Optimal control of a spatiotemporal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.svg" display="inline" id="d1e587"><mml:mi mathvariant="script">SIR</mml:mi></mml:math> model with reaction–diffusion involving <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg" display="inline" id="d1e592"><mml:mi>p</mml:mi></mml:math>-Laplacian operator

Abstract

The present paper’s primary goal is to make an analysis and study a reaction–diffusion SIR epidemic mathematical model expressed as a parabolic system of partial differential equations using the p-Laplacian operator. Immunity is compelled through vaccination distribution, which is seen as a control variable. Our main goal is to define an optimal control, which reduces the spread of infection and the cost of vaccination over a limited period of time and space. Existence and uniqueness of a positive solution and existence of an optimal control for the proposed model are proved. Then a description and characterization of the optimal control is provided in terms of state and adjoint functions. Optimality system is numerically resolved by a discrete iterative scheme pertained to and the forward–backward algorithm. Furthermore, using various p-values for the p-Laplaian operator, numerical results demonstrate the effectiveness of the suggested control strategy, which yields meaningful outcomes.