Solution of a Vaccination Based SIR Epidemic Model by Homotopy Analysis Method

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Solution of a Vaccination Based SIR Epidemic Model by

Homotopy Analysis Method

Sumera. K1, M. Rafiq2, M.O.Ahmed1, M.Usman3, M.I. Bhatti1

1. Department of Mathematics, Univ. of Engineering & Technology, Lahore, Pakistan 2. Faculty of Engineering, Univ. of Central Punjab, Lahore, Pakistan

3. Department of Basic Sciences & Humanities, Univ. of Engineering and Technology, Lahore (KSK Campus), Pakistan

Corresponding author: E.mail:[email protected]

Abstract

Modeling infectious diseases helped out to understand and overcome epidemics. This paper is based on epidemic model SIR, which fits well to many epidemiological diseases. Basic idea of Homotopy Analysis Method (HAM)is discussed and employed to compute an approximation to the solution of nonlinear system of differential equations. The effect of vaccination on the dynamics of childhood disease described by SIR model is monitored using HAM. The qualitative analysis reveals the vaccination reproduction number for disease control and eradication. MATLAB is used to carry out the computations. Graphical results are presented and discussed quantitatively.

Key Words

: Epidemiology, Fixed Points, Homotopy, Reproduction Number, Stability

1. Introduction

When we talk about epidemic diseases then importance of vaccination could not be ruled out. When West recommended practicing vaccination (measles case) to all infants of 15 month age and repeat at the age of 6 approximately, it resulted in an increase in immunization. This recommendation is based on continuous-time constant vaccination strategy. Similarly period observations show that vaccination has increased the level of permanent immunity against epidemic disease [1]. Many infectious diseases caught children easier than others and named as childhood disease. Such diseases attack within age of 5 years. One fact is that in this period children are very much attached with their fellows, so disease spread rapidly. Some most common childhood diseases are measles, mumps, chicken pox, etc. Vaccination proved to be very effective against childhood disease [2]. To prevent the spread of childhood disease we need to germinate a structure that would foretell superfine vaccine coverage level.

2. SIR

Model

SIR model is considered as a basic epidemic model. Most of the childhood diseases that propagate in population adjust quite simply into this model. Usually, diseases caused by a virus such as influenza,

measles, and chickenpox, are of SIR type. This model was proposed by Kermack and Mckendrick in 1927. Many epidemiological diseases could be described by SIR model. Consider the flow of SIR model with constant vaccination strategy [3].

Fig.1 SIR Model with constant vaccination where

S = Susceptible individuals I = Infected

R = Recovered people with permanent immunity

µ = Natural death rate β = average contact rate R = Recover rate π = Birth rate

P = new born vaccinated each year (0<P<1)

I

R

S

(1-P)π

P

π

µR

µI

(2)

Where µ, β, γ and π are considered as positive parameters. Furthermore, we assumed that vaccination is 100% effective and the natural death rates µ and birth rate π are not same, this cause N to be not constant [4]. A susceptible will move to I-compartment when comes in contact with an infected individual, an infected individual move to R compartment after recovery. Vaccinated individuals are also coming into R-compartment. Now SIR model can be formulated as

S N SI N P dt

dS ₌ ₋ _π ₋_β ₋_µ )

1

( (1)

I N

SI dt dl

) (γ µ

β − +

= (2)

R I N P dt

dR _π _γ _µ

− +

= (3)

We know N = S + I + R Adding (1) to (3), we have

N dt

dN

) (π−µ

= (4)

We have a case of varying total population.

2.1 Dimensionless Transformation

We want to convert the varying total population into a constant total population, for this we have to choose new variables

N R r N

I i N

S

s= , = , =

1

1⇒ =

= +

+i r N

s

Now total population is constant i.e., =0

dt dN _,

from equation (4), we have birth rate equal to death rate

µ π =

Putting respective values in (1), (2) & (3), new system is

s si P dt

ds _π _β _π

− − −

=(1 ) (5)

i si

dt di

) (γ π

β − +

= (6)

r i P dt

dr ₌ _π ₊ _γ ₋_π ₍₇₎

2.2 Qualitative Analysis

We will analyze system in two categories 1. Infection free equilibrium (i=0)

2. Endemic equilibrium (i≠0)

Subsystems in the closed set form are

} 1 0

| ) ,

{( ∈ + ≤ + ≤

=

Γ si R s i

To find fixed points, from eq (5) & (6)

s si p π−β −π

−

=(1 )

0 (8)

i si ( )

0=β − γ+π (9)

Case I. Infection free equilibrium

When disease dies out naturally then from eq (9)

0 &

0 )

(βs−γ−π ≠ i=

From eq (8); s=(1−P)

The solution comes on an infection free equilibrium E0 asymptotically

) 0 , 1 (

0 P

E = −

Reproduction number and basic reproduction numbers are:

π γ β π

γ β

+ − = +

= & (1 )

0 Rv P

R respectively

This is a threshold which determines the stability of equilibrium.

Case II. Endemic Equilibrium

An unstable disease free equilibrium i.e., Rv > 1 give rise to endemic equilibrium Eu.

Again from eq (9);

i

s )

(

0= β −γ −π

0 &

0 )

(βs−γ−π = i≠

Rv P s= (1− )

⇒

(3)

) 1 ( 1

) 1

( ₌ ₋

⎟ ⎠ ⎞ ⎜

⎝

⎛ − ₋

= or i Rv

s P i

β π β

π

So, we have endemic equilibrium of the form ⎟⎟

⎠ ⎞ ⎜⎜

⎝ ⎛

− −

= (1 ), (Rv 1) Rv

P Eu

β π

2.3 Stability Analysis

The infection free equilibrium E0 is locally

stable if Rv < 1 and endemic equilibrium Eu is

unstable [5]. Conversely for Rv > 1, endemic equilibrium Eu is stable and infection free equilibrium

E0 is unstable. In both cases local stability of

equilibrium give rise to Global stability in the particular domain of s and i [6]. An examination of local stability of the model’s equilibria reveals that there is a critical vaccination proportion

β π γ β− − − ⇒ −

= Pc

R Pc

0 1 1

Pc governs the system as follow

1. For relatively large vaccination level i.e., ,

P

Pc> infection free equilibrium is locally stable with the coordinates

0 &

1− =

= P i

s

While endemic equilibrium is unstable.

2. For relatively weak vaccination i.e., Pc<P, endemic equilibrium is locally stable with the coordinates

) 1 ( & ) 1

( − ₋

= Rv

Rv P s

β π

The Jacobian matrix at Endemic equilibrium Eu. ⎟⎟

⎠ ⎞ ⎜⎜

⎝ ⎛

−

+ − −

=

0 )

1 (

( Rv

Rv J

π

π γ π

Rv trcJ =−π

) ( ) (

detJ =πRv γ +π −π γ +π As we know

2

) (det 4 2 )

(trcJ J

trcJ ± −

On putting values, we have

2

)] ( ) ( [ 4 2 ) 2 ( 2

,

1 π π π γ π π γ π

λ = − Rv± − Rv − Rv + − + For small values of π &γ , we neglect the last term under the square root sign

) ( 4 2 2 1 2 2 ,

1 π π 2 π γ π

λ ≈− Rv± Rv_v − Rv +

For asymptotically stable, value under square root will be negative i.e.

π π

γ )

( 4 + ≤ Rv

The endemic equilibrium Eu is locally

asymptotically stable if π

π γ ) ( 4 1<Rv≤ +

We have complex eigenvalues with negative real part. So Eu can be treated as a spiral sink. This can be explained as initially susceptible are increasing and we have few infected. Then infection starts spreading and susceptible start to decrease. Disease spread more rapidly than increment in susceptible. As a result we are left with too small number of individuals who are susceptible to disease, the outbreaks ends and susceptible begins to increase again.

3. Homotopy Analysis Method (HAM)

This new analytical technique was proposed by S.J. Liao in 1992. Homotopy Analysis Method is a general analytical approach use to solve nonlinear equations and solutions are obtained in the form of series [7]. HAM has a great potential to solve strongly nonlinear problems in science and engineering such as the viscous flows of non-Newtonian fluids, nonlinear heat transfer, finance problems, Riemann problems related to nonlinear shallow water equations, projectile motion, Glauert-jet flow, nonlinear water waves, groundwater flows and Laplace equations with certain boundary conditions. HAM is different than all perturbation and non-perturbation techniques because of the following facts.

1. Large or small parameters are of no significance in HAM

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2. Convergence of solution can be ensured in a very simple way.

3. We are free to choose base function.

3.1 Underlying concept of HAM

We begin with a nonlinear algebraic equation 0

] ) ( [yt =

N (10)

N is operator which is nonlinear and y(t) exact solution is a function of the independent variable ‘t’. To construct homotopy we assume sy0(t) as the initial

value of y(t) and L is the auxiliary linear operator on the exact solution y(t) such that L[y(t)]=0 when y(t)=0. We construct such a homotopy [8].

) ; ( ˆ )] ( [ )] ( ) ( [ ) 1

( −q L yt −yot +qN yt =H t q ]

1 , 0 [

∈

q is called the homotopy parameter. We are

free to choose the initial value yo(t) and operator L. Enforcing the homotopy to be zero i.e., Hˆ(t;q)=0,

we have

0 )] ( [ )] ( ) ( [ ) 1

( −q L yt −yot +qN yt =

Above equation is not a single algebraic equation, it is a family. Homotopy parameter q plays an important role to solve it, now we can write above family as

0 )] ; ( [ )] ( ) ; ( [ ) 1

( −q Lϕ t q −yo t +qNϕyt q = (11) From eq (11) we find that as q increases from 0 to 1, ϕ(t;q) continuously changes from yo(t) to the y(t) of eq(10). It is continuous deformation, called Homotopy. Equation (11) is called deformation equation of order zero because of ϕ(t;q), Now we use Maclaurin series to expand it

qm t ym t

yo q t

m

) ( )

( ) ; (

1

∑

∞

=

+ =

ϕ (12)

where ϕ(t;0) is employed and ) ( )

; ( ! 1 )

( ϕ _q ₀ _m ϕ

m t q D

qm m m t

y =

∂ ∂

= ₌

We select initial approximation y0(t) and the

auxiliary linear parameter L in such a way that:

1. The solution ϕ(t;q) of the zero order deformation equation exist for all q∈ [0, 1]. 2. The deformation derivative

0 ) ; (

=

∂ ∂

q m t q

q m _ϕ

exist for m = 1, 2, …

3. The power series of ϕ(t;q) converges at q = 1.

So the solution series

) ( )

( ) ; (

1

t ym t

uo q t

m

∑

∞

=

+ =

ϕ

or () ( )

0

t y t

y _m

m

∑

∞ =

=

for briefness , we write in form of vector a

)} ( ... ) ( 2 ), ( 1 ), ( 0 { )

(t y t y t y t ymt

yr =

This analytical approach is liberated from any physical parameters; HAM is strong enough that its efficiency will not affect whether a nonlinear equation contains small physical parameter or large. Equations like (11) are not always convergent at q=1, it might be divergent. To overcome this difficulty Liao introduced an auxiliary parameter h≠0 so zeroth order deformation equation becomes

0 )] ; ( [ ) ( )] ( ) ; ( [ ) 1

( −q Lϕ t q −yot −qhH t N ϕ t q = (13) where H(t) is the auxiliary function independent of q and we have the great freedom to choose it. Now the modified equation (13) is our zero order deformation equation. Now we start differentiating equation (13) with respect to q;

⎥ ⎦ ⎤ ⎢

⎣ ⎡

− ∂ ∂ − + −

− [ (; ) ()] (1 ) (; ) 0 q

q t L q t

yo q t

Lϕ ϕ

)] ; ( [ ) ( )] ; ( [ ) (

q t q

N t qhH q t N t

hH ϕ _ϕ

∂

∂ =

= ∂ ∂ − + ∂ ∂ −

2 ) ; ( 2 ) 1 ( ) ; ( 2

q q t L q q

q t

L ϕ ϕ

2

)]) ; ( [ 2 ( ) ( )] ; ( [ ) ( 2

q q t N t qhH q

q t N t hH

∂ ∂ ∂

∂

= ϕ ϕ

= ∂ ∂ − + ∂ ∂ −

3 ) ; ( 3 ) 1 ( 2

) ; ( 2 3

q q t L q q

q t

(5)

3 )]) ; ( [ 3 ( ) ( 2 )] ; ( [ 2 ) ( 3 q q t N t qhH q q t N t hH ∂ ∂ ∂ ∂

= ϕ ϕ

At mth derivative

= ∂ ∂ − + − ∂ − ∂

− (; ) (1 ) (; )

1 1 q t qm m L q q t qm m

mXmL ϕ ϕ

)] ; ( [ ) ( )] ; ( [ 1 1 )

( N t q

qm m t qhH q t N qm m t

mhH ϕ ϕ

∂ ∂ − ∂ − ∂ = Where ⎩ ⎨ ⎧ > ≤ = 1 , 1 1 , 0 m if m if Xm Or ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ∂ − ∂ − ∂ ∂ − (; ) 1 1 ) ; ( ) 1

( t q

qm m mXm q t qm m q

L ϕ ϕ

)] ; ( [ ) ( )] ; ( [ 1 1 ) ( q t N qm m t qhH q t N qm t

mhH _ϕ _ϕ

∂ ∂ + − ∂ ∂ =

Substitute q=0 and divide by m!;

)] ; ( [ 1 1 ) ( )! 1 ( 1 ) ; ( 1 1 )! 1 ( 1 ) ; ( ! 1 q t N qm m t hH m q t q m m q t qm m m L ϕ ϕ ϕ − ∂ − ∂ − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ∂ − ∂ − − ∂ ∂

We know ym(t) 1_! (t;q)

qm m

m ∂ ϕ

∂

= in above equation, we have )) ( 1 ( ) ( )] ( 1 ) (

[ymt Xmym t hH t m ym t

L − − = ℜ − (14)

Where )] ; ( [ 1 1 )! 1 ( 1 )) ( 1

( N t q

QM M m t ym m ϕ − ∂ − ∂ − = −

ℜ (15)

⎩ ⎨ ⎧ > ≤ = 1 , 1 1 , 0 m m Xm

Obviously the higher order deformation equation (14) is governing by the linear operator L, and the term ℜm(ym−1(t)) which can be expressed for any nonlinear operator N. According to definition (15), the right hand side of eq (14) is only dependent upon ym−1(t). Thus, we gain y1(t), y2(t). By mean of solving the linear higher order deformation equation (14) one by one in order.

4. Solution of Model

We have to solve the equations 5 to 10.

To solve the system by Homotopy analysis method we assume the continuous mapping

) ; ( 3 ) ( ), ; ( 2 ) ( ), ; ( 1 )

(t t q it t q rt t q

s →ϕ →ϕ′ →ϕ′

Now we chose the auxiliary operator as

[

(; )

]

′(; ) =1,2,3 ∂

∂ =

′ it q i

t q t i

Liϕ ϕ

With the property 0

) (Ci =

Li where Ci is constant integral

Defining non-linear operators N1, N2, N3 for

equation (5), (6) & (7)

+ − − ′ ∂ ∂ =

′ ϕ π

ϕ1(; )] 1(; ) (1 ) [

1 t q P

t q t N ) ; ( 1 ) ; ( 2 ) ; (

1t qϕ t q πϕ t q ϕ

β ′ ′ + ′

− ′ ∂

∂ =

′2(; )] 2(; ) [

2 t q

t q t

N ϕ ϕ

) ; ( 2 ) ( ) ; ( 2 ) ; (

1t q ϕ t q γ π ϕ t q ϕ

β ′ ′ + + ′

− ′ ∂

∂ =

′3(; )] 3(; ) [

3 t q

t q t

N ϕ ϕ

) ; ( 3 ) ; (

2t q t q

Pπ−γϕ′ +πϕ′

Now we can write family of zero order deformation equations as follow

[

1(; ) 0( )

]

1 1 1[ 1(; )] )

1

( −q Lϕ′ t q −s t =qh H N ϕ′ t q (16)

[

2(; ) 0()

]

2 2 2[ 2(; )] )

1

( −q Lϕ′ t q −i t =qh H N ϕ′ t q (17)

[

3(; ) 0( )

]

3 3 3[ 3(; )] )

1

( −q Lϕ′ t q −r t =qh H N ϕ′ t q (18) Subject to initial conditions

0 ) ; 0 ( 3 , 0 ) ; 0 ( 2 , 0 ) ; 0 (

1 q =s ′ q =i ′ q =r

′ ϕ ϕ

ϕ

By Taylor’s theorem, we expand ϕ′i(t;q) by a power series of the embedding parameter q as follows: qm t sm t s q t m ) ( ) ( 0 ) ; ( 1 1

∑

∞ = + = ′ ϕ qm t im t i q t m ) ( ) ( 0 ) ; ( 2 1

∑

∞ = + = ′ ϕ qm t rm t r q t m ) ( ) ( 0 ) ; ( 3 1

∑

∞ = + = ′ ϕ

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where 0 ) ; ( 1 ! 1 ) ( ′ ₌ ∂ ∂

= t q _q

qm m m t sm ϕ 0 ) ; ( 2 ! 1 ) ( ′ ₌ ∂ ∂

= t q _q

qm m m t im ϕ 0 ) ; ( 3 ! 1 ) ( ′ ₌ ∂ ∂

= t q _q

qm m m t

rm ϕ

We know that zero order deformation equations (16), (17) and (18) can be written as

[

sm(t) Xmsm 1(t)

]

h1H1(t) m

(

sm 1(t)

)

L − − = ℜ −

[

im(t) Xmim 1(t)

]

h2H2(t) m

(

im 1(t)

)

L − − = ℜ −

[

rm(t) Xmrm 1(t)

]

h3H3(t) m

(

rm 1(t)

)

L − − = ℜ −

) 0 ( , 0 ) 0 ( , 0 ) 0

( sm rm

sm = =

where

(

)

1

[

1(; )

]

1 1 )! 1 ( 1 ) (

1 N t q

qm m m t sm m ϕ′ − ∂ − ∂ − = − ℜ

(

)

2

[

2(; )

]

1 1 )! 1 ( 1 ) (

1 N t q

qm m m t im m ϕ′ − ∂ − ∂ − = − ℜ

(

)

3

[

3(; )

]

1 1 )! 1 ( 1 ) (

1 N t q

qm m m t rm m ϕ′ − ∂ − ∂ − = − ℜ ⎩ ⎨ ⎧ > ≤ = 1 , 1 1 , 0 m m Xm As t

Li=_∂∂ then taking integration on both sides and substituting hi=−1,Hi=1 in above equations, we have

(

sm t

)dt

m

t Xmsm t

sm t

o 1()

) ( 1 )

( = − −

∫

ℜ − (19)

(

im t

)dt

m

t Xmim t

im t

o 1()

) ( 1 )

( = − −

∫

ℜ − (20)

(

rm t

)dt

m

t Xmrm t

rm t

o 1()

) ( 1 )

( = − −

∫

ℜ − (21)

as

(

)

1

[

1(; )

]

1 1 )! 1 ( 1 ) (

1 N t q

qm m m t sm m ϕ′ − ∂ − ∂ − = − ℜ

Putting value of N1

(

)

1 1 )! 1 ( 1 ) ( 1 − ∂ − ∂ − = − ℜ qm m m t sm m ⎥⎦ ⎤ ⎢⎣ ⎡ ₋ ₋ ₊ ₊ ∂ ∂ ) ; ( 1 ) ; ( 2 ) ; ( 1 ) 1 ( ) ; (

1t q P t q t q t q

tϕ π βϕ ϕ πϕ

(

1(; ) 1(; ) 2(; ) 1(; )

)

) (

1t t q im t q t q sm t q

m

s′ − + − + −

= β ϕ ϕ

Using initial conditions

0 ) ; 0 ( 2 , 0 ) ; 0 (

1 q =s ′ q =i

′ ϕ

ϕ

and ϕ′1(t;q)→s0(t), ϕ′2(t;q)→i0(t)

(

−

)

= ′ − +

ℜmsm 1(t) sm 1(t)

(

s0im−1+i0sm−1

)

+πsm−1(t)

β

(

−

)

= ′ − +

ℜmsm 1(t) sm 1(t)

) ( 1 ) ( 1 ) ( 1 0 t sm t k im t sk m k − + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − −

∑

− = π β

So, equation (19) becomes

) ( 1 )

(t Xmsm t

sm = −

dt t sm t k im t sk t m s m k t ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − − + − ′ −

∫

∑

− = ) ( 1 ) ( 1 ) ( ) ( 1 1 0

0 β π (22)

Similarly equation (20) & (21) becomes

) ( 1 )

(t Xmim t

im = −

⎢ ⎢ ⎣ ⎡ + − − − − −

∫

∑

− = ) ( 1 ) ( ) ( 1 1 0

0 ism t sk t im kt

m k t β dt t im _⎥ ⎦ ⎤ −

+ ) 1()

(γ π (23)

) ( 1 )

(t Xmrm t

rm = −

[

rm t im t rm

]

dt

t 1 ) ( 1 ) ( 1

0 − − − + −

−

∫

& γ π (24)

Equation (22), (23) & (24) form system of mth-order deformation equations for m>1.

Now we start iterating the above system of equations for different values of parameters and with the help of these parameters, we will observe the effect of high or low vaccination on childhood disease, hence stability will be discussed. We will discuss four cases [3];

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Table. 1 Initial values and parameters

Case s0 i0 r0 Β Π P Pc

1 1 0 0 0.8 0.03 0.04 0.9 0.4625

2 0.8 0.2 0 0.8 0.03 0.04 0.9 0.4625

3 0.8 0.2 0 0.8 0.03 0.04 0.3 0.4625

4 0.8 0.2 0 0.8 0.03 0.04 0.0 0.4624

Case 1

18604 . 0 ) 1

( ₌

+ − =

π γ

β P

Rv

Since Rv < 1, E0 is stable, Eu is unstable and

have disease eradication. For t = 0,

0 0 0 ) 1 ( 0

s i s P dt

ds

t

π β

π− −

− =

=

1 36 . 0 36

.

0 ⇒s=− t =s

− =

3 2 2

1 ₀_.₉₆ ₁₀

10 72 . 0 36 . 0 0 . 1 )

(t t t t

s = − + × − − × −

5 4 4

3 ₀_.₇₈₆ ₁₀

10 96 .

0 × − t − × − t

+

6 6 10 5688888892 .

0 × − t

−

0 ) (t =

i

− ×

+ ×

−

=0.36 0.72 10−1 2 0.96 10−2 3

)

(t t t t

r

6 5 5

4 4

3 ₀_.₇₆₈ ₁₀ ₀_.₅₁₂ ₁₀

10 96 .

0 × − t + × − t − × − t

0 ) ( 0 0

0

i i

s dt

ds

t

π γ

β − +

= =

1

0 i i= = ⇒

0 0

0

r i P dt

dr

t

π γ π+ − =

=

1

36 .

0 t r

r= =

⇒ For

4 3 2 4

3

2,s ,s ..., i ,i ,i

s and r₂,r₃,r₄..., We will use equation (22), (23) and (24) respectively.

Table.2 Approximations for case-1

Itr s(t) I(t) r(t)

0 1.0 ₀ 0

1 –0.36 ₀ 0.36

2 _0.72_×₁₀−1 ₀ _–0.72_×₁₀−1

3 _–0.96_×₁₀3 ₀ _0.96_×₁₀3

4 _0.96_×₁₀3 ₀ _–0.96_×₁₀3

5 _–0.768_×₁₀−4 ₀ _0.768_×₁₀4

6 _{–0.5688888892}_×₁₀−6 ₀ _–0.515_×₁₀−5

Case 2

Since Rv < 1, E0 is stable, Eu is unstable and have

disease eradication.

− ×

− +

−

=0.8 0.408 0.10082 8.224 10−3 3 )

(t t t t

s

− ×

+

×₁₀−2 4 ₀_._2838500158 ₁₀−3 5

1811776 .

0 t t

− ×₁₀−4 6

4866281149 .

0 t 0.1973168518

+ ×₁₀−5 7

t 0.1567280763×10−7t9+ ×

4557699387 .

0 ₁₀−9 9−

t

10 11

10 1747626667 .

0 × − t

= ) (t i

− ×

+₀_.₄₂ ₁₀−1 ₀_.₂₈₂₃ ₁₀−1 2

2 .

0 t t

− ×

+

×₁₀−2 3 ₀_._2759918751 ₁₀−2 4

11687 .

0 t t

4741940899 .

0 10 3762609484 .

0 × −3t5+

− ×

+

×₁₀−4 6 ₀_._199013997 ₁₀−5 7

t t

× −

×10− 0.4575903832 1540349563

.

0 7t9

10 11 9

9 ₀_._1747626667 ₁₀

10− t + × − t

− ×

+ ×

−

=0.366 0.7257 10−1 2 0.93937 10−2 3 )

(t t t t

r

× +

×10− 0.9241093251 94814275

.

0 3t4

− ×

− −

−43 5 ₀_._4445486401 ₁₀ 5 6

10 t t

+ ×

−

×10−7 7 0.269312 10−9 9 1697145904

.

0 t t

9 11

10 1820444445 .

0 × − t

Case 3.

Since Rv > 1, , E0 is stable, Eu is stable and we

don’t have disease eradication

− ×

+ −

=₀_.₈ ₀_.₁₆₈ ₀_.₃₃₆ ₁₀−1 2 ₀_.₂₄₆₄ ₁₀−2 3

)

(t t t t

s

− ×

+

×₁₀−3 4 ₀_._22308195 ₁₀−5 5

12521 .

0 t t

7 4 6

3 ₀_._7803698863 ₁₀

10 1932440964 .

0 × − t − × − t

× +

×

+0.4251830616 10−5t9 0.1094303622

10 7 9

5 ₀_._0293723614 ₁₀

(8)

Table.3

Approximations for case-2

Itr s(t) i(t) r

( )

t)

0 0.8 0.2 0

1 –0.408 _0.42_×₁₀−1 0.366

2 0.1008 _–0.2823_×₁₀−1_–0.7257_×₁₀−1

3 _–8.224_×₁₀−3_–0.11697_×₁₀−2_0.93937_×₁₀−2

4 _–0.1811776_×₁₀−2_0.2759918751_×₁₀−2_{–0.94814275}_×₁₀−3

5 _0.2838500158_×₁₀−3_{–0.3762609484}_×₁₀−3_0.9241093251_×₁₀−4

6 _{–0.4866281149}_×₁₀−4_0.4741940899_×₁₀−4_0.4445486401_×₁₀−5

7 _{–0.1973168518}_×₁₀−5_0.199013997_×₁₀−5_{–0.1697145904}_×₁₀−7

8 _0.1567280763_×₁₀−7_{–0.1540349563}_×₁₀−7_–0.269312_×₁₀−9

9 _0.4557699387_×₁₀−9_{–0.4575903832}_×₁₀−9_0.182044444_×₁₀−11

10 _{–0.1747626667}_×₁₀−11_0.1747626667_×₁₀−11 0

Itr s(t) i(t) r

( )

t)

0 0.8 0.2 0

1 –0.168 _0.42_×₁₀−1 0.126

2 _0.336_×₁₀−1_–0.903_×₁₀−2_–0.2457_×₁₀−1

3 _–0.2464_×₁₀−2_–0.7217_×₁₀−3_0.31857_×₁₀−2

4 _–0.125216_×₁₀−3_0.44919875_×₁₀−3_{–0.32398275}_×₁₀−3

5 _0.22308159_×₁₀−5_{–0.308446284} ₁₀ 4 5

t

−

× 0.286138125×10−4

6 _{–1932440964}_×₁₀−3_0.1860204709 ₁₀ 3 6

t

−

× 0.1534736535×10−5

7 _{–0.7803698868}_×₁₀−4_0.7869655811 ₁₀ 4 7

t

−

× –0.6595694941×10−6 8 _0.4251830616_×₁₀−5_{–0.41594566} ₁₀ 5 9

t

−

× –0.9237401597×10−7

9 _0.1094303622_×₁₀−5_{–0.1098674509} ₁₀ 5 9

t

−

× 0.4370887113×10−9

10 _{–0.293723614}_×₁₀−7_0.293723614 ₁₀ 7 10

t

−

× 0

Itr s(t) i(t) r

( )

t)

0 0.8 0.2 0

1 _–0.48_×₁₀−1_0.42_×₁₀−1_0.6_×₁₀−2

2 0 _0.57_×₁₀−3_–0.57_×₁₀−3

3 _0.416_×₁₀−3_–0.4977_×₁₀−3_0.817_×₁₀−4

4 _0.26864_×₁₀−4_–0.1496125_×₁₀−4_–0.1190275_×₁₀−4

5 _–0.7711584_×₁₀−5_0.68491314_×₁₀−5_0.86245251_×₁₀−6

6 _{–2076444349}_×₁₀−3_0.19838225_×₁₀−3_0.3573295998_×₁₀−5

7 _{–0.1402303147}_×₁₀−3_0.141400832_×₁₀−3_{–0.1170517333}_×₁₀−5

8 _0.1075760762_×₁₀−4_{–0.1048829562}_×₁₀−4_–0.269312_×₁₀−6

9 _0.4557699387_×₁₀−5_{–0.4575903832}_×₁₀−5_0.1820444445_×₁₀−7

(9)

+ ×

− ×

− × +

=₀_.₂ ₀_.₄₂ ₁₀−1 ₀_.₉₀₃ ₁₀−2 2 ₀_.₇₂₁₇ ₁₀−3 3

)

(t t t t

i

+ ×

−

×₁₀−3 4 ₀_._308446284 ₁₀−4 5

44919875 .

0 t t

7 4 6

3 ₀_._7869655811 ₁₀

10 1860204709 .

0 × − t + × − t

9 5 9

5 ₀_._1098674509 ₁₀

10 41594566 .

0 × − t − × − t

−

10 7

10 293723614 .

0 × − t

+

− ×

+ ×

−

=0.126 0.2457 10−1 2 0.31857 10−2 3 )

(t t t t

r

+ ×

+

×10−3 4 0.286138125 10−4 5 3239827

.

0 t t

7 6 6

5 ₀_._6595694941 ₁₀

10 1534736535 .

0 × − t − × − t

9 9 9

7 ₀_._4370887113 ₁₀

10 9237401597 .

0 × − t + × − t

−

Case 4

Since Rv > 1, E0 is stable, Eu is stable and we

don’t have have disease eradication. + ×

+ × −

=₀_.₈ ₀_.₄₈ ₁₀−1 ₀_.₄₁₆ ₁₀−3 3

)

(t t t

s

− ×

−

×₁₀−4 4 ₀_._7711584 ₁₀−5 5

26864 .

0 t t

7 3 6

3 ₀_._1402303147 ₁₀

10 2076444349 .

0 × − t − × − t

9 5 9

4 ₀_._4557699387 ₁₀

10 1075760762 .

0 × − t + × − t

+

10 6

10 1747626667 .

0 × − t

−

− ×

+ × +

=0.2 0.42 10−1 0.57 10−3 3 )

(t t t

i

+ ×

−

×10−3 3 0.1496125 10−4 4 4977

.

0 t t

+ ×

+

×10−5 5 0.19838225 10−3 6 68491314

.

0 t t

− ×

−

×10−3 7 0.1048829562 10−4 9 141400832

.

0 t t

10 6 9

5 ₀_._1747626667 ₁₀

10 4575903832 .

0 × − t + × − t

− ×

+ ×

− ×

=₀_.₆ ₁₀−2 ₀_.₅₇ ₁₀−3 2 ₀_.₈₁₇ ₁₀−4 3

)

(t t t t

r

+ ×

+

×₁₀−4 4 ₀_._86245251 ₁₀−6 5

1190275 .

0 t t

7 5 6

5 ₀_._117051333 ₁₀

10 3573295998 .

0 × − t − × − t

9 7 9

6 ₀_._1820444445 ₁₀

10 269312 .

0 × − t + × − t

−

0 1 2 3 4 5 6 7 8 9 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time

P

opu

la

ti

o

n

F

rac

ti

on

Susceptible Infected Recovered

Fig.2 Plot for case 1

0 1 2 3 4 5 6 7 8 9 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time

P

o

pu

la

ti

on

F

rac

ti

o

n

0 1 2 3 4 5 6 7 8 9 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time

P

o

pu

la

ti

on

F

rac

ti

o

n

0 1 2 3 4 5 6 7 8 9 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time

P

opul

a

ti

on F

rac

ti

on

(10)

5. Results and Discussion

In first two cases high level of vaccination coverage is given and finally we reached infection free stage in both cases. In case 1, we did not introduce infection, all susceptible are given vaccination and all are recovered. In other words all susceptible moved to R-compartment without passing through I-compartment. Hence not a single infection is shown in Fig.2. While in case 2, we introduced small number of infection, two kinds of individuals are coming into R-compartment, one from S-compartment after being vaccinated and other from I-compartment with permanent immunity. In last two cases low level vaccination is implied as a result we have endemic situation at the end; disease permanently exists and infection free equilibrium cannot be attained.

It is clear that we can achieve infection free equilibrium if (P>P_c) where P_c is threshold value, otherwise endemic situation occurs.

6. Conclusion

HAM generates series which converge speedily after some iteration. HAM nullifies much of computational work that arises in finite difference method and other parallel technique. Analytical approximations are time-tested and ensure the stability of HAM. Hence Homotopy Analysis Method (HAM) is reliable to solve non-linear system of differential equations.

References

[1] Roy M. Anderson & Robert M. May. 1985. Vaccination and Herd Immunity to Infectious Disease Nature, 318(6044):323-9.

[2] Shaban N, Anderson M, Svensson A, Britton T. 2008. Network, epidemics and vaccination through contact tracing. Math. Biosci, 216(1): 1-8

[3] O.D. Makinde. 2007. A domian decomposition approach to a SIR epidemic model with constant vaccination strategy. applied mathematics and competition 184, 842-848 [4] S. Busenberg, P. van den Driessche. 2990.

Analysis of a Disease Transmission Model in a Population with Varying Size. J. Math. Bio. 28, 257-270.

[5] Gul Zaman and I. Hoyo Jung. 2007. Stability Techniques in SIR Epidemic Model. PAMM Proc. Appl. Math. Mech. 7, 2030063– 2030064.

[6] Buonomo B, D'Onofrio A, Lacitignola D. 2008. Global stability of an SIR epidemic model with information dependent vaccination, Math Biosci. 216(1):9-16.

[7] Liao S.J. 1998. Homotopy Analysis Method: A New Analytical Method for Nonlinear Problems. Applied Mathematics and Mechanics (English Edition, Vol.19, No.10. [8] Liao, S.J. 2004. On the homotopy analysis

method for nonlinear problems. App Math Comput.; 174:499-513.