Sirs-Si Model of Malaria Disease with Application of Vaccines, Anti-Malarial Drugs, and Spra)'ing

ISSN: 2319-765X

Journal of

Mathematics

Volume 10. Issue 5. Version 2

September-October 2014

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e-/SSN: 2278-5728. p-ISSN: 2319-765X. Volume 10. Issue 5 Ver. 11 rSep-Ocr. 2014J. PP 66-72

ll'll'H'.

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Sirs-Si Model of Malaria Disease with Application of Vaccines,

Anti-Malarial Drugs, and Spra)'ing

Randita Gustian Putri, Jaharuddin, Toni Bakhtiar

(Departrnent oj.1\-lathe111a1ics. Bogur Agrh·ulturu/ L:ni1·ersity.

JI .. \>/eranti. Kan1p11s /PB Darn1aga. Bogor 16680, /ndonl!siaJ

Abstract: :\-la/aria is a dead(i· diseuse 1runsn1it1ed to J111111ans 1hro11gh the hite ofin/i!cred fenu1le 1nosquiroe.,· .//

can also be tran.wniuedji·o1n an ｩＱｾＯｩＺ｣ｴ･､＠ n1otht•r ｨﾷｮｮｧｴｮｩｴ｡ＯｾＱﾷＱ＠ ur thruuf!,h filoud trt11J.\/t1.,ion /11 1/Ji, ＯＧｴｬｦＧｾﾷｲＮ＠ Ill'

discussed thi! 1ransn1ission <!l111alariuji:ururing in theji·tuneH·ork '!lun .'\'/RS-S! n1t1de/ 11·i1h f1Vatn1t•11f\ arl' git"t·n

to hzunans and 1nosquitues. IVe here utili:ed thL' use

'!l

\'tu..·c:ines. 1h11 list.'

o/.

anti-1nalarial drug.\. and the 1oe 11/

spraying as 1reatn1en1 efforts. A stability analysis was then pe1j(Jrn111d und n11111ericul sitnulution H'US provided to

clarify the result. It is shou•n that 1reatn1enls a/feel 1he dynamics of hunu1n and n1osqui10 popula1ions. In

addilion, we proposed rhe Homoropy Analysis Merhod (HAM) 10 cons1111ct the approximate so/111ivn ｾｲ＠ rhe

model.

Keylvords: HAM, A1alaria 1nodt!l, SIRS-SJ nuxlel. S1abili1y ana(rsis. Treat111ent

ＭｾＭＭＭＭＧＭＭＭＭＭＭＭＭＭＭＧＭＭＭＭＭＭＭＭＧＭＭｾＭＧＭＢＭＺＺＮ｟ＮＮＮ｣ＭＧＭＭＭＭＭＭＧＭＧＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭﾷＭ

I. Introduction

Malaria is an infectious disease caused by a parasite kno\vn as plasn1odium. Carrier of plasmodiun1 parasite is the female anopheles mosquito that causes the destruction of red blood cells in humans and animals through bites. Malaria can be transmitted through blood transfusion, sharing needles, or congenital.

tv1any researchers have developed a mathematical n1odel of malaria transmission. SIR \vith nonlinear

infection rate and the use of vaccination against the human population \Vas analyzed by [I]. In this model it \\'as

assumed that the vaccination at the right time can cause susceptible hun1an beings vaccinated can 111ove direct I} to recovered class. The etTect of treatn1ent as a control variable on n1alaria trans1nission systc111 \\·as s1udicd in

the frame\vork of optimal control theory by [2]. ｾｬｯ､･ｬ＠ of ·n1alaria 1rans111ission \Vas ｩｮｶ｣ｳｴｩｧ｡ｴｾ､＠ in

l.lJ

by

considering the existence of human-to-human transmission through blood transfusions and through malaria-infected pregnant women (congenital).

In this current work. we discuss the effectiveness of the use of drugs in a malaria transmission model based on [3]. Modification of the model is done by considering the assumption that humans belong to recovered

class have possibility to be susceptible, i.e .. \Ve consider a SIRS-SI model [4]. Moreover. \\'C also consider the

application of vaccine and spraying as introduced in [5.6]. Stability analysis is then performed to reveal _the effects of treatments on population dynamics. Additionally \Ve also propose the use of the hon1otopy analysis method in providing an approximate solution of the model.

II.

Mathematical Model

In constructing the model we employ the following assumptions. We assume that the human population

is divided into three classes. namely susceptible human So. infected human lh, and recovered human Rh. while

the mosquito population is grouped into two classes: susceptible mosquito

S

111 and infected mosquito /111 •

Individuals who are born and migrate to the susceptible class has a constant rate of Ah. Humans in susceptible

class can move into the infected class due to blood transfusion at a rate of

ap,

(\vith

a

is average number of

blood transfusion per unit tin1e and

p

1 is the chances of ､ｩｳ｣｡ｳｾ＠ transn1ission 10 susceptible hun1an fro111 infr:ctt!d

human) or by an infected mosquito bite at a rate of bp2 (\vith b is average number of infected mosquito bites on

susceptible human per unit time and

P

₂ is the chances of disease transmission fro1n infected mosquitoes to

susceptible humans). Humans in susceptible class can move into the recovered class due to vaccination at a rate of8. Humans insusceptible class may die at a rate of µh. A ne\vborn baby can be infected malaria due to

congenital with a rate of 17. Humans in infected class can move to the recovered class due to the use of

anti-malarial drugs with a rate ofky (\vith k is the rate of human recovery and y is the effectiveness of anti-malarial

drugs). Humans in infected class can die at a rate of Jl1, and death due to 111alaria at a rate ofa. Hun1ans in

recovered class (R,) can die at a rate of 11,.

Furthermore, mosquitoes are born and migrate to susceptible class \vi th a constant rate of

A

111 •

Mosquitoes in susceptible class may move into the infected class for biting infected humans at a rate of

cp

3 (with c is average number of susceptible mosquito bites on infected humans per unit time and

P

3 is the

chances of disease transmission from infected humans to susceptible mosquitoes) or can die at a rate Jlm.

Mosquitoes in susceptible class and infected class can die because the use of spraying at a rate of p. Mosquitoes

in infected class can die at a rate Jt111 .Co1npartmen1al diagrain of the 1nodcl b illustratl!d in Fig. I ;ind ｩｴｾ＠

dynamical equations are formulated by system (I) as fol lows:

dSh

dt

="•+<JR• - (ap,1h + hP21,,,)S,, - (9+111,)S,,.

di,,

dt

= ｾ＠ lh + (ap, I,, + hP2 l11.)Sh - (111, +a + ky )11,.

dRh

dt

= ky I,, - (111,

+

<J)R,, +BS,,. 11)

dSm

dt

=Am - (t:P1l1i

+

Jlm

+

p)Slll'

dim

(it=

c{J3/hSm - (Jl 111

+

p)/111 .

Ill.

Stability Analysis

A. Eq11ilibri11m State

Equilibrium points can

be

obtained by simultaneously solving the tbllo\ving cqua1ions:

dS11 dlh dR1i dSm dim

dt=dt=dt= dt=dt=

O.

System (1) has t\vo types of equilibrium points. namely disease-free equilibriun1 point x₁₁₁t' and l!ndcn1ic

equilibrium point Xee. Jt is easy to verify that

Xi1f1!(Sh,Jh,Rh,Sm,lm) = (Sh.0.Rh.S,:,.O). (.2)

where

and

where

• A11_(µ1i +a) . ).1,H

s··

·-

A"'

s,,

= /th(9+<J+J••)' Rh= µh(9+<J+Jth)' "'--,,,-.. -+-p'

es;,·+

ky1;-11h

+

(J

B. Basic Reproduction Number

(J)

Basic reproductive number

R

0 is denoted by the expectation value of the number of infections per unit time. This infection occurs in a susceptible population produced by one infected individual.To determine the basic reproduction number we use the next generation matrix approach [7]. Based on system (I), we may define matrices F and V as follow

(

aPtAh(µh +a) hP2Ah(µh +a))

F

=

µh (9 +a+ 1<1,) /th (9 +a+ 11,,) • V = ＨＭｾ＠ + 111, + a + ky 0 ).

cp3Am . 0 I'm

+

p

- - 0

µm +p

Basic reproductive number R0 is the largest positive eigenvalues of the matrix K = Fv-1, from \vhich we

obtain

where

ｾｾｾＫ｡Ｉ＠ ｾａｨｾＫ｡Ｉ＠

b,

b,

=

---'--'--'--'-"----µh (

e

+a+ µ,)(-ry +I'•+ a+ ky)' (11111 +p)11,(e+<J+11.)'

cp,;."'

b,

(µ.,

+

p)(-ry

+

l'h +a+ ky)"

(4)

According to the basic reproduction number we say that: if R₀

<

1 then the number of infected individuals will

decrease with each generation, so that the disease will not spread, and if R0

>

1 then the number of infected

individuals will increase with each generation, so that the disease will spread.

ｓｩｲｳｾｓｩ＠ 1\;/odel o_/A'l<1h1ria Disease 11'ith Application

o/

ｬｾ｡｣｣ｩｮｬＧｓＮ＠ ａｮｴｩｾＬ｜ＧＱ｡ｬ｡ｲｩ｡ｬ＠ Drugs. and ＵｩｰｲｴｾＱﾷｩＱＱｧ＠

l: Sin111lutio11

In this simulation. the population dynan1ics are obser\·cd in conditions such 1hat R₀

<

1. ｔｨｵｾ＠ \\C

\YOuld like to sho'v the effect of vaccination, anti·malarial drug. and spraying in a situation \\·here the disease doesn't spread. The selection of parameters is relied on the studies conducted by various reliable sources. Based

on [2] we use the following values11h = 0.0004,11,., = 0.04,a = 1/730,a = 0.05. According 10 [SJ we ha,c:

Ah= 0.027,b = 0.13.P, = 0.010,p, = 0.072. and from

II

J "" ha'" k = 0.611. ·111c foll<ming parameter

\'alues are assun1ed based lln the n1os1 con1n1on si1u;uion ..1. • .,

=

0.13.a

=

0.038,c

=

0.022,/J₁

=

0.02,µ E

[O,l].8 E [O.l],ry = 0.005,y E [0.01.lJ.S,,(O) = 40./,,(0) = 2.R.,(O) = o.s .. ,(O) = 500./, .. (0) = 10.

/) Sin111/arion oj'rhe ejji!cti\·eness ｾｻｴｨ･＠ use t11Ui·11u1/arial dr11g.\

Simulation is performed to demons1ra1e the effectiveness of 1he use of an Ii· malaria drugs on hun1an

populations and mosquito populations. In this case. it \\"ill be shO\\'n that an increase or decrease;,; the \alue of

the parametery can alter the basic reproduction nun1ber R11 defined in (-l). The change in \aluc ofyand lhu:,

R0can be seen in Table I. In the human population, as sho\\11 in ｆｩｧＮｾＮ＠ if 1he ･ｮｾ｣ＱｩＬ･ｮ･ｳｳ＠ of thl!' use l)f 1.tnti·

malarial drugs is increased. then it increases the number of ｳｵｾＱＮＺｴＡｰＱｩ｢ｬ･＠ hun1ans as \\·ell a:, the nurnber of

recovered humans, but it decreases the number of infected hu1nans. The use of anli·n1alarial drugs gi,en 10

human has also an irnpact on the niosquilo population. as sho\\ll in Fig.3. !'he ｾＱＱＱＱＨＧ＠ 1rea1n1!.!'nl llll y ｩＺｾＱｵＭＮ･ｳ＠ ;1

decrease in the number of infected mosquitoes bu1 an increase in the number of suscep1iblc rnosquillli.!'S. Increasing or decreasing in the number of hun1ans and mosquitoes in each class tend to be equal Lo an) increa:,c in the effectiveness of the use of anti-malarial drugs. The maximum number of infected humans and mosquitoes

occurs at

t

= 25 days. At this point, the effectiveness level of20% can reduce the number of infected humans

up to 23.81°/o of the total human population and can reduce the number of intected mosquitoes up to 5.880/0 •

2) Silnu/ation qfthe e..(fecth·enes.\· <?fthe use 1·acci11''

In this part \VC den1onstratc the et1Ccth encss of 1hc us1..· llf 'accincs llll hu111a11 and 111l1:-qui10

populations. It is assun1ed that infCcted hun1ans treated \\'ith anti·n1alarial drug:, ｢ｾ＠ I0°!t1. In 1his case. it \\ill bl.!

sho\vn the effect of para1neters 9 and the basic reproduction nurnlxr R0 10 1hc population dyna111ics. The change

in value of the parameters8 andR0 can be seen in Table 2. In the human population, as shown in Fig.4. ifthe

effectiveness of the vaccine is improved and the other parameters are fixed, then the number of hu111ans in

susceptible and recovered classes are increased. It indirectly causes a decrease in the nun1bcr of hun1an in

infected class. While. in the mosquito population as sho\\·n in Fig.:'. i111pro\·e111ent on vaccine cfTcc1i>vcncss indirectly decreases the number of mosquito in infected class.

3) Sitnu/ativn qfthe use e.lfi!ctil·eness <!llht! use spraying

Now we demonstrate the effectiveness of the use of spraying on hun1an populations and n1osqui1n

populations. In this case \Ye still assumed that infected hu1nans consume a111i-111alarial drugs b) 10°/o and ｜｜ｾ＠

show the effect of parameter p and R₀ on population dynamics. The change in value of the parameter p and R0

can be seen in Table 3. In Fig.6

we

can

see

that an improvement on the effectiveness of spraying increases the

number of susceptible humans, but decreases the numbers of infected and recovered humans. Mean\vhile, toward the vector population, the same treatment decreases the population in both classes as spraying is ain1ed to mosquitoes.

IV.

Appiication Of The Homotopy Analysis Method

Based on [9], this section discusses the utilization of the so-called homotopy analysis method to derive an approximate solution of the SIRS-SI model of malaria disease. With respect to system (I). we define the

following linear operators£, (i

=

1,2,3,4,5) and nonlinear operators

N,

(i = 1,2,3,4,5):

- d<jJ,(t; q) .

£,(<jJ(t;q)j dt ,I= 1,2,3,4,5,

N

1

((i>)

=

d!,

-Ah -

aq,

₃

+

(a/3

1

q,,

+

b/32

'1>

5

)'1>1

+(Ii

+

µ,,

)¢,.

N,((i>)

=

d:C'

-

ry¢_{2 -}

(ap,

¢2

+

bP

2

<Psl'1>1

+(/th

+

ky

+

a)cp,.

- d<J>3

N

₃

(¢)

=

dt

-

kycp,

+

(µh

+

a)¢

3 -

ecp,,

- ､ｾ＠ )

N.C<Pl

=

dt-A,,,

+

(cp,cp,

+

µ,,,

+

P

<P •.

- d<f>s

N5(¢)

=

dt-

cp,cp,cp

4

+

(11,.,

+

p)¢5 .

"'\\'\\'. i osr j ourna Is. or g

(5)

(6)

Sirs-Si 1'vfoclel oj.1'vfalaria Dr:n.:ase u•i/h Applicalion ＼ｾＨｖ｡｣｣ｩｮ･ｳＮ＠ Anti-:\'1alarial D111g.,·. and Ｎ｜ｪｊｮｾｲｩｮｧ＠

In (5) q E[O,l] is a parameter and (ji = (<f>1,<f>2,<f>3,<f>,,<f>,) is a lltnction that depends on r andq.

Based on (5) and (6) \Ve then construct zero-order deformations as follO\\·:

(1-q)L,l<f>,(t.q)-</>,_0(r)j = qhN,[</>,(t,q)].i = 1.2.3.4.5. (7)

\Vith

h

is an auxiliary variable. In (7). if q ;:;; 0 and q ;:;; 1. 1hen frc•m (7) \\i..' ｨｾｶ･＠

cp,

(t, O) ;; <P. (r)

and</>, (r, 1) = </>, (C). Using the concept of Ta) lor series.</>,(:;</). i = 1.2,3.'l,5. «111 be <lc«>mp<»cd imu

<f>,(c.q) = <f>,0(r)

+

L

</>,,,(c)q".

n= I

\vhere

<t>,,,cc>

=

1 d"<t>,cr.q)I . · n!

dq"

,, l)

If

q

=

1, equation (8) becomes

</>, cr.1J

=

<1> •.• crJ

+

L

<1>,._{0 crJ.;}

=

1.2.3.4.5.

n=l

Next, for i

=

1,2,3.4,5. set then-th order equations as follows

L[</>;,n (t) - Xn</>;,n-1 (tJ] = hR,.0 (¢;.n-1), i = 1,2,3,4,5,

\vhere

(8)

(9)

11-I n-1

R1.o

=

:c </>1.0-1 - Ah (1 - Xo) - a </>1.0-1 + ap,

L

</>1.,</>1 ..

-1-•

+ b/!1

L

<f>u </>; ,,_, _, + (IJ +I'•• l</>1 .,_,.

ｫｾｯ＠ k=O

11-1 11-l

R2 ...

=

:t </>2.n-1 - ry<J>i.0-1 - ap,

L

<i>l.k

</>2.><-1-k - bp,

L

<l>u

<l>s.•<-1-k

+

(J<h

+

ky)</>2.•<-1 •

k=O k=O

d

R,_0

=

dt <f>J.n-1 - ky<f>2 ... -t

+

(µ,,

+

a)</>3•0_1 -IJ</>1.,,-t.

11-1

R,_,, = :t </>4.><-1 - Am (I - Xo)

+

cp,

L

</>.., </>2.•<-1-k

+

(11,,,

+

P )</>4 _,,_ 1,

k"'O

11-l

Rs.n

=

:t <f>s.n-1 - cp,

L

<f>4.k<f>2.o-l-k

+

(µ,,,

+

p)<f>s ... -1. k=O

with Xn

=

0 for

n

$ 1 and Xo

=

1 for

n

>

!.

We also employ the following initial values; <f>1•0(t) = 51, (O) = 40, <f>2.0(t) = Jh (0) = 2, <f>3.0(r) = R1, (O) =

O,<f>,,o(t)

=

Sm(O)

=

500,<f>s.o(t)

=

lm(O)

=

10. The solutioo to then-th order of (8) is

</>,_,,(t)=x,,</>;.n-i(c)+h

f

R,_,((ji,_,,_1). (10)

Thus, the approximate solutioo of the model up to the order of N is expressed as follows

N

</>;(t)

=

L

</>;.,,(t). (11)

n=O

Therefore an approximate solution of system (I) can

be

derived by homotopy analysis method

according to the following procedures;

I. For a given model represented by a systen1 of ditTercntial equations. de line a ｳ｣ｲｩ｣ｾ＠ l)f lin('<tr npcratnr:. (5) and nonlinear operators (6).

2. Determine a set ofn-th order equations given in (9) including their initial values and then solve (I 0). 3. Construct an approximate solution</>; according to ( 11) up to order N.

From (IO) and (11 ), the approximate solution obtained by homotopy analysis method depends not only

on a time variable t but also on an auxiliary variable h. The later variable should

be

selected in order to provide

a proper approximate solution. The selection of auxiliary variables h obtained by homotopy solution twice

lowered to q then evaluated at q = 0. Curves intersecting h interval would be the value of h is taken in the

solution of homotopy method. Based on Fig.8. the fifth curve is tangent to h interval -1.8 sh:::: 0 .. ,. ｂ｡ｳｾ＼ｬ＠ on this interval. a value of h can be chosen in order to obtain a solution \vith the absolute error is sn1all \\'hen compared with a numerical solution. In this case, is chosen h

=

-1. With the selection of h

=

-1. then the

homotopy solution will be obtained as at function. Fig.9 shows that the solution of homotopy analysis method (HAM) and numerical solution (NUM) Runge-Kutta up to I 0-th order which has a small absolute error. Seen at a distance of two curves are quite close to solution. This means that the homotopy method can be used to solve SIRS-SJ model of malaria disease.

Sirs-Si lvfvdel vj.i'vla/aria Diseuse u·irh Applicarion o.f V<1'·cinl's, Anti-Jlalariu/ /)rugs. and 5iprt{nng

V.

Figures and Tables

...

70'

µ,

l"ll:

ＧＬｾｉｦＬ＠

fl .,

-2...0."z.-:::.-:'.''

,

..

/ / (

" rr

,-,

,

..

'"

"

Human \1osquito

Fig. I: Compartmental diagram of malaria disease

:L

· ... ＧＺＭＬｾＭＮ｣｣｣Ｍ｣ＮＭＭＮ＠

·:·::'" =·•Y'

i

.1

I

• t ••• _____ : : •• :': .. '."..':". .. ＺＺＺＮ］］］ｾ＠

ﾷｾ＠

;

·,,

Ｍｾ＠ '';.,.

: . ·-.. .

··

. . . • . . .

.

"

.

'

·:.::•·· -!;,·,..:

[image:11.595.59.510.90.782.2]

- r =-o.os "' () I$ ｾＭＭ I "' (l ｾＵ＠

Fig .. l· Dyna1nic::. of mosquito popularwn ｮｮ､ｾＱｲｨｾＱＱＭＺｩｬｦｭｾＱＱｴ＠ ,,f ant1-mi1!ilr:,1! ＬｩｮＺｾｾ＠

.. ;

··,·i

[image:11.595.224.464.638.732.2]

.:::

...

·:

..

·,· ..

1 - 8 = o.05 8 =a.is -

a=

0.25

Fig.5: DynauUcs of u1osquiro population under the treat1nent of ,·acdnes

'-··

---

---

--i

'i

.;

/

i

ｾｩＯ＠

---

----""---.

---

---... : ':·'

ｾＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭ

[image:12.613.85.509.65.769.2]

-p=U.05 µ

=

•).15 - µ

=

V . .!5

Fig.6: ｄｾＱＱ｡ｵＱｩ｣ｳ＠ hu1nan popula1iou uudC"r the" trC"atn1C"n1 of $praying.

l

; 1_

=

..

ＧＡｾ｜Ｚ］＿＾］］］］］］ﾷ＠

... :.:

:::r:.;

､｡ｹｾ＠

- p = 0.05 p

=

0.15 - p

=

0.25

Fig."'.": Dyna1nics of 111osquito populalltl11 ｵＱＱ､ｾＱ＠ ｴｨｾ＠ IJ('aunc-111 of ｳｰｲ｡ｹＱＱＱｾ＠

·"

! •.

,.i=-·:-· ...

F1g.S: The h-curve up 10 I 0-th orde1

..

\

.

.

. . .

- - ,_ [image:12.613.191.447.253.356.2]

I·••

ｈａｾＱ＠

-

ｎｕｾＱ＠

Fig.9: The -:oluuon olho1no1op;. .u1aly$i:S ｵＱ･ｴＱＱｾＧ､＠ up!(• !11-th 01Je1

[image:12.613.201.440.382.631.2]

Sirs-Si A1ode/ oj'!vfaluriu Disease with App/i,·ution oj' Vaccini:s . .4nti-.\1a/orial Drll}.!.\, and Ｎ｜ＧｰｮｾＱﾷｮＱｧ＠

Table I: Paran1e1er \'alues u:;ed for sunulanon l'f rhe effectiveness 1,.lf an11-malarial drugs

Parame1ei r

)' = 0 10

r = 0 l.O

)' = 0 ill

Ba.>1;: 1epr0Juc11on _{... ___}

-

_.. 1nuube1

R;- = 0646

ＭＭﾷﾷｾﾷＭ

-·.

··---···---...

/(, ＭｾＭ 0 ·15-l

!i . . :.. \) ＮＧｬｾＮＧ＠ l ｔ｡｢ｬ･ｾＭ Par<t1nete1 \ :ilues useJ fu1 ::.111m!a11011vl1he cffe.;ll\ eue.>s '"'!

\ace in es

Paran1eter 8

8 = 0 10

tJ - ll.20

{J = 0 230

Basic reprodu.::non nu1uber

R,=OO<l-7

R.· - 0 039

Tabk 3: Parruneter ,·aJues used 1i.--ir suuulauon ｾｦｴｨ･＠ el1(.:u•;e11ess ｾＩｦ＠

spravuH.!

-ｐ｡ｲ｡ｮＱ･Ｑｾ＠ p ｂ｡ｳｩｾ＠ reproduciion 11u1nbe1 p

=

0.10

n,

=

0.499

p

=

0.20 R0

=

0 488

' '

J

I

p

=

0.30

:

_- Ro

=

0.485 _J

VI.

Conclusion

We have presented an SIRS-SJ model of malaria transmission equipped \\.·ith a st:rit:s of ｴｲ･｡ｵｮ｣ｮｴｾ＠

namely the use of vaccines, anti-malarial drugs and spraying. Toward disease-free and endemic equilibriu1n points, we perform a stability analysis characterized by a basic reproduction number. Through a seril!s of simulations we have shown the effects of treatments with different levels of effectiveness to human and mosquito populations. We have also proposed the use of so·called homotopy analysis n1ethod as an altt.!rnate

technique in deriving an approxin1ate solution of tht.! n1odt.!I. It has been sh0\\.-11 that such 111ethod can

approximate the numerical solution quite \vell.

References

[I] Laarabi H, ·Labriji EH. Rachik M. Kaddar A. 2012 Op111n;il C"\lnlrol of an ep1r.kmu: model \\llh a sa1urnh.'d 111i.:1di:ni:1..· ra1i:

Modelling and Control. Vo\.17, No.4, pp. 448-459.

(2) Agusto FB. Marcus N. Okosun KO, 2012. Application of optimal control to the epidemiology of malaria_ Electronic Journal or

Differential Equation. Vol. 2012. No 81. pp 1-22

[3) Abdullahi MB. Hasan YA. Abdullah FA 2013 A ma1hema111:ar modd or malaria and 1hc eff..:ct1\i:n..:ss of drugs :\pphcJ

Mathematical Sciences Vol 7. 2013. No 62. pp 3079-3095

(4 J Manda] S, Sarkar RR. Sinha S 2011 Mathemaucal mockls of malaria - a re\IC\\ 1\fol;1na Journal Vnl 1 u. pp 1 -19

(5) Schwnrtz L. Brown GV. <.icnton H. ｍｯｯｲｴｨｾﾷ＠ VS 2012 I\ ｦｬＧ｜ＧＱｾﾷｷ＠ nmlana "'ci:inc dm1cal pro11..·..:ts b<1!il'd on thl' n11nl'l1\\ 1al•k

Malana Journal 11 11

(6) Ra1ovonja10 cl aL 2014. Entomolog1cal and parasnologn::il tmpai:l:. of mdoor r1..·s1Jual ＺＺ［ｰｲ｡ｾｬｬｬｧ＠ \\1lh DDT. ｡ｬｰｨ｡ＮＮＺｾｰｩＺｴＱｮｬＧｴｨｲＱＱＱ＠ anJ

deltamethrin in the western foo1hill are<i of Madagascar. Malaria Journal 13.21

[7] Van Den Driessche P, Watmough J. 2008. Chapter 6. Funher Notes on the Basic Reproduction Number. In. Urauer

r.

V.in Oen

Dricssche P, Wu, J. Mathematical Epidemiology. 1945 Lecture Notes in Mathematics. Springer. pp. 159-178

[8] Chitnis NR. 2005. Using mathematical models in controlling the spread of malaria (d1ssenatmnl Arizona (lJS). The ｬｩｭｮＺＺｲｳＱＱｾ＠ of

Arizona.

[9] Liao. 2004 Beyond Perturbation. ln1roduction 10 the ｨｯｭｯＱｯｰｾ＠ ｡ｮ｡ｬｾﾷｳＱｳ＠ method New York (lJS )· Boca Raton

[image:13.595.128.501.106.365.2]

IOSR Journal of Mathematics

Volume

10,

Issue 5, Version

2

September-October 2014

ISSN: 2319-765X

Contents

1 The Degree of an Edge in Alpha Product, Beta Product and Gamma Product of Two

l -

19

Fuzzy Graphs,

K.

Radha, N. Kumaravel

DOI: 10.9790/5728-10520119

2

Oscillation Theorems for Second Order Neutral Differcnee Equations. P. Sundar, B.

20 - :io

Kishokkumar

DOI: 10.9790/5728-10522030

3

Counting Subgroups of Finite Nonmetacyclic 2-groups Ha,ing No Elemental)· Abclian

31 - 32

Subgroup of Order 8, M. Enioluwqfe

DOI: 10.9790/5728-10523132

4

Stochastic Analysis of Manpower System with Production and Sales, K.H. Kumar,

33 - 37

P.Sekar

DOI: 10.9790/5728-10523337

5 Two-Phase Multivariate Stratified Sampling with Travel Cost: A Fuzzy Programming

38 - 49

Approach, S. Iftekhar, M.J. Ahsan, Q.M. Ali

DOI: 10.9790/5728-10523849

6

Mathematical Model on Power Needed to Sustain Flight, T.E. Efor, D.I. Ajibo

50 - 55

DOI: 10.9790/5728-10525055

7 A SIR Mathematical Model of Dengue Transmission and Its Simulation, Asmaidi,

56 - 65

P. Sianturi, E.H. Nugrahani

DOI: 10.9790/5728-10525665

8

SIRS-SI Model of Malaria Disease with Application of Vaccines, Anti-Malarial Drugs,

66 - 72

and Spraying, R.G. Putri, Jaharuddin, T. Bakhtiar

DOI: 10.9790/5728-10526672

9 Theoretical Mathematical Model on Temperature Regulation during Wound Healing

73 - 82

after Plastic Surgeiy of Two Dimensional Human Peripheral Regions, M. Jain

DOI: 10.9790/5728-10527382

10 On Fractional Differential Operators Invoh·ing Multirnriable Generalized

Hypergeometric Function, R.M. Meghwal