Research Article | Open Access

Volume 2021 |Article ID 5584315 | https://doi.org/10.1155/2021/5584315

Driss Kada, Bouchaib Khajji, Omar Balatif, Mostafa Rachik, El Houssine Labriji, "Optimal Control Approach of Discrete Mathematical Modeling of the Spread of Gaming Disorder in Morocco and Cost-Effectiveness Analysis", Discrete Dynamics in Nature and Society, vol. 2021, Article ID 5584315, 12 pages, 2021. https://doi.org/10.1155/2021/5584315

# Optimal Control Approach of Discrete Mathematical Modeling of the Spread of Gaming Disorder in Morocco and Cost-Effectiveness Analysis

Academic Editor: Manuel De la Sen
Revised26 May 2021
Accepted10 Jun 2021
Published19 Jun 2021

#### 1. Introduction

Mathematical modeling and control theory are considered as two of the most necessary tools to represent, simulate, and control the evolution of some phenomena including ecological, social, and economic ones. These tools help convert the phenomenon into mathematical equations and also to formulate study, analyze, and interpret their results. For example, Brida and Cayssials [8] generalize the classical model of determination of production prices for two commodities by introducing dynamics generated by the possibility that the port rate can be computed using prices of different stages. Wang and Petrosian [9] consider and describe the class of cooperative differential games with nontransferable utility and the process of construction of the optimal Pareto strategy with continuous updating. Guo and Li [10] establish a new online game addiction model with low- and high-risk exposure and use the optimal control theory to study the optimal solution problem with three kinds of control measures (isolation, education, and treatment). The work by Akanni et al. [11] formulated and analyzed a mathematical model for population dynamics of financial crime with optimal control measures. Kouidere et al. [12] studied an optimal control approach of mathematical modeling with multiple delays of the negative impact of delays in applying preventive precautions on the spread of the COVID-19 pandemic with a case study of Brazil. Other models from optimal control problems and population dynamics can be found in [1320].

In this research, we will adopt discrete-time modeling where statistical data are collected at a discrete time (day, week, month, and year). Therefore, it is more direct, more convenient, and more accurate to describe a phenomenon using discrete-time modeling compared to continuous-time modeling. Also, we mainly shed light on the category of addicts and make a differentiation between practitioners and addicts of electronic games. Eventually, we propose strategies for optimal control of the spread of this addiction. We add the cost-effectiveness ratio (ICER) to be able to choose between strategies based on obtaining better results at a lower cost.

In this work, we propose a new model that describes the dynamics of electronic game addiction. The population that we study consists of children and adolescents aged less than 24 years old within Morocco. The population under study is divided into four compartments, which are potential gamers, engaged gamers, addicted gamers, and recovered gamers. Our main objective is to propose an optimal control strategy that will minimize the number of addicted gamers. To achieve this objective, we adopt three controls that represent awareness through media, guidance on alternative educational and recreational methods, and creating rehabilitation centers for electronic game addiction.

This paper is organized as follows. In Section 2, we propose a PEAR mathematical model that describes the dynamics of a population that reacts to the spread of the E-game infection. In Section 3, we present an optimal control problem for the proposed model where we give some results concerning the existence of the optimal control, and we characterize the optimal controls using the Pontryagin maximum principle in discrete time. Numerical simulations through MATLAB software and the cost-effectiveness analysis are given in Section 4. Finally, we conclude the paper in Section 5.

#### 2. Mathematical Model and Numerical Simulation of Gaming Disorder

##### 2.1. Description of the Model

We consider a discrete mathematical model PEAR that describes the dynamics of a population having gaming disorder. We divide the population into four compartments. The following illustration will show disease trends in the compartments in Figure 2.

###### 2.1.1. The Potential Gamers P

It represents children and youth who are vulnerable to infection or who are more likely to become addicted to electronic games. This compartment is increased by the recruitment rate denoted by and the risk factor for transmission from recovered persons to potential gamers . It is decreased by the rate coefficient of transmission from potential persons to engaged gamers due to the effects of advertising for electronic games through media. Also, it is decreased by an effective contact with engaged gamers at rate (the rate of patients who become engaged gamers because of the negative contact with the other engaged gamers) and with addicted gamers at rate (the rate of patients who become engaged gamers because of the negative contact with the addicted gamers).

###### 2.1.2. The Engaged Gamers E

It represents children and youth who are interested in electronic games and play more than four hours a day without secondary effects on the body and on the individual’s behaviour in the social environment. This compartment is increased by , , and . Also, it is increased by (the transmission factor from recovered persons to engaged gamers) and by (the transmission factor from addicted gamers to engaged gamers). The compartment of engaged gamers is decreased by that represents the rate of the engaged gamers who have become addicted gamers.

###### 2.1.3. The Addicted Gamers A

It represents children and youth who are addicted to electronic games, suffer from gaming disorders, and have no control over their gaming habits. They prioritize gaming over other interests and activities and continue to game despite its negative consequences. This compartment is increased by and it is decreased by . Also, it is decreased by the rate of the addicted gamers who have become recovered gamers denoted by .

###### 2.1.4. The Recovered Gamers R

It represents children and youth recovering from their addiction to electronic games. This compartment is increased by . It is decreased by and by the risk factor for transmission from recovered persons to engaged gamers .

##### 2.2. Model Equations

By adding the rates at which the steps of gaming disorder enter the compartment and also by subtracting the rates at which people leave a compartment, we obtain a system of difference equations for the rate at which patients change in each compartment during separate times. Therefore, we present the gaming disorder model with the following system of difference equations:where , and are given initial states. is the recruitment rate of potential gamers. is coefficient of transmission from potential persons to engaged gamers due to the effects of advertising for electronic games through media. is the rate of the engaged gamers who become addicted gamers. is the rate of the addicted gamers who become recovered gamers. is the rate of patients who become engaged gamers because of the negative contact with the other engaged gamers. is the rate of patients who become engaged gamers because of the negative contact with the addicted gamers. is the transmission factor from recovered persons to engaged gamers. is the transmission factor from addicted gamers to engaged gamers. is the risk factor for transmission from recovered persons to potential gamers.

In order to demonstrate the efficiency of the model, we propose a numerical simulation (see Figure 3) which allows us to see how the growth results adapt to reality to some extent. We calculated transactions , and based on the statistical results for Morocco included in [7], and we used the Moroccan population census data (see [7, 21]) to determine the , and .

Figure 3 shows that there is a rapid development of the number of the engaged gamers and on the other hand, there is a significant increase in the number of addicted gamers in a few months. It is clearly seen from Figure 3 that the phenomenon of gaming is in constant increase.

At present and in the light of the rapid development of computer technology and consequently the development and spread of electronic games, the world including Morocco has witnessed an increase in the number of addicts to electronic games. In fact, the addiction of children and young people leads to a sharp decline in the level of academic achievement, productivity at work and introspection, and a strong tendency to isolation. To address this phenomenon and mitigate its effects on individuals and society, we have proposed a set of practical strategies, which we present in the following paragraph.

#### 3. The Optimal Control Problem

Our objective in the proposed control strategy is to minimize the number of engaged gamers and the number of addicted gamers. Therefore, in model (1), we include the controls: , which represents the effort to raise awareness of the dangers of electronic gaming and the dangers of addiction through written and visual media. Hence, the term is used to reduce the number of the affected individuals. In order to provide harmless alternatives for gaming which encompass educational games that help children and young people learn innovative programming skills, benefit themselves and society, and help them overcome their addiction to electronic games, we propose the control that represents the effort to provide these alternatives to children and young people to compensate for their addiction to electronic games. Some games reach a dangerous level of addiction which necessarily requires joining gaming rehabilitation centers that offer comprehensive treatment for such disorders. Hence, we propose the control that represents the effort for creating rehabilitation center for addicts to quit electronic game addiction.

Thus, the controlled mathematical system is given by the following system of difference equations:where , and are given initial states.

Then, the problem is to minimize the objective functional:where the parameters , , and for are the cost coefficients.

They are selected to weigh the relative importance of , , and at time k.

T is the final time. In other words, we seek the optimal controls u, , and such thatwhere is the set of admissible controls defined by and , and ; .

The sufficient condition for the existence of the optimal controls u, , and for the problems (2) and (3) comes from Theorem 1.

Theorem 1. There exist the optimal controls , and such thatsubject to the control system (2) with initial conditions.

Proof. Since the coefficients of the state equations are bounded and there are a finite number of time steps, and are uniformly bounded for all in the controls set and thus J(u, v, w) is bounded for all since is bounded. is finite, and there exists a sequence such that and corresponding sequences of states ; since there are a finite number of uniformly bounded sequences, there exist and , and such that sequences , and .Finally, due to the finite-dimensional structure of system (2) and the objective function J(u,v,w), , , and are the optimal controls with corresponding states , and . Therefore, is achieved.In order to derive the necessary condition for the optimal controls, we use the discrete version of Pontryagin’s maximum principle [2230]. The idea is to introduce the adjoint function to attach the system of difference equations to the objective function resulting in the formation of a function called the Hamiltonian. This principle converts into a problem of minimizing à Hamiltonian at time step k defined bywhere is the right side of the system of difference equation (2) of the state variable at time step k + 1. Using Pontryagin’s maximum principle in discrete time [3133], we state Theorem 2.

Theorem 2. Given the optimal controls , and the solutions , and of the corresponding state system (2), there exist adjoint variables , and satisfyingWith the transversality conditions at time T, ; ; ; .
Furthermore, for the optimal controls , , and are given by

Proof. The Hamiltonian at time step k is given byfor , the adjoint equations and transversality conditions can be obtained by using Pontryagin’s maximum principle in discrete time given in [2230] such thatand with the transversality conditions at time T, ; ; ; . For , the optimal controls , and can be solved from the optimality condition , and ; that is,so we haveand by the bounds in of the controls, it is easy to obtain , , and in the form (8)–(10).

#### 4. Numerical Simulation and Cost-Effectiveness Analysis

In this section, we present the results obtained by numerically solving the optimality system. As explained in [1420], the optimality system is a two-point limit value problem system with separate boundary conditions at the times when the step k = 0 and k = T. We solve the optimality system by an iterative method with forward solving of the state system followed by backward solving of the adjoint system. We start with an initial guess for the controls at the first iteration and then before the next iteration, we update the controls by using the characterization. We continue until convergence of successive iterates is achieved. A code is written and compiled in MATLAB; we use the data in Table 1.To obtain results that simulate reality to some extent, we calculated transactions , and based on the statistical results for Morocco included in [7], and we used the Moroccan population census data (see [7, 21]) to determine the , and .

 Parameter Description Estimated value Source The number of children and youth in Morocco 15000000 [21] Initial number of potential gamers 9585000 [7] Initial number of engaged gamers 1290000 [7] Initial number of addicted gamers 4125000 [7] Initial number of recovered gamers 0 Assumed The incidence of potential gamers 7988 [21] The rate of the potential gamers who become engaged gamers 0.36 [7] The rate of the engaged gamers who become addicted gamers 0.76 [7] The rate of the addicted gamers who become recovered gamers 0.1 Assumed The rate of the P who become engaged gamers by contact with engaged gamers 0.46 [7] The rate of the P who become engaged gamers by contact with addicted gamers 0.26 [7] The rate of the recovered gamers who become engaged gamers 0.2 Assumed The rate of the addicted gamers who become engaged gamers 0.1 Assumed The rate of the recovered gamers who become potential gamers 0.4 Assumed
##### 4.1. Discussion

In this section, we offer a digital simulation to highlight the effectiveness of our strategy to combat the impact of addiction to electronic games on children and young people and reduce their addiction. Different simulations can be carried out using various values of parameters as shown in Table 1. We introduce our control strategy which consists of using three types of controls: the first one is symbolized by and represents awareness of the dangers of electronic games through written and visual media, the second one is symbolized by and represents the effort to directing children and adolescents to educational and entertaining alternative means, and the third one is symbolized by and represents the effort for creating rehabilitation center for addicts to quit electronic game addiction. Furthermore, we investigate numerically the impact of each of the following optimal control strategies.

Scenario 1. In this scenario, we simulate the case where we apply a single control , raising awareness of the dangers of electronic games through written and visual media. In Figure 4, we see that, after 12 months, the number of addicted gamers decreased slightly from (without control) to (with control). Also, for the number of engaged gamers as shown in Figure 5, it decreased slightly from (without control) to (with control).

Scenario 2. In this scenario, applying the control which represents providing harmless alternatives as means for education and entertainment for children and adolescents, we see that, after 12 months as shown in Figure 4, the number of addicted gamers decreased slightly from (without control) to (with control). In Figure 5, the number of engaged gamers increased from (without control) to (with control).

Scenario 3. In this scenario, we apply the control , creating gaming rehabilitation centers for the addicts to electronic games. We see that, after 12 months as shown in Figure 4, the number of addicted gamers decreased from (without control) to (with control), which demonstrates the effectiveness of this intervention in reducing the number of the addicted gamers. In Figure 5, the number of the engaged gamers increased from (without control) to (with control).

Scenario 4. In this scenario, we combine Scenarios 1 and 2. We see that, after 12 months as shown in Figure 6, the number of the addicted gamers decreased from (without control) to (with controls). In Figure 7, the number of the engaged gamers increased from (without control) to (with controls).

Scenario 5. In this scenario, we combine Scenarios 1 and 3 by using the controls and . We see that, after 12 months, as shown in Figure 6, the number of the addicted gamers decreased from (without control) to (with controls). In Figure 7, the number of the engaged gamers decreased from (without control) to (with controls), which shows the effectiveness of the combination of these controls.

Scenario. 6In this scenario, we combine Scenario 2 and Scenario 3 by using the controls and . We see that, after 12 months, as shown in Figure 6, the number of the addicted gamers decreased from (without control) to (with controls). In Figure 7, the number of the engaged gamers increased from (without control) to (with controls).

Scenario 7. In this scenario, we combine all the three controls , , and . We see that, after 12 months, as shown in Figure 8, the number of the addicted gamers decreased from (without controls) to (with controls), demonstrating the effectiveness of this intervention in reducing the number of the addicted gamers. In Figure 9, the number of engaged gamers increased slightly from (without controls) to (with controls).
Finally, in Figures 10(a)10(c), we present the optimal control variables , , and used in the previous scenarios.

##### 4.2. Cost-Effectiveness Analysis

In this section, we analyze the cost-effectiveness ratio (ICER) of the previous seven scenarios by comparing them to determine the most cost-effectiveness ratio (ICER). Following the method as applied in several studies [3436], this ratio was used to compare the differences between the costs and health outcomes of two competing strategies. The ICER is defined as the quotient of the difference in costs in strategies i and j, by the difference in infected averted in strategies i and j. Given two competing strategies i and j, where strategy j has higher effectiveness than strategy i, the ICER values are calculated as follows:where the total costs (TC) and the total cases averted (TA) are defined, in our study, during a given period for scenario i for i = 1, 2, 3, 4, 5, 6, 7 bywhere , , and correspond to the person unit cost of the three possible interventions, while and are the optimal solution associated to the optimal controls , , and . Based on the model simulation results, we ranked in Table 2 our control strategies in order of increased numbers of averted infections.

 Strategy Total averted infections (TA) Total cost (TC) 2 4 1 6 3 7 5

First, we compared the cost-effectiveness of Scenarios 2 and 4:

We note that ICER (2) is higher than ICER (4). This means that Scenario 2 is dominated by Scenario 4. Therefore, strategy 2 is excluded from the set of alternatives.

Second, we compared the cost-effectiveness of Scenarios 4 and 1:

By comparing Scenarios 1 and 4, the lower ICER for Scenario 1 indicates that Scenario 4 is strongly dominated by Scenario 1. That is, Scenario 4 is more costly and less effective than Scenario 1. Therefore, Scenario 4 is excluded from the set of alternatives.

Next, Scenario 1 is compared with Scenario 6:

Since , then Scenario 1 is less effective than Scenario 6. Therefore, Scenario 1 is excluded from the set of alternatives.

Next, we compare the cost-effectiveness of Scenario 6 and Scenario 3:

The comparison of ICER (3) and ICER (6) indicates that Scenario 6 is more costly than Scenario 3; this means that Scenario 6 is more expensive and less effective than Scenario 3. Therefore, Scenario 1 is excluded from the set of alternatives.

Now, we compare the cost-effectiveness of Scenario 3 and Scenario 7:

The lower ICER obtained for Scenario 3 is an indication that Scenario 3 strongly dominates Scenario 7 and this simply indicates that Scenario 7 is more costly to implement in comparison to Scenario 3. Therefore, it is best to exclude Scenario 7 from the set of control strategies and alternative interventions to implement in order to preserve limited resources.

Finally, Scenario 3 is compared with Scenario 5:

The comparison reveals that Scenario 3 is less costly than Scenario 5 by saving 3.737; therefore, Scenario 3 is the best strategy among all the compared strategies due to its cost-effectiveness.

#### 5. Conclusion

In this research, we proposed a new mathematical model that describes the dynamics of gaming disorder. The model will allow us to understand the phenomenon of the spread of this disorder and to deal with it. Thus, we applied three control strategies. As a result, we were able to study and combine several scenarios in order to see the impact and effect of each of these controls on the spread of this disorder. To obtain results that relatively simulate reality, we calculated transactions , and based on the statistical results from Morocco included in [7], and we used the Moroccan population census data (see [7, 21]) to determine , and . The numerical resolution of the system with the difference equations as well as the numerical simulation allowed us to compare and see the difference between each scenario in a concrete way. The numerical results prove the effectiveness of our strategy and its importance in the fight against the spread of gaming disorders in society, particularly among children and young people. Using ICER cost-effectiveness analysis, we showed that Scenario 3 is the most effective strategy. In the next research in this area, we will try to shed light on the social and economic causes and conditions that contribute to the spread of e-game addiction and to rely in this study on realistic statistics related to proliferation coefficients as well as the use of mathematical model using Atangana–Baleanu–Caputo fractional derivative.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### References

1. D. J. Kuss, M. D. Griffiths, and H. M. Pontes, “Chaos and confusion in DSM-5 diagnosis of Internet Gaming Disorder: issues, concerns, and recommendations for clarity in the field,” Journal of Behavioral Addictions, vol. 6, no. 2, pp. 103–109, 2017. View at: Publisher Site | Google Scholar
2. D. A. G. Pathological, “Video-game use among youth ages 8 to 18: a national study,” Psychological Science, vol. 20, no. 5, pp. 594–602, 2009. View at: Google Scholar
3. P. K. Bender and D. A. Gentile, “Internet gaming disorder: relations between needs satisfaction in-game and in life in general,” Psychology of Popular Media, vol. 9, no. 2, pp. 266–278, 2020. View at: Publisher Site | Google Scholar
4. World Health Organization, “Sharpening the focus on gaming disorder,” Bulletin of the World Health Organization, vol. 97, pp. 382-383, 2019. View at: Google Scholar
5. J. B. Funk, D. D. Buchman, and J. N. Germarm, “preference for violent electronic games, self concept, and gender differences in young children,” American Journal of Orlhopsychiaty, vol. 70, no. 2, 2000. View at: Publisher Site | Google Scholar
6. J. B. Funk, J. Hagan, J. Schimming, W. A. Bullock, D. D. Buchman, and M. Myers, “Aggression and psychopathology in adolescents with a preference for violent electronic games,” Aggressive Behavior, vol. 28, no. 2, pp. 134–144, 2002. View at: Publisher Site | Google Scholar
7. F. A. Etindele Sosso, D. J. Kuss, C. Vandelanotte et al., “Insomnia, sleepiness, anxiety and depression among different types of gamers in African countries,” Scientific Reports, vol. 10, 2020. View at: Publisher Site | Google Scholar
8. J. G. Brida and G. Cayssials, “A dynamic extension of the classical model of production prices determination,” Journal of Dynamics and Games, vol. 7, 2020. View at: Publisher Site | Google Scholar
9. Z. Wang and O. Petrosian, “On class of non transferable utility cooperative differential games with continuous updating,” Journal of Dynamics and Games, vol. 7, 2020. View at: Publisher Site | Google Scholar
10. Y. Guo and T. Li, “Optimal control strategies for an online game addiction model with low and high risk exposure,” Discrete and Continuous Dynamical Systems-B, vol. 26, 2021. View at: Publisher Site | Google Scholar
11. J. O. Akanni, F. O. Akinpelu, S. Olaniyi, A. T. Oladipo, and A. W. Ogunsola, “Modelling financial crime population dynamics: optimal control and cost-effectiveness analysis,” International Journal of Dynamics and Control, vol. 8, 2020. View at: Google Scholar
12. A. Kouidere, D. Kada, O. Balatif, M. Rachik, and M. Naim, “Optimal control approach of a mathematical modeling with multiple delays of the negative impact of delays in applying preventive precautions against the spread of the COVID-19 pandemic with a case study of Brazil and cost-effectiveness,” Solitons and Fractals, vol. 142, Article ID 110438, 2021. View at: Publisher Site | Google Scholar
13. M. A. Aba Oud, A. Ali, H. Alrabaiah, S. Ullah, M. Altaf Khan, and S. Islam, “A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load,” Advances in Difference Equations, vol. 2021, 2021. View at: Publisher Site | Google Scholar
14. D. Kada, A. Kouidere, O. Balatif, M. Rachik, and E. H. Labriji, “Mathematical modeling of the spread of COVID-19 among different age groups in Morocco: optimal control approach for intervention strategies,” Chaos, Solitons & Fractals, vol. 141, Article ID 110437, 2020. View at: Publisher Site | Google Scholar
15. D. Kada, A. Labzai, O. Balatif, M. Rachik, and H. Labriji, “Spread of COVID-19 in Morocco discrete mathematical modeling: optimal control strategies and cost-effectiveness analysis,” Journal of Mathematical And Computational Science, vol. 10, no. 5, pp. 2070–2093, 2020. View at: Publisher Site | Google Scholar
16. B. Khajji, D. Kada, O. Balatif, and M. Rachik, “Amulti-region discrete time mathematical modeling of the dynamics of COVID-19 virus propagation using optimal control,” Journal of Applied Mathematics and Computing, vol. 64, 2020. View at: Publisher Site | Google Scholar
17. Y. Guo and T. Li, “Optimal control and stability analysis of an online game addiction model with two stages,” Mathematical Methods in the Applied Sciences, vol. 43, pp. 4391–4408, 2020. View at: Google Scholar
18. T. Li and Y. Guo, “Stability and optimal control in a mathematical model of online game addiction,” Filomat, vol. 33, 2019. View at: Google Scholar
19. O. Balatif, A. Labzai, and M. Rachik, “A discrete mathematica modeling and optimal control of the electoral behavior with regard to a political party,” Discrete Dynamics in Nature and Society, vol. 2018, Article ID 9649014, 14 pages, 2018. View at: Publisher Site | Google Scholar
20. J. Karrakchou, M. Rachik, and S. Gourari, “Optimal control and infectiology: application to an HIV/AIDS model,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 807–818, 2006. View at: Publisher Site | Google Scholar
21. https://www.hcp.ma/Population-par-groupes-d-ages-fonctionnels-1960-2050_a681.html.
22. J. Workman and S. Lenhart, Optimal Control Applied to Biological Models, Chapmal Hall/CRC, Boca Raton, FL, USA, 2007.
23. W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics,” Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, vol. 115, no. 772, pp. 700–721, 1927. View at: Publisher Site | Google Scholar
24. D. C. Zhang and B. Shi, “Oscillation and global asymptotic stability in a discrete epidemic model,” Journal of Mathematical Analysis and Applications, vol. 278, no. 1, pp. 194–202, 2003. View at: Publisher Site | Google Scholar
25. Z. Hu, Z. Teng, and H. Jiang, “Stability analysis in a class of discrete SIRS epidemic models,” Nonlinear Analysis: Real World Applications, vol. 13, no. 5, pp. 2017–2033, 2012. View at: Publisher Site | Google Scholar
26. M. D. Rafal and W. F. Stevens, “Discrete dynamic optimization applied to on-line optimal control,” AIChE Journal, vol. 14, no. 1, pp. 85–91, 1968. View at: Publisher Site | Google Scholar
27. L. L. Cui, H. G. Zhang, D. Liu, and Y. S. Kim, “Constrained optimal control of affine nonlinear discrete-time systems using ghjb method,” in Proceedings of the 2009 IEEE International Symposium on Adaptive Dynamic Programming and Reinforcement Learning, ADPRL, pp. 16–21, Nashville, TN, USA, April 2009. View at: Google Scholar
28. D. Liu, D. Wang, D. Zhao, Q. Wei, and N. Jin, “Neural-networkbased optimal control for a class of unknown discrete-time nonlinear systems using globalized dua heuristic programming,” IEEE Transactions on Automation Science and Engineering, vol. 9, 2012. View at: Publisher Site | Google Scholar
29. A. Kouidere, B. Khajji, A. El bhih, O. Balatif, and M. Rachik, “A mathematical modeling with optimal control strategy of transmission of COVID-19 pandemic virus,” Communications in Mathematical Biology and Neuroscience, vol. 2020, 2020. View at: Google Scholar
30. A. Labzai, O. Balatif, and M. Rachik, “Optimal control strategy for a discrete time smoking model with specific saturated incidence rate,” Discrete Dynamics in Nature and Society, vol. 2018, Article ID 5949303, 10 pages, 2018. View at: Publisher Site | Google Scholar
31. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, New York, NY, USA, 1962.
32. V. Guibout and A. M. Bloch, “A discrete maximum principle for solving optimal control problems,” in Proceedings of the 2004 43rd IEEE Conference on Decision and Control (CDC), pp. 1806–1811, IEEE, Nassau, Bahamas, December 2004. View at: Google Scholar
33. W. Ding, R. Hendon, B. Cathey, E. Lancaster, and R. Germick, “Discrete time optimal control applied to pest control problems,” Involve, vol. 7, no. 4, pp. 479–489, 2014. View at: Publisher Site | Google Scholar
34. K. O. Okosun, O. Rachid, and N. Marcus, “Optimal control strategies and cost-effectiveness analysis of a malaria model,” Biosystems, vol. 111, no. 2, pp. 83–101, 2013. View at: Publisher Site | Google Scholar
35. F. B. Agusto and I. M. ELmojtaba, “Optimal control and cost-effective analysis of malaria/visceral leishmaniasis co-infection,” PLoS One, vol. 12, no. 2, Article ID e0171102, 2017. View at: Publisher Site | Google Scholar
36. N. Hidayat and R. B. E. Wibowo, “Optimal control and cost-effectiveness analysis of HIV model with educational campaigns and therapy,” Matematika: MJIAM, vol. 35, 2019. View at: Google Scholar