Page 43 - IJOCTA-15-3

Modeling and analysis of the dynamics of an excessive gambling problem with modified fractional operator
it is easy to prove that the remaining state vari- Theorem 6. The gambling problem-free equilib-
ables are non-negative for all t > 0. Therefore, rium point E 0 is locally asymptotically stable when
the solutions of system (7) are non-negative for R 0 < 1 and unstable when R 0 > 1.
all t ≥ 0.

Proof. To prove the local stability, we consider
3.2. Stability analysis of the model
the Jacobian matrix of the system (7) at E 0 as
In this section, the basic reproduction number follows:
and stability analyses of endemic and no problem J (E 0 ) =
 
gambling equilibrium points are discussed. a 11 a 12 a 13 a 14 0
a 21 a 23 a 24
 0 0 
 
 0 0 a 33 a 34 0  , (25)
3.2.1. No problem gambling equilibrium point  
0 0 0 0
 a 44 
To find the gambling problem-free equilibrium 0 a 52 a 53 a 54 a 55
point of the model, we set the derivatives of all
compartments to zero and assume that there are
no individuals with gambling problem ( M =
0, P = 0 and R = 0). The system reduces to:
where

γ 1 A
 Λ − α 1 N − ψN = 0,


 T
(24) a 11 = −(α 1 γ 1 − ℘), a 12 = −(℘ + ψ)
γ 1 A


 α 1 N − (℘ + ψ) A = 0.

T ℘ + ψ ℘ + ψ
a 13 = −γ 2 , a 14 = −γ 3
Solving these equations yield: γ 1 γ 1

α 1 γ 1 − ℘ − ψ
E 0 = s, s, 0, 0, 0 . a 21 = α 1 γ 1 − ℘ − ψ,
℘ + ψ
Λ ℘ + ψ α 1 γ 1 − ℘ − ψ
where s = and α 1 γ 1 > ℘ to ensure posi- a 23 = γ 2 + λ − α 2 γ 2
α 1 γ 1 − ℘
γ 1 α 1 γ 1
tivity of N 0 and α 1 γ 1 > ℘+ψ for positivity of A 0 .
This equilibrium represents a state where there ℘ + ψ α 1 γ 1 − ℘ − ψ
are no individuals with gambling problems and a 24 = γ 3 + κ − α 2 γ 3 ,
γ 1 α 1 γ 1
the population consists only of non-gamblers (N)
and non-problem gamblers (A). a 25 = 0, a 52 = ℘, a 53 = ς, a 54 = ζ

α 1 γ 1 − ℘ − ψ
3.2.2. The basic reproduction number a 33 = α 2 γ 2 − (λ + ς + ψ)
α 1 γ 1
The basic reproduction number R 0 is a critical
threshold in epidemiological models, representing α 1 γ 1 − ℘ − ψ
the average number of new ”infections” (in this a 34 = α 2 γ 3 + δ,
α 1 γ 1
case, new problem gamblers) generated by a sin-
gle addicted gambler in a fully susceptible popu- a 44 = −(κ + δ + ζ + ψ), a 55 = −ψ
lation.
In this Jacobian matrix, three of the eigenvalues
are negative, that is m 1 = −ψ, m 2 = −(κ+δ+ζ+

α 2 γ 2 q α 2 γ 3 q u −δ
F = , V = ψ) and m 3 = − (λ + ς + ψ) (1 − R 0 ) for R 0 < 1.
0 0 0 v
The remaining eigenvalues can be obtained from
where the characteristic equation:
α 1 γ 1 − ℘ − ψ 2
q = , u = λ + ς + ψ, m + d 1 m + d 2 = 0. (26)
α 1 γ 1
Since the coefficients d 1 = α 1 γ 1 − ℘ > 0 for
v = κ + δ + ζ + ψ.
α 1 γ 1 > ℘ and d 2 = (℘ + ψ)(α 1 γ 1 − ℘ − ψ) > 0 for
Using the next-generation matrix method, the ba- α 1 γ 1 > ℘+ψ. The solutions of Equation (26) have
sic reproduction number of the proposed model is negative real parts. Since the Im(m i = 0, i =
the dominant eigenvalue of FV −1 . Therefore, υπ
1, 2, 3, 4, then clearly | arg(m i )| = π > , υ ∈
2
α 1 γ 1 − ℘ − ψ (0, 1). According to the Matignon criterion, 38 the
R 0 = α 2 γ 2 .
α 1 γ 1 (λ + ς + ψ) equilibrium point E 0 is stable when R 0 < 1.
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