Vaccination-Based Modelling for the Dynamics of COVID-19

Muhammad Masood; Zia Ullah Afridi; Atiqa Sarfaraz; Siraj Amad

Vaccination-Based Modelling for the Dynamics of COVID-19

Muhammad Masood, Zia Ullah Afridi, Atiqa Sarfaraz, Siraj Ahmad

Vol. 13, Issue 10, PP. 61-71 October 2026

Keywords: Mathematical modelling, vaccination, stability analysis, basic reproduction number, numerical simulation, nonlinear dynamical systems

Download PDF

Coronavirus disease (COVID-19) was a serious health issue in late 2019 and caused a huge number of deaths worldwide. Various mathematical models were constructed to predict and analyse the dynamics of COVID-19. In this study, we formulate a deterministic model that incorporates a vaccination compartment and some other important parameters. The disease-free and disease-present equilibria were analysed, and the analytical expression for the basic reproduction number was derived using the next-generation matrix approach. Stability of the system was checked using Jacobian and eigenvalue analysis. The analytical results establish that the system preserves biological feasibility, with all state variables remaining positive and bounded over time. The model dynamics and effect of key parameters on the transmission of COVID-19 are demonstrated through numerical simulations using the MATLAB ODE45 solver. We found that vaccination has a positive impact on controlling COVID-19 spread.

Muhammad Masood, , University of Engineering and Technology Peshawar, Pakistan.
Zia Ullah Afridi, , University of Engineering and Technology Peshawar, Pakistan.
Atiqa Sarfaraz, , University of Engineering and Technology Peshawar, Pakistan.
Siraj Amad, , University of Engineering and Technology Peshawar, Pakistan.

Muhammad Masood Zia Ullah Afridi Atiqa Sarfaraz Siraj Ahmad “Vaccination-Based Modelling for the Dynamics of COVID-19” International Journal of Vol. 13 Issue 10 PP. 61-71 October 2026. https://doi.org/10.5281/zenodo.23320775.

[1]. Aronna, M. S., Guglielmi, R., & Moschen, L. M. (2021). A model for COVID-19 with isolation, quarantine and testing as control measures. Epidemics, 34, 100437.

[2]. Biala, T. A., & Khaliq, A. Q. M. (2021). A fractional-order compartmental model for the spread of the COVID-19 pandemic. Communications in Nonlinear Science and Numerical Simulation, 98, 105764.

[3]. Khan, A., Zarin, R., Hussain, G., Ahmad, N. A., Mohd, M. H., & Yusuf, A. (2020). Stability analysis and optimal control of COVID-19 with convex incidence rate in Khyber Pakhtunkhawa (Pakistan), Results Phys., 20 (2021), 103703. Crossref, Web of Science.

[4]. Esteban, E. P., & Almodovar-Abreu, L. (2021). Assessing the impact of vaccination in a COVID-19 compartmental model. Informatics in medicine unlocked, 27, 100795.

[5]. Iboi, E., Sharomi, O. O., Ngonghala, C., & Gumel, A. B. (2020). Mathematical modelling and analysis of COVID-19 pandemic in Nigeria. MedRxiv, 2020-05.

[6]. Rattanakul, C., & Chaiya, I. (2024). A mathematical model for predicting and controlling COVID-19 transmission with impulsive vaccination. AIMS Mathematics, 9(3), 6281-6304.

[7]. Trawicki, M. B. (2017). Deterministic seirs epidemic model for modelling vital dynamics, vaccinations, and temporary immunity. Mathematics, 5(1), 7.

[8]. Meyerowitz, E. A., Richterman, A., Gandhi, R. T., & Sax, P. E. (2021). Transmission of SARS-CoV-2: a review of viral, host, and environmental factors. Annals of internal medicine, 174(1), 69-79.

[9]. Jdid, T., Benbrahim, M., Kabbaj, M. N., & Naji, M. (2024). A vaccination-based COVID-19 model: Analysis and prediction using Hamiltonian Monte Carlo. Heliyon, 10(19).

[10]. Li, C. Z., Lü, X., Gong, J. J., & Lei, Y. (2025). Extended SEIR model of COVID-19 spread focusing on compartmental flow in England: C. Li et al. Nonlinear Dynamics, 113(1), 971-988.