Mathematical models are critical in provision of information to the development of infections. Not with standing the effectiveness of vaccines, some vaccinated individuals nonetheless get infected. To deal with this, non-pharmaceutical measures inclusive of social distancing and handwashing are encouraged. This study offers a mathematical version that combines the effects of vaccination and social distancing, utilizing Kermack-McKendrick compartments and ordinary differential equations (ODE's). The study determines the basic reproduction number ($R_0$) by the use of the Next Generation Matrix (NGM). If $R_0$ is less than 1, the ailment will in the end die out; if $R_0$ is more than 1, the ailment will continue to spread. Python simulations show that while vaccination and social distancing can reduce transmission, they may not be sufficient to eliminate the disease entirely. Isolation is critical for reducing transmission similarly, the efficacy of vaccines and the vaccination rate are crucial additives of a vaccination strategy. These techniques provide extra time for public health officers to put in force further measures, supplementing current processes. As we continue to come upon with new and evolving health challenges, the mixing of most reliable management strategies into epidemic modelling may be important. Further studies and interdisciplinary collaboration will enhance our capability to combat infectious sicknesses.
[1] F. B. Agusto, Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model, Biosystems, 113(3) (2013), 155–164.
[2] A. Ayalew, Y. Molla, T. Tilahun, and T. Tesfa, Mathematical Model and Analysis on the Impacts of Vaccination and Treatment in the Control of the COVID-19 Pandemic with Optimal Control, Journal of Applied Mathematics, 2023 (2023).
[3] Y. Bai, L. Yao, T. Wei, F. Tian, D. Y. Jin, L. Chen, and M. Wang, Presumed asymptomatic carrier transmission of COVID-19, JAMA, 323(14) (2020), 1406-1407.
[4] A. Babaei, H. Jafari, S. Banihashemi, and M. Ahmadi, Mathematical analysis of a stochastic model for the spread of Coronavirus, Chaos, Solitons & Fractals, 145 (2021), 110788.
[5] G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Blaisdell Pub. Co., Waltham, Mass, (1969).
[6] S. Biswas, A. Subramanian, I. M. ELMojtaba, J. Chattopadhyay, and R. R. Sarkar, Optimal combinations of control strategies and cost-effective analysis for visceral leishmaniasis disease transmission, PLoS One, 12(2) (2017), e0172465.
[7] S. P. Brand, R. Aziza, I. K. Kombe, C. N. Agoti, J. Hilton, K. S. Rock, and E. W. Barasa, Forecasting the scale of the COVID-19 epidemic in Kenya, MedRxiv, 2020(04) (2020).
[8] K. M. Bubar, K. Reinholt, S. M. Kissler, M. Lipsitch, S. Cobey, Y. H. Grad, and D. B. Larremore, Model-informed COVID-19 vaccine prioritization strategies by age and serostatus, Science, 391(6352) (2021), 916-921.
[9] B. Buonomo, D. Lacitignola, and C. Vargas-De-Le´on, Qualitative analysis and optimal control of an epidemic model with vaccination and treatment, Mathematics and Computers in Simulation, 100 (2014), 88-102.
[10] C. Connelly, Epidemiology Through the Lens of Differential Equations, (2023).
[11] K. Dehingia, A. A. Mohsen, S. A. Alharbi, R. D. Alsemiry, and S. Rezapour, Dynamical behavior of a fractional order model for within-host SARS-CoV-2, Mathematics, 10(13) (2022), 2344.
[12] C. T. Deressa, Y. O. Mussa, and G. F. Duressa, Optimal control and sensitivity analysis for transmission dynamics of Coronavirus, Results in Physics, 19 (2020), 103942.
[13] M. L. Diagne, H. Rwezaura, S. Y. Tchoumi, and J. M. Tchuenche, A mathematical model of COVID-19 with vaccination and treatment, Computational and Mathematical Methods in Medicine, 2021 (2021).
[14] O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 395-392.
[15] K. Dietz, The estimation of the basic reproduction number for infectious diseases, Statistical Methods in Medical Research, 2 (1993), 23–41.
[16] K. D. Elgert, Immunology: understanding the immune system, John Wiley & Sons, (2009).
[17] L. X. Feng, S. L. Jing, S. K. Hu, D. F. Wang, and H. F. Huo, Modelling the effects of media coverage and quarantine on the COVID-19 infections in the UK, Math. Biosci. Eng., 17(4) (2020), 3918-3939.
[18] M. Fudolig and R. Howard, The local stability of a modified multi-strain SIR model for emerging viral strains, PLoS One, 15(12) (2020), e0243408.
[19] Y. Gu, S. Ullah, M. A. Khan, M. Y. Alshahrani, M. Abohassan, and M. B. Riaz, Mathematical modeling and stability analysis of COVID-19 with quarantine and isolation, Results in Physics, 34 (2022), 105284.
[20] S. D. Gupta, R. Jain, and S. Bhatnagar, COVID-19 pandemic in Rajasthan: Mathematical modelling and social distancing, Journal of Health Management, 22(2) (2020), 129-139.
[21] E. Hansen and T. Day, Optimal control of epidemics with limited resources, Journal of Mathematical Biology, 62 (2011), 423-451.
[22] D. La Torre, D. Liuzzi, T. Malik, O. Sharomi, and R. Zaki, Dynamics and optimal control for a spatially-structured environmental-economic model, Electronic Journal of Differential Equations, 2015, 1-15.
[23] S. Lenhart and J. T. Workman, Optimal control applied to biological models, Chapman and Hall/CRC, 2007.
[24] Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, ..., and Z. Feng, Early transmission dynamics in Wuhan, China, of novel coronavirus–infected pneumonia, New England Journal of Medicine, 392(13) (2020), 1199-1207.
[25] E. T. Lofgren, M. E. Halloran, C. M. Rivers, J. M. Drake, T. C. Porco, B. Lewis, ..., and S. Eubank, Mathematical models: A key tool for outbreak response, Proceedings of the National Academy of Sciences, 111(51) (2014), 1809518096.
[26] S. Mwalili, M. Kimathi, V. Ojiambo, D. Gathungu, and R. Mbogo, SEIR model for COVID-19 dynamics incorporating the environment and social distancing, BMC Research Notes, 13(1) (2020), 1-5.
[27] H. J. Namawejje, L. S. Luboobi, D. Kuznetsov, and E. Wobudeya, Modeling optimal control of rotavirus disease with different control strategies, Journal of Mathematical and Computational Science, 4(5) (2014), 892.
[28] E. G. Nepomuceno, M. L. Peixoto, M. J. Lacerda, A. S. Campanharo, R. H. Takahashi, and L. A. Aguirre, Application of optimal control of infectious diseases in a model-free scenario, SN Computer Science, 2(5) (2021), 405.
[29] G. Oliveira, Refined compartmental models, asymptomatic carriers and COVID-19, arXiv preprint, arXiv:2004.14780, (2020).
[30] B. Pantha, F. B. Agusto, and I. M. Elmojtaba, Optimal control applied to a visceral leishmaniasis model, (2020).
[31] C. A. Pearson, F. Bozzani, S. R. Procter, N. G. Davies, M. Huda, H. T. Jensen, and R. M. Eggo, Health impact and cost-effectiveness of COVID-19 vaccination in Sindh Province Pakistan, medRxiv, (2021).
[32] Z. H. Shen, Y. M. Chu, M. A. Khan, S. Muhammad, O. A. Al-Hartomy, and M. Higazy, Mathematical modeling and optimal control of the COVID-19 dynamics, Results in Physics, 31 (2021), 105028.
[33] B. Tang, X. Wang, Q. Li, N. L. Bragazzi, S. Tang, Y. Xiao, and J. Wu, Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, Journal of Clinical Medicine, 9(2) (2020), 462.
[34] P. Van Den Driessche and J. Watmough, Further notes on the basic reproduction number, Mathematical Epidemiology, (2008), 159-178.
[35] H. M. Wanjala, M. O. Okongo, and J. O. Ochwach, Impact of Media Awareness and Use of Face-Masks on Infectious Respiratory Disease, Pan-American Journal of Mathematics, 2 (2023), 15.
[36] H. M. Wanjala, M. Okongo, and J. Ochwach, Mathematical Model of the Impact of Home-Based Care on Contagious Respiratory Illness Under Optimal Conditions, Jambura Journal of Biomathematics (JJBM), 5(2) (2024), 83-94.
[37] I. M. Wangari, S. Sewe, G. Kimathi, M. Wainaina, V. Kitetu, and W. Kaluki, Mathematical modelling of COVID19 transmission in Kenya: a model with reinfection transmission mechanism, Computational and Mathematical Methods in Medicine, 2021 (2021).
[38] L. Yao, T. Wei, F. Tian, D. Y. Jin, L. Chen, and M. Wang, Presumed asymptomatic carrier transmission of COVID-19, JAMA, 323(14) (2020), 1406-1407.
[39] X. M. Zhang and Q. L. Han, New Lyapunov-Krasovskii functionals for global asymptotic stability of delayed neural networks, IEEE Transactions on Neural Networks, 20(3) (2009), 533-539.
Wanjala, H. M. , Okongo, M. O. and Ochwach, J. O. (2026). Mathematical modelling with optimal control of infectious diseases with vaccination. Computational Methods for Differential Equations, 14(2), 616-632. doi: 10.22034/cmde.2024.62350.2743
MLA
Wanjala, H. M., , Okongo, M. O., and Ochwach, J. O.. "Mathematical modelling with optimal control of infectious diseases with vaccination", Computational Methods for Differential Equations, 14, 2, 2026, 616-632. doi: 10.22034/cmde.2024.62350.2743
HARVARD
Wanjala, H. M., Okongo, M. O., Ochwach, J. O. (2026). 'Mathematical modelling with optimal control of infectious diseases with vaccination', Computational Methods for Differential Equations, 14(2), pp. 616-632. doi: 10.22034/cmde.2024.62350.2743
CHICAGO
H. M. Wanjala , M. O. Okongo and J. O. Ochwach, "Mathematical modelling with optimal control of infectious diseases with vaccination," Computational Methods for Differential Equations, 14 2 (2026): 616-632, doi: 10.22034/cmde.2024.62350.2743
VANCOUVER
Wanjala, H. M., Okongo, M. O., Ochwach, J. O. Mathematical modelling with optimal control of infectious diseases with vaccination. Computational Methods for Differential Equations, 2026; 14(2): 616-632. doi: 10.22034/cmde.2024.62350.2743
)