DOI : 10.17577/IJERTV11IS020062
- Open Access
- Authors : M. Abubakar , D. G. Yakubu , M. Abdulhamid , A. M. Kwami, Auwal Abdullahi
- Paper ID : IJERTV11IS020062
- Volume & Issue : Volume 11, Issue 02 (February 2022)
- Published (First Online): 03-03-2022
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
A Semi Analytic Solution of the Caputo-Fabrizio Fractional Order Mathematical Model for Cancer Treatment by Stem Cells and Chemotherapy
1M. Abubakar, 2D. G. Yakubu, 3M. Abdulhameed, 4A. M. Kwami and 5Auwal Abdullahi
1,2,4Abubakar Tafawa Balewa University, (ATBU), Bauchi,
3Federal Polytechnic Bauchi, 5Federal University Kashere
Abstract:- In this paper we develop an algorithm for the semi analytic solution of the Caputo-Fabrizio fractional order mathematical model for cancer treatment by stem cell and chemotherapy using Laplace Adomian decomposition method. The recursive iterative schemes for the model were solved using the model parameters and the given initial conditions, the result obtained is in good agreement with the one presented by Alqudah (2020) validating LADMs efficiency and accuracy in finding the approximate analytic solution of the Caputo Fabrizio fractional order mathematical model for cancer treatment by stem cell and chemotherapy.
Keywords: Cancer disease, stem cells, effector cells, tumor cell, chemotherapy, Caputo-Fabrizio.
1.0 INTRODUCTION
Cancer is a disease which is non communicable but lethal, it constitutes 12% of all deaths globally where new cases and death from the disease keep rising [1]. According to [2] Cancer is responsible for 72,000 deaths in Nigeria every year, with an estimated 102,000 new cases of cancer annually, [3] added that about 116,000 new cases were recorded in only 2018 and about 41,000 died of cancer-related in Nigeria. They also added that if proper care is not taken against cancer risk factors, the cancer burden in Africa is projected to double from 1,055,172 new cancer cases in 2018 to 2,123,245 by 2040. The overall age- standardized cancer incidence rate is almost 25% higher in men than in women, with rates of 205 and 165 per 100,000, respectively. Male incidence rates vary almost five-fold across the different regions of the world, with rates ranging from 79 per 100,000 in Western Africa to 365 per 100,000 in Australia/New Zealand." (NNCCP 2018 2022).
There are more than two hundred (200) different kind of cancer that exist worldwide but the common among them include breast and cervical cancer which are common cancer in the world and the most frequent cancer among women, prostate cancer and liver cancer common in men, colorectal cancer which is common cancer in men and in women worldwide [4]. Some of the common signs and symptoms of cancer may include pyrexia, ail, tiredness, changes in skin appearance (redness, sores that won't heal, jaundice, darkening), unplanned weight loss or weight gain, lumps or tumors (mass), inconvenient swallowing, changes and difficulties with bowel or bladder function, never ceasing cough or throatiness, curtly of breath, chest pains, bleeding and discharges that cant be explained [5], [6] and [7].
Cell mutations and other factors that assist in damaging the DNA eventually led to cancer, and these factors include air pollution, smoking, heavy alcohol drinking, eating a poor diet, obesity, exposure to chemicals, and other toxins [8]. And if these triggering factors are prevented it will go a long way in minimizing the risk of effecting with cancer [9]. Different clinical procedures are used in the treatment and management of cancer include radiotherapy, chemotherapy, surgery, and immunotherapy which depend upon the patient's condition, location of the tumor, and the stage of cancer [10].
It has been for many years back biological processes are being modeled mathematically to provide means of a better understanding of the processes. [11], [12], [13], [14], [13], [15], [16], [17], [18], [16] and [19] have developed several linear, nonlinear and stochastics classical mathematical models that provided a comprehensive understanding of hematopoietic stem cell dynamic, the coexistence between the healthy and mutated cells, the rapid and uncontrolled proliferation of immature cells that invade the circulating blood which result to cancer disease.
In understanding more on the dynamics of cancer illness, and in particular to find theoretical conditions for efficient delivery of drugs under various treatments such as chemotherapy, immune therapy, radiotherapy, and to choose suitable dosages, durations and frequencies so many works have been carried out which include [20], [21], [20], [22], [23] and [24]. A new classical mathematical model of cancer treatment by stem cells and chemotherapy was developed [25] which modify the patient immune system and subsequently provide hope of a cure for the patient.
Thus, the existing model
dS
dt
1 S ks MS
dE E
dt
p1 ES
(S 1)
-
p2
T M E
(1)
dT
dT
r1 bT T p3 E kT
dt
M T
dM
dt
2 M V t
where S t E t T t , and M t are the concentration of stem cells, effector cells, tumor cells, and chemotherapy concentration
drug with the initial conditions S0 S
, E0 E , T 0 T , and M 0 0 ifV
0 . 0 t
0 0 0 0
Table 1: Model Parameters and their description
PARAMETER
DESCRIPTION
VALUE
S
Stem Cells
1
E
Effector Cells
1
T
Tumor Cells
1
1
The Decay Rate of Concentration of the Stem Cells
-0.02825
Rate of Produced the Effector Cells
0.17
The Natural Death Rate of the Effector Cells
0.03
b
Carrying Capacity of the Tumor Cells
10-9
ks
Fractional Stem Cells killed by Chemotherapy
1
p1
Maximum Proliferation Rate of the Effector Cells
0.1245
r
Tumor Growth Rate
0.18
p2
Decay Rate of the Effector Cells killed by Tumor Cells and Chemotherapy
1
kT
Fractional Tumor Cells killed by Chemotherapy
0.9
p3
Decay Rate of the Tumor Cells killed by the Effector Cells
0.9
2
Decay Rate of Chemotherapy Drug
6.4
V (t)
The time dependent external influx of Chemotherapy Drug
1
Many areas of nowadays researches such as physics, engineering, biological sciences, finance, economics and other related areas have been observed to make a shift from classical based models that were based on integer order derivatives to fractional order derivatives. Because fractional derivatives can provide a better agreement between measured and simulated data than the classical derivatives, the fractional derivatives have a special characteristics of memory effect that depends not only upon its current state but also upon all of its historical states which does not apply to classical derivatives [26], [27] and [28]. These additional properties increase the accuracy and reliability of fractional order systems than the ordinary or classical order systems as testified by the works of [26]. [27]. [28], [29], [30], [31], [32], [33], [34], [35], [36], [37].
2.0 FRACTIONAL DIFFERNTIAL EQUATIONS
Fractional Calculus is one of the branches of mathematics that investigate the properties of integrals and derivatives of non integer orders which involves the notion and methods of solving the differential equations that involves fractional derivatives of the unknown function. There are three important definitions of fractional differential equations, they are the Riemann-Liouville, the Grunwald-Letnikov, and the Caputo definitions.
()() = lim ( )
(1)
(+1)
( ()) Grunvald Letnikov fractional derivative.
0
=0
1 x
!(+1)
I ( ) ( f )
(x u) 1 f (u)du
( ) a
Reiman Liovelle Fractional Integral
1 x
a x (1 )
c D f (x) (x t) f ' (t)dt
a
Caputo Fractional Derivative
Now let [0,1] f (x) H1 (a,b) for a b such that
M (a) x
( x t )
a x 1
CF D f (x) e
0
1
f ' (t)dt
Caputo Fabrizio Fractional Derivative
where
M ( ) is called Normalized function,
H 1 (a,b)
is a space called SOBOLEV SPACE, it contains all those functions
{ f L2 (a,b) : f ' L2 (a,b)}where
b
L2 (a,b) is all those function which were defined on this interval
(a, b) such that
f (x) 2 dt integrable function.
a
Thus, the existing model can be extended to Caputo Fabrizio sense as
CF Dq S (t) q S (t) k q M (t)S (t)
CF q
1 s
q q
p1 E(t)S (t) q
D E(t)
E(t)
(S 1)
p2 T (t)
M (t) E(t)
q
q
(2)
CF DqT (t) r q 1 bqT (t)T (t) pq E(t) k q M (t)T (t)
3 T
2
2
CF Dq M (t) q M (t) V t
-
ADOMIAN DECOMPOSITION METHOD
Among the several semi analytic methods for the analytical approximation of both linear and nonlinear problem is Adomian Decomposition Method (ADM). A great mathematician George Adomian from America developed the method to serve as a semi analytical method for solving numerous mathematical problems of partial and ordinary differential equation in linear and nonlinear system [38]. Now with extension to fractional order derivatives [39]. ADM provides efficient algorithm for the approximate analytical solutions [38]. ADM warrants the solution to both nonlinear initial value problems and boundary value problems without unphysical restrictive assumptions such as required by linearization, perturbation, adhoc assumptions, guessing the initial terms or a set of basic functions and so forth ( [40], [39], [41]). Hence ADM solves nonlinear operator equation for analytic non linearity by providing an easily computable, rapidly verifiable and rapidly convergent sequence of analytic approximate functions [42].
Many analytical approximations have been carried out using ADM to provide an approximate semi analytic solution of real-life problems, among which are application of Adomian Decomposition method on Mathematical model of malaria [41], solution of typhoid fever model by Adomian decomposition method [43].
-
Adomian Decomposition Method in solving system ODEs
A system of ordinary differential equations of the first order as defined by [42], can be considered as
y ' f x, y ,…y
1 1 1 n
y
y
f
f
'
'
2 2 x, y1 ,…yn
.
.
.
y
y
f
f
'
'
n n x, y1 ,…yn
(3)
where each equation represents the first derivative of one of the unknown functions as a mapping depending on the independent
variable x , and n unknown functions
f1,… fn .
[42] has considered the solution of (3) by presenting the system (3) in the ith equation asx
x
Lyi f1 x, y1,…, yn
i 1,2,…,n
(4)
where L is the linear operator
d with inverse
L1
0 ()dx . Applying the inverse operator on (2) the following canonical
dx
dx
form are suitable for applying ADM.
x
yi yi (0) 0 fi (x, y1 ,…,yn )dx
i 1,2,…,n
(5)
As usual in ADM the solution of (3) is considered to be as the sum of a series
yi fij
j0
And the integrand in (3) as the sum of the following series
fi (x, y1 ,…,yn ) Aij ( fi0 ,…, fij )
j0
(6)
(7)
where
Aij ( fi0 , fi1,…, fin ) are called Adomian polynomials.
Substituting (6) and (7) into (5) we get
x
fij
j0
yi (0) 0 Aij ( fi0 ,…, fij )
j0
(8)
x
From which
yi (0) 0 Aij ( fi0 ,…, f
j0
fi,0 yi (0)
x
x
fi,n1 0 Aij ( fi,0 ,…, fi, j )dx
n 0,1,2,…
(9)
4.0 CAPUTO FABRIZIO FRACTIONAL DERIVATIVE BY LAPLACE ADOMIAN DECOMPOSITION METHOD Fractional order mathematical models are also solved using the Laplace Adomian Decomposition techniques. This has
been proved efficient through researches as [43] who carried out numerical solution of integro differential equations of fractional order by Laplace decomposition method. [39], on modelling of epidemic childhood diseases with Caputo Fabrizio derivative by using the Laplace Adomian Decomposition method.
Taking Laplace of both side of the Caputo Fabrizio fractional derivative as
cf
1 x
x t
1
'
L 0 Dx f x L e f t dt
(10)
Since
1 0
1 is constant, we shift it outside
1
1 x
x t
1
'
0
0
1 L e f
t dt dt
(11)
x
Since e
0
xt
1
f ' t is a convolution of two function, therefore it is written as
1
1
Lcf D f x 1 Le t f ' t
(12)
0 x 1
By the property of convolution of two functions
1
1
1 Le t Lf ' t
(13)
1
Using the properties of Laplace transform
1 1 SLf t f 0
(14)
1 s
1
Simplifying (12) we get
1 1
SF S F 0
(15)
1 S 1
Hence the Laplace transform of Caputo Fabrizio fractional derivative is given as
Lcf D f x SFS F0
(16)
0 x S 1 S
Applying (16) the Laplace transform of the Caputo Fabrizio on both side of (2)
SLS S 0 L q S (t) k q M (t)S (t)
S q1 S
SLT T 0
1 s
q q
q
q
p1 E(t)S (t) q
S q1 S
L
E(t)
(S 1)
p2 T (t)
M (t) E(t)
(18)
SLE E0 Lr q 1 bqT (t)T (t) pq E(t) k q M (t)T (t)
S q1 S 3 T
SLM M 0 L q M (t) V t
S q1 S 2
By using the boundary conditions and taking inverse Laplace transform of (18)
S t S 0 L1 S q1S L q S (t) k q M (t)S (t)
s 1 s
Et E0 L1 S q1S L q q E(t) pq E(t)S (t) pq T (t) M (t)E(t)
s 1 2
(19)
s
s
3
3
T
T
T t T 0 L1 S q1S Lr q 1 bqT (t)T (t) pq E(t) k q M (t)T (t)
M t M 0 L1 S q1S L q M (t) V t
s 2
Assume that the solutions S , E , T and M are in the form of infinite series
S(t) Sn (t) , E(t) En (t) , T (t) Tn (t) and M (t) Mn (t)
n0
n0
n0
n0
S t S 0 L1 S q1S L q S (t) k q M (t)S (t)
s 1 s
Et E0 L1 S q1S L q q E(t) pq E(t)S (t) pqT (t)E(t) pq M (t)E(t)
s 1 2 2
(20)
s
s
3
3
T
T
T t T 0 L1 S q1S Lr q 1 bqT (t)T (t) pq E(t)T (t) k q M (t)T (t)
M t M 0 L1 S q1S L q M (t) V t
s 2
By the Adomian polynomials, the nonlinear terms T t Et , M t Et , Et S t , M t S t and M t T t as follows
An t T (t)E(t) , Bn t M t Et , Cn t Et St , Dn t M t St and Gn t M tT t
n0
n0
n0
n0
(21)
n0
where
An (t) , Bn (t) , Cn (t) , Dn (t) and Gn (t) are Adomian polynomials and defines as
1 d n n
n k
An (t) n 1 dn Tk (t)
Ek (t)
(22)
1 d n
k 0
n
k 0
n
k
0
Bn (t) n 1 dn M k (t)
Ek (t)
(23)
1 d n
k 0
n
k 0
n
k
0
Cn (t) n 1 dn Ek (t)
Sk (t)
(24)
1 d n
k 0
n
k 0
n
k
0
Dn (t) n 1 dn M k (t)
Sk (t)
(25)
1 d n
k 0
n
k 0
n
k
0
Gn (t) n 1 dn M k (t) Tk (t)
(26)
k 0
k 0
0
Now by applying (19-24) into (18) and taking the Laplace transform of both side we have
L S
t S0 S q1S L q S
t k q
D t
n
n0
s s
1 n
n0
s n
n0
L
E t E0 S q1S L q q
E t pq C
t pq
A t pq
B t
n
n0
s s
n n0
-
n n0
-
n n0
2 n
n0
L T
t T 0 S q1S Lrq 1 bqT
(t) T
t ) pq
A t k q
G t
s
s
n s
n0
n n n0
3 n
n0
T n n0
L M
t M 0 S q1S L q M
t V t
n
n0
s s
2 n
n0
(27)
From (27) we get
L(S ) L(S ) .L(S
) … S0 S q1S q L(S ) L(S ) .L(S
) … k q L(D ) L(D ) .L(D ) …
0 1 2
s s 1 0 1 2
s 0 1 2
On equating we have obtain
S0
L(S0 ) s
L(S ) S q1 S q L(S ) k q L(D )
1 s 1 0 s 0
.L(S ) S q1 S q L(S ) k q L(D )
2 s 1 1 s 1
.
.
.
(28)
L(S
) S q1 S q L(S
) k q L(D )
n1
s 1 n s
n
q
-
-
q LE
LE
…. pq LC
LC
…
LE
LE . … E 0
S q1S s 0 0
1 0 0
0 1 s
s
pq LA
LA
… pq LB
LB
…
LE0
E0
s
2 0 0
2 0 0
(29)
S q1S q
-
q
q
q
q
L E1 .
s s
L E0
p1 L C0
p2 L A0
p2 L B0
S q1S q
-
q
q
q
q
L E2
.
.
.
s s
L E1
p1 L C1
p2 L A1
p2 L B1
(30)
S q1S q
-
q
q
q
q
q q
q q
L En1 s s
L En
p1 L Cn
p2 L An
p2 L Bn
T 0 S q1S r LT LT …) (b LN LN …
0 1 0 1
0 1 0 1
L T0 L T1
…
s s
pq (LA
LA ) … pq k q LG
LG …
3 0 1
3 T 0
1
(32)
LT0
T 0
s
LT S q1S r q LT bq LN pq (LA k q LG
1 s 0 0 3 0 T 0
LT S q1S r q LT ) (bq LN pq (LA k q LG
2 s 1
.
.
.
1 3 1 T 1
(31)
LT
S q1S r q LT
) (bq LN
pq (LA
k q LG
n1 s n
.
n 3 n T n
LM
LM LM
… M 0 S q1S q LM LM LM
… V t
(32)
0
LM 0
1 2
M 0
s
s s 2 0 1 2
LM
S q1S q LM
V t
1 s 2 0
s
LM
S q1S q LM
V t
2 s 2 1
.
.
.
s
(33)
LM
S q1S q LM
V t
n1
s 2
n1
s
Summary of the Laplace inverse of the system of equation and the iterative schemes are as follows
S0 S0 , E0 E0 , T0 T0 , M0 M 0 = 1 (34)
1
1
1
1
0
0
S q S
-
k q M
0 S0
1 q(t 1)
(35)
s
s
E q q M pq E S pqT E
1 q(t 1)
(36)
1 0 1 0 0 2 0 0
T r q bqT T pq (T E k q M T )(1 q(t 1))
(37)
2
2
1 0 0
3 0 0
T 0 0
2
2
1
1
0
0
M q M
qV (t)(1 q(t 1))
(38)
-
q q S
-
k q M S k q M S
-
k q M S
1 q(t 1)
2 1 1 0
s 0 0
s 0 1
s 1 0
q q S
-
k q M S
k q M S
-
k q M S
1 q(t 1)
(39)
S 1 1 0
s 0 0
s 0 1
s 1 0
1 q(t 1)
2 ( q M qV (t))1 q(t 1)S
2 0 2 0
E q q M pq C pq A 1 q(t 1)
2 1 1 1 2 1
E q q q M qV (t)1 q(t 1) pq E S
-
E S
pq T E
-
T E
1 q(t 1)
2 2 0 2
1 0 1 1 0
2 0 1 1 0
1
1
1
1
0
0
0
0
s
s
0
0
0
0
0
0
q q q M qV (t)1 q(t 1) pq q E S
k q E M S 1 q(t 1)
2 0 2
0
0
S
q q M
-
pq E S
-
-
pqT E
1 q(t 1)
E2
2
0
2
0
0 1 0 0
0
1
0
1
2 0 0
0
2
0
2
0
0
0
0
1 q(t 1)
pq T q q M
-
pq E S pqT E
1 q(t 1)
0
0
E r q bqT T
-
-
pq T E k q M
T 1 q(t 1)
0 0 0
3 0 0
t 0 0
(40)
T r q bqT T pq A k q G 1 q(t 1)
2 1 1 3 1 T 1
r q bq r q bqT T pq T E
<> k q M T 1 q(t 1)r q bqT T
0 0 3 0 0 T 0 0
0 0
T
q q M
-
pq E S pqT E
pq T E k M T 1 q(t 1) pq 0
0 1 0 0 2 0 0
-
-
3 0 0
T 0 0
3 1 q(t 1)
1 q(t 1)
2 E r q bqT T
-
pq T E
k q M T 1 q(t 1)
0 0 0 3 0 0
T 0 0
r q bqT T
pq T E k q M T 1 q(t 1)
k q M
0
0
0 0 3 0 0
T 0 0
T
q M
q 1 q(t 1)T
2 0 2 0
(41)
2
2
2
2
1
1
2
2
M q M qV (t)1 q(t 1)
2
2
2
2
0
2
2
2
0
2
(42)
M 2 M V (t) 1 q(t 1)
q q q
2
2
0
2
2
2
0
2
qV (t) 1 q(t 1)
In a similar passion we obtain the recursive iterative formula for Caputo Fabrizio fractional order mathematical model for cancer treatment by stem cell and chemotherapy as
S 1 q(t 1) q S k q D
n1
1 n s n
n
n
n
n
z n
z n
E 1 q(t 1) q q E
-
pq C
1
1
-
-
pq A
-
pq B
z
n
z
n
n1
n
n
3
n
T
n
n
n
3
n
T
n
(43)
2
2
n
n
Tn1
1 q(t 1) r q bqT T pq A k q G
Mn1
1 q(t 1) q M
V (t)
4.0 RESULTS
Concentration of Cells
Concentration of Cells
In order to validate our semi analytic solution of the Caputo Fabrizio Fractional order mathematical model by the derived recursive iterative scheme through Laplace Adomian Decomposition method we consider the value of q = 1 corresponding to the classical model of Alqudah (2020) which is shown in figure 1a. where the present result presented in figure 1b is in good agreement with the result presented by Algudah (2020).
1.2
1
0.8
0.6
0.4
0.2
0
S(t)
E(t)
T(t)
M(t)
1.2
1
0.8
0.6
0.4
0.2
0
S(t)
E(t)
T(t)
M(t)
1 2 3 4 5 6 7 8 9 10 11
Time(t)
1 2 3 4 5 6 7 8 9 10 11
Time(t)
Fig. 1a: Numerical solution of the classical by Alqudah (2020) Fig. 1b: Semi Analytic Solution by the LADM
We now give the numerical simulation of the individual compartment with varying values of the fractional order; the result is presented below.
Concentration of Stem
Cells
Concentration of Stem
Cells
1.2
1
0.8
0.6
q = 1
q = 0.85
1.2
Concentration of Effectors
Cells
Concentration of Effectors
Cells
1
0.8
0.6
q = 1
q = 0.85
0.4 q = 0.90
0.2 q = 0.95
0.4 q = 0.90
0.2 q = 0.95
0
1 3 5 7 9 11
Time(t)
0
q = 0.98
1 3 5 7 9 11
Time(t)
q = 0.98
Figure 2: Concentration of Stem Cells with varying values of q Figure 3: Concentration of Effector Cells with varying values of q
Concentration of Tumor
Cells
Concentration of Tumor
Cells
1.2
1
0.8
0.6
q = 1
q = 0.85
1.2
Concentration of
Chemotheraphy
Concentration of
Chemotheraphy
1
0.8
0.6
q = 1
q = 0.85
0.4 q = 0.90
0.2 q = 0.95
0
0.4 q = 0.90
0.2 q = 0.95
0
1 3 5 7 9 11
Time(t)
q = 0.98
1 3 5 7 9 11
Time(t)
q = 0.98
Figure 4: Concentration of Tumor Cells Figure 5: Concentration of Chemotherapy Drug
5.0 CONCLUSION
In this paper we develop a recursive iterative formula for the solution of Caputo-Fabrizio fractional order mathematical model for cancer treatment by stem cell and chemotherapy. The result obtained by this method is in good agreement with classical solution of Alqudah (2020) as shown in figure 1a and 1b. Furthermore, numerical simulations were carried out at different value of q for each of the individual cells were presented in both tables and figures.
REFERENCE
| [1] |
Abimbola S., Maryna V., Lucinda B. and Trevor K., "Cytotoxicity Evaluation of Selected Nigerian Plants used in Traditional Cancer Treatment," Journal od Medicinal Plants Research, vol. Vol 5, no. issue 11, pp. 2442 – 2444, 2011. |
| [2] |
Joseph O. O., "New Plan will Help Improve cancer treatment and save lives in Nigeria," Journal of Medicinal Plan, pp. 157-169, 2018. |
| [3] |
WHO, "Cancer Killed 41,000 People in Nigeria in 2008," Premum Times, Nigeria, June 6, 2020. |
| [4] |
Abubakar I. B., Ukwuani-Kwaja A. N., Garba A. D., "Ethnobotanical Study of Medicinal Plants used for Cancer Treatment in Kebbi State, North West Nigeria," Acta Ecologica Sinica, 2020. |
| [5] |
Abubakar M. S., Musa A. M., Ahmed A., and Hussaini I. M., "The Perception and Practice of Traditional Medicine in the Treatment of Cancers and Inlammations by the Hausa and Fulani tribes of Northern Nigeria.," Journal of Ethnopharmeology, vol. 111, pp. 625 – 629, 2017. |
| [6] |
Alison I. A., Sharon A. V., Alan B. T. nd Ann F. C., "Tumor Dormancy and Cancer Stem Cells: Implications for the Biology and Treatment of Breast Cancer Metastasis.," Breast Disease, vol. 26, pp. 87 – 98, 2007. |
| [7] |
Aliyu U. M., Awosan K. J., Oche M. O., Taiwo A. O., Okuofo E. C. , "Prevalence and Correlates of Complementary and Alternative Medicine use among Cancer Patient in Usman Danfodiyo University Teaching Hospital, Sokoto, Nigeria," Niger J. Clin Pract., vol. 20, pp. 1576 – 83, 2017. |
| [8] |
Awadalla M., Yemeni Y. Y. and Abuasaba K. , "A New Fractional Model for Cancer Treatment by Radiotherapy using the Hadamard Fractional Derivative.," Online Mathematics, pp. 14 – 18, 2019. |
| [9] |
Belostotski G., and Freedman H. I. , "A Control Theory Model for Cancer Treatment by Radiotherapy.," International Journal of Pure and Applied Mathematics., vol. 25, no. 4, pp. 447 – 480, 2005. |
| [10] |
G. D. I., "Combination of Chemotherapy and Immunotherapy for Cancer: A Paradigm Revisited," The Lancet Oncology, vol. 8, no. 1, pp. 2 – 3, 2007. |
| [11] |
C. M. Mackay, "Unified Hypothesis for the Origin of Aplastic Anemia and Period Hematopoiesis Blood," Volume 51. No 5, 1978. |
| [12] |
Catherine F. and Mackey M. C., "Dynamic Hematological Disease: A Review," Journal of Mathematical Biology Volume 58, pp. 285 – 322, 2009. |
| [13] |
Mostapha A., Fabien C. and Abderrahim E., "Discrete – Maturity Structured Model of Cell Differentiation with Application to Acute Myelogenous Leukemia," Journal of Biological Systems. 16(3), pp. 395 – 424, 2008. |
| [14] |
Avila J. L., Bonnet C., Clairambault J., Ozbay H., Niculescu S. I., Merhi F., Tang R., and Marie J. P., "A New Model of Cell Dynamic in Acute Myloid Leukemia involving Distributed Delay's;Proceedings of the 10th IFAC workshop on Time Delay Systems.," The International Federation of Automatic Control Northeasthern University, Boston, USA, pp. 22 – 24, 2012. |
| [15] |
Avila J. L., Bonnet C., Ozbay H., Clairambau J., Niculeescu S. I., Merhi F., Tang R., and Marie J. P., "A Coupled Model for Healthy and Cancerous Cells Dynamic in Acute Myloid Leukemia.," Proceedings of the 19th World Congress. The International Federation of Automatic Control Cape Town, South Africa. , pp. 24 – 29, 2014. |
| [16] |
Mostafa A., Abdennasser C., and Tarik – Mohammed T. , "Age – Structured and Delay Differential Difference Model of Hematopoietic Stem Cell Dynamics.," Discrete and Continous Dynamical Systems – Series B. American Institute of Mathemtical Science. 20 (9),, p. 27, 2015. |
| [17] |
Alexander B., Ivan Y. and Daniil Y., "Dynamic Mathematical Models of Therepy Processes against Glioma and Leukemia under Stochastic Uncertainties.," Meccanica dei Materiali e delle structure 4(1)., pp. 131 – 138, 2016. |
| [18] |
Djema W., Bonnet C., Clairambault J., Mazanc F., Hirsch P. and Delhommeau F., "Stability of a Delay System Coupled to a Delay Differential Difference System Describing the Coexistence of Ordinary and Mutated Hematopoietic Stem Cells," 2016. |
| [19] |
Djema W., Frederic M. and Catherine B., "Stability Analysis and Robustness for a Nonlinear System with Distributed Delays Describing Hematopoiesis," System and Control Letters, Elsevier., 2017. |
| [20] |
G. W. F. K. R. H. T. a. M. K. De Pillis L. G., "Chemotherapy for Tumors: An Analysis of the Dynamic and a Study of Quadratic and Linear Optimal Controls," pp. 292 – 315, 2007. |
| [21] |
Agarwal M., Bhadauria A. S., and Al E. T., "Mathematical Modelling and Analysis of Tumor Therapy with Oncolytic Virus.," pp. 131 – 140, 2011. |
| [22] |
Baloni J. and Us R. , "Mathematical Modelling of Strategic Treatments on Tumor Growth," Applied and Computational Mathematics, 2013. |
| [23] |
Alexei T., Simeone M and Denise E. K. , "A Mathematical Model of Gene Therepy for the Treatment of Cancer," 2013. |
| [24] |
Pang L. Shen L., and Zhao Z., "Mathematical Modelling and Analysis of the Tumor Treatment Regiments with pulsed Immunotherapy and Chemotherapy," 2016. |
| [25] |
Alqudah M. A., "Cancer Treatment by Stem Cells and Chemotherapy as a Mathematical Model with Numerical Simulations.," Alexandra Engineering Journal, 2020. |
| [26] |
K. Diethelm, "A Fractional Calculus Based Model for the Simulation of an outbreak of Dengue Fever," NonLinear Dyn vol. 17, pp. 613 – 619, 2013. |
| [27] |
Eric O., Francis T. O., Samuel K. A., Isaac K. D., and Nana K. F., "Fractional Order SIR Model with Constant Population.," British Journal of Mathematics and Computer Science 14(2), pp. 1 – 12, 2016. |
| [28] |
Musiliu F. F., Sharidan S., Fuaada M. S., and Ilyas K., "Mathematical Modeling of Radiotherapy Cancer Treatment using Caputo Fractional Derivatives,," Compter Methods and Programs in Biomedicine., 2019. |
| [29] |
Nur'Izzati H. and Adem K. , "A Fractional Order SIR Epidemic Model for Dengue Transmission.," Chaos, Solitons and Fractals V(114), pp. 55 – 62, 2018. |
| [30] |
Rashid J. Muhammad A. K., Poom K., and Phatiphat T., "Modelling the Transmission of Dengue Infection through Fractional Derivatives.," Chaos, Solitons and Fractals V(127), pp. 189 – 216, 2019. |
| [31] |
Amin J., Sadia A., and Dumitru B., "A New Fractional Modelling and Control Strategy for the Outbreak of Dengue Fever.," Physica A 535, p. 122524, 2019. |
| [32] |
Musiliu F. F., Sharidan S., Fuaada M. S. and Ilyas K., "Numerical Simulation of Normal and Cancer Cell's Populations with Fractional Derivative under Radiotherapy," Computer Methods and Programs in Biomedicine, 2019. |
| [33] |
Kolade M. O., and Abdon A. , "Computational Study of Multi – Species Fractional Reaction Diffusion System with ABC Operator," Chaos, Solitons and Fractals (128), pp. 280 – 289, 2019. |
| [34] |
Sania Q. and Atangana A., "Mathematical Analysis of Dengue Fever Outbreak by Novel Fractional Operators with Field Data," Physica A 526, p. 121127, 2019. |
| [35] |
Ilknur K., "Modeling the Heat Flow Equation with Fractional – Fractal Differentiation.," Chaos, Soliton and Fractals 128, pp. 83 – 91, 2019. |
| [36] |
Abdon A. and Muhammad A. K., "Validity of Fractal Dervative to Capturing Chaotic Attractors.," Chaos, Soliton and Fractals 126 , pp. 50 – 59, 2019. |
| [37] |
Zafar I., Nauman A., Dumitru B., Waleed A., Muhammad R., Muhammad A. and Ali S. A., "Positivity and Boundedness Preserving Numerical Algothrim for the Solution of Fractionals Nonlinear Model of HIV/AIDS Transmission.," Chaos, Soliton and Fractals 134, p. 109706, 2020. |
| [38] |
G. Adomian, "A Review of the Decomposition Method in Applied Mathematics," Journal of Mathematical Analysis and Applications 135, pp. 501 – 544, 1988. |
| [39] |
Dumitru B., Babolian E., and Islam R. , "On Modelling of Epidemic Childhood Diseases with the Caputo – Fabrizio Derivative by Using Laplace Adomian Decomposition Method.," Alexandria University 59, pp. 3029 – 3039, 2020. |
| [40] |
A. Wazwaz, "A New Algorithm for Calculating Adomian Polynomials for Nonlinear Operators," Applied Mathematics and Computation 111, pp. 53 – 69, 2000. |
| [41] |
Abioye A. I., Peter O. J., Ayaade A. A., Uwaheren O. A. and Ibrahim M. O. , "Application of Adomian Decomposition Method on a Mathematical Model of Maleria.," Advances in Mathematics: Scientific Journal 9(1), pp. 417 – 435, 2020. |
| [42] |
Biazar J., Babolian E. and Islam R. (2004), "Solution of the System of Ordinary Differential Equations by Adomian Decomposition Method.," Applied Mathematics and Computation 147, pp. 713 – 719, 2004. |
| [43] |
Changquing Y. and Jianhua, "Numerical Solution of Integro – Differential Equations of Fractional Order by Laplace Decomposition Method.," WSEAS Transaction on Mathematics issues 12, volume 12, pp. 1173-1183, 2013. |