The role of imperfect vaccination and treatment in containing an epidemic for a delayed SVITR epidemic model

This research is concerned with studying the global dynamics of a delayed SVIR epidemic model with two distributed delays. The vaccination considered in this research is imperfect, which means the vaccinated person can be infected after vaccination. The basic reproduction number [Formula: see text] is successfully identified and interpreted. The threshold role of this number is also determined, wherein for [Formula: see text], we established the existence of a globally stable disease-free equilibrium. Conversely, for [Formula: see text], the model demonstrates permanence, exhibiting a distinct endemic equilibrium. Furthermore, we demonstrate the global asymptotic stability of this unique endemic equilibrium. A different perspective is also cited, which is determining the effect of extensive measures, vaccination, and treatment on containing the epidemic and determining the required public health intervention force for stopping the epidemic outbreak. The findings are checked numerically using graphical representations, with a brief discussion on the biological interpretation of the results.