Hand, foot, and mouth disease (HFMD), associated with more than 20 disease-causing enteroviruses, is one of the major public health problems in mainland China, and the unique vaccine available is for enterovirus 71 (EV71). In this paper, we propose a new epidemic model to investigate the effect of EV71 vaccination on the spread of HFMD with multiple pathogenic viruses in mainland China. In addition, suitable periodic transmission functions are designed, with a two-year period and taking into consideration the effects of opening and closing of schools. After defining the basic reproduction number R0, we prove that the disease-free equilibrium is globally asymptotically stable if R0<1, and there exists at least one positive periodic solution and the disease is uniformly persistent if R0>1. We use the model to simulate the HFMD reported data in mainland China from January 2008 to June 2019. The numerical experiments show that increasing the vaccinated rate can effectively control the spread of HFMD in mainland China, yet the disease does not become extinct. Moreover, if we can control the baseline contact rate of infectious individuals and the recovery rate of symptomatic infectious individuals under certain conditions, which can be achieved by improving protective measures and medical conditions, then the disease will be eliminated.