# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022071
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## Optimal control of a fractional smoking system

 1 School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong, China 2 School of Information and Electronic Engineering, Shandong Technology and Business University, Yantai, Shandong, China 3 School of Computer Science and Technology, Shandong Technology and Business University, Yantai, Shandong, China

Received  September 2021 Revised  March 2022 Early access April 2022

Fund Project: This work was supported by the Natural Science Foundation of China (No. 11771008), and the Natural Science Foundation of Shandong Province, China (Nos. ZR2019MA031 and ZR2019QA009)

In this paper, considering the contact transmission between heavy smokers and potential smokers, we propose a novel fractional smoking model. Moreover, we introduce two intervention measures, i.e., media campaign and adjuvant drug treatment, to the fractional smoking model in order to control the spread of smoking. For this controlled fractional smoking model, some important properties are discussed. Then, we propose an optimal control problem involving the controlled fractional system. The existence of the optimal control for this problem is also established. By applying the calculus of variation, we derive the necessary optimality conditions for the fractional optimal control problem. Furthermore, we develop a numerical solution approach based on Adams-type predictor-corrector method to solve the derived optimality system. Finally, numerical results show that the obtained optimal control strategies are effective in reducing the susceptible individuals and smokers.

Citation: Chongyang Liu, Wenjuan Sun, Xiaopeng Yi. Optimal control of a fractional smoking system. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022071
##### References:
 [1] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dyn., 38 (2004), 323-337.  doi: 10.1007/s11071-004-3764-6. [2] D. Baleanu, S. S. Sajjadi, J. H. Asad, A. Jajarmi and E. Estirl, Hyperchaotic behaviors, optimal control, and synchronization of a nonautonomous cardiac conduction system, Adv. Differ. Equ., 2021 (2021), 157, 24 pp. doi: 10.1186/s13662-021-03320-0. [3] D. Baleanu, S. S. Sajjadi, A. Jajarmi and Ö. Defterli, On a nonlinear dynamical system with both chaotic and nonchaotic behaviors: A new fractional analysis and control, Adv. Differ. Equ., 2021 (2021), 234, 17 pp. doi: 10.1186/s13662-021-03393-x. [4] D. Baleanu, S. Zibaei, M. Namjoo and A. Jajarmi, A nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system, Adv. Differ. Equ., 2021 (2021), 1-19. [5] M. Bergounioux and L. Bourdin, Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 35, 38 pp. doi: 10.1051/cocv/2019021. [6] M. C. Caputo and D. F. M. Torres, Duality for the left and right fractional derivatives, Signal Process., 107 (2015), 265-271.  doi: 10.1016/j.sigpro.2014.09.026. [7] C. Castillo-Garsow, G. Jordan-Salivia and A. R. Herrera, Mathematical models for dynamics of tobacco use, recovery and relapse, WHO Tech. Rep. Ser., (2000). [8] J. Danane, Z. Hammouch, K. Allali, S. Rashid and J. Singh, A fractional-order model of coronavirus disease 2019 (COVID-19) with governmental action and individual reaction, Math. Meth. Appl. Sci., (2021), 1-4. [9] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, 2010. [10] K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3-22.  doi: 10.1023/A:1016592219341. [11] V. S. Ertürk, G. Zaman and S. Momani, A numeric-analytic method for approximating a giving up smoking model containing fractional derivatives, Comput. Appl. Math., 64 (2012), 3065-3074.  doi: 10.1016/j.camwa.2012.02.002. [12] Z. Gong, C. Liu, K. L. Teo, S. Wang and Y. Wu, Numerical solution of free final time fractional optimal control problems, Appl. Math. Comput., 405 (2021), 126270, 15 pp. doi: 10.1016/j.amc.2021.126270. [13] Z. Gong, C. Liu, K. L. Teo and X. Yi, Optimal control of nonlinear fractional systems with multiple pantograph-delays, Appl. Math. Comput., 425 (2022), 127094.  doi: 10.1016/j.amc.2022.127094. [14] F. Guerrero, F. J. Santonja and R. J. Villanueva, Analysing the Spanish smoke-free legislation of 2006: A new method to quantify its impact using a dynamic model, Int. J. Drug Policy, 22 (2011), 247-251.  doi: 10.1016/j.drugpo.2011.05.003. [15] O. K. Ham, Stages and processes of smoking cessation among adolescents, West J. Nurs. Res., 38 (2004), 417-433. [16] F. Haq, K. Shah, G. U. Rahman and M. Shahzad, Numerical solution of fractional order smoking model via laplace Adomian decomposition method, Alex. Eng. J., 57 (2018), 1061-1069.  doi: 10.1016/j.aej.2017.02.015. [17] World Health Organization report on the global tobacco epidemic, 2009. [18] C. Ionsecu, A. Lopes, D. Copot, J. A. T. Machado and J. H. T. Bates, The role of fractional calculus in modeling biological phenomena: A review, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 141-159.  doi: 10.1016/j.cnsns.2017.04.001. [19] A. Jajari, D. Baleanu and K. Z. Vahid, A general fractional formulation and tracking control for immunogenic tumor dynamics, Math. Meth. Appl. Sci., 45 (2022), 667-680.  doi: 10.1002/mma.7804. [20] A. Jajarmi, N. Pariz, S. Effati and A. V. Kamyad, Infinite horizon optimal control for nonlinear interconnected large-scale dynamical systems with an application to optimal attitude control, Asian J. Control, 14 (2012), 1239-1250.  doi: 10.1002/asjc.452. [21] L. Jódar, R. J. Villanueva, A. J. Arenas and G. C. Gonzoalez, Nonstandard numerical methods for mathematical model for influenza disease, Math. Comput. Simul., 79 (2008), 622-633.  doi: 10.1016/j.matcom.2008.04.008. [22] H. Kheiri and M. Jafari, Optimal control of a fractional-order model for the HIV/AIDS epidemic, Int. J. Biomath., 11 (2018), 1850086.  doi: 10.1142/S1793524518500869. [23] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [24] A. Labzai, O. Balatif and M. Rachik, Optimal control strategy for a discrete time smoking model with specific saturated incidence rate, Discrete Dyn. Nat. Soc., 2018 (2018), 1-10.  doi: 10.1155/2018/5949303. [25] W. Li, S. Wang and V. Rehbock, Numerical solution of fractional optimal control, J. Optim. Theory Appl., 180 (2019), 556-573.  doi: 10.1007/s10957-018-1418-y. [26] Y. Li, Y. Chen and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965-1969.  doi: 10.1016/j.automatica.2009.04.003. [27] C. Liu, Z. Gong, K. L. Teo and S. Wang, Optimal control of nonlinear fractional-order systems with multiple time-varying delays, J. Optim. Theory Appl., (2021). [28] C. Liu, Z. Gong, C. Yu, S. Wang and K. L. Teo, Optimal control computation for nonlinear fractional time-delay systems with state inequality constraints, J. Optim. Theory Appl., 191 (2021), 83-117.  doi: 10.1007/s10957-021-01926-8. [29] C. Liu, X. Yi and Y. Feng, Modelling and parameter identification for a two-stage fractional dynamical system in microbial batch process, Nonlinear Anal. Model., 27 (2022), 350-367.  doi: 10.15388/namc.2022.27.26234. [30] A. M. Mahdy, K. A. Gepreel, K. Lotfy and A. A. El-bary, A numerical method for solving the Rubela ailment disease model, Int. J. Mod. Phys. C, 32 (2021), 2150097, 15 pp. doi: 10.1142/S0129183121500972. [31] A. M. S. Mahdy, M. Higazy, K. A. Gepreel and A. A. A. El-dahdouh, Optimal control and bifurcation diagram for a model nonlinear fractional SIRC, Alex. Eng. J., 59 (2020), 3481-3501. [32] A. M. S. Mahdy, M. Higazy and M. S. Mohamed, Optimal and memristor-based control of a nonlinear fractional tumor-immune model, Comput. Mater. Contin., 67 (2021), 3463-3486. [33] A. M. S. Mahdy, M. S. Mohamed, A. Y. Al Amiri and K. A. Gepreel, Optimal control and spectral collocation method for solving smoking models, Intell. Autom. Soft. Co., 31 (2022), 899-915. [34] A. M. S. Mahdy, M. S. Mohamed, K. A. Gepreel, A. AL-Amiri and M. Higazya, Dynamical characteristics and signal flow graph of nonlinear fractional smoking mathematical model, Chaos Solitons Fractals, 141 (2020), 110308, 13 pp. doi: 10.1016/j.chaos.2020.110308. [35] A. M. S. Mahdy, M. S. Mohamed, K. Lotfy, M. Alhazmi, A. A. El-Bary and M. H. Raddadi, Numerical solution and dynamical behaviors for solving fractional nonlinear Rubella ailment disease model, Results Phys., 24 (2021), 104091.  doi: 10.1016/j.rinp.2021.104091. [36] A. M. S. Mahdy, N. H. Sweilam and M. Higazy, Approximate solution for solving nonlinear fractional order smoking model, Alex. Eng. J., 59 (2020), 739-752.  doi: 10.1016/j.aej.2020.01.049. [37] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 1976. [38] A. B. Salati, M. Shamsi and D. F. M. Torres, Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 334-350.  doi: 10.1016/j.cnsns.2018.05.011. [39] J. Singh, Analysis of fractional blood alcohol model with composite fractional derivative, Chaos Solitons Fractals, 140 (2020), 110127, 6 pp. doi: 10.1016/j.chaos.2020.110127. [40] J. Singh, B. Ganbari, D. Kumar and D. Baleanu, Analysis of fractional model of guava for biological pest control with memory effect, J. Adv. Res., 32 (2021), 99-108.  doi: 10.1016/j.jare.2020.12.004. [41] J. Singh, D. Kumar, M. A. Qurashi and D. Baleanu, A new fractional model for giving up smoking dynamics, Adv. Differ. Equ., 2017 (2017), 1-16.  doi: 10.1186/s13662-017-1139-9. [42] R. Ullah, M. Khan, G. Zaman, S. Islam, M. A. Khan and T. Gul, Dynamical features of a mathematical model on smoking, J. Appl. Environ. Biol. Sci., 6 (2016), 92-96. [43] V. Verma, Optimal control analysis of a mathematical model on smoking, Model. Earth Syst. Environ., 6 (2020), 2535-2542.  doi: 10.1007/s40808-020-00847-1. [44] C. Wang, H. Zhang and S. Wang, Positive solution of a nonlinear fractional differential equation involving Caputo derivative, Discrete Dyn. Nat. Soc., 2012 (2012), Art. ID 425408, 1–16. doi: 10.1155/2012/425408. [45] G. Zaman, Qualitative behavior of giving up smoking models, Bull. Malays. Math., Sci. Soc., 34 (2011), 403-415. [46] Y. Zhou, J. Wang and L. Zhang, Basic Theory of Fractional Differential Equations, 2$^{nd}$ edition, World Scientific, New Jersey, 2017.

show all references

##### References:
 [1] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dyn., 38 (2004), 323-337.  doi: 10.1007/s11071-004-3764-6. [2] D. Baleanu, S. S. Sajjadi, J. H. Asad, A. Jajarmi and E. Estirl, Hyperchaotic behaviors, optimal control, and synchronization of a nonautonomous cardiac conduction system, Adv. Differ. Equ., 2021 (2021), 157, 24 pp. doi: 10.1186/s13662-021-03320-0. [3] D. Baleanu, S. S. Sajjadi, A. Jajarmi and Ö. Defterli, On a nonlinear dynamical system with both chaotic and nonchaotic behaviors: A new fractional analysis and control, Adv. Differ. Equ., 2021 (2021), 234, 17 pp. doi: 10.1186/s13662-021-03393-x. [4] D. Baleanu, S. Zibaei, M. Namjoo and A. Jajarmi, A nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system, Adv. Differ. Equ., 2021 (2021), 1-19. [5] M. Bergounioux and L. Bourdin, Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 35, 38 pp. doi: 10.1051/cocv/2019021. [6] M. C. Caputo and D. F. M. Torres, Duality for the left and right fractional derivatives, Signal Process., 107 (2015), 265-271.  doi: 10.1016/j.sigpro.2014.09.026. [7] C. Castillo-Garsow, G. Jordan-Salivia and A. R. Herrera, Mathematical models for dynamics of tobacco use, recovery and relapse, WHO Tech. Rep. Ser., (2000). [8] J. Danane, Z. Hammouch, K. Allali, S. Rashid and J. Singh, A fractional-order model of coronavirus disease 2019 (COVID-19) with governmental action and individual reaction, Math. Meth. Appl. Sci., (2021), 1-4. [9] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, 2010. [10] K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3-22.  doi: 10.1023/A:1016592219341. [11] V. S. Ertürk, G. Zaman and S. Momani, A numeric-analytic method for approximating a giving up smoking model containing fractional derivatives, Comput. Appl. Math., 64 (2012), 3065-3074.  doi: 10.1016/j.camwa.2012.02.002. [12] Z. Gong, C. Liu, K. L. Teo, S. Wang and Y. Wu, Numerical solution of free final time fractional optimal control problems, Appl. Math. Comput., 405 (2021), 126270, 15 pp. doi: 10.1016/j.amc.2021.126270. [13] Z. Gong, C. Liu, K. L. Teo and X. Yi, Optimal control of nonlinear fractional systems with multiple pantograph-delays, Appl. Math. Comput., 425 (2022), 127094.  doi: 10.1016/j.amc.2022.127094. [14] F. Guerrero, F. J. Santonja and R. J. Villanueva, Analysing the Spanish smoke-free legislation of 2006: A new method to quantify its impact using a dynamic model, Int. J. Drug Policy, 22 (2011), 247-251.  doi: 10.1016/j.drugpo.2011.05.003. [15] O. K. Ham, Stages and processes of smoking cessation among adolescents, West J. Nurs. Res., 38 (2004), 417-433. [16] F. Haq, K. Shah, G. U. Rahman and M. Shahzad, Numerical solution of fractional order smoking model via laplace Adomian decomposition method, Alex. Eng. J., 57 (2018), 1061-1069.  doi: 10.1016/j.aej.2017.02.015. [17] World Health Organization report on the global tobacco epidemic, 2009. [18] C. Ionsecu, A. Lopes, D. Copot, J. A. T. Machado and J. H. T. Bates, The role of fractional calculus in modeling biological phenomena: A review, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 141-159.  doi: 10.1016/j.cnsns.2017.04.001. [19] A. Jajari, D. Baleanu and K. Z. Vahid, A general fractional formulation and tracking control for immunogenic tumor dynamics, Math. Meth. Appl. Sci., 45 (2022), 667-680.  doi: 10.1002/mma.7804. [20] A. Jajarmi, N. Pariz, S. Effati and A. V. Kamyad, Infinite horizon optimal control for nonlinear interconnected large-scale dynamical systems with an application to optimal attitude control, Asian J. Control, 14 (2012), 1239-1250.  doi: 10.1002/asjc.452. [21] L. Jódar, R. J. Villanueva, A. J. Arenas and G. C. Gonzoalez, Nonstandard numerical methods for mathematical model for influenza disease, Math. Comput. Simul., 79 (2008), 622-633.  doi: 10.1016/j.matcom.2008.04.008. [22] H. Kheiri and M. Jafari, Optimal control of a fractional-order model for the HIV/AIDS epidemic, Int. J. Biomath., 11 (2018), 1850086.  doi: 10.1142/S1793524518500869. [23] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [24] A. Labzai, O. Balatif and M. Rachik, Optimal control strategy for a discrete time smoking model with specific saturated incidence rate, Discrete Dyn. Nat. Soc., 2018 (2018), 1-10.  doi: 10.1155/2018/5949303. [25] W. Li, S. Wang and V. Rehbock, Numerical solution of fractional optimal control, J. Optim. Theory Appl., 180 (2019), 556-573.  doi: 10.1007/s10957-018-1418-y. [26] Y. Li, Y. Chen and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965-1969.  doi: 10.1016/j.automatica.2009.04.003. [27] C. Liu, Z. Gong, K. L. Teo and S. Wang, Optimal control of nonlinear fractional-order systems with multiple time-varying delays, J. Optim. Theory Appl., (2021). [28] C. Liu, Z. Gong, C. Yu, S. Wang and K. L. Teo, Optimal control computation for nonlinear fractional time-delay systems with state inequality constraints, J. Optim. Theory Appl., 191 (2021), 83-117.  doi: 10.1007/s10957-021-01926-8. [29] C. Liu, X. Yi and Y. Feng, Modelling and parameter identification for a two-stage fractional dynamical system in microbial batch process, Nonlinear Anal. Model., 27 (2022), 350-367.  doi: 10.15388/namc.2022.27.26234. [30] A. M. Mahdy, K. A. Gepreel, K. Lotfy and A. A. El-bary, A numerical method for solving the Rubela ailment disease model, Int. J. Mod. Phys. C, 32 (2021), 2150097, 15 pp. doi: 10.1142/S0129183121500972. [31] A. M. S. Mahdy, M. Higazy, K. A. Gepreel and A. A. A. El-dahdouh, Optimal control and bifurcation diagram for a model nonlinear fractional SIRC, Alex. Eng. J., 59 (2020), 3481-3501. [32] A. M. S. Mahdy, M. Higazy and M. S. Mohamed, Optimal and memristor-based control of a nonlinear fractional tumor-immune model, Comput. Mater. Contin., 67 (2021), 3463-3486. [33] A. M. S. Mahdy, M. S. Mohamed, A. Y. Al Amiri and K. A. Gepreel, Optimal control and spectral collocation method for solving smoking models, Intell. Autom. Soft. Co., 31 (2022), 899-915. [34] A. M. S. Mahdy, M. S. Mohamed, K. A. Gepreel, A. AL-Amiri and M. Higazya, Dynamical characteristics and signal flow graph of nonlinear fractional smoking mathematical model, Chaos Solitons Fractals, 141 (2020), 110308, 13 pp. doi: 10.1016/j.chaos.2020.110308. [35] A. M. S. Mahdy, M. S. Mohamed, K. Lotfy, M. Alhazmi, A. A. El-Bary and M. H. Raddadi, Numerical solution and dynamical behaviors for solving fractional nonlinear Rubella ailment disease model, Results Phys., 24 (2021), 104091.  doi: 10.1016/j.rinp.2021.104091. [36] A. M. S. Mahdy, N. H. Sweilam and M. Higazy, Approximate solution for solving nonlinear fractional order smoking model, Alex. Eng. J., 59 (2020), 739-752.  doi: 10.1016/j.aej.2020.01.049. [37] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 1976. [38] A. B. Salati, M. Shamsi and D. F. M. Torres, Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 334-350.  doi: 10.1016/j.cnsns.2018.05.011. [39] J. Singh, Analysis of fractional blood alcohol model with composite fractional derivative, Chaos Solitons Fractals, 140 (2020), 110127, 6 pp. doi: 10.1016/j.chaos.2020.110127. [40] J. Singh, B. Ganbari, D. Kumar and D. Baleanu, Analysis of fractional model of guava for biological pest control with memory effect, J. Adv. Res., 32 (2021), 99-108.  doi: 10.1016/j.jare.2020.12.004. [41] J. Singh, D. Kumar, M. A. Qurashi and D. Baleanu, A new fractional model for giving up smoking dynamics, Adv. Differ. Equ., 2017 (2017), 1-16.  doi: 10.1186/s13662-017-1139-9. [42] R. Ullah, M. Khan, G. Zaman, S. Islam, M. A. Khan and T. Gul, Dynamical features of a mathematical model on smoking, J. Appl. Environ. Biol. Sci., 6 (2016), 92-96. [43] V. Verma, Optimal control analysis of a mathematical model on smoking, Model. Earth Syst. Environ., 6 (2020), 2535-2542.  doi: 10.1007/s40808-020-00847-1. [44] C. Wang, H. Zhang and S. Wang, Positive solution of a nonlinear fractional differential equation involving Caputo derivative, Discrete Dyn. Nat. Soc., 2012 (2012), Art. ID 425408, 1–16. doi: 10.1155/2012/425408. [45] G. Zaman, Qualitative behavior of giving up smoking models, Bull. Malays. Math., Sci. Soc., 34 (2011), 403-415. [46] Y. Zhou, J. Wang and L. Zhang, Basic Theory of Fractional Differential Equations, 2$^{nd}$ edition, World Scientific, New Jersey, 2017.
Flow chart of controlled smoking system (1)
The optimal control strategies when $\alpha = 0.6$
The optimal state trajectories when $\alpha = 0.6$
The optimal control strategies when $\alpha = 0.7, 0.8, 0.9$ and $1.0$
The optimal state trajectories and total population density when $\alpha = 0.7, 0.8, 0.9$ and $1.0$
Parameter values used in system (3)
 Parameter $b$ $\mu$ $\beta_1$ $\beta_2$ $r$ Value $0.035$ (Assumed) $0.04$ ([24]) $0.6$ ([24]) $0.1$ (Assumed) $0.7$ ([24]) Parameter $e$ $m$ $d$ $\eta$ Value $0.5$ ([24]) $0.7$ ([24]) $0.05$ ([24]) $0.04$ (Assumed)
 Parameter $b$ $\mu$ $\beta_1$ $\beta_2$ $r$ Value $0.035$ (Assumed) $0.04$ ([24]) $0.6$ ([24]) $0.1$ (Assumed) $0.7$ ([24]) Parameter $e$ $m$ $d$ $\eta$ Value $0.5$ ([24]) $0.7$ ([24]) $0.05$ ([24]) $0.04$ (Assumed)
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