Analysis of Hepatitis C model within the scope of
fractional calculus
1
Ilknur Koca and 2 Abdon Atangana
1
Department of Mathematics, Faculty of Sciences, Mehmet Akif Ersoy
University, 15100,Burdur, Turkey,
email: ikoca@mehmetakif.edu.tr
2
Institute for Groundwater Studies, Faculty of Natural and Agricultural
Sciences, University of the Free State, 9300, Bloemfontein, South Africa,
email : abdonatangana@yahoo.fr
Abstract: A model of Hepatitis C was considered using the concept of derivative with fractional order. Using the bene…ts associated to Caputo derivative
with fractional order, we studied the existence and uniqueness of the system
solutions with the help of …xed-point theorem. We derived a special solutions
using an iterative method. To see the e¢ ciency of the used method, we presented
in detail the stability analysis of this method together with the uniqueness of
the special solutions.
Keywords and Phrases: Hepatitis C model, special solution, …xed point
theorem, iterative method.
1
Introduction
In the recent decade the world has wetness arrival of strange contagious diseases.
One of these strange contagious disease is hepatitis C that attacks the liver of
a human-kind .These diseases occur to human-kind as a surprise, however, to
handle these disease, human-kind reply on the combination of theoretical and
experimental studies. The theoretical version required a conversion of physical observation into mathematical equations. The mathematical equations are
therefore used to predict the behavior of the disease. The variation or the
spread of the disease is expressed as function of time, which in mathematical interpretation is known as derivative. Nonetheless, the big challenge faced
by mathematician is the process of modeling the physical occurrence. The real
question is, what derivative is suitable for modeling. In the literature nowadays,
one can …nd more than two type of derivatives with most commonly used one
being the local derivative and the derivative with fractional order [6-10]. With
proof in the literature, the derivatives with fractional order were found suitable
mathematical tools for modeling [6-10,12] although the real physical interpretation of these derivatives has not been properly provided. In many papers in the
literature, some researchers say, the fractional derivative portrays the memory
1
e¤ect, which has not been proven in practice. For instant, it is possible for the
groundwater ‡ow model with fractional derivative to remember it path via the
geological formation? Obviously no. On the other hand, a fractional derivative can be viewed as convolution. In image processing …eld, the convolution
is usually employed as a …lter to remove impurities in a given blurred image.
If therefore, one borrows this idea, one can say the fractional derivative is a
…lter of the local derivative. With the Caputo type, we have a …lter of local
derivative with the function x1
which has a problem of singularity, but with
Caputo-Fabrizio, we have a …lter of local derivative with the function e 1 t
this version has no singularity [11]. To avoid singularity while predicting the
spread of hepatitis C, in this paper, we employ the Caputo-Fabrizio derivative
with fractional order to accurately model the spread of this disease. To accommodate readers that are not aware of the new derivative, we present …rst some
properties of the Caputo-Fabrizio derivative in the next section.
2
Model derivation with Caputo-Fabrizio derivative with fractional order
We …rst give the de…nitions of Caputo-Fabrizio derivative with fractional order
.Caputo-Fabrizio derivative with fractional order has been considered with no
singular kernel [2-3].
De…nition 1 Let f 2 H 1 (a; b); b > a;
of fractional order is de…ned as:
M( )
Dt (f (t)) =
1
Zt
2 [0; 1] then the new Caputo derivative
t
1
0
f (x) exp
x
dx
(1)
a
where M ( ) is a normalization function such that M (0) = M (1) = 1 [2-3].
But, if the function does not belongs to H 1 (a; b) then, the derivative can be
reformulated as
M( )
Dt (f (t)) =
1
Zt
(f (t)
t
1
f (x)) exp
x
dx
(2)
a
Remark 2 The instigators observed that, if
= 1
2 [0; 1);
=
[0; 1]; then new Caputo derivative of fractional order assumes the form
Dt (f (t)) =
N( )
Zt
t
0
f (x) exp
x
dx; N (0) = N (1) = 1
a
In addition,
lim
!0
1
exp
t
x
2
= (x
t)
1
1+
2
(3)
At this instant subsequent to the preface of the novel derivative, the connected
anti-derivative turns out to be imperative; the connected integral of the derivative
was proposed by Nieto and Losada [3].
De…nition 3 Let 0 <
is de…ned by
2(1
t
I (f (t)) =
< 1: The fractional integral of order
(2
2
)
f (t) +
)M ( )
(2
)M ( )
Zt
of a function f
f (s)ds; t
0
(4)
0
Remark 4 Note that, according to above de…nition, the fractional integral of
Caputo type of function of order 0 < < 1 is an average between function f
and its integral of order one. This therefore imposes
2(1
(2
)
+
)M ( ) (2
2
)M ( )
=1
(5)
The above expression yields an explicit formula for
M( ) =
2
(2
)
; 0
1
Because of the above, Nieto and Losada proposed that the new Caputo derivative
of order 0 < < 1 can be reformulated as
Dt (f (t)) =
1
1
Zt
0
f (x) exp
t
1
x
dx:
(6)
a
2.1
Hepatitis C model with Caputo-Fabrizio fractional derivative
The model that we study in this paper is a Caputo-Fabrizio derivative and
satisfying 0 <
1 with fractional order is presented via following system:
CF
0 Dt T (t)
CF
0 Dt I(t)
CF
0 Dt V (t)
= s dT (t) (1
) V (t)T (t)
= (1
) V (t)T (t)
I(t)
= (1 ")pI(t) cV (t)
(7)
where T represents uninfected hepatocytes, I represents infected hepatocytes
and V represents virus. The model assumes that uninfected hepatocytes are
produced at a constant rate s, die at rate d, per cell and are infected at constant
rate : Infected hepatocytes are lost at a rate per cell. Viral particles (virions)
are produced at rate p per infected hepatocyte and cleared at rate c per virion.
3
Let
2 (0; 1] and consider the system
8 CF
< 0 Dt T (t) = f1 (T; I; V )
CF
0 Dt I(t) = f2 (T; I; V )
: CF
0 Dt V (t) = f3 (T; I; V )
with the initial conditions T1 (0) = T01 , I1 (0) = I01 and V1 (0) = V01 : Also the
e¢ cacy of treatment in blocking virion production and reducing new infections
are described by two parameters, 0 < " < 1 and 0 < < 1 respectively.
Here
f1 (T; I; V ) = s dT (t) (1
) V (t)T (t)
f2 (T; I; V ) = (1
) V (t)T (t)
I(t)
f3 (T; I; V ) = (1 ")pI(t) cV (t)
3
Derivation of the special solution
The aim of this section is to provide a special solution of the above equation (7)
applying the Sumudu transform on both sides of equation (7) together with an
iterative method. We shall give the Sumudu transform in the following theorem.
Theorem 5 Let f (t) be a function for which the Caputo-Fabrizio exists, then
the Sumudu transform of the Caputo-Fabrizio fractional derivative of f (t) is
given as:
SF (f (t)) f (0)
ST CF
0 Dt ((f (t)) = M ( )
1
+ u
The proof of the theorem 5 can be found in [11].
To solve the above equation (7), we apply the Sumudu transform on both
sides of equation (7), we obtain
SF (T (t))
1
+
SF (I(t))
M( )
1
+
SF (V (t))
M( )
1
+
M( )
T (0)
s
I(0)
s
V (0)
s
= SL fs
= SL f(1
= SL f(1
dT (t)
(1
) V (t)T (t)g
) V (t)T (t)
")pI(t)
I(t)g
cV (t)g
(8)
Rearranging, we obtain
+ s)
SL fs dT (t) (1
) V (t)T (t)g
M( )
(1
+ s)
SF (I(t)) = I(0) +
SL f(1
) V (t)T (t)
I(t)g
M( )
(1
+ s)
SF (V (t)) = V (0) +
SL f(1 ")pI(t) cV (t)g
(9)
M( )
SF (T (t))
= T (0) +
(1
4
Now applying the inverse Sumudu transform on both sides of equation (9), we
obtain;
8
8
99
s
< (1
<
==
+
s)
dT (t)
T (t) = T (0) + SL 1
SL
(10)
:
:
;;
M( )
(1
) V (t)T (t)
I(t)
= I(0) + SL
V (t)
= V (0) + SL
(1
1
+ s)
SL f(1
M( )
(1
+ s)
SL f(1
M( )
1
) V (t)T (t)
")pI(t)
We next obtain the following recursive formula;
8
8
<
< (1
+ s)
SL
T(n+1) (t) = T(n) (0) + SL 1
:
:
M( )
I(n+1) (t)
= I(n) (0) + SL
1
V(n+1) (t)
= V(n) (0) + SL
1
(1
(1
I(t)g
cV (t)g
99
s
==
dT(n) (t)
;;
) V(n) (t)T(n) (t)
+ s)
(1
) V(n) (t)T(n) (t)
SL
I(n) (t)
M( )
(1
+ s)
SL (1 ")pI(n) (t) cV(n) (t)
M( )
(11)
And the solution of (11) is provided by
3.1
T (t)
=
I(t)
=
V (t)
=
lim Tn (t)
n!1
lim In (t)
n!1
lim Vn (t)
n!1
Application of …xed-point theorem for stability analysis of iteration method
Let (X; k:k) be a Banach space and H a self-map of X. Let yn+1 = g(H; yn ) be
particular recurcive procedure. Suppose that, F (H) the …xed-point set of H has
at least one element and that yn converges to a point p 2 F (H): Let fxn g X
and de…ne en = kxn+1 g(H; xn )k : If lim en = 0 implies that lim xn = p;
n!1
n!1
then the iteration method yn+1 = g(H; yn ) is said to be H-Stable. Without
any loss of generality, we must assume that, our sequence fxn g has an upper
boundary; otherwise we cannot expect the possibility of convergence. If all these
conditions are satis…ed for yn+1 = Hyn which is known as Picard’s iteration,
consequently the iteration will is H-Stable. We shall then state the following
theorem.
Theorem 6 Let (X; k:k) be a Banach space and H a self-map of X satisfying
kHx
for all x; y in X where 0
[5].
Hy k
C; 0
C kx
Hx k + c kx
yk ;
c < 1: Suppose that H is Picard H-Stable
5
Let us take into account the following recursive formula equation (11) connected to equation (7).
T(n+1) (t)
= T(n) (0) + SL
1
I(n+1) (t)
= I(n) (0) + SL
1
V(n+1) (t)
= V(n) (0) + SL
1
where
1+(s 1)
M( )
(1
+ s)
s dT(n) (t)
SL
(1
) V(n) (t)T(n) (t)
M( )
(1
+ s)
(1
) V(n) (t)T(n) (t)
SL
I(n) (t)
M( )
(1
+ s)
SL (1 ")pI(n) (t) cV(n) (t) (12)
M( )
is the fractional Lagrange multiplier.
Theorem 7 Let P be a self-map de…ned as
8
P (T(n) (t)) = T(n+1) (t) = T(n) (t)
>
>
>
>
s dT(n) (t)
>
(1
+ s)
1
>
+SL
>
M ( ) SL
>
(1
) V(n) (t)T(n) (t)
>
>
>
<
P (I(n) (t)) = I(n+1) (t) = I(n) (t)
(1
) V(n) (t)T(n) (t)
>
+SL 1 (1 M (+ ) s) SL
>
>
I(n) (t)
>
>
>
>
P
(V
(t))
=
V
(t)
=
V(n) (t)
>
(n)
(n+1)
>
o
n
>
>
: +SL 1 (1 + s) SL (1 ")pI(n) (t) cV(n) (t)
(13)
M( )
is P stable in L1 (a; b) if
8
1) (K + S)g( )g < 1
< f1 df ( ) + (
f1 + (1
) (K + S)h( ) sQ( )g < 1
:
f1 + (1 ")pL( ) cN ( )g < 1
Proof. The …st step of the proof will consist on showing that P has a …xed
point. To achieve this, we evaluate the followings for all (n; m) 2 N N.
P (T(n) (t))
+SL
=
SL
P (T(m) (t))
T(n) (t)
n
1+(s 1)
n M( )
1 1+(s 1)
M( )
1
P (I(n) (t))
=
+SL
SL
n
SL s
SL s
T(m) (t)
dT(n) (t)
(1
) V(n) (t)T(n) (t)
dT(m) (t)
(1
) V(m) (t)T(m) (t)
o
P (I(m) (t))
1+(s 1)
n M( )
1 1+(s 1)
M( )
1
(14)
o
(15)
I(n) (t)
SL (1
SL (1
6
I(m) (t)
) V(n) (t)T(n) (t)
I(n) (t)
) V(m) (t)T(m) (t)
I(m) (t)
o
o
and
P (V(n) (t))
+SL
=
SL
P (V(m) (t))
V(n) (t)
n
1+(s 1)
n M( )
1 1+(s 1)
M( )
1
(16)
V(m) (t)
SL (1
")pI(n) (t)
cV(n) (t)
SL (1
")pI(m) (t)
cV(m) (t)
o
o
Let consider the equality (14) and applying norm on both sides and without
loss of generality
P (T(n) (t))
P (T(m) (t))
(17)
T(n) (t)
=
SL
1+(s 1)
M( )
1
SL
(1
)
T(m) (t)+
d T(n) (t) T(m) (t)
V(n) (t)T(n) (t) V(m) (t)T(m) (t)
using the properties of the norm in particular the triangular inequality, the right
hand side of equation (17) is converted to
P (T(n) (t))
(18)
P (T(m) (t))
T(n) (t)
T(m) (t)
8
>
>
8
<
1
<
+ SL
>
SL
>
:
:
1+(s 1)
M( )
(1
)
d T(n) (t) T(m) (t)
V(n) (t) T(n) (t) T(m) (t)
+T(m) (t) V(n) (t) V(m) (t)
9
>
9 >
=
=
>
>
; ;
The above can further be transformed using the property of norm and integral
as follows
P (T(n) (t))
T(n) (t)
+SL
+SL
1
1
P (T(m) (t))
T(m) (t)
1 + (s 1)
SL
M( )
8
<
: SL
(
1)
d T(n) (t)
T(m) (t)
1+(s 1)
M( )
V(n) (t) T(n) (t) T(m) (t)
+T(m) (t) V(n) (t) V(m) (t)
9
=
(19)
;
Since both the solutions play the same role, we shall assume in this case that
T(n) (t)
T(m) (t)
=
V(n) (t)
V(m) (t)
T(n) (t)
T(m) (t)
=
I(n) (t)
I(m) (t)
7
Replacing this in equation (19), we obtain following relation
P (T(n) (t))
T(n) (t)
+SL
+SL
1
1
P (T(m) (t))
T(m) (t)
1 + (s 1)
SL
M( )
8
<
: SL
(
d T(n) (t)
T(m) (t)
(20)
1+(s 1)
M( )
V(n) (t) T(n) (t) T(m) (t)
+T(m) (t) T(n) (t) T(m) (t)
1)
9
=
;
Since T(n) (t), T(m) (t) and V(n) (t) are bounded, we can …nd three di¤erent positive constants, R; S; K such that for all t:
T(m) (t) < S,
T(n) (t)
<
R,
V(n) (t)
<
K, (n; m) 2 N
(21)
N:
Now considering equation (20) with (21), we obtain the following
P (T(n) (t))
P (T(m) (t))
where f; g are functions from SL
In the same way, we get
P (I(n) (t))
P (I(m) (t))
1 df ( )
1) (K + S)g( )
+(
1
n
1+(s 1)
M( )
1 + (1
T(n) (t)
(22)
o
SL .
) (K + S)h( )
Q( )
T(m) (t)
I(n) (t)
I(m) (t)
(23)
and
P (V(n) (t))
P (V(m) (t))
f1 + (1
")pL( )
cN ( )g V(n) (t)
V(m) (t)
(24)
For
f1 df ( ) + (
1) (K + S)g( )g < 1
f1 + (1
) (K + S)h( )
Q( )g < 1
f1 + (1 ")pL( ) cN ( )g < 1
Then the nonlinear P -self mapping has a …xed point. We next show that, P
satis…es the conditions in theorem (6). Let (22)-(23)-(24) holds thus putting
8
1) (K + S)g( )
< 1 df ( ) + (
1 + (1
) (K + S)h( )
Q( )
c = (0; 0; 0), C =
:
1 + (1 ")pL( ) cN ( )
then the above shows that condition of theorem (6) hold for the nonlinear mapping P . Therefore since all condition in theorem (6) hold for the de…ned nonlinear mapping P , then P is Picards P stable. This completes the proof of
theorem (7).
8
4
Uniqueness of the special solution
In this section, we show that the special solution of equation (7) using the
iteration method is unique. We shall …rst assume that, equation (7) has an
exact solution via which, the special solution converges for a large number m.
We consider the following Hilbert space H = L2 ((a; b) (0; T )) that can be
de…ned as the set of those functions.
ZZ
v : (a; b) [0; T ] ! R;
uvdudv < 1
We now, consider the following operator
8
) V (t)T (t)
< s dT (t) (1
(1
) V (t)T (t)
I(t)
P (T; I; V ) =
:
(1 ")pI(t) cV (t)
The aim of this part is to prove that the inner product of
(P (X11
where (X11
However,
X12 ; X21
X12 ) ; (X21
X22 ; X31
X22 ) and (X31
X32 ); (w1 ; w2 ; w3 ))
X32 ) are special solution of system.
(P (X
X12 ; X21 X22 ; X31 X32 ); (w1 ; w2 ; w3 ))
8 11
(
d
(X
X12 ) (1
) (X31 X32 ) (X11 X12 ) ; w1 )
<
11
((1
) (X31 X32 ) (X11 X12 )
(X21 X22 ) ; w2 )
=
:
((1 ")p (X21 X22 ) c (X31 X32 ) ; w3 )
(25)
We shall evaluate the …rst equation in the systhem without lost of generality
( d (X11
( d (X11
X12 ) (1
) (X31 X32 ) (X11 X12 ) ; w1 )
(26)
X12 ) ; w1 ) + ( (1
) (X31 X32 ) (X11 X12 ) ; w1 )
Since both solutions play almost the same role, we can assume that,
X11
X12 = X21
X22 = X31
X32
then the equation (26) becomes
( d (X11
= ( d (X11
X12 )
(1
) (X31
X12 ) ; w1 ) + (1
X32 ) (X11
) (X11
X12 ) ; w1 )
2
X12 ) ; w1
Nevertheless, using the relationship between the norm and the inner function,
we obtain the following inequality
( d (X11
(d (X12
d kX12
= (d + (1
X12 )
(1
) (X31
X11 ) ; w1 ) + (1
X32 ) (X11
) (X12
X11 k kw1 k + (1
) kX12
) w) kX12 X11 k kw1 k
9
X12 ) ; w1 )
2
X11 ) ; w1
2
X11 k kw1 k
(27)
Using the same routine, the second equation of the system can be evaluated as
follows
((1
) (X31
X32 ) (X11
X12 )
(X21
2
(1
= (1
) kX22 X21 k kw2 k + kX22
) f + kX22 X21 k kw2 k
X22 ) ; w2 )
(28)
X21 k kw2 k
Finally, third equation of the system is considered below
((1 ")p (X21 X22 ) c (X31 X32 ) ; w3 )
(" 1)p kX32 X31 k kw3 k + c kX32 X31 k kw3 k
((" 1)p + c) kX32 X31 k kw3 k
(29)
Replacing equation (27)-(28) and (29) into equation (25), we obtain
(P (X11 X12 ; X21 X22 ; X31 X32 ); (w1 ; w2 ; w3 ))
8
) w) kX12 X11 k kw1 k
< (d + (1
(1
) f + kX22 X21 k kw2 k
:
((" 1)p + c) kX32 X31 k kw3 k
(30)
But, for large number m; n and k, both solution converges to the exact solution,
using the topology concept; we can …nd three very small positive parameters
ln ; lm ; lk such that
ln
kT
X11 k ; kT
X12 k <
3 (d + (1
kI
X21 k ; kI
X22 k <
3(1
X31 k ; kV
X32 k <
3 (("
kV
) w) kw1 k
lm
) f + kw2 k
lk
1)p + c) kw3 k
Thus introducing the exact solution in the right hand side of equation (30), using
the triangular inequality and …nally taking M = max(n; m; k); l = max(ln ; lm ; lk ):
8
8
) w) kX12 X11 k kw1 k
< (d + (1
< l
l
(1
) f + kX22 X21 k kw2 k <
:
:
l
((" 1)p + c) kX32 X31 k kw3 k
Since l is a very small positive parameter, we conclude on the based of the
topology idea that
8
8
) w) kX12 X11 k kw1 k
< 0
< (d + (1
0 ;
(1
) f + kX22 X21 k kw2 k <
:
:
0
((" 1)p + c) kX32 X31 k kw3 k
But
(d + (1
) w) ; (1
) f + ; ((" 1)p + c) =
6
0;
kw1 k ; kw2 k ; kw3 k =
6
0)
10
kX12
X11 k = kX22 X21 k = kX31 X32 k = 0 )
X11 = X12 ; X21 = X22 and X31 = X32
This completes the proof of uniqueness.
5
Conclusion
A model of Hepatitis C was considered using the concept of derivative with
fractional order.We used the newly proposed derivative with fractional order.
The design of the de…nition allows the description of the memory without any
singularity like in the case of Caputo and Riemann-Liouville derivative. With
the bene…ts of the new derivative, we presented the stability of the equilibrium
points. We derived the solution of the new equation using an iterative scheme.
In order to show the e¢ ciency of the used method, we employed the …xedpoint theorem to study the stability analysis of the method for solving the new
equation. We presented in detail the uniqueness of the special derived solution.
References
[1] M. Chong, M. Shahrill, L. Crossley and A. Madzvamuse, The Stability
Analyses of the Mathematical Models of Hepatitis C Virus Infection, Modern Applied Science, Vol. 9, No. 3, (2015)
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