Abstract:
In this paper a mathematical model that investigates how vaccination affects the dynamics of COVID-19 was
considered. More particularly the model takes into account the waning rate of immunity after vaccination as well as
administration of booster vaccine. Posititivity and boundedness of solutions of the model were proved. The disease free
equilibrium of the model was determined and by using the next generation matrix method both the basic and effective
reproduction numbers of the model were determined. Further, from the effective reproduction number, the minimum critical
value of individuals to be vaccinated for containment of the diseases was determined. It was found that the value is less for a
perfect vaccine compared to an imperfect vaccine. Numerical simulation of the model was done to determine how the
parameters of interest in the study (waning rate of immunity, vaccination rate, administration of booster vaccine and efficacy of
the vaccine) affect the effective reproduction number. The results show that increasing the rates of vaccination, administering
booster vaccine will decrease the effective reproduction number while an increase in waning rate of immunity increases the
effective reproduction number. The disease persist in the population due to the declining of immunity after vaccination which
increases the effective reproduction number.
Keywords: Vaccination, Reproduction Number, COVID-19, Mathematical Model, Re-infection, Waning of Immunity
