Social dilemma analysis on vaccination game accounting for the effect of immunity waning

A new SEIRS-based ODE (Ordinary Differential Equation) model is built, where asymptomatic and symptomatic infectious compartments, hospitalized state and immune state are additionally introduced, which is dovetailed with a behavior model that defines the vaccination rate as time-variable. We also apply Social Efficiency Deficit; SED, to quantitatively identify whether or not social dilemma taking place behind the model dynamics. Numerical results prove that, depending on the fraction of asymptomatic infected individuals and the discount ratio of infected force for asymptomatic state compared with that for symptomatic one, whether or not social dilemma appearing is different, which is also substantially affected by the vaccination cost and the immunity waning rate. The behavior model tends to incur a slight social dilemma brought by wasting vaccination even in the region of disease free, identified by R0<1. Meanwhile, a relatively strong social dilemma is indicated in the parameter region where the behavior model shows less amount of vaccinated population while the social optimal state needs much more vaccinated for complete remission of disease.