Post-peak dynamics and epidemic overshoot in SIR-type frameworks

Most existing studies on disease transmission models concentrate on identifying control measures required to reduce the reproduction number below unity, with a primary focus on epidemic dynamics preceding the infection peak. However, the cumulative numbers of infections and deaths occurring after the epidemic becomes controllable (that is, the effective reproduction number drops to one) remain critical yet insufficiently explored. In this study, we systematically investigate post-peak epidemic indices by deriving explicit analytical results for the cumulative numbers of infected and deceased individuals following the infection peak within the SIR (Susceptible-Infected-Recovered) model framework. We present closed-form expressions for the epidemic peak and its timing, and rigorously establish sharp upper bounds for cumulative infections and deaths after the peak. The theoretical results are validated through numerical simulations based on three representative real-world epidemic scenarios. These findings advance the theoretical understanding of epidemic dynamics in SIR models with diverse incidence rates and offer practical tools for accurately assessing post-peak risks, thereby informing more effective public health interventions and resource planning. We finally explore, within some extended general model frameworks, the characteristics of cumulative numbers of infected and deceased individuals following the infection peak, and discuss the intrinsic limitations that prevent exact analytical expressions and upper bounds.