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Optimal Immunity Control and Final Size Minimization by Social Distancing for the SIR Epidemic Model.

Pierre-Alexandre Bliman1, Michel Duprez2, Yannick Privat3

  • 1Inria, Sorbonne Université, Université Paris-Diderot SPC, CNRS, Laboratoire Jacques-Louis Lions, équipe Mamba, Paris, France.

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Summary

This study reveals optimal social distancing strategies to minimize epidemic final size using the SIR model. Implementing maximum containment early, before herd immunity is reached, significantly reduces total infections.

Keywords:
Epidemic final sizeHerd immunityLockdown policyOptimal controlSIR epidemic model

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Area of Science:

  • Epidemiology
  • Mathematical Modeling
  • Public Health Interventions

Background:

  • Understanding epidemic dynamics is crucial for effective public health responses.
  • Minimizing the final size of an epidemic is a key objective in disease control.

Purpose of the Study:

  • To determine optimal containment strategies for the SIR epidemic model within a finite time frame.
  • To minimize the cumulative number of infected cases (epidemic final size).

Main Methods:

  • Application of partial or total containment strategies to the SIR epidemic model.
  • Mathematical analysis to prove the existence and uniqueness of an optimal strategy.
  • Development of a simple algorithm to identify optimal intervention start dates.

Main Results:

  • The optimal strategy involves maximal social distancing until the end of the intervention period.
  • Intervention should commence before the susceptible population reaches the herd immunity threshold.
  • Numerical simulations confirm a significant reduction in epidemic final size with the proposed strategy.

Conclusions:

  • An optimal, unique strategy exists for minimizing epidemic final size through social distancing.
  • Early and maximal intervention is more effective than delayed or partial measures.
  • The optimal intervention threshold is always below the herd immunity level.