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Post-peak dynamics and epidemic overshoot in SIR-type frameworks
Zhimin Li1, Yijun Lou2, Zhen Jin3
1Complex Systems Research Center, Shanxi University, Taiyuan, 030006, China; School of Science, Chang'an University, Xi'an, 710064, China.
Mathematical Biosciences
|June 19, 2026
Summary
This study provides analytical results for cumulative infections and deaths post-epidemic peak using the SIR model. It offers tools for assessing post-peak risks and planning public health interventions.
Area of Science:
- Epidemiology
- Mathematical Biology
- Public Health
Background:
- Existing disease transmission models focus on pre-peak dynamics and reducing the reproduction number.
- Post-peak cumulative infections and deaths are critical but understudied aspects of epidemic control.
Purpose of the Study:
- To systematically investigate post-peak epidemic indices within the Susceptible-Infected-Recovered (SIR) model.
- To derive analytical results for cumulative infections and deaths after the epidemic peak.
- To provide practical tools for assessing post-peak risks and informing public health interventions.
Main Methods:
- Derivation of explicit analytical results for cumulative infections and deaths in the SIR model.
- Development of closed-form expressions for the epidemic peak and its timing.
- Establishment of rigorous upper bounds for post-peak cumulative infections and deaths.
- Validation of theoretical findings using numerical simulations on real-world epidemic data.
Main Results:
- Closed-form expressions for epidemic peak timing and magnitude derived.
- Sharp upper bounds established for cumulative infections and deaths post-peak.
- Theoretical results validated against real-world epidemic scenarios, demonstrating applicability across diverse incidence rates.
Conclusions:
- The study advances the theoretical understanding of post-peak epidemic dynamics in SIR models.
- Provides practical tools for accurate post-peak risk assessment and resource planning.
- Highlights limitations in deriving exact analytical expressions for cumulative post-peak outcomes in extended models.
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