathematical model for infectious disease transmission”. Control and Optimization in Applied Mathematics, 9(1), 35–48. https://doi.org/10.30473/coam.2023. 64849.1210
[4] Almuallem, N. A., Ali, H. M., Elsaid, E. M., Eid, M. R., Hassanin, W. S. (2025). “Mathematical simulation of optimal control measures to avoid chickenpox infection”. International Journal of Mathematics and Mathematical Sciences, 2025(1), 3238188. https://doi.org/10.1155/ijmm/3238188
[5] Ali, A., Iqbal, Q., Asamoah, J. K. K., Islam, S. (2022). “Mathematical modeling for the transmission potential of Zika virus with optimal control strategies”. The European Physical Journal Plus, 137(1), 146. https://doi.org/10.1140/epjp/ s13360-022-02368-5
[6] Alshehri, A., Ullah, S. (2023). “Optimal control analysis of Monkeypox disease with the impact of environmental transmission”. AIMS Mathematics, 8(7), 16926–16960. https: //doi.org/10.3934/math.2023865
[7] Anderson, R. M., May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford. https://doi.org/10.1093/oso/ 9780198545996.001.0001
[8] Baig, A., Zhouxin, L. (2025). “Modeling infectious diseases: From SIR models to diffusion-based approaches and numerical solutions”. arXiv preprint arXiv:2502.15439. https://doi.org/10.48550/arXiv.2502.15439
[9] Baleanu, D., Hasanabadi, M., Mahmoudzadeh Vaziri, A., Jajarmi, A. (2023). “A new intervention strategy for an HIV/AIDS transmission by a general fractional modeling and an optimal control approach”. Chaos, Solitons & Fractals, 167, 113078. https://doi. org/10.1016/j.chaos.2022.113078
[10] Barbieri, E., Cocchio, S., Furlan, P., Scamarcia, A., Cantarutti, L., Donà, D., Giaquinto, C., Baldo, V. (2024). “A population database analysis to estimate the varicella vaccine effectiveness in children <14 years in a high vaccination coverage area from 2004 to 2022”. Vaccine, 42(26), 126387. https://doi.org/10.1016/j.vaccine.2024.126387
[11] Batista, M. (2021). “On the reproduction number in epidemics”. Journal of Biological Dynamics, 15(1), 623–634. https://doi.org/10.1080/17513758.2021.2001584
[12] Behncke, H. (2000). “Optimal control of deterministic epidemics”. Optimal Control Applications and Methods, 21(6), 269–285. https://doi.org/10.1002/oca.678
[13] Blayneh, K., Cao, Y., Kwon, H. D. (2009). “Optimal control of vector-borne diseases: Treatment and prevention”. Discrete and Continuous Dynamical Systems B, 11(3), 587– 611. https://doi.org/10.3934/dcdsb.2009.11.587
[14] Boulaaras, S., Yavuz, M., Alrashedi, Y., Bahramand, S., Jan, R. (2025). “Modeling the co-dynamics of vector-borne infections with the application of optimal control theory”. Discrete and Continuous Dynamical Systems – Series S, 18(5), 1331–1352. https://doi.org/10.3934/dcdss.2024109
[15] Brauer, F., van den Driessche, P., Wu, J. (Eds.). (2008). Mathematical Epidemiology. Lecture Notes in Mathematics, Vol. 1945. Springer, Berlin. https://doi.org/10.1007/ 978-3-540-78911-6
[16] Brauer, F., Castillo-Chavez, C., Feng, Z. (2019). Mathematical Models in Epidemiology. Texts in Applied Mathematics, Vol. 69. Springer, New York. https://doi.org/ 10.1007/978-1-4939-9828-9
[17] Brisson, M., Edmunds, W. J., Gay, N. J., Law, B., De Serres, G. (2000). “Modelling the impact of immunization on the epidemiology of varicella zoster virus”. Epidemiology and Infection, 125(3), 651–669. https://doi.org/10.1017/S0950268800004714
[18] Diekmann, O., Heesterbeek, J. A. P., Roberts, M. G. (2010). “The construction of next-generation matrices for compartmental epidemic models”. Journal of the Royal Society Interface, 7(47), 873–885. https://doi.org/10.1098/rsif.2009.0386
[19] Foutel-Rodier, F., Charpentier, A., Guérin, H. (2025). “Optimal vaccination policy to prevent endemicity: A stochastic model”. Journal of Mathematical Biology, 90(1), 10. https://doi.org/10.1007/s00285-024-02171-z
[20] Funk, S., Salathè, M., Jansen, V. A. A. (2010). “Modelling the influence of human behaviour on the spread of infectious diseases: A review”. Journal of the Royal Society Interface, 7(50), 1247–1256. https://doi.org/10.1098/rsif.2010.0142
[21] Garnett, G. P., Grenfell, B. T. (1992). “The epidemiology of varicella–zoster virus infections: A mathematical model”. Epidemiology and Infection, 108(3), 495–511. https: //doi.org/10.1017/S0950268800050007
[22] Grassly, N. C., Fraser, C. (2008). “Mathematical models of infectious disease transmission”. Nature Reviews Microbiology, 6(6), 477–487. https://doi.org/10.1038/nrmicro1845
[23] Hassan, A. H., Aldila, D., Noor Aziz, M. H. (2024). “Optimal control and stability analysis of monkeypox transmission dynamics with the impact of contaminated surfaces”. Frontiers in Applied Mathematics and Statistics, 10, 1372579. https://doi.org/10.3389/ fams.2024.1372579
[24] Heininger, U., Seward, J. F. (2006). “Varicella”. The Lancet, 368(9544), 1365–1376. https://doi.org/10.1016/S0140-6736(06)69561-5
[25] Hethcote, H. W. (2000). “The mathematics of infectious diseases”. SIAM Review, 42(4), 599–653. https://doi.org/10.1137/S0036144500371907
[26] Heydari Dastjerdi, R., Ahmadi, G., Dadkhah, M., Yari, A. (2023). “Optimal control of infectious diseases using artificial neural networks”. Control and Optimization in Applied Mathematics, 8(2), 17–32. https://doi.org/10.30473/coam.2023.64776.1208
[27] Holmdahl, I., Buckee, C. (2020). “Wrong but useful—what COVID-19 epidemiologic models can and cannot tell us”. New England Journal of Medicine, 383(4), 303–305. https://doi.org/10.1056/NEJMp2016822
[28] Jajarmi, A., Ebrahimzadeh, A., Khanduzi, R. (2025). “Coronavirus metamorphosis optimization algorithm and collocation method for optimal control problem in COVID-19 vaccination model”. Optimal Control Applications and Methods, 46(1), 292–306. https: //doi.org/10.1002/oca.3215
[29] Jose, S. A., Raja, R., Dianavinnarasi, J., Baleanu, D., Jirawattanapanit, A. (2023). “Mathematical modeling of chickenpox in Phuket: Efficacy of precautionary measures and bifurcation analysis”. Biomedical Signal Processing and Control, 84, 104714. https: //doi.org/10.1016/j.bspc.2023.104714
[30] Kalachev, L., Graham, J., Landguth, E. L. (2024). “A simple modification to the classical SIR model to estimate the proportion of under-reported infections using case studies in flu and COVID-19”. Infectious Disease Modelling, 9(4), 1147–1162. https://doi.org/10. 1016/j.idm.2024.06.002
[31] Kermack, W. O., McKendrick, A. G. (1927). “A contribution to the mathematical theory of epidemics”. Proceedings of the Royal Society of London, Series A, 115(772), 700–721. https://doi.org/10.1098/rspa.1927.0118
[32] Khanduzi, R., Jajarmi, A., Ebrahimzadeh, A., Shahini, M. (2024). “A novel collocation method with a coronavirus optimization algorithm for the optimal control of COVID-19: A case study of Wuhan, China”. Computers in Biology and Medicine, 178, 108680. https://doi.org/10.1016/j.compbiomed.2024.108680
[33] Kim, S., Choi, J. K., Suh, J., Park, S. H., Lee, J. (2024). “The epidemiologic and economic impact of varicella and herpes zoster vaccination in South Korea: A mathematical modelling study”. Vaccine, 42(19), 4046–4055. https://doi.org/10.1016/j.vaccine. 2024.05.016
[34] Lenhart, S., Workman, J. T. (2007). Optimal Control Applied to Biological Models. Chapman & Hall/CRC. https://doi.org/10.1201/9781420011418
[35] Li, T., Guo, Y. (2022). “Modeling and optimal control of mutated COVID-19 (Delta strain) with imperfect vaccination”. Chaos, Solitons & Fractals, 156, 111825. https://doi. org/10.1016/j.chaos.2022.111825
[36] Nisar, K. S., Farman, M., Abdel-Aty, M., Ravichandran, C. (2024). “A review of fractional order epidemic models for life sciences problems: Past, present and future”. Alexandria Engineering Journal, 95, 283–305. https://doi.org/10.1016/j.aej.2024.03. 059
[37] Nkeki, C., Mbarie, I. (2025). “On a mathematical model and the efficacy of control measures on the transmission dynamics of chickenpox”. Bulletin of Biomathematics, 3(1), 37– 61. https://doi.org/10.59292/bulletinbiomath.1707079
[38] Omoloye, M. A., Adewale, S. O. (2021). “Optimal control analysis on mathematical model of dynamical transmission of Ebola–Malaria co-infections”. Journal of Information and Computational Science, 11(9), 174–195.
[39] Ouakka, A., Alla Hamou, A., El Azzouzi, A., Agarwal, P. (2026). “A computational model of varicella dynamics with age-dependent waning immunity: Stability, parameter estimation, and optimal control via finite difference schemes”. Journal of Applied Mathematics and Computing, 72(4), 152. https://doi.org/10.1007/s12190-026-02806-y
[40] Peter, O. J., Madubueze, C. E., Ojo, M. M., Oguntolu, F. A., Ayoola, T. A. (2023). “Modeling and optimal control of monkeypox with cost-effective strategies”. Modeling Earth Systems and Environment, 9(2), 1989–2007. https://doi.org/10.1007/ s40808-022-01607-z
[41] Shi, L., Lu, J., Sun, X., Li, Z., Zhang, L., Lu, Y., Yao, Y. (2023). “Impact of varicella immunization and public health and social measures on varicella incidence: Insights from surveillance data in Shanghai, 2013–2022”. Vaccines, 11(11), 1674. https://doi.org/ 10.3390/vaccines11111674
[42] Siettos, C. I., Russo, L. (2013). “Mathematical modeling of infectious disease dynamics”. Virulence, 4(4), 295–306. https://doi.org/10.4161/viru.24041
[43] Skrip, L. A., Townsend, J. P. (2019). “Modeling approaches toward understanding infectious disease transmission”. In P. J. Krause, P. B. Kavathas, N. H. Ruddle (Eds.), Immunoepidemiology (pp. 227–243). Springer, Cham. https://doi.org/10.1007/ 978-3-030-25553-4_14
[44] Teklu, S. W. (2024). “Impacts of optimal control strategies on the HBV and COVID-19 co-epidemic spreading dynamics”. Scientific Reports, 14, 5328. https://doi.org/10. 1038/s41598-024-55111-8
[45] World Health Organization. (2014). “Varicella and herpes zoster vaccines: WHO position paper, June 2014”. Weekly Epidemiological Record, 89(25), 265–288. https: //www.who.int/publications/i/item/who-wer-8925-265-288
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