Page 105 - IJOCTA-15-2

A. Ebrahimzadeh, R. Khanduzi, A. Jajarmi / IJOCTA, Vol.15, No.2, pp.294-310 (2025)
where D (1) = [d ij ] is the (N + 1) × (N + 1) oper-
ational matrix of derivative given by: T (1)
S D ϕ(t) =
T T T
0, i ≤ j, Λ − (U ϕ(t))(S ϕ(t)) − µ(S ϕ(t))
1
d ij = (21)
1, i ≥ j.
By utilizing equations (17) and (20), we ob- T T T
tain: − (1 − U ϕ(t))β(S ϕ(t))(I ϕ(t))
2
v
′ T ′ T (1)
S (t) = S ϕ (t) = S D ϕ(t), (22) β(S ϕ(t))(1 − U ϕ(t))(I ϕ(t))
T
T
T
− 3 c , (38)
T
1 + α(I ϕ(t))
c
′ T ′ T (1)
V (t) = V ϕ (t) = V D ϕ(t), (23)
T
V D (1) ϕ(t) =
′ T ′ T (1)
I (t) = I ϕ (t) = I D ϕ(t), (24)
c
c
c
T
T
T
− (1 − U ϕ(t))β(I ϕ(t))(V ϕ(t))
′ T ′ T (1) 2 v
I (t) = I ϕ (t) = I D ϕ(t), (25)
v
v
v
T
T
T
− µ(V ϕ(t)) + (U ϕ(t))(S ϕ(t)), (39)
1
′ T ′ T (1)
R (t) = R ϕ (t) = R D ϕ(t), (26)
T
I D (1) ϕ(t) =
c
T
u 1 (t) = U ϕ(t), (27)
1
T
− (µ + c c + ρw c + (1 − ρ)w c )(I ϕ(t))
c
T
u 2 (t) = U ϕ(t), (28)
2
T
T
T
β(1 − U ϕ(t))(S ϕ(t))I ϕ(t)
3
c
T
u 3 (t) = U ϕ(t), (29) + 1 + αI ϕ(t), , (40)
T
3
c
where the vectors S, V , I c , I v , R, U 1 , U 2 , and U 3
are defined as follows:
T
T
S = [c 0 , . . . , c n ] , (30) I D (1) ϕ(t) =
v
T
T
T
T
V = [c n+1 , . . . , c 2n+1 ] , (31) β(1 − U ϕ(t))(I ϕ(t))(V ϕ(t))
v
2
T
I c = [c 2n+2 , . . . , c 3n+2 ] , (32) T T T
+ β(1 − U ϕ(t))(S ϕ(t))(I ϕ(t))
2
v
T
I v = [c 3n+3 , . . . , c 4n+3 ] , (33)
T
T
+ (1 − ρ)w c (I ϕ(t) − (µ + c v + w v )I ϕ(t),
c v
(41)
T
R = [c 4n+4 , . . . , c 5n+4 ] , (34)
T
R D (1) ϕ(t) =
T
U 1 = [c 5n+5 , . . . , c 6n+5 ] , (35)
T
T
T
− µR ϕ(t) + w v I ϕ(t) + ρw c I ϕ(t). (42)
v
c
T
U 2 = [c 6n+6 , . . . , c 7n+6 ] , (36)
T
U 3 = [c 7n+7 , . . . , c 8n+7 ] . (37)
Using equations (20) and (22)-(29), we sim- Also, the initial conditions yield the following
plify the system dynamics as follows: five linear equations:
300