# Optimal Implementation of Intervention Strategies for Elderly People with Ludomania

## Article information

## Abstract

### Objectives

Now-a-days gambling is growing especially fast among older adults. To control the gratuitous growth of gambling, well-analyzed scientific strategies are necessary. We tried to analyze the adequacy of the health of society mathematically through immediate treatment of patients with early prevention.

### Methods

The model from Lee and Do was modified and control parameters were introduced. Pontryagin's Maximum Principle was used to obtain an optimal control strategy.

### Results

Optimal control can be achieved through simultaneous use of the control parameters, though it varies from society to society. The control corresponding to prevention needed to be implemented in full almost all the time for all types of societies. In the case of the other two controls, the scenario was greatly affected depending on the types of societies.

### Conclusion

Prevention and treatment for elderly people with ludomania are the main intervention strategies. We found that optimal timely implementation of the intervention strategies was more effective. The optimal control strategy varied with the initial number of gamblers. However, three intervention strategies were considered, among which, preventing people from engaging in all types of gambling proved to be the most crucial.

**Keywords:**basic reproduction number; epidemiological model; gambling; maximal principle; optimal control

## 1 Introduction

Problem gambling or ludomania is a type of disorder that consists of an urge to continuously gamble despite harmful negative consequences or a desire to stop and that is associated with both social and family costs. Problem gambling and wider gambling-related harm constitute a significant health and social issue [1]. To study problems associated with gambling, Shaffer and Korn [2] used the classic public health model for communicable disease, which examines the interaction among host, agent, environment, and vector. Moreover, some sociologists [3–6] have shown that a significant predictor of the occurrence of ludomania is peer pressure; in the sense that the occurrence depends on the number of individuals involved, the number of individuals who might be involved, as well as the frequency, duration, priority, and intensity of association with peers. Therefore, ludomania might be considered as a contagious disease. Recently, from the point of view of a communicable disease, Lee and Do [7,8] used a mathematical modeling approach to study the dynamics of problem gambling.

In this study, we adopted the optimal control theory to their model and tried to find optimal strategies for intervention. A variety of policies and services have been developed with the intent of preventing and reducing problem gambling and related harm. The prevalence and consequences of problem gambling as well as approaches to treatment can be found in the book by Petry [9]. We considered a basic model [7] to incorporate some important epidemiological features, such as time-dependent control functions. The extended model can then be used to determine cost-effective strategies for combating the spread of problem gambling in a given population; a mathematical modeling approach to study the dynamics of problem gambling.

## 2 Materials and methods

### 2.1 Basic model

We considered the model of Lee and Do [7] without demographic effect as follows:

The whole population

### 2.2 Optimal control

Using sensitivity analysis, Lee and Do [7] showed that the best way to reduce gambling problems among elderly people is to minimize the value of

We formulated an optimal control problem for the transmission dynamics of gambling by adding control terms to the basic model (1) as follows:

Here, we noted that

The control variables

We defined our control set to be:

An optimal control problem with the objective cost functional can be given by

*u*

_{1},

*u*

_{2}and

## 3 Results

Let

Furthermore, the optimal controls

(See Appendix 1 for the detailed derivation)

For numerical simulation, the forward–backward sweep method [13] based on 4th order Runge–Kutta algorithm was used to treat the problem. The problem consisted of eight ordinary differential equations describing states and adjoint variables along with three controls. The parameters in Table 1 were adopted from a previous study [7] and used for our simulation. A natural shortcoming was that the controls were not 100% effective, so the upper boundary of the controls

Figure 1 depicts the variation of the maximum for the three controls, which illustrates that the control

Figure 2 depicts the numerical simulation that was carried out in the time interval

On the other hand, the control scenario would not be similar in all societies. The control scenario might be greatly affected by the number of gamblers and pathological gamblers, that is to say, the control scenario may vary depending on the initial conditions. To analyze the effect of the number of gamblers in society, keeping the total population unchanged, we varied the total number of gamblers and pathological gamblers from 5% to 35%, among which gamblers and pathological gamblers were in the ratio 7:3, and the proportion of treated gamblers was 5% of the total gambling population. Simulation results have been plotted in Figure 3, which illustrates that the control

## 4 Discussion

An optimal control strategy was analyzed with the help of Pontryagin's Maximum Principle for three control factors. The control scenario would not be similar in all societies. The control scenario might have been affected by its impact on society, and the impact of gamblers on society was introduced into the model through the coefficients

In conclusion, it was conspicuous that simultaneous implementation of all the controls gave the most effective result. However, the control

## Conflicts of interest

All contributing authors declare no conflicts of interest.

## References

## Appendix 1 The derivation of optimal controls.

Let

Furthermore, the optimal controls

To determine the adjoint equations and the transversality conditions, we used the Hamiltonian (6). By Pontryagin's Maximum Principle [12], setting

To obtain the optimality conditions (8), we also differentiated the Hamiltonian

Solving for optimal controls, we obtained:

To determine an explicit expression for the optimal controls for

On the set

On the set

On the set

Combining these three cases above, we found a characterization of

Using similar arguments, we also obtained the second and third optimal control function

## Acknowledgments

This work was supported by Kyungpook National University Research Fund, 2012.

## Notes

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.