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 and Continuous Dynamical Systems - S, 2022, 15 (3) : 621-637. doi: 10.3934/dcdss.2021155
References:
[1]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag–Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098-1107.  doi: 10.22436/jnsa.010.03.20.

[2]

O. P. Agrawal, Formulation of Euler–Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), 368-379.  doi: 10.1016/S0022-247X(02)00180-4.

[3]

R. Almeida, What is the best fractional derivative to fit data?, Appl. Anal. Discrete Math., 11 (2017), 358-368.  doi: 10.2298/AADM170428002A.

[4]

R. T. Alqahtani, Atangana–Baleanu derivative with fractional order applied to the model of groundwater within an unconfined aquifer, J. Nonlinear Sci. Appl., 9, (2016), 3647–3654. doi: 10.22436/jnsa.009.06.17.

[5]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non–singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.

[6]

A. Atangana and J. F. Gómez–Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus, 133 (2018), Art. 166, 22pp. doi: 10.1140/epjp/i2018-12021-3.

[7]

G. M. Bahaa, Fractional optimal control problem for variable–order differential systems, Fract. Calc. Appl. Anal., 20 (2017), 1447-1470.  doi: 10.1515/fca-2017-0076.

[8]

R. K. Biswas and S. Sen, Numerical method for solving fractional optimal control problems, In Proceedings of the ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, San Diego, CA, USA, (2010), 1205–1208. doi: 10.1115/DETC2009-87008.

[9]

R. K. Biswas and S. Sen, Fractional optimal control problems: A pseudo-state-space approach, J. Vib. Control, 17 (2011), 1034-1041.  doi: 10.1177/1077546310373618.

[10]

R. K. Biswas and S. Sen, Fractional optimal control problems with specified final time, J. Comput. Nonlinear Dyn., 6 (2011), 021009.  doi: 10.1115/1.4002508.

[11]

R. K. Biswas and S. Sen, Fractional optimal control within Caputo's derivative, In Proceedings of the ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Washington, DC, USA, (2012), 353–360. doi: 10.1115/DETC2011-48045.

[12]

R. K. Biswas and S. Sen, Free final time fractional optimal control problems, J. Frankl. Inst., 351 (2014), 941-951.  doi: 10.1016/j.jfranklin.2013.09.024.

[13]

K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dynam., 71 (2013), 613-619.  doi: 10.1007/s11071-012-0475-2.

[14]

Y. DingZ. Wang and H. Ye, Optimal control of a fractional-order HIV-immune system with memory, IEEE Trans. Control Syst. Technol., 20 (2011), 763-769.  doi: 10.1109/TCST.2011.2153203.

[15]

J. D. DjidaG. M. Mophou and I. Area, Optimal control of diffusion equation with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel, J. Optim. Theory Appl., 182 (2019), 540-557.  doi: 10.1007/s10957-018-1305-6.

[16]

T. L. Guo, The necessary conditions of fractional optimal control in the sense of Caputo, J. Optim. Theory Appl., 156 (2013), 115-126.  doi: 10.1007/s10957-012-0233-0.

[17]

A. A. LaaroussiR. GhazzaliM. Rachik and S. Benrhila, Modeling the spatiotemporal transmission of Ebola disease and optimal control: A regional approach, Int. J. Dyn. Control, 7 (2019), 1110-1124.  doi: 10.1007/s40435-019-00525-w.

[18] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall, CRC Press, 2007. 
[19]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris 1969.

[20]

G. M. Mophou, Optimal control of fractional diffusion equation, Comput. Math. Appl., 61 (2011), 68-78.  doi: 10.1016/j.camwa.2010.10.030.

[21]

G. M. Mophou and G. M. N'Guérékata, Optimal control of fractional diffusion equation with state constraints, Comput. Math. Appl., 62 (2011), 1413-1426.  doi: 10.1016/j.camwa.2011.04.044.

[22]

E. Okyere, F. T. Oduro, S. K. Amponsah and I. K. Dontwi, Fractional order optimal control model for malaria infection, arXiv preprint, https://arXiv.org/abs/1607.01612, 2016.

[23] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999. 
[24]

S. Rosa and D. F. M. Torres, Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infectio, Chaos Solitons Fractals, 117 (2018), 142-149.  doi: 10.1016/j.chaos.2018.10.021.

[25]

M. R. Sidi AmmiM. Tahiri and D. F. M. Torres, Global stability of a Caputo fractional SIRS model with general incidence rate, Math. Comput. Sci., 15 (2021), 91-105.  doi: 10.1007/s11786-020-00467-z.

[26]

M. R. Sidi Ammi and D. F. M. Torres, Optimal control of a nonlocal thermistor problem with ABC fractional time derivatives, Comput. Math. Appl., 78 (2019), 1507-1516.  doi: 10.1016/j.camwa.2019.03.043.

[27]

C. J. Silva and D. F. M. Torres, Optimal control for a tuberculosis model with reinfection and post-exposure interventions, Math. Biosci., 244 (2013), 154-164.  doi: 10.1016/j.mbs.2013.05.005.

[28]

Q. Tang and Q. Ma, Variational formulation and optimal control of fractional diffusion equations with Caputo derivatives, Adv. Diff. Equa., 2015 (2015), Art. 283, 14pp. doi: 10.1186/s13662-015-0593-5.

[29]

S. YadavR. K. Pandey and A. K. Shukla, Numerical approximations of Atangana–Baleanu Caputo derivative and its application, Chaos, Solitons and Fractals, 118 (2019), 58-64.  doi: 10.1016/j.chaos.2018.11.009.

[30]

J. YuanB. ShiD. Zhang and S. Cui, A formulation for fractional optimal control problems via left and right Caputo derivatives, The 27th Chinese Control and Decision Conference, 2515 (2015), 816-821.  doi: 10.1109/CCDC.2015.7162031.

show all references

References:
[1]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag–Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098-1107.  doi: 10.22436/jnsa.010.03.20.

[2]

O. P. Agrawal, Formulation of Euler–Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), 368-379.  doi: 10.1016/S0022-247X(02)00180-4.

[3]

R. Almeida, What is the best fractional derivative to fit data?, Appl. Anal. Discrete Math., 11 (2017), 358-368.  doi: 10.2298/AADM170428002A.

[4]

R. T. Alqahtani, Atangana–Baleanu derivative with fractional order applied to the model of groundwater within an unconfined aquifer, J. Nonlinear Sci. Appl., 9, (2016), 3647–3654. doi: 10.22436/jnsa.009.06.17.

[5]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non–singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.

[6]

A. Atangana and J. F. Gómez–Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus, 133 (2018), Art. 166, 22pp. doi: 10.1140/epjp/i2018-12021-3.

[7]

G. M. Bahaa, Fractional optimal control problem for variable–order differential systems, Fract. Calc. Appl. Anal., 20 (2017), 1447-1470.  doi: 10.1515/fca-2017-0076.

[8]

R. K. Biswas and S. Sen, Numerical method for solving fractional optimal control problems, In Proceedings of the ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, San Diego, CA, USA, (2010), 1205–1208. doi: 10.1115/DETC2009-87008.

[9]

R. K. Biswas and S. Sen, Fractional optimal control problems: A pseudo-state-space approach, J. Vib. Control, 17 (2011), 1034-1041.  doi: 10.1177/1077546310373618.

[10]

R. K. Biswas and S. Sen, Fractional optimal control problems with specified final time, J. Comput. Nonlinear Dyn., 6 (2011), 021009.  doi: 10.1115/1.4002508.

[11]

R. K. Biswas and S. Sen, Fractional optimal control within Caputo's derivative, In Proceedings of the ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Washington, DC, USA, (2012), 353–360. doi: 10.1115/DETC2011-48045.

[12]

R. K. Biswas and S. Sen, Free final time fractional optimal control problems, J. Frankl. Inst., 351 (2014), 941-951.  doi: 10.1016/j.jfranklin.2013.09.024.

[13]

K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dynam., 71 (2013), 613-619.  doi: 10.1007/s11071-012-0475-2.

[14]

Y. DingZ. Wang and H. Ye, Optimal control of a fractional-order HIV-immune system with memory, IEEE Trans. Control Syst. Technol., 20 (2011), 763-769.  doi: 10.1109/TCST.2011.2153203.

[15]

J. D. DjidaG. M. Mophou and I. Area, Optimal control of diffusion equation with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel, J. Optim. Theory Appl., 182 (2019), 540-557.  doi: 10.1007/s10957-018-1305-6.

[16]

T. L. Guo, The necessary conditions of fractional optimal control in the sense of Caputo, J. Optim. Theory Appl., 156 (2013), 115-126.  doi: 10.1007/s10957-012-0233-0.

[17]

A. A. LaaroussiR. GhazzaliM. Rachik and S. Benrhila, Modeling the spatiotemporal transmission of Ebola disease and optimal control: A regional approach, Int. J. Dyn. Control, 7 (2019), 1110-1124.  doi: 10.1007/s40435-019-00525-w.

[18] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall, CRC Press, 2007. 
[19]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris 1969.

[20]

G. M. Mophou, Optimal control of fractional diffusion equation, Comput. Math. Appl., 61 (2011), 68-78.  doi: 10.1016/j.camwa.2010.10.030.

[21]

G. M. Mophou and G. M. N'Guérékata, Optimal control of fractional diffusion equation with state constraints, Comput. Math. Appl., 62 (2011), 1413-1426.  doi: 10.1016/j.camwa.2011.04.044.

[22]

E. Okyere, F. T. Oduro, S. K. Amponsah and I. K. Dontwi, Fractional order optimal control model for malaria infection, arXiv preprint, https://arXiv.org/abs/1607.01612, 2016.

[23] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999. 
[24]

S. Rosa and D. F. M. Torres, Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infectio, Chaos Solitons Fractals, 117 (2018), 142-149.  doi: 10.1016/j.chaos.2018.10.021.

[25]

M. R. Sidi AmmiM. Tahiri and D. F. M. Torres, Global stability of a Caputo fractional SIRS model with general incidence rate, Math. Comput. Sci., 15 (2021), 91-105.  doi: 10.1007/s11786-020-00467-z.

[26]

M. R. Sidi Ammi and D. F. M. Torres, Optimal control of a nonlocal thermistor problem with ABC fractional time derivatives, Comput. Math. Appl., 78 (2019), 1507-1516.  doi: 10.1016/j.camwa.2019.03.043.

[27]

C. J. Silva and D. F. M. Torres, Optimal control for a tuberculosis model with reinfection and post-exposure interventions, Math. Biosci., 244 (2013), 154-164.  doi: 10.1016/j.mbs.2013.05.005.

[28]

Q. Tang and Q. Ma, Variational formulation and optimal control of fractional diffusion equations with Caputo derivatives, Adv. Diff. Equa., 2015 (2015), Art. 283, 14pp. doi: 10.1186/s13662-015-0593-5.

[29]

S. YadavR. K. Pandey and A. K. Shukla, Numerical approximations of Atangana–Baleanu Caputo derivative and its application, Chaos, Solitons and Fractals, 118 (2019), 58-64.  doi: 10.1016/j.chaos.2018.11.009.

[30]

J. YuanB. ShiD. Zhang and S. Cui, A formulation for fractional optimal control problems via left and right Caputo derivatives, The 27th Chinese Control and Decision Conference, 2515 (2015), 816-821.  doi: 10.1109/CCDC.2015.7162031.

Figure 1.  Dynamic of the system without control for $ \alpha = 1 $
Figure 2.  Dynamic of the system without control for $ \alpha = 0.95 $
Figure 3.  Dynamic of the system without control for $ \alpha = 0.9 $
Figure 4.  Dynamic of the system with control for $ \alpha = 1 $
Figure 5.  Dynamic of the system with control for $ \alpha = 0.95 $
Figure 6.  Dynamic of the system with control for $ \alpha = 0.9 $
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
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
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} $
$ \alpha $ 0.9 0.95 1
J $ 7.4350 e^{+04} $ $ 7.1586 e^{+04} $ $ 7.7019 e^{+04} $
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} $
$ \alpha $ 0.9 0.95 1
J $ 4.9157 e^{+04} $ $ 4.7489 e^{+04} $ $ 5.2503 e^{+04} $
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