This repository contains the Julia code and numerical results supporting the paper:

On the problem of minimizing the epidemic final size for SIR model by social distancing

Authors: Pierre-Alexandre Bliman, Anas Bouali, Patrice Loisel, Alain Rapaport, Arnaud Virelizier

We address the optimal control problem of minimizing the final epidemic size under an $L^1$ constraint on the confinement effort that reduces the infection rate.

We consider the following optimal control problem $\mathcal{P}_S$:

• $S(t)$, $I(t)$: susceptible and infected populations at time $t$.
• Control $u(t)$: social distancing effort, $0 \le u(t) \le \bar u$.
• Budget $K$: total effort constraint, tracked by auxiliary state $C(t)$ with

We augment the classical SIR dynamics to include social distancing:

• $\beta$: transmission rate
• $\gamma$: recovery rate

We solve $\mathcal{P}_S$ using the OptimalControl.jl package in Julia and we present the following cases:

1. Clone this repository:

   git clone https://github.com/AnasXbouali/SIR-final-size-minimization.git
   cd SIR-final-size-minimization

2. Install dependencies in Julia REPL:

   import Pkg; Pkg.instantiate(); Pkg.precompile();

This project is implemented in a Julia Jupyter notebook.

Use the function:

SIRocp(tf, S0, I0, K, β, γ, umax)

• tf: final time
• S0, I0: initial susceptible and infected fractions
• K: total social distancing budget
• β: constant transmission rate
• γ: recovery rate
• umax: maximum control effort

Use the function:

SIRocp2beta(T, S0, I0, K, τ, β1, β2, γ, umax, k)

• T: final time
• S0, I0: initial conditions
• K: budget value
• τ: time when transmission rate changes
• β1, β2: transmission rates before and after τ
• γ: recovery rate
• umax: max control level
• k: regularization parameter