On the control of a reaction–diffusion system: a class of SIR distributed parameter systems

• Published in: International journal of dynamics and control 2019-09, Vol.7 (3), p.1021-1034
• Main Authors: Ghazzali, Rachid, Laaroussi, Adil El Alami, EL Bhih, Amine, Rachik, Mostafa
• Format: Article
• Language: English
• Subjects: Boundary conditions; Complexity; Control; Control and Systems Theory; Distributed parameter systems; Dynamical Systems; Engineering; Epidemics; Mathematical models; Optimal control; Optimization; Vaccines; Vibration
• ISSN: 2195-268X; 2195-2698
• DOI: 10.1007/s40435-019-00553-6

In this paper, two types of distributed control functions, vaccine and treatment have been applied to a spatiotemporal SIR model with no-flux boundary conditions. The spatiotemporal SIR epidemic model is formulated from existing SIR epidemic model by including a diffusion term in his different compartments to study the impact of spatial heterogeneity of disease transmission in dense regions. Our main objective to find the optimal control pair that minimizes the number of infected individuals, the corresponding vaccination and treatment costs. The existence of the positive solution for the state system and the existence of a distributed optimal control pair are proved. Techniques of optimal control are used to characterize optimal control pair in terms of state and adjoint functions. The optimality system is solved numerically; the numerical results show that the control effect is effective if the treatment and vaccine strategies are used simultaneously.