A new modification of the SIR epidemic model incorporating vaccination is proposed in the present paper. The recent trend of vaccinating against COVID-19 pandemic reveals a strong control of infectious disease. On the other hand, it is observed in some countries that, the vaccine application offers less control over the spread of virus, since some portion of vaccinated people is not totally protected/immuned and viable to infection again after a while due to weak/loss immunity offered by the vaccine. This requires transition from vaccinated department to infected for COVID-19. This character of COVID-19 helps us reconsideration of the vaccinated department by letting some part of it being exposed to the infection again. Taking this into account, as a result of modification of the SIR model, the epidemiology is now governed with three main epidemic dimensionless parameters, having provided an initial fraction of infected individuals. The dimensionless model with these parameters is analyzed initially from the stability point of view. The effects of weak immunity are then illustrated numerically on some chosen parameter range. How some of the countries applying the COVID-19 vaccine programs affected by weak/loss immunity is eventually examined with the modified model. The rate of vaccination as well as the basic Reproduction number are found to affect the epidemic demography of the population subject to weak or loss of immunity. In the case of a high vaccination rate, the countries are not anticipated to be highly influenced by the weak immunity of low level, whereas weak immunity prolongs the contagious disease by appearance of secondary multiple peaks in the epidemic compartments with relatively small vaccination rates and basic Reproductive numbers. (C) 2022 Elsevier B.V. All rights reserved.