Abstract
In this work, we are interested in studying a spatiotemporal two-dimensional SIR epidemic model, in the form of a system of partial differential equations (PDE). A distribution of a vaccine in the form of a control variable is considered to force immunity. The purpose is to characterize a control that minimizes the number of susceptible, infected individuals and the costs associated with vaccination over a nite space and time domain. In addition, the existence of the solution of the state system and the optimal control is proved. The characterization of the control is given in terms of state function and adjoint function. The numerical resolution of the state system shows the effectiveness of our control strategy.
Highlights
The numerical resolution of the state system shows the effectiveness of our control strategy
We have adopted two situations: In the first, the disease starts from the middle (1) and in the second, the disease starts at the corner (2).Figures 1, 2, and 3 present numerical results for susceptible, infected, and recovered individuals.Results show that in both situations, susceptible individuals become infected after an incubation period, and after a period of time, the disease spreads throughout the population.In order to fight against the spread of the disease we adopted a strategy based on the introduction of a vaccine in the form of a control variable
(2.4-2.6), and we find the characterization of optimal control
Summary
V(x, t) represents the vaccination rate at time and position x. We seek to minimize the functional objective. The cost of vaccination is a nonlinear function of v, we choose a quadratic function indicating the additional costs associated with high vaccination rates. The parameter α2 , with the units P opuvlaactcioinn2/km[2 ], balances the cost squared of the vaccine with the cost associated with the infected population. Our objective is to find control functions such that. We put H (Ω) = L2 (Ω) 3, we denote by W 1,2 ([0, T ] , H (Ω)) the space of all absolutely continuous functions y : [0, T ] → H (Ω) having the property that ∂y ∈ L2 ([0, T ] , H (Ω)).
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