Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
11,
No.
2,
April
2021,
pp.
1460
1468
ISSN:
2088-8708,
DOI:
10.11591/ijece.v11i2.pp1460-1468
r
1460
An
analytic
study
of
the
fractional
order
model
of
HIV
-1
virus
and
CD4+
T
-cells
using
adomian
method
Kamel
Al-Khaled,
Maha
Y
ousef
Department
of
Mathematics
and
Statistics,
Jordan
Uni
v
ersity
of
Science
and
T
echnology
,
Irbid,
Jordan
Article
Inf
o
Article
history:
Recei
v
ed
Apr
29,
2020
Re
vised
Apr
8,
2020
Accepted
Jun
17,
2020
K
eyw
ords:
Approximate
solutions
Fractional
calculus
Adomian
decomposition
Laplace
transform
Fractional
model
for
HIV
infection
of
CD4+
T
-cells
ABSTRA
CT
In
this
article,
we
study
the
fractional
mathematical
model
of
HIV
-1
infection
of
CD4+
T
-cells,
by
studying
a
system
of
fra
ctional
dif
ferential
equations
of
first
order
with
some
initial
conditions,
we
study
the
changing
ef
fect
of
man
y
parameters.
The
frac-
tional
deri
v
ati
v
e
is
described
in
the
caputo
sense.
The
adomian
decomposition
method
(Shortly
,
ADM)
method
w
as
used
to
calculate
an
approximate
solution
for
the
system
under
study
.
The
nonlinear
term
is
dealt
with
the
help
of
Adomian
polynomials.
Nu-
merical
results
are
presented
with
graphical
justifications
to
sho
w
the
accurac
y
of
the
proposed
methods.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Kamel
Al-Khaled
Jordan
Uni
v
ersity
of
Science
and
T
echnology
Irbid,
P
.O.Box
3030,
Jordan
Email:
kamel@just.edu.jo
1.
INTR
ODUCTION
There
is
a
strong
correlation
coef
ficient
between
a
person’
s
immune
system
and
an
infection
caused
by
virus,
there
there
are
man
y
dif
ferent
f
actors
and
restrictions
that
af
fect
this
relationship,
ne
g
ati
v
ely
or
positi
v
ely
.
When
studying
such
a
case,
we
cannot
see
the
rules
or
foundations
that
control
the
dynamics
or
ho
w
infection
occurs
by
studying
the
data
or
statistical
dra
wings
resulting
from
that
study
.
By
studying
some
mathematical
models
that
simulate
reality
,
we
found
that
these
models
were
important
in
dealing
with
a
set
of
h
ypotheses
related
to
the
study
through
which
the
understanding
of
the
study
w
as
reached
in
a
logical
and
correct
w
ay
.
As
a
result
of
that,
we
can
gi
v
e
find
ne
w
h
ypotheses
or
scenarios
which
leads
to
the
calculation
of
some
numerical
constants
that
ha
v
e
a
significant
impact
in
the
study
.
In
this
research,
our
main
concern
will
be
limited
to
the
study
of
HIV
.
The
study
of
the
ef
fect
of
dynamic
processes
such
as
mo
v
ement
and
speed
between
the
human
immune
system
and
HIV
disease
is
a
more
comple
x
f
act
compared
to
infection
from
other
sides.
The
immune
response
then
has
the
susceptibility
and
ability
to
fight
the
virus
as
HIV
infects
the
auxiliary
cells
kno
wn
as
CD4+T
[1].
It
is
one
of
the
most
ab
undant
cells
in
the
immune
system,
which
controls
the
production
of
specific
immune
responses
to
the
same
goal.
It
must
be
noted
here
that
HIV
can
af
fect
some
other
immune
cells
that
ha
v
e
a
n
ef
fecti
v
e
role
in
producing
and
generating
anti-virus
immunity
.
This
means
that
the
weak
ened
immune
response
gro
ws
significantly
and
early
during
the
first
and
adv
anced
stage
of
the
disease
infection
and
this
leads
to
viral
tolerance
and
the
ability
to
mutate
and
gro
w
the
virus.
As
for
the
de
v
elopment
of
AIDS,
it
occurs
as
a
result
of
infection
with
the
disease
i
nitially
,
the
symptoms
are
not
clear
and
for
periods
that
may
e
xtend
for
se
v
eral
years,
before
the
viral
load
increases
and
the
number
of
cells
called,
CD4+
T
-cells
decreases
to
a
lo
w
le
v
el,
which
leads
to
a
decrease
and
weak
resistance
of
the
immune
system
in
humans
and
as
a
result
the
disease
spreads
to
the
rest
of
the
cells
J
ournal
homepage:
http://ijece
.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1461
in
the
body
.
The
de
v
elopment
and
spread
of
t
he
disease
depends
on
the
de
v
elopment
of
some
v
ariables
related
to
the
virus
itself,
that
is
spr
eading
quickly
.
There
are
some
anti-retro
viral
therapies
that
ha
v
e
pro
v
en
successful
and
ha
v
e
had
a
major
impact
in
limiting
the
multiplication
of
the
virus
or
postponing
the
de
v
elopment
of
the
disease
for
long
periods
of
time,
this
is
pro
v
en
true
for
man
y
who
ha
v
e
contracted
the
disease.
These
days,
there
are
tw
o
w
ays
to
recei
v
e
the
medicine.
The
first
is
through
re
v
erse
transcriptase
inhibitors,
where
the
y
o
v
erlap
and
pre
v
ent
the
infection
of
the
health
y
cell.
As
for
the
second
mechanism,
i
t
pre
v
ents
protease
inhibitors
from
manuf
acturing
a
ne
w
infectious
virus
by
the
infected
cell,
meaning
that
the
issue
in
general
is
either
delaying
time
or
Pre
v
enting
the
spread
of
the
virus
to
pre
v
ent
infection
of
health
y
cells.
Most
of
the
infected
cells
ha
v
e
a
relati
v
ely
short
lifespan,
and
therefore
these
cells
li
v
e
long.
This
means
that
eliminating
the
virus
through
medication
is
impossible.
T
aking
medicines
for
this
disease
has
se
v
eral
ne
g
ati
v
e
side
ef
fects,
the
most
important
of
which
is
generating
resistance
ag
ainst
these
drugs.
This
led
to
the
researchers’
k
een
interest
in
finding
treatment
re
gimens
that
strengthen
the
immune
responses
to
the
virus.
Thus,
thinking
about
a
specific
mechanism
is
to
find
therapeutic
systems
that
support
the
immune
responses
to
the
virus.
Mathematical
models
ha
v
e
been
de
v
oted
to
understanding
the
transmission
of
HIV
infection.
There
are
man
y
pre
vious
studies
that
were
de
v
oted
to
the
study
of
t
ransmission
of
HIV
infection
in
humans,
as
these
pedals
were
basically
based
on
establishing
a
mathematical
model,
it
is
often
made
up
of
a
system
of
ordinary
dif
ferential
equations
with
se
v
eral
parameters,
researchers
were
able
to
find
e
xact
v
alues
for
the
parameters
to
use
after
solving
the
problem.
Perelson,
Kirschner
and
De
Boer
[2]
the
y
b
uilt
a
mathematical
model
that
is
a
system
of
re
gular
dif
ferential
equations
representing
the
spread
of
a
virus
that
does
not
contain
cells
to
HIV
in
a
closed
place
that
represents
the
bloodstream.
The
mathematical
model
consists
of
four
main
parts:
the
total
number
of
he
alth
y
cells,
then
the
total
number
of
uninfected
cells,
the
infected
and
infected
cells
with
acti
vity
,
and
finally
the
free
virus
particles.
This
mathematical
model
is
most
popular
for
its
HIV
transmis-
sion.
There
are
man
y
related
mathematical
models
that
ha
v
e
been
mentioned
in
the
follo
wing
pre
vious
studies
[3–8],
where
the
model
of
Perelson,
Kirschner
and
De
Boer
,
w
as
used
as
a
pillar
in
b
uilding
these
mathematical
models.
It
has
been
observ
ed
that
the
mathematical
model
can
be
used
to
predi
ct
and
kno
w
the
symptoms
of
AIDS
clinically
,
which
is
a
good
characteristic
of
the
model.
In
the
article
[2],
the
researchers
simplified
the
model
that
the
y
proposed
and
wrote
the
model
in
the
form
of
three
dif
ferential
dif
ferential
equations
of
the
first
order
through
the
h
ypothesis
that
all
infected
cells
are
the
only
ones
capable
of
creating
and
producing
cells
infected
with
the
virus.
A
mathematical
model
consi
sting
of
four
equations
w
as
studied
in
a
pre
vious
research
by
the
first
author
and
others
[9]
in
t
hat
dif
ferential
equations
were
used
to
be
of
ordinary
deri
v
ati
v
es
without
e
xamining
the
fractional
de
ri
v
ati
v
es,
that
is
ne
w
in
this
paper
.
W
ang
and
Song
in
[10]
found
the
equilibrium
points
of
the
mathematical
system
and
study
the
stability
of
solutions
that
represent
a
c
yclical
solution
to
the
model
b
uild
for
HIV
.
W
e
can
say
that
a
lar
ge
number
of
pre
vious
studies
that
dealt
with
the
de
v
elopment
of
a
mathematical
model
consisting
of
ordinary
dif
ferential
equations
most
of
these
studies
were
of
the
type
that
the
deri
v
ati
v
es
are
normal
and
not
fractional
order
as
in
thi
s
research.
In
[11],
Araf
a
studied
an
HIV
model
system
of
fractional
order
for
CD+4
T
-cells
using
the
generalized
euler
method
(GEM).
Where
in
the
study
,
the
components
were
di
vided
into
three
main
parts,
namely
,
the
concentration
of
sensiti
v
e
cells,
the
concentration
of
HIV
-infected
cells,
and
finally
HIV
particles
that
are
denoted
respecti
v
ely
by
T
(
t
)
;
I
(
t
)
and
V
(
t
)
.
These
days,
fractional
dif
ferentiation
and
inte
gration
has
been
e
xtensi
v
ely
used
in
man
y
fields
wi
th
practical
applications
in
science,
engineering,
and
other
kno
wledge
[12–16].
Mathematicians
were
able
to
de
v
elop
a
fractional
mathematical
model
related
to
the
human
root
in
[17].
Fractional
deri
v
ati
v
es
embody
the
basic
properties
of
cell
rheological
beha
vior
and
is
highly
successful
in
the
subject
that
is
related
to
rheology
[18].
T
alking
about
ordinary
dif
ferential
equations
with
fractional
(FODE)
deri
v
ati
v
es
is
lik
e
talking
about
the
memory
that
relates
to
that
phenomenon
in
relation
to
biological
systems
[19].
In
this
paper
,
we
will
study
a
system
of
FODE
which
is
a
v
alid
model
for
studying
the
de
v
elopment
and
gro
wth
of
HIV
,
as
we
be
gin
to
define
fractional
deri
v
ati
v
es
and
fractional
inte
gration.
[20,
21].
Pre
vious
studies
ha
v
e
sho
wn
that
there
are
man
y
definitions
of
fractional
deri
v
ati
v
es,
and
here
only
we
mention
the
method
that
we
will
use
which
is
Caputo’
s
definition,
which
is
suitable
for
solving
dif
ferential
equations
with
initial
v
alues.
The
focus
of
this
article
is
as,
we
present
in
section
2
the
mathematical
model
under
study
relat
ed
to
HIV
.
In
the
third
section,
we
briefly
introduce
some
definitions
of
the
fractional
calculus.
In
section
4,
we
e
xtend
the
application
of
Adomian
decomposition
to
b
uild
our
analytical
approximate
solutions
for
the
HIV
-1
fractional
system.
Finally
,
graphical
justification
are
displayed
to
v
alidate
the
obtained
solution.
The
paper
ends
with
some
concluding
remarks.
An
analytic
study
of
the
fr
actional
or
der
model
of
HIV
-1
virus
and
CD4+...
(Kamel
Al-Khaled)
Evaluation Warning : The document was created with Spire.PDF for Python.
1462
r
ISSN:
2088-8708
2.
FRA
CTION
AL
D
YN
AMICAL
MODEL
HIV
-1
INFECTION
OF
CD4+
T
-CELLS
Here
we
sho
w
the
mathematical
model
of
HIV
virus
infection
of
CD4+-cells
in
its
general
form
with
fractional
deri
v
ati
v
es.
This
model
is
written
in
the
form
of
a
system
of
re
gular
non-linear
dif
ferential
equations
that
tak
es
the
follo
wing
formulas.
8
>
<
>
:
d
T
dt
=
T
+
r
T
(1
T
+
I
T
max
)
K
V
T
d
I
dt
=
K
V
T
I
;
m
1
<
m:
d
V
dt
=
N
I
V
(1)
that
has
suitable
conditions
T
(0)
=
T
0
;
I
(0)
=
I
0
;
V
(0)
=
v
0
;
(2)
where
T
(
t
)
,
I
(
t
)
and
V
(
t
)
symbolizes
to
the
concentration
of
sensiti
v
e
CD4+
T
-cells,
the
concentration
of
diseased
CD4+
T
-cells
by
the
HIV
viruses
and
the
well
HIV
virus
particles
in
the
body
,
respecti
v
ely
.
r
T
(1
T
+
I
T
max
)
is
logistic
model
of
the
cell
gro
wth
that
are
free
of
disease
CD4
T
-cells,
and
proliferation
of
infected
CD4+
T
-cells
is
ignored,
r
is
the
rate
of
doubling
the
T
-cells
that
are
mitosis.
T
max
represent
the
lar
gest
s
tage
of
CD4+
T
-cells
in
the
infected
body
.
The
function
K
V
T
represents
an
infected
of
HIV
infection
of
disease-
free
T
-cells,
here
K
>
0
is
the
infection
change.
N
is
the
general
rate
of
the
number
of
diseas
e-transmitting
molecules
that
are
produced
from
diseased
cells
throughout
the
entire
cell
life.
The
body
is
e
xcreted
CD4+
T
-
cells
from
precursors
in
the
bone
marro
w
and
th
ymus
at
a
fix
ed
change
of
.
per
capita
death
rate
of
infected
virus
particles.
N
is
the
rate
of
production
of
virions
by
infected
cells.
It
is
noted
that
the
set
that
follo
ws
represents
some
constraints
with
the
appropriate
constants
for
the
aid
of
numerical
solutions
[2,
10].
T
(0)
=
1
:
0
10
1
;
I
(0)
=
0
;
V
(0)
=
1
:
0
10
1
;
=
2
:
0
10
2
;
=
3
:
0
10
1
,
=
2
:
4
;
=
1
:
0
10
1
;
K
=
2
:
7
10
3
;
r
=
3
;
T
max
=
1500
;
N
=
10
:
3.
B
ASICS
OF
FRA
CTION
AL
CALCULUS
Fractional
calculus
and
its
applications
encounter
e
xpeditious
de
v
elopments
with
more
and
more
con-
vincing
applications
in
real
w
orld.
Here
we
will
present
some
of
the
characteristics
related
to
the
subject
of
fractional
calculus
where
there
are
man
y
definitions
in
pre
vious
studies,
that
ha
v
e
been
proposed
and
we
will
mention
here
only
the
important
part,
that
we
will
need
to
formulate
the
approximate
solution
to
the
issue
that
we
study
.
These
definiti
on
s
include,
Riemman-Liouvil
le,
Reize,
Caputo,
Rabotno
v
,
Caputo-F
abrizio
and
Atang
ana-Baleanu
fractional
operator
.
The
reason
for
finding
fractional
deri
v
ati
v
es
in
relation
to
time
is
the
characteristic
representation
of
the
k
ernel,
in
which
each
has
its
o
wn
dif
ficulties,
lik
e
the
singularities
that
might
accrues
in
the
k
ernel,
the
memory
ef
fect
caused
by
the
locality
of
the
k
ernel
w
as
major
problem
in
the
Caputo-F
abrizio
fractional
deri
v
ati
v
e
w
as
the
best
operator
for
dealing
with
systems
in
v
olving
the
electro-
osmotic
magneto
h
ydrodynamic
(MHD)
free
con
v
ectional
flo
w
of
W
alters’-B
fluid
in
addition
to
the
heat
and
mass
produced
[22].
Another
fractional-e
xponential
function
is
Y
ang-Abdel-Aty-Cattani
FC,
namely
the
Y
ang-
Abdel-Aty-Cattani
FC,
is
updated
and
addressed
on
propag
ation
equations
[23].
The
main
matter
under
study
that
is
related
to
relation
to
fractional
deri
v
ati
v
es,
is
that
we
will
re-present
the
material
in
section
2
of
the
research
paper
[24].
Here
we
will
introduce
the
definition
of
Riemann-Liouville
J
a
as
follo
ws:
Definition
1
F
or
a
positi
v
e
real
number
.
The
function
J
,
realized
on
the
normal
Lebesgue
room
L
1
[
a;
b
]
as
J
a
f
(
x
)
=
1
(
)
R
x
a
(
x
t
)
1
f
(
t
)
dt;
J
0
a
f
(
x
)
=
f
(
x
)
when
a
x
b
,
is
purport
to
be
the
Riemann-Liouville
fractional
inte
gral
function
with
sort
.
Some
of
the
pairing
properties
of
the
funct
ion
J
can
be
found
in
the
reference
[14],
and
here
we
mention
the
important
ones:
F
or
f
2
L
1
[
a;
b
]
;
;
0
and
>
1
J
a
f
(
x
)
e
xists
for
almost
e
v
ery
x
2
[
a;
b
]
J
a
J
a
f
(
x
)
=
J
+
a
f
(
x
)
J
a
J
a
f
(
x
)
=
J
a
J
a
f
(
x
)
Int
J
Elec
&
Comp
Eng,
V
ol.
11,
No.
2,
April
2021
:
1460
–
1468
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1463
J
a
x
=
(
+1)
(
+
+1)
(
x
a
)
+
As
seen
in
the
reference
[13],
Riemann
deri
v
ati
v
es
ha
v
e
some
dra
wbacks
when
we
solv
e
realisti
c
problems
of
fractional
deri
v
ati
v
es.
Hence,
we
present
an
adjustment
for
the
fractional
dif
ferentiation
function
D
presented
by
Caputo
in
his
paper
about
the
subject
of
visco-elasticity
[25].
Definition
2
W
e
define
the
fractional
deri
v
at
i
v
e
of
a
gi
v
en
function
f
(
x
)
in
the
Caputo
meaning
in
the
form
D
f
(
x
)
=
J
m
D
m
f
(
x
)
=
1
(
m
)
Z
x
0
(
x
t
)
m
1
f
(
m
)
(
t
)
dt;
(3)
for
v
alues
of
such
that
m
1
<
m;
here
m
2
N
;
and
x
>
0
:
Here
we
mention
tw
o
main
characteristics,
because
we
need
them
and
we
will
use
them
later
Lemma
1
F
or
al
pha
to
satisfy
m
1
<
m
,
and
the
funtcion
f
to
be
in
the
class
L
1
[
a;
b
]
,
we
ha
v
e
D
a
J
a
f
(
x
)
=
f
(
x
)
,
also
J
a
D
a
f
(
x
)
=
f
(
x
)
m
1
X
k
=0
f
(
k
)
(0)
(
x
a
)
k
k
!
;
x
>
0
:
One
of
the
reasons
for
using
the
definition
related
to
Caputo
definition,
is
that
in
order
to
obtain
a
solution
to
the
dif
ferential
equation,
additional
conditions
must
be
established
in
order
to
obtain
a
single
unique
solution.
F
or
the
sense
of
ha
ving
fractional
dif
ferential
equations
of
Caputo
form,
the
mentioned
supplementary
conditions
are
just
the
usual
constrains,
that
are
tak
en
to
those
of
traditional
equations.
W
ith
comparis
on
with
Riemann-Liouville
fractional
dif
ferential
equations,
these
supplementary
terms
gi
v
e
some
fractional
deri
v
ati
v
es
at
the
starting
point
x
=
0
of
the
solution
to
be
found.
Primiti
v
e
conditions
are
not
perceptible
in
addition
to
not
kno
wing
the
amount
of
quantities
to
be
calculated
through
the
e
xperiment,
so
that
we
can
as
sign
them
perfectly
in
the
analysis
of
the
solution.
There
are
man
y
geometric
and
ph
ysical
interpretations
of
fractional
deri
v
ati
v
es
defined
by
the
tw
o
methods
Riemann-Liouvil
le
and
Caputo
types,
it
can
be
vie
wed
by
the
reader
in
the
research
[13]
due
to
its
importance.
Definition
3
Let
the
number
m
,
represent
the
smallest
inte
ger
that
is
bigger
than
by
1
,
then
the
fractional
deri
v
ati
v
es
of
order
>
0
,
according
to
Caputa
sense
can
be
presented
in
the
form
D
u
(
x;
t
)
=
@
u
(
x;
t
)
@
t
=
8
>
<
>
:
1
(
m
)
R
t
0
(
t
)
m
1
@
m
u
(
x;
)
@
m
d
;
m
1
<
<
m
@
m
u
(
x;t
)
@
t
m
;
=
m
2
N
F
or
more
information
and
to
kno
w
the
characteri
stics
and
proofs
related
to
fractional
deri
v
ati
v
es,
we
recommend
the
reader
to
refer
to
the
specialized
references
mentioned
at
the
end
of
this
research.
4.
IMPLEMENT
A
TION
OF
ADOMIAN
DECOMPOSITION
METHOD
In
order
to
find
our
approximate
solution
for
the
nonlinear
system
of
fractional
dif
ferential
in
(1),
via
the
use
of
the
proposed
methodology
that
is
kno
wn
as
Adomian
decomposition
method.
W
e
assume
that
functions
T
(
t
)
;
I
(
t
)
;
V
(
t
)
are
suf
ficiently
dif
ferentiable
and
a
unique
solution
for
these
equations
e
xists.
Start
by
taking
Laplace
transform
that
is
denoted
by
L
of
both
sides
of
(1)
Lf
d
T
dt
g
=
Lf
g
Lf
T
g
+
Lf
r
T
g
r
T
max
Lf
T
2
g
r
T
max
Lf
I
T
g
Lf
K
V
T
g
;
Lf
d
I
dt
g
=
Lf
K
V
T
g
Lf
I
g
;
(4)
Lf
d
V
dt
g
=
Lf
N
I
g
Lf
V
g
;
0
<
1
:
An
analytic
study
of
the
fr
actional
or
der
model
of
HIV
-1
virus
and
CD4+...
(Kamel
Al-Khaled)
Evaluation Warning : The document was created with Spire.PDF for Python.
1464
r
ISSN:
2088-8708
Using
the
formulas
of
Laplace
transform
on
the
abo
v
e
equation
s
Lf
T
g
T
0
s
1
=
s
Lf
T
g
+
r
Lf
T
g
r
T
max
Lf
T
2
g
r
T
max
Lf
I
T
g
K
Lf
V
T
g
;
s
Lf
I
g
I
0
s
1
=
K
Lf
V
T
g
Lf
I
g
;
s
Lf
V
g
V
0
s
1
=
N
Lf
I
g
Lf
V
g
;
0
<
1
:
(5)
Substituting
the
initial
conditions,
we
arri
v
e
at
s
Lf
T
g
=
0
:
1
s
1
+
s
Lf
T
g
+
r
Lf
T
g
r
T
max
Lf
T
2
g
r
T
max
Lf
I
T
g
K
Lf
V
T
g
;
s
Lf
I
g
=
K
Lf
V
T
g
Lf
I
g
;
s
Lf
V
g
=
0
:
1
s
1
+
N
Lf
I
g
Lf
V
g
;
0
<
1
:
or
Lf
T
g
=
0
:
1
s
+
s
+1
s
Lf
T
g
+
r
s
Lf
T
g
r
s
T
max
Lf
T
2
g
r
s
T
max
Lf
I
T
g
K
s
Lf
V
T
g
;
Lf
I
g
=
K
s
Lf
V
T
g
s
Lf
I
g
;
Lf
V
g
=
0
:
1
s
+
N
s
Lf
I
g
s
Lf
V
g
;
0
<
1
:
(6)
T
o
obtain
the
approximate
solution,
we
follo
w
the
Adomian’
s
method,
which
depends
on
writing
the
solution
as
an
infinite
bounded
series,
according
to
the
follo
wing:
T
(
t
)
=
1
X
n
=0
T
n
(
t
)
;
I
(
t
)
=
1
X
n
=0
I
n
(
t
)
;
V
(
t
)
=
1
X
n
=0
V
n
(
t
)
;
(7)
which
will
be
determined
recursi
v
ely
according
to
a
recursi
v
e
relation.
Moreo
v
er
,
to
o
v
ercome
the
non
linearity
problem,
the
method
defines
the
nonlinear
functions
A
=
T
2
;
B
=
T
I
;
C
=
V
T
by
an
infinite
sum
of
polynomials,
defined
by
A
=
1
X
n
=0
A
n
;
B
=
1
X
n
=0
B
n
;
C
=
1
X
n
=0
C
n
;
(8)
where
the
A
n
;
B
n
;
C
n
are
the
Adomian’
s
polynomials
which
are
generated
according
to
some
formulas
set
by
Adomian
[26,
27],
or
as
in
the
modified
form
done
by
W
azw
az
[28].
A
0
=
T
2
0
;
A
1
=
2
T
0
T
1
;
A
2
=
2
T
0
T
2
+
T
2
1
;
A
3
=
2
T
0
T
3
+
2
T
1
T
2
;
A
4
=
2
T
0
T
4
+
2
T
1
T
3
+
(
T
2
)
2
;
B
0
=
T
0
I
0
;
B
1
=
T
0
I
1
+
T
1
I
0
;
B
3
=
T
0
I
3
+
T
1
I
2
+
T
2
I
1
+
T
3
I
0
;
B
4
=
T
0
I
4
+
T
1
I
3
+
T
2
I
2
+
T
3
I
1
+
T
4
I
0
;
C
0
=
V
0
T
1
+
V
1
T
0
;
C
1
=
V
0
T
1
+
V
1
T
0
;
C
3
=
V
0
T
3
+
V
1
T
2
+
V
2
T
1
+
V
3
T
0
;
C
4
=
V
0
T
4
+
V
1
T
3
+
V
2
T
2
+
V
3
T
1
+
V
4
T
0
;
(9)
Int
J
Elec
&
Comp
Eng,
V
ol.
11,
No.
2,
April
2021
:
1460
–
1468
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1465
Substituting
(5)
and
(6)
into
(4)
yields
Lf
P
1
n
=0
T
n
(
t
)
g
=
0
:
1
s
+
s
+1
s
Lf
1
X
n
=0
T
n
(
t
)
g
+
r
s
Lf
1
X
n
=0
T
n
(
t
)
g
r
s
T
max
Lf
1
X
n
=0
A
n
(
t
)
g
r
s
T
max
Lf
1
X
n
=0
B
n
(
t
)
g
K
s
Lf
1
X
n
=0
C
n
(
t
)
g
;
Lf
P
1
n
=0
I
n
(
t
)
g
=
K
s
Lf
1
X
n
=0
C
n
(
t
)
g
s
Lf
1
X
n
=0
I
n
(
t
)
g
;
Lf
P
1
n
=0
V
n
(
t
)
g
=
0
:
1
s
+
N
s
Lf
1
X
n
=0
I
n
(
t
)
g
s
Lf
1
X
n
=0
V
n
(
t
)
g
;
(10)
It
is
to
be
observ
ed
that
the
iterati
v
e
relation
is
constructed
on
the
basis
that
the
first
iteration
of
the
solution
L
[
T
0
]
,
L
[
I
0
]
,
and
L
[
V
0
]
are
obtained
those
v
alues
that
generated
from
the
initial
conditions,
i.e.,
Lf
T
0
(
t
)
g
=
0
:
1
s
+
s
+1
;
Lf
T
1
(
t
)
g
=
s
Lf
T
0
(
t
)
g
+
r
s
Lf
T
0
(
t
)
g
r
s
T
max
Lf
A
0
(
t
)
g
r
s
T
max
Lf
B
0
(
t
)
g
K
s
Lf
C
0
(
t
)
g
;
Lf
T
2
(
t
)
g
=
s
Lf
T
1
(
t
)
g
+
r
s
Lf
T
1
(
t
)
g
r
s
T
max
Lf
A
1
(
t
)
g
r
s
T
max
Lf
B
1
(
t
)
g
K
s
Lf
C
1
(
t
)
g
;
Lf
T
n
+1
(
t
)
g
=
s
Lf
T
n
(
t
)
g
+
r
s
Lf
T
n
(
t
)
g
r
s
T
max
Lf
A
n
(
t
)
g
r
s
T
max
Lf
B
n
(
t
)
g
K
s
Lf
C
n
(
t
)
g
;
(11)
Lf
I
0
(
t
)
g
=
0
;
Lf
I
1
(
t
)
g
=
K
s
Lf
C
0
(
t
)
g
s
Lf
I
0
(
t
)
g
;
Lf
I
2
(
t
)
g
=
K
s
Lf
C
1
(
t
)
g
s
Lf
I
1
(
t
)
g
;
Lf
I
n
+1
(
t
)
g
=
K
s
Lf
C
n
(
t
)
g
s
Lf
I
n
(
t
)
g
;
(12)
Lf
V
0
(
t
)
g
=
0
:
1
s
;
Lf
V
1
(
t
)
g
=
N
s
Lf
I
0
(
t
)
g
s
Lf
V
0
(
t
)
g
;
Lf
V
2
(
t
)
g
=
N
s
Lf
I
1
(
t
)
g
s
Lf
V
1
(
t
)
g
;
Lf
V
n
+1
(
t
)
g
=
N
s
Lf
I
n
(
t
)
g
s
Lf
V
n
(
t
)
g
;
(13)
As
a
result,
the
components
are
identified
by
appl
ying
in
v
erse
Laplace
transform
of
the
abo
v
e
equations,
to
obtain
T
0
(
t
)
=
0
:
1
+
L
1
s
+1
;
I
0
(
t
)
=
0
;
V
0
(
t
)
=
0
:
1
;
An
analytic
study
of
the
fr
actional
or
der
model
of
HIV
-1
virus
and
CD4+...
(Kamel
Al-Khaled)
Evaluation Warning : The document was created with Spire.PDF for Python.
1466
r
ISSN:
2088-8708
and
T
n
+1
(
t
)
=
L
1
s
Lf
T
n
(
t
)
g
+
r
s
Lf
T
n
(
t
)
g
r
s
T
max
Lf
A
n
(
t
)
g
r
s
T
max
Lf
B
n
(
t
)
g
K
s
Lf
C
n
(
t
)
g
;
I
n
+1
(
t
)
=
L
1
K
s
Lf
C
n
(
t
)
g
s
Lf
I
n
(
t
)
g
;
V
n
+1
(
t
)
=
L
1
N
s
Lf
I
n
(
t
)
g
s
Lf
V
n
(
t
)
g
;
(14)
Thus,
the
solutions
in
the
form
of
an
infinite
series
are
entirely
founded.
Ho
we
v
er
in
man
y
cases
(especially
when
is
an
inte
ger)
the
e
xact
solution
in
a
more
compact
form
may
be
obtained.
The
n
th
term
approxi-
mation
n
=
P
n
1
k
=0
T
k
,
n
=
P
n
1
k
=0
I
k
,
n
=
P
n
1
k
=0
V
k
can
be
used
to
approximate
the
solution.
The
w
ay
we
choose
the
initial
solution
al
w
ays
leads
to
noise
oscillation
during
the
iteration
technique.
In
addition,
the
choice
of
T
0
(
t
)
;
I
0
(
t
)
;
V
0
(
t
)
containing
fe
w
number
of
terms
gi
v
es
more
fle
xibility
to
solv
e
complicated
non-
linear
equations
,
especially
when
e
v
aluating
the
in
v
erse
of
the
laplace
transform.
A
trustw
orth
y
alter
form
of
the
adomian’
s
method
has
been
mentioned
and
discussed
by
W
azw
az
[28].
The
construction
of
the
zeroth
term
of
the
decomposition
series
can
be
defined
by
splitting
the
zeroth
term
into
the
sum
of
tw
o
terms,
then
the
first
term
assigned
to
T
0
(
t
)
,
while
the
other
remaining
term
for
T
0
(
t
)
be
assigned
to
the
second
com
po
ne
n
t
T
1
(
t
)
.
T
o
obtain
the
in
v
erse
laplace
tranform
of
(13),
we
perform
Mathematica
simple
calculations.
The
approximate
solution
is
gi
v
en
by
T
(
t
)
=
P
3
k
=0
T
k
,
I
(
t
)
=
P
3
k
=0
I
k
and
V
(
t
)
=
P
3
k
=0
V
k
.
The
numerical
results
sho
wn
in
Figures
1
and
2
imply
the
ef
ficienc
y
of
the
proposed
method
discussed
here.
This
method
gi
v
es
highly
accurate
results
in
v
ery
fe
w
iterations.
α
=
0.5
α
=
0.65
α
=
0.75
α
=
0.85
α
=
1
1
2
3
4
5
days
50
100
150
cells
Uninfected
Figure
1.
Dynamics
of
uninfected
CD4+
T
-cells
for
v
arious
v
alues
of
α
=
0.5
α
=
0.65
α
=
0.75
α
=
0.85
α
=
1
1
2
3
4
5
days
0.001
0.002
0.003
0.004
0.005
cells
Infected
α
=
0.5
α
=
0.65
α
=
0.75
α
=
0.85
α
=
1
1
2
3
4
5
days
-
20
-
15
-
10
-
5
particles
Figure
2.
Dynamics
of
infected
CD4+
T
-cells
and
dynamics
of
density
of
particles
in
plasma
for
v
arious
v
alues
of
Int
J
Elec
&
Comp
Eng,
V
ol.
11,
No.
2,
April
2021
:
1460
–
1468
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1467
5.
CONCLUSION
The
elementary
goal
of
this
paper
is
to
suggest
a
systematic
algorithm
for
the
solution
of
fract
ional
ordinary
dif
ferential
system
of
HIV
-1
virus
and
CD4+
T
-Cells
.
This
goal
has
been
achie
v
ed
by
using
adomian
decomposition
method
with
the
successful
help
of
laplace
tranform.
The
v
alidity
and
accurac
y
of
our
approach
is
e
xamined
by
focusing
on
the
changing
beha
vior
of
in
the
pre
vious
illustrati
v
e
plots
that
guarantee
that
the
adomian
decomposit
ion
method
is
v
ery
helpful
and
ef
fecti
v
e
method
to
produce
analytical
solutions.
Fi-
nally
,
The
abo
v
e
method
can
solv
e
the
nonlinear
cases
with
no
dif
ficulty
,
without
linearization,
perturbation
or
discretization.
A
CKNO
WLEDGEMENT
This
research
w
as
supported
by
deanship
of
research
at
Jordan
Uni
v
ersity
of
Science
and
T
echnology
(JUST),
Grant
No.
168-2019.
W
e
thank
our
Uni
v
ersity
(JUST)
and
colleagues
who
pro
vided
insight
and
e
xpertise
that
greatly
assisted
the
research.
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