A game theoretic approach to discuss the positive secondary effect of vaccination scheme in an infinite and well-mixed population

Pre-emptive vaccination policy used in controlling the rapid spreading of infectious diseases is considered as one of the most challenging issues imposed to mankind, causing enormous death tolls over the years. This paper dedicatedly studies the dilemma effect coming from the failure of getting perfect immunity to those individuals who committed vaccination earlier. Therefore, we propose a new theoretical model that slows down the infection spreading and also facilitates quicker recovery time than what the previous model does even if a vaccinator fails to obtain perfect immunity. We name this effect as the “positive secondary effect” of vaccination as it gives a second chance to the vaccinators which in return subdues the rapid spreading that helps in producing better social average payoff as well as keeping the final epidemic size smaller. Moreover, to address the positive secondary effect more precisely, we introduce two different parameters; namely, relaxation parameter (η) and foster parameter (δ) in two different directions to quantify the individual effects resulting from each of the parameter space as well as their superposition effect. An in-depth discussion focuses on the influential role played by our proposed model via discounting and faster recovery effects while a second chance is given to the vaccinators. In addition, we also examine the situation when discounting effect brought by η outperforms very much than its faster recovery controlled by δ as well as the superposition effects. Unlike all previous studies dealing with vaccination game, we pay much attention to investigating the secondary effect of imperfect vaccination policy. Our proposed theoretical scheme completely reproduces the decision-making process of choosing an imperfect provision based on evolutionary game theory entailed with the widely used SIR (Susceptible–Infected–Recovered) epidemic model. Without considering any spatial structure and perfect vaccination policy, our model presumes the population being infinite and well-mixed to represent the infection spreading dynamics mathematically. This study is conducted throughout using the so-called theoretical approach. Besides that, three different updating rules based on evolutionary game theory have also been considered to investigate all possible situations. Later on, we draw 2D full phase diagrams showing the final epidemic size, vaccination coverage, and average social payoff quantitatively. Finally, our theoretical result is compared with the counterpart result obtained from the multi-agent simulation (MAS) approach and a good agreement is found, hence the appropriateness of the proposed model is fully justified.