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Modeling and analysis of the dynamics of an excessive gambling problem with modified fractional operator
where
  υ
1 − υ  A = (n − i + 2 + υ) (n + 1 − i)
 m n+1 (t) = m (0) + f (t n , m (t n )) 

B (υ) 
 

 
  υ
 B = (n − i + 2 + 2υ) (n − i)
 
 

  P n R υ−1 
 t i+1
 (t n+1 − ς)
υ i=1 t i

+    υ+1
 C = (n + 1 − i)

B (υ) Γ (υ) 
 
 ×f (ς, m (ς)) dς 
 
  υ
D = (n − i + 1 + υ) (n − i)
 




 1 − υ γ υ υ
 − f (0, m (0)) 1 + t n .


B (υ) Γ (υ + 1)
(29) 5. Graphical results and discussion
The graphical results validate the theoretical pro-
The function f (ς, m (ς)) can be approximated cesses. The choice of the fractional order υ in the
over (t i , t i+1 ) using the Lagrange’s interpolation modified ABC fractional derivative, which is in
polynomial as: the range (0, 1], is primarily owing to its math-
ematical consistency with integer-order deriva-
 
f (t i , m (t i )) tives. In this section, we implement the proposed
(t − t i−1 )
 h  technique to solve system (7) with the initial solu-
 
tions and model parameters provided in Table 1.
f (ς, m (ς)) =  
f (t i−1 , m (t i−1 ))
 
− (t − t i ) The approximate results were obtained using the
h numerical scheme presented in Section 4. Due to
the lack of organized data on excessive gambling,
Substituting it into Equation (29), we obtain: no real data have been provided for our model re-
sults. Therefore, the approximate values are not
1 − υ

 m n+1 (t) = m (0) + f (t n , m (t n )) estimated based on real data, and the results are

 B (υ)
 obtained using assumed parameter values and ini-


tial conditions. The gambling problem-free equi-


  
f (t i , m (t i ))


 librium point exists when α 1 γ 1 > ℘ + ψ. The
 I 1
υ P n fractional order υ influences transient dynamics

  h 
+ i=1  
B (υ) Γ (υ) but not equilibrium point stability. The basic re-


  f (t i−1 , m (t i−1 )) 

−

 I 2 production number is then computed to be R 0 =

 h
 1.4826 and unstable gambling problem-free equi-



 librium point is E 0 = (2.7583, 15.6306, 0, 0, 0).
1 − υ

 γ υ υ
 − f (0, m (0)) 1 + t n . R 0 > 1 means that on average each problem gam-

B (υ) Γ (υ + 1)
bler spreads addiction to more than 1 person. In-
(30)
terventions must reduce R 0 below one. Some key
R υ−1 strategies are to reduce transmission α 2 &γ 2 , in-
where I 1 = t i+1 (t n+1 − ς) (ς − t i−1 ) dς and
t i crease recovery rates, λ, & ς and raise awareness.
R t i+1 υ−1
I 2 = (t n+1 − ς) (ς − t i ) dς. Computing
t i
these integrals, we finally get the approximate
solution as:
5.1. Optimal control analysis
1 − υ

 m n+1 (t) = m (0) + f (t n , m (t n )) We consider the system (7) by adding the con-

 B (υ)

 trol variable c(t), which ranges from 0 to 1, where



  υ  zero corresponds to the absence of application of
h f (t i , m (t i ))



 (A − B) any control mechanism, and one refers to the case

 
 Γ (υ + 2)

   where there is a fully controlled scenario. The in-
υ P n  

+  υ  termediate values c(t) ∈ (0, 1) quantify the effect
B (υ) i=1  h f (t i−1 , m (t i−1 )) 

 −  of applying the intervention mechanisms. The


  Γ (υ + 2) 
 control directly reduces the progression to addic-
(C − D)



tion by a factor of 1 − c. The control variable:







 1 − υ
 γ υ υ • Represents interventions like counseling,
 − f (0, m (0)) 1 + (nh) .

B (υ) Γ (υ + 1) self-exclusion programs, and early treat-
(31) ment
• Reduces the transition rate from M to P.
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