In this paper, a susceptible–infectious–recovered infectious disease model with the game inoculation process is established. First, the basic regeneration number $R_0$ of the model is obtained, and the positive and bounded connections of the system are proved. second, the stability of each equilibrium point is discussed by analyzing the characteristic equations of the system. Next, by taking $\omega$ as the parameter, the conditions for the occurrence of Hopf bifurcation are obtained. Furthermore, we use the method of parameter estimation to analyze the influence of each parameter on this system. Finally, through a numerical simulation, we verify all previous conclusions. Of course, symmetry provides a tractable framework for analyzing vaccination games but may overlook heterogeneity-driven phenomena.