Ci a ion: Azeem, M.; Fa man, M.;
Akgül, A.; De la Sen, M. F ac ional
O de Ope a o o Symme ic
Analysis o Cance Model on S em
Cells wi h Chemo he apy. Symme y
2023,15, 533. h ps://doi.o g/
10.3390/sym15020533
Academic Edi o s: Cemil Tunç,
Jen-Chih Yao, Mou ak Benchoh a
and Ahmed M. A. El-Sayed
Recei ed: 12 Janua y 2023
Re ised: 13 Feb ua y 2023
Accep ed: 14 Feb ua y 2023
Published: 16 Feb ua y 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
symme y
S
S
A icle
F ac ional O de Ope a o o Symme ic Analysis o Cance
Model on S em Cells wi h Chemo he apy
Muhammad Azeem 1,† , Muhammad Fa man 2,3,4,† , Ali Akgül 3,4,5,*,† and Manuel De la Sen 6,†
1Depa men o Ma hema ics and S a is ics, The Uni e si y o Laho e, Laho e 54590, Pakis an
2Ins i u e o Ma hema ics, Khwaja Fa eed Uni e si y o Enginee ing and In o ma ion Technology,
Rahim Ya Khan 64200, Pakis an
3Depa men o Compu e Science and Ma hema ics, Lebanese Ame ican Uni e si y, Bei u 5053, Lebanon
4Ma hema ics Resea ch Cen e , Depa men o Ma hema ics, Nea Eas Uni e si y, Nea Eas Boule a d,
Nicosia, Me sin 99138, Tu key
5Depa men o Ma hema ics, A and Science Facul y, Sii Uni e si y, Sii 56100, Tu key
6Depa men o Elec ici y and Elec onics, Ins i u e o Resea ch and De elopmen o P ocesses,
Facul y o Science and Technology, Uni e si y o he Basque Coun y, 48940 Leioa, Spain
*Co espondence: [email p o ec ed]
† These au ho s con ibu ed equally o his wo k.
Abs ac :
Cance is dange ous and one o he majo diseases a ec ing no mal human li e. In his pape ,
a ac ional-o de cance model wi h s em cells and chemo he apy is analyzed o check he e ec s o
in ec ion in indi iduals. The model is in es iga ed by he Sumudu ans o m and a e y e ec i e
nume ical me hod. The posi i i y o solu ions wi h he ABC ope a o o he p oposed echnique is
e i ied. Fixed poin heo y is used o de i e he exis ence and uniqueness o he solu ions o he
ac ional o de cance sys em. Ou de i ed solu ions analyze he ac ual beha io and e ec o cance
disease in he human body using di e en ac ional alues. Mode n ma hema ical con ol wi h
he ac ional ope a o has many applica ions including he complex and c ucial s udy o sys ems
wi h symme y. Symme y analysis is a powe ul ool ha enables he use o cons uc nume ical
solu ions o a gi en ac ional di e en ial equa ion in a ai ly sys ema ic way. Such an analysis will
p o ide a be e unde s anding o con ol he o cance disease in he human body.
Keywo ds:
cance model; exis ence; uniqueness; Sumudu ans o m; ac ional ope a o ; A angana–
Tou ik me hod
MSC: 37M05; 92B05
1. In oduc ion
Cance is conside ed o cause di e en diseases wi h di e en cha ac e is ics o a
complex sys em. To be e unde s and he dynamics o cance , many esea che s a e
a emp ing o use a a ie y o me hodologies o in es iga e he ela ionships be ween
immune cells and umo cells [
1
,
2
]. Diseases a e ca ego ized; cance is one o hem which
ep esen s ou -o -con ol cell g ow h. When wo hings a ise, malignan umo s o ex a
haza ds appea in he human body. The i s one is when cance cells mo e h ough he
blood o lymph sys em h ough he body and des oy lou ishing issues, which is known
as in asion. The second issue is when cance ous cells y o spli and p oli e a e o nou ish
hemsel es in a p ac ice p oducing new blood essels, which is known as angiogenesis.
Hence a umo des oys he o he heal hy issues and can sp ead h oughou he body in a
p ocess called me as asizing. This phase is mo e di icul o ea , and his p ocess is called
me as asis [3].
Because o i s signi icance, ac ional calculus is use ul o depic dynamic p ocesses
in se e al ields oge he wi h physics, economics and inance [
4
,
5
], enginee ing, biology
and medicine, and plen y o o he cu en ields [
6
]. The inclusion o eminiscence and
Symme y 2023,15, 533. h ps://doi.o g/10.3390/sym15020533 h ps://www.mdpi.com/jou nal/symme y
Symme y 2023,15, 533 2 o 13
gene ic ma ke s, which gi e a ex a a ional app oach o he cance emedy epidemic
e sion, highligh s he equi emen o managing ac ional o de ou pu s. This was in
pa assis ed by Saeedian e al. who buil he epidemic model and lea ning [
7
], which
s udy he beha io and he e ec o memo y on he disease’s sp ead. Conside ing he ABC
ac ional ope a o s, he powe o he smoking model ype and i s communi y was gi en
by Uca e al. [
8
]. The amous Ge man chemis Paul Ehelich began o de elop he apeu ic
d ugs o in ec ious diseases in he ea ly 1900s. He began o use he e m “chemo he apy”,
which he desc ibed as he use o chemicals o ea cance o i s powe ul an i-in lamma o y
chemicals. He was he i s pe son o do so. Al hough Eh lich was no op imis ic abou
he u u e, he was pa icula ly in e es ed in cance ea men , including aniline dyes and
ea ly alkyla ing agen s. He w o e “Gi e up hope you who en e ”. In he 1960s su ge y and
adio he apy domina ed in he ield o cance ea men un il he s anda d o ea men
a e all local he apies had s abilized when he 33A s a is ical ac ional-o de model
p o ec i e es me hod o he body became clea by using he amewo k o ac ional
di e en ial equa ions (FDEs) [
9
]. Ma hema ical models a e used o analyze he in e ac ion
be ween di e en umo cells and an ibodies and dengue based on he sys em o ac ional
di e en ial equa ions [
10
]. FDEs and a pa ial ma hema ical model ha e been used as
an al e na i e ope a o o discuss he clinical e ec s o diabe es and he coexis ence o
ube culosis [11].
The ac ional de i a i e is ini ially di ided in o wo majo ypes. The ac ionals
wi h a singula ke nel a e Riemann–Liou ille (RL) and Capu o [
12
]. The ac ionals wi h-
ou singula ke nels a e ABC (Mi ag–Le le ) and Capu o–Fab izio (exponen ial) [
13
,
14
].
F ac ional calculus is used in inance, chemical, biological, pha maceu ical, physical, and en-
ginee ing ields as i has many applica ion in ou daily li e [
15
,
16
]. Fu he mo e, se e al ap-
plica ions a e gi en in [
17
–
22
] o ac ional o de e sions. Wi h he aid o Capu o–Fab izio,
ac ional-o de equal-wid h equa ions we e sol ed using he homo opy pe u ba ion ans-
o m app oach in [
23
,
24
]. A bo h o he ac ional-o de epidemic model’s s eady s a es,
local and global s abili y a e examined. A e analy ical ea men , he ac ional-o de
epidemic model is nume ically sol ed using a empla e ha p ese es he s uc u e [
25
,
26
].
This pape aims o examine he ac ional o de cance model wi h s em cells and
chemo he apy. Fu he mo e, we e i ied he esul s using he ad anced echnique o
A angana–Tou ik. Posi i i y o he p oposed model was also de i ed. In e ms o unique-
ness o ou s udy, one impo an poin was he s abili y analysis o a scheme u nished wi h
he aid o ixed poin heo em.
2. Basic Concep s
De ini ion 1
(Re . [
27
])
.
A angana–Baleanu in he Liou ille–Capu o sense (ABC) de i a i e is
as ollows
ABC
γDγ
{ ( )}=AB(γ)
1−γZ
γ
d
dw (w)Eγ[−γ( −w)γ
1−γ]dw,n−1<γ<n, (1)
whe e
Eγ
ep esen s he unc ion as Mi ag–Le le , and
AB(γ)
ep esen s he unc ion as no mal-
iza ion and AB(0) = AB(1) = 1. The Laplace ans o m is gi en o abo e as
[ABC
γDγ
( )](s) = AB(γ)
1−γ
sγL[ ( )](s)−sγ−1 (0)
sγ+γ
1−γ
. (2)
By applying he Sumudu ans o m (ST), we ob ain he ollowing esul :
ST[ABC
γDγ
( )](s) = AB(γ)
1−γ(γΓ(γ+1)Eγ(−1
1−γνγ)) ×[ST( ( )) − (0)]. (3)
Symme y 2023,15, 533 3 o 13
3. Ma e ials and Me hod
In o de o ea he majo i y o cance s by using he handies s em cellula he apy,
we de eloped a ac ional o de ma hema ical model ha conside ed h ee popula ions:
T( )
umo cells,
E( )
e ec o cells, and
S( )
s em cells. Fu he mo e, we added an ampli i-
ca ion A a he ealiza ion o he simpli ied ODEs ha desc ibed he in e ac ion among all
h ee popula ions.
M( )
is a chemo he apeu ic a en ion medica ion, and speci ics o he
pa ame e s and hei alues a e gi en in [
28
,
29
]. The subsequen equa ions deli e an ABC
de i a i e-based ac ional o de e sion o cance .
ABC
0Dγ
S( ) = γ1S−kSMS,
ABC
0Dγ
E( ) = α−µE+p1ES
(S+1)−p2(T+M)E,
ABC
0Dγ
T( ) = (1−bT)T−(p3E+kTM)T,
ABC
0Dγ
M( ) = −γ2M+V( ), (4)
wi h beginning condi ions as
S0( ) = S(0),E0( ) = E(0),T0( ) = T(0),M0( ) = M(0). (5)
By applying Sumudu ans o m ope a o on bo h sides, we ob ain
OγEγ(−1
1−γωγ)ST[S( )) −S(0)]=ST[γ1S−kSMS],
OγEγ(−1
1−γωγ)ST[E( )) −E(0)]=ST[α−µE+p1ES
(S+1)−p2(T+M)E],
OγEγ(−1
1−γωγ)ST[T( )) −T(0)]=ST[ (1−bT)T−(p3E+kTM)T],
OγEγ(−1
1−γωγ)ST[M( )) −M(0)]=ST[−γ2M+V( )], (6)
whe e Oγ=B(γ)γΓ(γ+1)
1−γsys em (7) becomes
ST[S( )] = S(0) + 1
OγEγ(−1
1−γωγ)×ST[γ1S−kSMS],
ST[E( )] = E(0) + 1
OγEγ(−1
1−γωγ)×ST[α−µE+p1ES
(S+1)−p2(T+M)E],
ST[T( )] = T(0) + 1
OγEγ(−1
1−γωγ)×ST[ (1−bT)T−(p3E+kTM)T],
ST[M( )] = M(0) + 1
OγEγ(−1
1−γωγ)×ST[−γ2M+V( )]. (7)
Using in e se Sumudu T ans o m, we ha e
S( ) = S(0) + ST−11
OγEγ(−1
1−γωγ)×ST[γ1S−kSMS],
E( ) = E(0) + ST−11
OγEγ(−1
1−γωγ)×ST[ (1−bT)T−(p3E+kTM)T],
Symme y 2023,15, 533 4 o 13
T( ) = T(0) + ST−11
OγEγ(−1
1−γωγ)×ST[ (1−bT)T−(p3E+kTM)T],
M( ) = M(0) + ST−11
OγEγ(−1
1−γωγ)×ST[−γ2M+V( )]. (8)
The e o e, we ob ain
S(j+1)( ) = Sj(0) + ST−1{1
OγEγ(−1
1−γωγ)×ST[γ1Sj( )−kSj( )Mj( )Sj( )]},
E(j+1)( ) = Ej(0) + ST−1{1
OγEγ(−1
1−γωγ)×ST[α−µEj( ) + p1Em( )Sj( )
(Sj( ) + 1)
−p2(Tj( ) + Mj( ))Ej( )]},
T(j+1)( ) = Tj(0) + ST−1{1
OγEγ(−1
1−γωγ)×ST[ (1−bTj( ))Tj( )
−(p3Ej( ) + kTj( )Mj( ))Tj( )]},
M(j+1)( ) = Mj(0) + ST−1{1
OγEγ(−1
1−γωγ)×ST[−γ2Mj( ) + V( )]}. (9)
The solu ion o sys em (4) is ep esen ed as
S=lim
j→∞
Sj;E=lim
j→∞
Ej;T=lim
j→∞
Tj( );M=lim
j→∞
Mj. (10)
Posi i i y o Solu ions wi h ABC Ope a o
All solu ions a e posi i e i all o he beginning condi ions a e ue o nonlocal
ope a o s. We need o de ine he no m
kΠk∞=Sup ∈DΠ|Π( )|, (11)
such ha
DΠ
is he domain o
Π
. By applying his no m, we ob ain o he A angana–
Baleanu de i a i e
S( )≥S0Eγ−γkSkMk∞−γ1
AB(γ)−(1−γ)(kSkMk∞−γ1) ,∀ >0 (12)
E( )≥E0Eγ−γµ −p1kSk∞
kSk∞+1+p2(kTk∞+kMk∞
AB(γ)−(1−γ)(µ−p1kSk∞
kSk∞+1+p2(kTk∞+kMk∞) ,∀ >0 (13)
T( )≥T0Eγ−γp3kEk∞+kTkMk∞
AB(γ)−(1−γ)(p3kEk∞+kTkMk∞) ,∀ >0 (14)
M( )≥M0Eγ−γγ2
AB(γ)−(1−γ)(γ2) ,∀ >0 (15)
Theo em 1.
Assume
(X
,
|·|)
o be a Banach space and conside
H
o be a sel -map o
X
sa is ying
kH 1−Hxk ≤ θkX−H 1k+θk 1−xk, (16)
o e e y 1,xeX, whe e 0≤θ<1.
Suppose ha he sys em (4), and we ob ain he ollowing esul as
1−γ
B(γ)γΓ(γ+1)Eγ(−1
1−γωγ). (17)
Symme y 2023,15, 533 5 o 13
P oo . De ining Kas a sel -map, we may hen w i e his as
K[S(j+1)] = S(j+1)=Sj(0) + ST−1[1
OγEγ(−1
1−γωγ)×ST[γ1Sj( )−kSj( )Mj( )Sj( )],
K[E(j+1)] = E(j+1)=Ej(0) + ST−1[1
OγEγ(−1
1−γωγ)
×ST[α−µEj( ) + p1Ej( )Sj( )
(Sj( ) + 1)−p2(Tj( ) + Mj( ))Ej( )],
K[T(j+1)( )] = T(j+1)( ) = Tj(0) + ST−1[1
OγEγ(−1
1−γωγ)
×ST[ (1−bTj( ))Tj( )−(p3Ej( ) + kTj( )Mj( ))Tj( )],
K[M(j+1)( )] = M(j+1)( ) = Mj(0) + ST−1[1
OγEγ(−1
1−γωγ)
×ST[−γ2Mj( ) + V( )]. (18)
Using he no m’s aspec s along wi h iangula inequali y,
kK[Sj( )] −K[Si( )]k ≤ kSj( )−Si( )k+kST−1{1−γ
BEγ(−1
1−γωγ)
×ST[γ1Sj( )−KSj( )Mj( )Sj( )]} − ST−1{1−γ
BEγ(−1
1−γωγ)
×ST[γ1Si( )−KSi( )Mi( )Si( )]}k,
kK[Ej( )] −K[Ei( )]k ≤ kEj( )−Ei( )k+kST−1[1−γ
BEγ(−1
1−γωγ)
×ST[α−µEj( ) + p1Ej( )Sj( )
(Sj( ) + 1)−p2(Tj( ) + Mj)( )Ej( )]
−ST−1[1−γ
BEγ(−1
1−γωγ)×ST[α−µEi( ) + p1Ei( )Si( )
(Si( ) + 1)−p2(Ti( ) + Mi( ))Ei( )]}k,
kK[Tj( )] −K[Ti( )]k ≤ kTj( )−Ti( )k+kST−1[1−γ
BEγ(−1
1−γωγ)
×ST[ (1−BTj( ))Tj( )−(p3Ej( ) + KTj( )Mj( ))Tj( )]} − ST−1[1−γ
BEγ(−1
1−γωγ)
×ST[ (1−BTi( ))Ti( )−(p3Ei( ) + KTi( )Mi( ))Ti(T)]}k,
kK[Mj( )] −K[Mi( )]k ≤ kMj( )−Mi( )k+kST−1[1−γ
BEγ(−1
1−γωγ)
×ST[−γ2Mj( ) + V( )]} − ST−1[1−γ
BEγ(−1
1−γωγ)
×ST[−γ2Mi( ) + V( )]}k, (19)
whe e B=B(γ)γΓ(γ+1)K sa is ied when
Symme y 2023,15, 533 6 o 13
θ= (0, 0, 0, 0) =
kSj( )−Si( )k × k − Sj( ) + Si( )k
+γ1kSj( )−Si( )k − kkSj( )−Si( )kkMj( )−Mi( )kkSj( )−Si( )k,
×kEj( )−Ei( )k × k − Ej( ) + Ei( )k
+α−µkEj( )−Ei( )k+p1kEj( )−Ei( )kkSj( )−Si( )k
(kSj( )−Si( )k+1),
−p2(kTj( )−Ti( )k+kMj( )−Mi( )k)kEj( )−Ei( )k
×kTj( )−Ti( )k × k − Tj( ) + Ti( )k
+ (1−bkTj( )−Ti( )k)kTj( )−Ti( )k − (p3k − Ej( ) + Ei( )k
+kkTj( )−Ti( )kkMj( )−Mi( )k)kTj( )−Ti( )k,
×kMj( )−Mi( )k × k − Mj( ) + Mi( )k
−γ2kMj( )−Mi( )k+V( ),
(20)
and we ind ha Kis Pica d K-s able.
Theo em 2. Sys em (9) is a unique and dis inc solu ion ound by u ilizing he i e a ion me hod
P oo . Le us conside he Hilbe space,
H=L2((q,p)×(0, T))
h:(q,p)×[0, T]→R,Z Z ghdgdh <∞. (21)
Fo his pu pose, he ollowing ope a o s a e used
θ= (0, 0, 0, 0) =
γ1S−kSMS,
α−µE+p1ES
(S+1)−p2(T+M)E,
(1−bT)T−(p3E+kTM)T,
−γ2M+V( ).
(22)
We ha e
T(S11( )−S12( ),E21( )−E22( ),T31( )−T32( ),M41( )−M42( ),( 1, 2, 3, 4), (23)
whe e
(
S
11( )−
S
12( )
,E
21( )−
E
22( )
,T
31( )−
T
32( )
,M
41( )−
M
42( )
, which displays he
sys em’ Sspecial solu ions. We may ob ain his by using he inne unc ion and no m.
{γ1A−kADA, 1} ≤ γ1kAkk 1k − kkAkk 1kkDkk 1kkAkk 1k,
{α−µB+p1A
(A+1)−p2(C+D)B, 2} ≤ α−µkBkk 2k
+p1kBkkAkk 2k
(kAkk 2k+1)−p2(kCkk 2k+kDkk 2k)kBkk 2k,
{ (1−bC)C−(p3(B+kCkD)C, 3} ≤ (1−bkCkk 3k)Ck 2k − (p3kBkk 3k
+kkBkk 3kkDkk 2k)kCkk 3k,
{−γ2D+V( ), 4} ≤ −γ2kDkk 4k+V( )k 4k,
whe e
A=
S
11( )−
S
12( )
,
B=
E
21( )−
E
22( )
,
C=
T
31( )−
T
22( )
and
D=
M
41( )−
M
42( )
. In he case o a la ge numbe E
1
,E
2
,E
3
, and E
4
, all o he esul s con e ge o an
exac solu ion. By u ilizing ou posi i e and e y small pa ame e s and he opology
concep , we ha e χE1,χE2,χE3,χE4.
Symme y 2023,15, 533 7 o 13
kS( )−S11( )k,kS( )−S12( )k<χE1
ω,
kE( )−E21( )k,kE( )−E22( )k<χE2
ς,
kT( )−T31( )k,kT( )−T32( )k<χE3
ϑ,
and
kM( )−M41( )k,kM( )−M32( )k<χE4
$,
whe e
ω=4(γ1kAk − kkAkk 1kkDkkAk)k 1k,
ς=4(α−µkBk+p1kBkkAk
(kAkk 2k+1)−p2(kCk+kDk)kBk)k 2k,
ϑ=4( (1−bkCkk 3k)kCk − (p3kBk+kkBkkDk)kCk)k 3k,
$=4(−γ2kDk+V( ))k 4k,
whe e
γ1kAk − kkAkk 1kkDkkAk 6=0,
α−µkBk+p1kBkkAk
(kAkk 2k+1)−p2(kCk+kDk)kBk 6=0,
(1−bkCkk 3k)kCk − (p3kBk+kkBkkCk)kDk 6=0,
−γ2kDkk +V( )) 6=0,
whe e
k 1k,k 2k,k 3k,k 4k 6=0; kS11( )−S12( )k,
kE21( )−E22( )k,kT31( )−T32( )k,kM41( )−M42( )k=0.
S11( ) = S12( );E21( ) = E22( );T31( ) = T32( );M41( ) = M42( ). (24)
The uniqueness p oo is now comple e.
4. Nume ical Scheme wi h A angana–Tou ik
In his a icle, an ad anced scheme was applied o nonlinea FD equa ions on accoun
o FD wi h nonsingula ke nel and non-nea by ac ional de i a i e. Fo his pu pose,
ecollec he nonlinea sys em gi en in (5) and apply he echnique we ha e used.
S( )−S(0) = (1−γ)
ABC(γ){γ1S( )−kS( )M( )S( )}
+γ
Γ(γ)×ABC(γ)Z
0{γ1S(τ1)−kS(τ1)M(τ1)S(τ1)}( −τ1)γ−1dτ1,
E( )−E(0) = (1−γ)
ABC(γ){α−µE( ) + p1E( )S( )
(S( ) + 1)−p2(T( ) + M( ))E( )}
+γ
Γ(γ)×ABC(γ)Z
0{α−µE(τ1) + p1E(τ1)S(τ1)
(S(τ1) + 1)−p2(T(τ1) + M(τ1))E(τ1)}
( −τ1)γ−1dτ1,
T( )−T(0) = (1−γ)
ABC(γ){ (1−bT( ))T( )−(p3E( ) + kT( )M( ))T( )}
Symme y 2023,15, 533 8 o 13
+α1
Γ(γ)×ABC(γ)Z
0{ (1−bT(τ1))T(τ1)−(p3E(τ1) + kT(τ1)M(τ1))T(τ1)}
( −τ1)γ−1dτ1,
M( )−M(0) = (1−γ)
ABC(γ){−γ2M( ) + V( )}
+γ
Γ(γ)×ABC(γ)Z
0{−γ2M(τ1) + V( )}( −τ1)γ−1dτ1.
As gi en M+1,M=0, 1, 2, 3 . . ., hen he abo e equa ion can be e o mula ed as
S( M+1)−S(0) = (1−γ)
ABC(γ){γ1S( M)−kS( M)M( M)S( M)}
+γ
Γ(γ)×ABC(γ)
M
∑
k1=0Z k1+1
k1
{γ1S(τ1)−kS(τ1)M(τ1)S(τ1)}
( M+1−τ1)γ−1dτ1,
E( M+1)−E(0) = (1−γ)
ABC(γ){α−µE( M) + p1E( M)S( M)
(S( M) + 1)−p2(T( M) + M( M))E( M)}
+γ
Γ(γ)×ABC(γ)
M
∑
k1=0Z k1+1
k1
{α−µE(τ1) + p1E(τ1)S(τ1)
(S(τ1) + 1)−p2(T(τ1) + M(τ1))E(τ1)}
( M+1−τ1)γ−1dτ1,
T( M+1)−T(0) = (1−γ)
ABC(γ){ (1−bT( M))T( M)−(p3E( M) + kT( M)M( M))T( M)}
+γ
Γ(γ)×ABC(γ)
M
∑
k1=0Z k1+1
k1
{ (1−bT(τ1))T(τ1)−(p3E(τ1) + kT(τ1)M(τ1))T(τ1)}
( M+1−τ1)γ−1dτ1,
M( M+1)−M(0) = (1−γ)
ABC(γ){−γ2M( M) + V( )}
+γ
Γ(γ)×ABC(γ)
M
∑
k1=0Z k1+1
k1
{−γ2M(τ1) + V( )}( M+1−τ1)γ−1dτ1.
By using he abo e equa ion, we ha e
SM+1=S0+(1−γ)
ABC(γ){γ1S( M)−kS( M)M( M)S( M)}
+γ
Γ(γ)×ABC(γ)
M
∑
k1=0
(γ1Sk1−kSk1
Mk1Sk1
hB1
−γ1Sk1−1−kSk1−1Mk1−1Sk1−1
hAγ,k1,2),
EM+1=E0+(1−γ)
ABC(γ){α−µE( M) + p1E( M)S( M)
(S( M) + 1)−p2(T( M) + M( M))E( M)}
+γ
Γ(γ)×ABC(γ)
M
∑
k1=0
(
α−µEk1+p1Ek1Sk1
(Sk1+1)−p2(Tk1+Mk1)Ek1
hB1
Symme y 2023,15, 533 9 o 13
−
α−µEk1−1+p1Ek1−1Sk1−1
(Sk1−1+1)−p2(Tk1−1+Mk1−1)Ek1−1
hAγ,k1,2),
TM+1=T0+(1−γ)
ABC(γ){ (1−bT( M))T( M)−(p3E( M) + kT( M)M( M))T( M)}
+γ
Γ(γ)×ABC(γ)
M
∑
k1=0
( (1−bTk1)Tk1−(p3Ek1+kTk1
Mk1)Tk1
hB1
− (1−bTk1−1)Tk1−1−(p3Ek1−1+kTk1−1Mk1−1)Tk1−1
hAγ,k1,2),
MM+1=M0+(1−γ)
ABC(γ){−γ2M( M) + V( )}
+γ
Γ(γ)×ABC(γ)
M
∑
k1=0
(−γ2Mk1+V( )
hB1
−−γ2Mk1−1+V( )
hAγ,k1,2),
whe e
Aγ,k1,2 =R k1+1
k1(τ1− k1)( M+1−τ1)γ−1dτ1
and
B1=R k1+1
k1(τ1− k1−1)( M+1−
τ1)γ−1dτ1. By in eg a ing he abo e and pu ing in a sys em o equa ions, we ha e
SM+1=S0+(1−γ)
ABC(γ){γ1S( M)−kS( M)M( M)S( M)}
+γ
ABC(γ)
M
∑
k1=0
(hγ{γ1Sk1−kSk1
Mk1Sk1}
hB1
−hγ{γ1Sk1−1−kSk1−1Mk1−1Sk1−1}
hAγ,k1,2),
EM+1=E0+(1−γ)
ABC(γ){α−µE( M) + p1E( M)S( M)
(S( M) + 1)−p2(T( M) + M( M))E( M)}
+γ
ABC(γ)
M
∑
k1=0
(
hγ{α−µEk1+p1Ek1Sk1
(Sk1+1)−p2(Tk1+Mk1)Ek1}
hB1
−
hγ{α−µEk1−1+p1Ek1−1Sk1−1
(Sk1−1+1)−p2(Tk1−1+Mk1−1)Ek1−1}
hAγ,k1,2),
TM+1=T0+(1−γ)
ABC(γ){ (1−bT( M))T( M)−(p3E( M) + kT( M)M( M))T( M)}
+γ
ABC(γ)
M
∑
k1=0
(hγ{ (1−bTk1)Tk1−(p3Ek1+kTk1
Mk1)Tk1}
hB1
−hγ{ (1−bTk1−1)Tk1−1−(p3Ek1−1+kTk1−1Mk1−1)Tk1−1}
hAγ,k1,2),
MM+1=M0+(1−γ)
ABC(γ){−γ2M( M) + V( )}
+γ
ABC(γ)
M
∑
k1=0
(hγ{−γ2Mk1+V( )}
hB1
