### Introduction

### Materials and methods

### 2.1 Basic model

*N*(

*t*) =

*S*(

*t*)+

*A*(

*t*)+

*P*(

*t*)+

*R*(

*t*) is constant. The variable

*N*(

*t*) includes only adolescents and young adults between the ages 12 years and 23 years and is divided into four classes: susceptible,

*S*(

*t*); addicted,

*A*(

*t*); in treatment

*P*(

*t*); and recovered,

*R*(

*t*). Individuals of

*S*(

*t*) who try DSH move to

*A*(

*t*) with the per capita transition rate,

*α*, which is peer pressure on susceptible individuals in

*A*(

*t*) and

*P*(

*t*). Individuals repeating DSH remain in

*A*(

*t*), but individuals who stop DSH move to

*R*(

*t*) at the rate

*η*. This is the rate at which individuals in

*A*(

*t*) stop DSH without any treatment program or individuals who tried DSH only once and transferred to

*R*(

*t*). When individuals in

*A*(

*t*) seek treatment, they go to

*P*(

*t*) at the rate of

*β*(

*P*+

*R*)/

*N*+

*θ*. In this equation,

*β*is peer pressure due to individuals in

*P*(

*t*) and

*R*(

*t*) to the individuals in

*A*(

*t*), and

*θ*is the intervention rate at which addicted individuals seek treatment. If the treatment fails, individuals may go back to

*A*(

*t*) from

*P*(

*t*) at the rate

*ω*. Individuals in

*P*(

*t*) recover at rate

*ρ*and move to

*R*(

*t*). The values of

*α*and

*β*may be different, but in this study they are considered the same for homogeneous mixing. Among all of these parameters, the system is most sensitive to

*α*and

*η*[19]. The values of

*η*may also increase or decrease, depending on the positive or negative influence of the Internet [26]. Furthermore, the individual who performs DSH once, seeks more serious injury for the next DSH episode [27]. Therefore, a control strategy should be concerned with prevention through controlling peer pressure

*α*and early intervention

*η*.

### 2.2 Optimal control

*α*and increasing early intervention (

*η*)]. However, maintaining constant control over time is impractical. Therefore, our aim is to show that it is possible to implement time-dependent control techniques while minimizing the addicted population with minimum cost of implementation of the control measures.

*N*=

*S*(

*t*)+

*A*(

*t*)+

*P*(

*t*)+

*R*(

*t*) is constant.

*u*

_{1}(

*t*) and

*u*

_{2}(

*t*), represent the quantity of intervention associated with the parameters

*α*and

*η*, respectively at time

*t*. The factor of 1−

*u*

_{1}(

*t*) reduces the per capita transition rate

*α*from

*S*(

*t*) to

*A*(

*t*). The per capita transition rate

*η*from

*P*to

*R*increases at a rate that is proportional to

*u*

_{2}(

*t*) in which

*μ*> 0 is the proportionality constant.

*A*

_{c},

*B*

_{1}and

*B*

_{2}represent the weight constants. The costs associated with the controls of the transition rates are described by the terms

*A*

_{c}represents the degree of negative influence on the society by each addicted individual. The goal is to minimize the population

*A*(

*t*) of addicted individuals and the implementation cost of the controls. Therefore, we looked for optimal control functions

*u*

_{1}and

*u*

_{2}. The integrand of the objective functional along with the four right hand sides of the state equations constitutes the Hamiltonian for our problem. So the Hamiltonian is given by,in which

**X**(

*t*) = (

*S*(

*t*),

*A*(

*t*),

*P*(

*t*),

*R*(

*t*)),

**u**(

*t*) = (

*u*

_{1}(

*t*),

*u*

_{2}(

*t*)) and

**Λ**(

*t*) = (

*λ*

_{1}(

*t*),

*λ*

_{2}(

*t*),

*λ*

_{3}(

*t*),

*λ*

_{4}(

*t*)).

*S*

^{∗}(

*t*),

*A*

^{∗}(

*t*),

*P*

^{∗}(

*t*),

*R*

^{∗}(

*t*) be optimal state solutions with associated optimal control variables

*λ*

_{1}(

*t*),

*λ*

_{2}(

*t*),

*λ*

_{3}(

*t*),

*λ*

_{4}(

*t*) would then exist that satisfywith the transversality condition (or the boundary condition)

### Results

*S*; “addicted”,

*A*; “in treatment”,

*P*; and “recovered”,

*R;*the control

*u*

_{1}is associated with reducing peer pressure and the control

*u*

_{2}is associated with early intervention. As a general shortcoming, full efficiency of the controls is unfeasible. To choose an upper bound for the controls, we considered the study of Dunlop et al [29] in which they found that 79% of young people learned about suicide from the newspaper or from friends and family, and 59% of them learned from an online source. We assumed the upper bound of each of the controls was 0.6. The rate constant

*μ*is chosen to be 0.01 in accordance with the value of

*η*. Using the parameter values summarized in Table 1, the problem is solved numerically by the forward-backward sweep method [30], along with the fourth order Runge-Kutta algorithm, which is subject to a wide range of plausible values of weight factors

*A*

_{c},

*B*

_{1}and

*B*

_{2}because the weights should vary from group to group. For an institutional setting, we considered that the total population is

*N*(0) = 10000 with

*S*(0) = 8700,

*A*(0) = 900,

*P*(0) = 100,

*R*(0) = 300. Time span for the simulation is [0, T], in which T = 60 months (i.e., 5 years).

*A*

_{c}= 1,

*B*

_{1}= 500,

*B*

_{2}= 500. The rightmost graphs in Figure 1 show the time-dependent control strategy in which we see that the controls

*u*

_{1}and

*u*

_{2}should be implemented at maximum for a long period and then gradually decreased to zero. The controls work fairly well for reducing the number of addicted population.

*t*

_{1}and

*t*

_{2}be the period of time for maximum implementation of the optimal controls

*u*

_{1}and

*u*

_{2}, respectively. The time

*t*

_{1}and

*t*

_{2}may depend on the weights

*A*

_{c},

*B*

_{1},

*B*

_{2}and the initial conditions as well. Figure 2 depicts the changes of

*t*

_{1}and

*t*

_{2}with

*B*

_{1}= 100–1000 and

*A*

_{c}= 1–100 while keeping

*B*

_{2}= 200 fixed. Figure 2A shows that for

*A*

_{c}> 60, the time

*t*

_{1}is the same for all

*B*

_{1}; however, for smaller

*A*

_{c}; the effect of

*B*

_{1}to the change of

*t*

_{1}is more pronounced. A smaller

*B*

_{1}results in a higher

*t*

_{1}and vice versa. Figure 2B shows that

*t*

_{2}increases with

*A*

_{c}but it is not affected by

*B*

_{1}. Figure 3 depicts the changes of

*t*

_{1}and

*t*

_{2}with

*B*

_{2}= 100–1000 and

*A*

_{c}= 1–100 while keeping

*B*

_{1}= 200 fixed. The change of weight

*B*

_{2}does not affect the change of

*t*

_{1}for all

*A*

_{c}and also does not affect the change of

*t*

_{2}for

*A*

_{c}> 40. Figure 4 depicts the changes of

*t*

_{1}and

*t*

_{2}with

*B*

_{1}= 100–1000 and

*B*

_{2}= 100–1000 while keeping

*A*

_{c}= 1 fixed. Figure 4A illustrates that changes in

*B*

_{1}and

*B*

_{2}negatively affect changes in

*t*

_{1}and

*t*

_{2}, as we have already seen in Figures 2 and 3. In addition, for

*B*

_{1}> 100

*t*

_{1}increases with

*B*

_{2}. However,

*B*

_{1}has no noticeable effect in the change of

*t*

_{2}. Figure 5 depicts the changes of

*t*

_{1}and

*t*

_{2}with

*B*

_{1}= 100–1000 and

*B*

_{2}= 100–1000 while keeping

*A*

_{c}= 10 fixed. In this case,

*B*

_{1}, and

*B*

_{2}have no effect on changes in

*t*

_{2}and

*t*

_{1}, respectively.

*B*

_{1}and

*B*

_{2}reduce the implementation of the controls and consequently

*B*

_{1},

*B*

_{2}which include higher values for both. Figure 6B shows the same phenomena for different initial conditions

*S*(0) = 7400,

*A*(0) = 1800,

*P*(0) = 200,

*R*(0) = 600. In this case

*B*

_{1},

*B*

_{2}.

### Discussion

*B*

_{1}and

*B*

_{2}, respectively). The control problem is solved using Pontryagin's Maximum Principle. In this circumstance, the negative effect of an addicted individual is parameterized by

*A*

_{c}. The simultaneous use of both controls reduces the self-harm epidemic by increasing susceptible individuals and reducing the addicted individuals remarkably. But the costs associated with control strategies and the weight

*A*

_{c}may not be the same in all groups of young people. Depending on the groups, the costs and the weight may be varied so that different control strategies are needed. For a higher weight of addicted individuals, we used nearly the same control strategy for the groups, even with different control costs, which agrees with our intuition that a greater weight requires greater effort from the controls, irrespective of the control cost. However, if the weight is low, a great effort by the controls is no longer necessary. As a result, the strategy varies from group to group, depending on the control costs associated with the groups. Controls are implemented in smaller numbers in groups with a high control cost and vice versa.