Article Contents

2022, Volume 15, Issue 3: 621-637. Doi: 10.3934/dcdss.2021155

This issue Previous Article Set-valued problems under bounded variation assumptions involving the Hausdorff excess Next Article Stability and optimal control of a delayed HIV/AIDS-PrEP model

Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives

a.
Department of Mathematics, AMNEA Group, Laboratory MAIS, Faculty of Sciences and Techniques, Moulay Ismail University of Meknes, Morocco
b.
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

^* Corresponding author: M. R. Sidi Ammi
^* Corresponding author: M. R. Sidi Ammi

Received: February 2020

Revised: June 2021

Early access: December 2021

Published: March 2022

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Abstract

The main aim of the present work is to study and analyze a reaction-diffusion fractional version of the SIR epidemic mathematical model by means of the non-local and non-singular ABC fractional derivative operator with complete memory effects. Existence and uniqueness of solution for the proposed fractional model is proved. Existence of an optimal control is also established. Then, necessary optimality conditions are derived. As a consequence, a characterization of the optimal control is given. Lastly, numerical results are given with the aim to show the effectiveness of the proposed control strategy, which provides significant results using the AB fractional derivative operator in the Caputo sense, comparing it with the classical integer one. The results show the importance of choosing very well the fractional characterization of the order of the operators.

Keywords:

Epidemic model,
Optimality conditions,
Reaction-diffusion equations,
Atangana–Baleanu–Caputo fractional derivatives,
Numerical simulations.

Mathematics Subject Classification: 34A08, 49K20; 35K57, 47H10.

Citation:

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Figure 1. Dynamic of the system without control for $ \alpha = 1 $

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Figure 2. Dynamic of the system without control for $ \alpha = 0.95 $

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Figure 3. Dynamic of the system without control for $ \alpha = 0.9 $

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Figure 4. Dynamic of the system with control for $ \alpha = 1 $

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Figure 5. Dynamic of the system with control for $ \alpha = 0.95 $

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Figure 6. Dynamic of the system with control for $ \alpha = 0.9 $

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Table 1. Values of initial conditions and parameters

Symbol	Description (Unit)	Value

$ S_0(x, y) $	Initial susceptible population $ (people/km^2) $	$ 43 $ for $ (x, y)\in\Omega_1 $ $ 50 $ for $ (x, y)\notin\Omega_1 $
$ I_0(x, y) $	Initial infected population $ (people/km^2) $	$ 7 $ for $ (x, y)\in\Omega_1 $ $ 0 $ for $ (x, y)\notin\Omega_1 $
$ R_0(x, y) $	Initial recovered population $ (people/km^2) $	$ 0 $ for $ (x, y)\in\Omega_1 $ $ 0 $ for $ (x, y)\notin\Omega_1 $
$ \lambda_1=\lambda_2=\lambda_3 $	Diffusion coefficient ($ km^2/day $)	0.6
$ \mu $	Birth rate $ (day^{-1}) $	0.02
$ d $	Natural death rate $ (day^{-1}) $	0.03
$ \beta $	Transmission rate $ ((people/km^2)^{-1}.day^{-1}) $	0.9
$ r $	Recovery rate $ (day^{-1}) $	0.04
$ T $	Final time $ (day) $	20

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Table 2. Values of the cost functional $ J $ without control for different $ \alpha $

$ \alpha $	0.9	0.95	1
J	$ 7.4350 e^{+04} $	$ 7.1586 e^{+04} $	$ 7.7019 e^{+04} $

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Table 3. Values of the cost functional $ J $ with control for different $ \alpha $

$ \alpha $	0.9	0.95	1
J	$ 4.9157 e^{+04} $	$ 4.7489 e^{+04} $	$ 5.2503 e^{+04} $

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