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In contemporary society, social networks accelerate decision dynamics causinga rapid switch of opinions in a number of fields, including the prevention ofinfectious diseases by means of vaccines. This means that opinion dynamics cannowadays be much faster than the spread of epidemics. Hence, we propose aSusceptible-Infectious-Removed epidemic model coupled with an evolutionaryvaccination game embedding the public health system efforts to increase vaccineuptake. This results in a global system ``epidemic model + evolutionary game''.The epidemiological novelty of this work is that we assume that the switchingto the strategy ``pro vaccine'' depends on the incidence of the disease. As aconsequence of the above-mentioned accelerated decisions, the dynamics of thesystem acts on two different scales: a fast scale for the vaccine decisions anda slower scale for the spread of the disease. Another, and more methodological,element of novelty is that we apply Geometrical Singular Perturbation Theory(GSPT) to such a two-scale model and we then compare the geometric analysiswith the Quasi-Steady-State Approximation (QSSA) approach, showing acriticality in the latter. Later, we apply the GSPT approach to the diseaseprevalence-based model already studied in (Della Marca and d'Onofrio, Comm NonlSci Num Sim, 2021) via the QSSA approach by considering medium-large values ofthe strategy switching parameter.

A geometric analysis of the impact of large but finite switching rates on vaccination evolutionary games