Analysis of Measles Transmission and Vaccination Coverage Effects in Afghanistan Using the SIRV Mathematical Model

Said Omar Saidi; Mohammad Khan  Haidary

doi:10.62810/jnsr.v4i2.454

Authors

Said Omar Saidi Shaikh Zayed University, Department of Mathematics, Faculty of Education, Khost, Afghanistan
Mohammad Khan Haidary Kabul University, Department of Algebra, Faculty of Mathematics, Kabul, Afghanistan

DOI:

https://doi.org/10.62810/jnsr.v4i2.454

Keywords:

Basic reproduction number (r₀); , Disease-free equilibrium (dfe), Endemic equilibrium (ee)Measles, Sirv model

Abstract

In this study, the transmission dynamics of measles in Afghanistan and the effects of vaccination coverage are analyzed using an SIRV mathematical model, in which the transmission coefficient (β), the vaccination rate (v), and the basic reproduction number (R₀) play essential roles. The model's nonlinear differential equations are solved numerically using the fourth-order Runge–Kutta method (RK4), and solutions are computed in the Python programming environment to accurately investigate the behavior of different population compartments over time. The basic reproduction number (R₀) is derived using the Next Generation Matrix (NGM) method. Numerical and graphical results indicate that increasing the vaccination rate (v) significantly reduces R₀ and the number of infected individuals. The findings further demonstrate that reducing the transmission coefficient and improving vaccination coverage are effective strategies for controlling measles outbreaks. In addition, the minimum vaccination coverage required for disease control is calculated. The significance of this study lies in providing a scientific and quantitative framework for understanding measles transmission dynamics and evaluating vaccination strategies in Afghanistan. Overall, the proposed model and numerical approach offer valuable insights for infectious disease analysis and support evidence-based public health policies.

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References

Anderson, R. M., & May, R. M. (1991). Infectious diseases of humans: Dynamics and control. Oxford University Press.Link

Deng, Z.-Q., Singh, V. P., & Bengtsson, L. (2020). Numerical solution of fractional advection–dispersion equation. Applied Mathematics and Computation, 371, 124943.Link

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. Link

Fine, P. E., & Clarkson, J. A. (1982). Measles in England and Wales I: An analysis of factors underlying seasonal patterns. International Journal of Epidemiology, 11(1), 5–14.Link

Griffin, D. E. (2018). Measles virus. In D. M. Knipe & P. M. Howley (Eds.), Fields virology (7th ed., pp. 1944–1961). Wolters Kluwer.Link

Heesterbeek, J. A. P. (2002). A brief history of R₀ and a recipe for its calculation. Acta Biotheoretica, 50(3), 189–204. Link

Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Review, 42(4), 599–653.Link

Kumar, S., Jain, S., & Gupta, S. (2019). Mathematical modeling and analysis of infectious disease with vaccination strategy. Chaos, Solitons & Fractals, 129, 1–10.Link

Kumar, S., & Srivastava, P. K. (2021). Stability analysis of an epidemic model with vaccination and nonlinear incidence rate. Mathematical Methods in the Applied Sciences, 44(5), 3953–3970. Link

Patel, M., Lee, A. D., Clemmons, N. S., Redd, S. B., & Gastañaduy, P. A. (2019). National update on measles cases and outbreaks—United States, January 1–October 1, 2019. MMWR Morbidity and Mortality Weekly Report, 68, 893–896. Link

PopulationPyramid.net. (n.d.). Population pyramids of the world from 1950 to 2100: Afghanistan (2001–2024) [Data set]. Retrieved January 8, 2026, Link

Chatterjee, A. N., Sharma, S. K., & Al Basir, F. (2024). A compartmental approach to modeling the measles disease: A fractional order optimal control model. Fractal and Fractional, 8(4), 446.Link

Kröger, M., & Schlickeiser, R. (2023). On the analytical solution of the SIRV-model for the temporal evolution of epidemics for general time-dependent recovery, infection and vaccination rates. Preprints.Link

Schlickeiser, R., & Kröger, M. (2024). Mathematics of epidemics: On the general solution of SIRVD, SIRV, SIRD, and SIR compartment models. Mathematics, 12(7), 941.Link

Perasso, A. (2018). An introduction to the basic reproduction number in mathematical epidemiology. ESAIM: Proceedings and Surveys, 62, 123–138.Link

Fosu, G. O., Akweittey, E., & Adu-Sackey, A. (2020). Next-generation matrices and basic reproductive numbers for all phases of the coronavirus disease. Mathematical Sciences.Link

Jones, J. H. (2007). Notes on R₀. Stanford University.Link

Shirazian, M., Ammarloo, Z., & Janrian, Z. (2016). Optimal control strategy for rumor spreading using a modified SIR model. Proceedings of the National Seminar on Control and Optimization. (In Persian).

van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180, 29–48. Link

Peter, O. J., & Oguntolu, F. A. (2025). Mathematical model of measles epidemiology with control strategies. Mathematical Methods and Models in Biosciences Conference, Sofia.Link

Ishikawa, M. (2012). Optimal strategies for vaccination using the stochastic SIRV model. Journal of the Society of Instrument and Control Engineers, 25(12), 343–348.

Seydou, M., & Moussa Tessa, O. (2023). A stochastic SIRV model to estimate the effective reproductive number for measles epidemic in Niger. Journal of Advances in Mathematics and Computer Science, 38(10), 218–232. Link

Dai, Q., & Wang, Z. (2024). SIRV fractional epidemic model of influenza with vaccine game theory and stability analysis. Electronic Research Archive, 32(12), 6792–6821. Link

Grela, E., Stich, M., & Chattopadhyay, A. K. (2018). Epidemiological impact of waning immunization on a vaccinated population. arXiv preprint.Link

Sinha, A. (2023). A predictive analysis of measles outbreak dynamics using SIRV modeling (Preprint).

Oke, M. O., Ogunmiloro, O. M., Akinwumi, C. T., & Raji, R. A. (2019). Mathematical modeling and stability analysis of a SIRV epidemic model with nonlinear force of infection and treatment. Communications in Mathematics and Applications, 10(4), 717–731. link

Tang, L., Zhou, Y., Wang, L., Purkayastha, S., Zhang, L., He, J., Wang, F., & Song, P. X.-K. (2020). A review of multi-compartment infectious disease models. International Statistical Review, 88(2), 462–513.Link

Bjørnstad, O. N., Shea, K., Krzywinski, M., & Altman, N. (2020). The SEIRS model for infectious disease dynamics. Nature Methods, 17, 557–558.Link

Hou, Y., & Bidkhori, H. (2024). Multi-feature SEIR model for epidemic analysis and vaccine prioritization. PLOS ONE, 19(3), e0298932.Link

Tegegn, T. A. (2024). Age-structured SEIR model with vaccination for modeling the spread of measles in Africa. BIOMATH.

El-Doma, M. (2006). Stability analysis for an age-structured SEIR epidemic model under vaccination. Applications and Applied Mathematics, 1(2), 96–111.

World Health Organization. (2019, October). SAGE meeting documents: Measles and rubella vaccines [Presentation]. Link

World Health Organization. (2020). Measles fact sheet. Retrieved January 8, 2026. Link

World Health Organization. (2025a). Measles vaccines: WHO position paper. World Health Organization.Link

World Bank.(2024).Life expectancy at birth, total (years)-Afghanistan. Link

World Health Organization. (2025b). Afghanistan infectious disease outbreaks situation report: Epidemiological week 42, 2025 (Acute watery diarrhea cases). WHO Regional Office for the Eastern Mediterranean. Link