University of Tabriz Computational Methods for Differential Equations 2345-3982 14 2 2026 04 01 Mathematical modelling with optimal control of infectious diseases with vaccination 616 632 19469 10.22034/cmde.2024.62350.2743 EN Henry M. Wanjala Chuka University, Chuka, Kenya. Mark O. Okongo Chuka University, Chuka, Kenya. Jimrise O. Ochwach Mama Ngina University College, Gatundu, Kenya. Journal Article 2024 07 04 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. https://cmde.tabrizu.ac.ir/article_19469_9ffc84c2254b11dc55a38cb1a99aab96.pdf