The role of advanced and late provisions in a co-evolutionary epidemic game model for assessing the social triple-dilemma aspect

In the context of voluntary vaccination, we consider two additional provisions as well as pre-emptive vaccination for a unified model over epidemiology and evolutionary game theory to assess the extent to which advanced and late provisions restrict the spread of disease. To circumvent infection, people can be vaccinated pre-emptively before the epidemic season, but the imperfectness of vaccination or an unwillingness to be vaccinated may cause people instead to either be late-vaccinated or use self-protection. Here, self-protection corresponds to actions such as wearing a mask, washing hands, or using a mosquito net and is defined as the third strategy after pre-emptive vaccination (the first strategy) and late-vaccination (the second strategy). Our model can reproduce multiple social dilemma situations resulting from what is known as the vaccination dilemma (originating from preemptive vaccination), which works on a global time scale (i.e., repeated seasons approaching social equilibrium), and also from two other dilemmas due to late provisions, which work on a local time scale (i.e., every time step in a single season). To reproduce how an individual can acquire information for adaptation from neighbors or the society for a suitable provision, we introduce several strategy-updating rules for both global and local time scales and this behavioral feedback has a significant effect to reducing a transmissible disease. We also establish the social efficiency deficit (SED) indicator for a triple-dilemma game to quantify the existence of a social dilemma. Relying fully on a theoretical framework, our model provides a new perspective for evaluations: (i) how much more advantageous and effective pre-emptive vaccination is in eradicating a communicable disease compared with late provisions such as late vaccination and self-protection, and (ii) the extent of the social dilemma resulting from each of the three provisions, given the new idea of SED. The main effect of the triple-dilemma is that expensive provision displays no SED (no dilemma) until the efficiency or effectiveness of provisions reaches a certain level.