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Time-optimal control strategies in SIR epidemic models
Luca Bolzoni1, Elena Bonacini2, Cinzia Soresina3
1Risk Analysis Unit, Istituto Zooprofilattico Sperimentale della Lombardia e dell'Emilia Romagna, Via dei Mercati 13, Parma 43126, Italy.
Mathematical Biosciences
|August 13, 2017
Summary
Optimal control strategies for SIR epidemic models reveal that delaying interventions like vaccination or isolation, then applying them maximally, is often time-optimal. This contrasts with minimizing overall disease burden, highlighting trade-offs between speed and impact.
Area of Science:
- Epidemiology
- Mathematical Biology
- Control Theory
Background:
- Susceptible-Infected-Recovered (SIR) models are fundamental in understanding epidemic dynamics.
- Optimal control theory seeks to minimize disease spread and duration using interventions.
- Previous studies often focused on minimizing total infectious burden, not necessarily outbreak duration.
Purpose of the Study:
- To investigate time-optimal control strategies for SIR epidemic models.
- To analyze the effectiveness of different control policies: vaccination, isolation, culling, and transmission reduction.
- To determine if minimizing epidemic duration conflicts with minimizing total infectious burden.
Main Methods:
- Application of Pontryagin's Minimum Principle (PMP) to unconstrained SIR models.
- Analysis of bang-bang control strategies with at most one switch.
- Numerical simulations to explore complex scenarios and unexpected outcomes.
Main Results:
- For time-optimal control, bang-bang strategies with a single switch are optimal across various policies.
- The optimal strategy involves delaying control application and then implementing it at maximum rate.
- This contrasts with strategies for minimizing total infectious burden, which favor continuous maximal control.
- Numerical simulations revealed delayed optimal control even with a reproduction number below one, and switching post-peak infection.
Conclusions:
- Minimizing epidemic duration and total infectious burden may be conflicting objectives in SIR models.
- Delayed, maximal control application is a key finding for time-optimal epidemic management.
- Results have significant implications for livestock disease control, where minimizing outbreak duration is critical due to economic restrictions.
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