To simulate hidden epidemic dynamics connected with asymptomatic and unregistered patients, a new general SIR model was proposed. For some cases, the analytical solutions of the set of 5 differential equations were found, which allow simplifying the parameter identification procedure. Two waves of the pertussis epidemic in England in 2023 and 2024 were simulated with the assumption of zero hidden cases. The accumulated and daily numbers of cases and the duration of the second wave were predicted with rather high accuracy. If the trend will not change, the monthly figure of 9 new pertussis cases (as it was in January-February 2023) can be achieved only in May 2025. The proposed approach can be recommended for both simulations and predictions of different epidemics.