Dynamic analysis of mean-field and fractional-order epidemic vaccination strategies by evolutionary game approach

Individual's perception of the participant in a vaccine program reflects their intrinsic appreciation of the trade-off between vaccine behavior, risk of infection, and memory effect. Here we present a mean-field approximation and fractional-order model embedded with the evolutionary game theory (EGT) process for susceptible-vaccinated-infected-recovered (SVIR) epidemic dynamics, where the two types of immunity, artificial and natural immunity, are considered. Besides the well-known vaccination game models, we successfully establish a theoretical approach of fractional-order dynamics for vaccination games in which epidemic spread and individual decisions are supposed to evaluate social behavior. The EGT governs the strategy adoption, while the fractional-order process directs the memory effect state. Our analytical forecasts are validated by numerical simulation of the finite difference method and Adams-Bashforth-Moulton algorithm for the mean-field and Caputo fractional-order derivative that assesses various graphs at varying parameters. Our results show that an effective vaccine may meaningfully minimize the risk of infection. However, despite vaccines' obvious impacts and advantages on a community, many individuals choose not to vaccinate for various reasons, leading to anti-vaccination groups and vaccine apprehension. In this aspect, people are interested in gaining natural immunity. Notably, the individual's risk perception is fundamental for controlling the infection, while the fractional-order dynamics mainly dictate the degree of freedom with memory effect. Higher fractional order with EGT leads to an improved vaccination intake, while a cheaper and reliable vaccine ensures to lessening of contagious disease. The proposed model captures relevant disease behavior and the memory effect of a synchronized pandemic event, emphasizing the underlying role of social schemes.