# American Institute of Mathematical Sciences

March  2022, 15(3): 621-637. doi: 10.3934/dcdss.2021155

## 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

Received  February 2020 Revised  June 2021 Published  March 2022 Early access  December 2021

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.

Citation: Moulay Rchid Sidi Ammi, Mostafa Tahiri, Delfim F. M. Torres. Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2022, 15 (3) : 621-637. doi: 10.3934/dcdss.2021155
##### References:

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##### References:
Dynamic of the system without control for $\alpha = 1$
Dynamic of the system without control for $\alpha = 0.95$
Dynamic of the system without control for $\alpha = 0.9$
Dynamic of the system with control for $\alpha = 1$
Dynamic of the system with control for $\alpha = 0.95$
Dynamic of the system with control for $\alpha = 0.9$
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
 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
Values of the cost functional $J$ without control for different $\alpha$
 $\alpha$ 0.9 0.95 1 J $7.4350 e^{+04}$ $7.1586 e^{+04}$ $7.7019 e^{+04}$
 $\alpha$ 0.9 0.95 1 J $7.4350 e^{+04}$ $7.1586 e^{+04}$ $7.7019 e^{+04}$
Values of the cost functional $J$ with control for different $\alpha$
 $\alpha$ 0.9 0.95 1 J $4.9157 e^{+04}$ $4.7489 e^{+04}$ $5.2503 e^{+04}$
 $\alpha$ 0.9 0.95 1 J $4.9157 e^{+04}$ $4.7489 e^{+04}$ $5.2503 e^{+04}$
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