https://doi.org/10.1140/epjp/s13360-025-07092-4
Regular Article
Mathematical modeling and symmetry analysis of the impact of vaccination on the dynamics of influenza virus disease
1
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan
2
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University(IMSIU), Riyadh, Saudi Arabia
3
Department of Mathematics, C. K. Tedam University of Technology and Applied Sciences(IMSIU), Navrongo, Ghana
4
Department of Anatomy and Physiology, College of Medicine, Imam Mohammad Ibn Saud Islamic University(IMSIU), 13317, Riyadh, Saudi Arabia
5
Mathematics Department, Faculty of Science, Taibah University, 41411, Al-Madinah Al-Munawarah, Saudi Arabia
6
Basic Sciences Research Center (BSRC), Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University(IMSIU), Riyadh, Saudi Arabia
Received:
2
October
2025
Accepted:
16
November
2025
Published online:
29
November
2025
Influenza is a transmissible respiratory disease that has proven catastrophic throughout history. Its rapid spread poses serious challenges to public health, as existing vaccines often provide limited protection. Mathematical modeling serves as a vital quantitative tool for predicting the dynamics and control of such epidemics. In this study, a Susceptible–Vaccinated–Exposed–Infected–Recovered (SVEIR) model is developed and analyzed for influenza transmission. The model guarantees the positivity and boundedness of all state variables, ensuring biological feasibility. Two equilibrium points, the influenza-free equilibrium (IFE) and influenza-endemic equilibrium (IEE), are established, and the basic reproduction number 
is derived to assess disease persistence. Analytical results show that the IFE and IEE are locally and globally stable when 
and 
, respectively. Sensitivity analysis indicates that the vaccination rate for newborns is the most influential parameter; a 25–30
increase in this rate leads to an approximately 
reduction in infection prevalence. Bifurcation analysis confirms a forward (supercritical) bifurcation, implying that maintaining below unity ensures disease elimination. Numerical simulations, performed using the Nonstandard Finite Difference (NSFD) scheme, validate the analytical findings and illustrate how increasing vaccination rates significantly reduce the susceptible and infected populations.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
