Mathematical Analysis of a SIR Epidemic Model with Vaccination and Awareness

Abstract

This study focuses to use mathematics to better understand how infectious diseases propagate. It suggests a SIR epidemic model with immunization and awareness campaigns. The population is split into three categories in this model: susceptible, infected, and recovered. While awareness campaigns seek to lower the rate of disease transmission, vaccination attempts to lower the number of susceptible people. The basic reproduction number and equilibrium points are found by analysing the model. The requirements for disease extinction are revealed via stability analysis of the disease-free equilibrium. The effects of vaccination and awareness on disease dynamics are illustrated through numerical simulations. According to the study, raising vaccination knowledge and coverage can successfully lower illness rates.

Introduction

The text discusses a mathematical study of infectious disease spread using an improved SIR model that includes vaccination and public awareness.

It begins by explaining that traditional SIR models divide a population into susceptible, infected, and recovered groups, but they do not account for important real-world factors like vaccination and awareness campaigns. To address this, the study introduces an extended SIR model where vaccination reduces the susceptible population and awareness lowers the transmission rate.

The literature review shows that previous research has expanded epidemic models in different ways, including adding awareness compartments, media influence, memory effects, and combined vaccination–awareness structures, all of which generally help reduce infection levels.

The main objectives of the study are to build this enhanced model, analyze its mathematical properties (such as equilibrium points and the basic reproduction number), determine stability conditions, and study how vaccination and awareness affect disease spread through simulations.

In the model formulation, the population is divided into susceptible, infected, and recovered individuals. The transmission rate is reduced by awareness, and vaccination moves individuals directly from susceptible to recovered. The key parameter is the basic reproduction number R0=β(1−δ)γ+νR_0 = \frac{\beta(1-\delta)}{\gamma + \nu}R0?=γ+νβ(1−δ)?, which determines whether the disease dies out (R0<1R_0 < 1R0?<1) or spreads (R0>1R_0 > 1R0?>1).

Stability analysis shows that the disease-free equilibrium is stable when R0<1R_0 < 1R0?<1, meaning vaccination and awareness help eliminate the disease. Numerical simulations further demonstrate how parameter values affect infection dynamics.

Conclusion

An SIR epidemic model that included vaccination and awareness effects was created and examined in this study. Stability conditions, equilibrium analysis, and model design were investigated. The findings demonstrate that when the reproduction number is less than one, raising vaccination coverage and awareness considerably lowers the spread of disease and aids in its eradication. The model can help public health officials and policymakers develop strategies that effectively manage infectious diseases. More intricate models including exposed populations, treatment plans, and time-dependent characteristics might be included in future research.

References

[1] Kabir, K. A., Kuga, K., & Tanimoto, J. (2019). Analysis of SIR epidemic model with information spreading of awareness. Chaos, Solitons & Fractals, 119, 118-125. [2] Majee, S., Barman, S., Khatua, A., Kar, T. K., & Jana, S. (2023). The impact of media awareness on a fractional-order SEIR epidemic model with optimal treatment and vaccination. The European Physical Journal Special Topics, 232(14), 2459-2483. [3] Shanta, S. S., & Biswas, M. H. A. (2020). The Impact of Media Awareness in Controlling the Spread of Infectious Diseases in Terms of SIR Model. Mathematical Modelling of Engineering Problems, 7(3). [4] Baba, I. A., Sani, M. A., Rihan, F. A., & Hincal, E. (2025). Modeling the impact of vaccination efficacy and awareness programs on the dynamics of infectious diseases. Journal of Applied Mathematics and Computing, 71(2), 1649-1671.

Copyright

Copyright © 2026 Rithika S, Dr. Mary Tency E L , Monicasri L S , Shaline S , Gayathri B , Harish M . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.