Page 38 - IJOCTA-15-3

M. Aychluh et.al. / IJOCTA, Vol.15, No.3, pp.407-425 (2025)
using a parameter α 1 , which is directly related to

υ
L mABC D m (t) (η) the proportion of non-problem gamblers, minor
0
0
risk gamblers, and excessive problem gamblers al-
(4)
υ υ−1
B (υ) η L {m (t)} (η) − η m (0) ready participating in gambling activities.
= .
υ
1 − υ η + γ υ
ϑ 1 = γ 1 A + γ 2 M + γ 3 P .
1
Definition 2. ( 27–32 ) Let m (t) ∈ L (0, T) be a α 1 T
function. The modified ABC integral is then ex- Individuals in stage A may transfer to stage R
press as follows: by stopping any gambling activities before ad-
ψ (1 − υ) diction, measured using a parameter ℘, or they
mAB υ [m (t) − m (0)]
I m (t) =
0
B (υ) may develop minor symptoms of problem gam-
(5)
bling. This step could be induced by the influ-

RL υ ence rate, measured in terms of the proportion
+γ υ I (m (t) − m (0)) .
0
of already existing problematic gambling stages:
where, the Riemann-Liouville integral operator is γ 2 M + γ 3 P
given by multiplied by a proportionality pa-
T
rameter α 2 . Therefore, the overall problem gam-
Z t
1 s (η) bling rate is:
RL υ
I m (t) = dη. (6)
Γ (υ) 0 (t − η) 1−υ
ϑ 2 γ 2 M + γ 3 P
= .
3. Model formulation α 2 T
There are three possible transition directions from
In this section, we consider a newly created model stage M. Individuals in stage M may return to
of excessive gambling based on modified ABC stage A by minimizing the frequency and inten-
fractional differential equations. We focus on the sity of their gambling, measured using a param-
NAMPR model, which addresses the problem of eter λ, transfer to stage R by stopping gambling
gambling within a population. This model com- activities, denoted by the stopping rate ς, or tran-
prises the following five types. sition to the excessive gambling problem stage.
• N: The number of individuals who know The latter step can be introduced by the influ-
nothing about gambling (non-gamblers). ence rate, measured in terms of the proportion
• A: The number of individuals who are γ 3 P
in the excessive gambling stage: multiplied
aware of the risks of gambling and gam- T
ble without problems but are exposed to by a constant α 3 . Therefore, the overall problem
γ 3 P
problem gambling (no problem gamblers). of the excessive gambling rate is: ϑ 3 = α 3 .
T
• M: The number of individuals who have
minor symptoms of problem gambling and
are at risk of addiction (minor-risk gam-
blers).
• P: The number of individuals who are
addicted to gambling and gamble perma-
nently (permanent gamblers).
• R: The number of individuals who have
stopped or recovered from problem gam-
bling (recovered gamblers).
The total population is computed as follows:
Figure 1. Problem gambling model flow diagram
T(t) = N(t) + A(t) + M(t) + P(t) + R(t).
All parameters in the model are assumed to have There are also three possible transition directions
values greater than zero and are defined as fol- from Stage P. Addicted gamblers may recover
lows: all new recruits, assumed to have no aware- naturally or with the help of religion, social me-
ness of the risks of gambling and never having dia awareness, school education, and professional
gambled before, are recruited at a rate of Λ. An help. When an addicted gambler returns to be-
individual in stage N is not introduced to gam- ing a non-problem gambler, a transition to stage
bling, but contacts from gamblers urge them to A occurs, modeled using κ. Individuals in the
gamble with no problems, leading them to transi- excessive problem gambling stage will either re-
tion to stage A at time t. This contact is modeled cover or stop all gambling activities, resulting in
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