Page 42 - IJOCTA-15-3

M. Aychluh et.al. / IJOCTA, Vol.15, No.3, pp.407-425 (2025)
υ
Proof. Differentiating the total population and for the case α = υ, β = υ + 1 and x = −ηt ,
T(t) = N(t) + A(t) + M(t) + P(t) + R(t) with we have
respect to time and using equations in system (7), υ 1 υ
we obtain E υ, υ+1 (−ηt ) = ηt υ (1 − E υ, 1 (−ηt )) . (21)
The Mittag-Leffeler function is bounded for all
mABC υ t > 0, possess an asymptotic behavior, 37 intro-
D T (t) ≤ Λ − ψT (t) . (19)
0
ducing the relation (21) in inequality (20), it is ob-
Applying the Laplace transform on two sides of Λ
the above inequality, we obtain: vious that T (t) ≤ as t → ∞. Thus, N (t) and
ψ
Λ
mABC υ all other variables of the system (7) are bounded
L D T (t) ; s ≤ − ψL {T (t) ; s}
0
s in the region Ω.
Theorem 5. If the set of initial conditions
υ
B (υ) s L {T (t) ; s} {N(0) ≥ 0, A(0) ≥ 0, M(0) ≥ 0, P(0) ≥
⇒ + ψL {T (t) ; s} 0, R(0) ≥ 0} ∈ R 5 then, the solutions of
υ + (1 − υ) s υ +
N(t), A(t), M(t), P(t), and R(t) are non-
B (υ) T (0) s υ−1
υ−(υ+1)
≤ Λs + . negative for all t ≥ 0.
υ + (1 − υ) s υ
Proof. Assume that all the state variables of the
Implies
model are continuous. Consider the first equation
Λυ
L {T (t) ; s} ≤ of the system (7):
B (υ) + (1 − υ) ψ
γ 1 A + γ 2 M + γ 3 P
υ
mABC D N (t) = Λ−α 1 N −ψN.
s υ−(υ+1) 0 T
×
υψ (22)
υ
s + Since, all the solutions are bounded, let
B (υ) + (1 − υ) ψ
γ 1 A + γ 2 M + γ 3 P
is bounded by σ. Then,
T
Λ (1 − υ) B (υ) T (0)
+ + mABC υ
B (υ) + (1 − υ) ψ B (υ) + (1 − υ) ψ D N (t) ≥ −aN, (23)
0
where a = α 1 σ + ψ. Applying the Laplace trans-
s υ−1
× υψ . form in Eq. (23) and using Eq. (4), we have:
υ
s +
υ
B (υ) + (1 − υ) ψ " s L {N (t) ; s} − N (0) s υ−1 #
B (υ)
Applying the inverse Laplace, we arrive at: υ υ
1 − υ s +
Λυ 1 − υ
υ
T (t) ≤ E υ, υ+1 (−ηt )
B (υ) + (1 − υ) ψ ≥ −aL {N (t)} .
Further simplification gives:
1 (20) υ υ−1
+ B (υ) s L {N (t) ; s} B (υ) N (0) s
B (υ) + (1 − υ) ψ −
υ + (1 − υ) s υ υ + (1 − υ) s υ
υ
× (Λ (1 − υ) + B (υ) T (0)) E υ, 1 (−ηt ) ,
≥ −aL {N (t)} .
υψ υ υ
where η = and E α, β (.) [B (υ) s + aυ + a (1 − υ) s ] L {N (t) ; s}
B (υ) + (1 − υ) µ υ−1
is the Mittag-Leffler function of two parameters ≥ B (υ) s N (0)
α > 0 and β > 0 defined as:
∞ B (υ) N (0) s υ−1
X x i L {N (t) ; s} ≥ .
E α, β (χ) = , B (υ) + a (1 − υ) υ aυ
Γ (αi + β) s +
i=0 B (υ) + a (1 − υ)
whose Laplace transform is Taking the inverse Laplace transform in the above
n o s α−β inequality, we obtain:
α
L χ β−1 E α, β (∓ηχ ) ; s = υ
α
s ± η B (υ) N (0) −aυt
N (t) ≥ E υ, 1 .
provided that s > |η| 1/α . The Mittag-Leffler func- B (υ) + a (1 − υ) B (υ) + a (1 − υ)
tion satisfies aυ
Since E υ, 1 − t υ > 0 and
E α, β−α (χ) 1 B (υ) + a (1 − υ)
E α, β (χ) = − ,
χ χΓ (β − α) N (0) > 0, then N (t) ≥ 0. In the same way,
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