Boundary optimal control of time-space SIR model with nonlinear Robin boundary condition

Spatio-Temporal SIR Model with Robin Boundary Condition and Automatic Lockdown Policy

This paper presents the SIR space-time model, which is a coupled reaction-diffusion system with nonlinear Robin boundary conditions. These boundary conditions are supposed to lock the border (no outflow neither immigration nor migration) when the number of infected individuals explode, and this may be considered as an automatic containment or lock-down. In practice, we can precise some threshold for the number of infected individuals and when it is reached the model locks the region automatically. This work provides a thorough study of the presented model, including the existence and uniqueness of the solution, its boundedness and its asymptotic behaviour. We end with some numerical experiments performed on the basis of the finite difference approach and Newton's method to highlight and validate the theoretical results.