Novel strains emerge every year and cause seasonal flu epidemic. These variants exhibit differences in antigenicity and may escape immunity from vaccines and thus invade the population. Here, we devise a discrete population game model of vaccination and transmission of two strains–a resident and a relatively rare mutant–to explain the influence of social behaviour, specifically vaccinating behaviour on the emergence of the mutant strain in the population. Our integrated framework of behaviour-prevalence model exhibits time evolution of two strategic interactions such as individuals' perceived vaccination risk and probability of infection, where both declines as more and more individuals vaccinate in the population. Numerical simulation and analysis of the model show that the predominant Nash strategy allows invasion of the mutant strain under very disparate individuals perceived disease risks from these two pathogens and relatively lower vaccine efficacy against the mutant. The model also demonstrates that invasion of the mutant has a nonlinear dependence on epidemiological characteristics such as the relative force of infection, cross-immunity, perceived cost of treatment and vaccine efficacy. This study exemplifies that adaptive social behaviour is an important component of public health strategic decision-making during the control of the flu-like epidemic. © 2021 Informa UK Limited, trading as Taylor & Francis Group.