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

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 compa...

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Bibliographic Details
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
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Summary: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.
ISSN:2195-268X
2195-2698
DOI:10.1007/s40435-019-00553-6