Homotopy Perturbation Method For A Seir Model With Varying Total Population Size
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HOMOTOPY PERTURBATION METHOD FOR A SEIR
MODEL WITH VARYING TOTAL POPULATION SIZE
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.Jaharuddin
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Bogor Agricultural University
Jalan Meranti, Kampus !PB Dramaga
Bogor 16680, Indonesia
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Abstract
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We consider a SElR model with varying total population size. We find
the analytical solution of the proposed model
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solution of the nonlinear problem. By using this 1ncthod, we solve the
problem analytically and then compare the nu1nerical result with other
standard 1nethods. We also justify the numerical simulation and their
results. The comparison reveals that our approximate solutions are in
very good agreement with those by numerical 1nethod. Moreover, the
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I. Introduction
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Mathematical modelling has become important tools in analyzing the
spread and control of infectious diseases. The model helps us to understand
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by homotopy
perturbation method which is one of the best 1nethods for finding the
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different factors such as the transmission and recovery rates and to predict
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Keywords and phrases: epidemic models, homotopy perturbation method, system of nonlinear
differential equation.
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188
Homotopy Perturbation Method for a SEIR Model ...
Jaharuddin
189
.: how the diseases will spread over a period of time I 11. In recent years, many
method, this method is independent of small parameters and can overcome
: aitcmpts have been made to develop realistic mathematical models for
the restrictions of the perturbation methods. The homotopy perturbation
' investigating the transmission dynamic of infectious diseases. One of them is
method provides a universal technique to introduce a perturbative parameter.
: a ·model of epidemic which is usually given as a system of differential
This method has been introduced in 1999 by He [6] and was further
equations. To understand the behavior of epidemic model, we need to know
developed and improved by him [4]. The method has been used by many
the analysis of steady state and their stability 18 I. Studies of epidemic models
authors [3] in a wide variety of scientific and engineering applications to
that incorporate disease caused death and varying total population have
solve different types of governing differential equations. In this paper, the
become one of the in1portant areas in the mathematical theory of
basic idea of the homotopy perturbation method is introduced and then, the
epidemiology. Most of the research literatures on these types of models
nonlinear equation of SE!R model is solved through the homotopy
assume that the disease incubation is negligible so that once infected, each
perturbation method. The purpose of this paper is to extend the homotopy
susceptible individual instantaneously becomes infected and later recovered
perturbation method for computing the approximate analytical solution of a
with a permanent or tcrnporary acquired immunity. A compartmental model
SEIR model with varying total population size and then see how these
based on these assumptions is customarily called a SIR model. Models that
solutions compare with the solutions by numerical method.
arc more general than the SIR types need to be studied to investigate the role
This paper is organized as follows: Sections 2 and 3 are devoted to a
short description of the SEIR models with varying total population size and
the analysis of homotopy perturbation method, respectively. In Section 4, we
present the analytical approximate solutions obtained by implementing the
homotopy perturbation method to the SEIR model with varying total
of incubation in disease transmission I 101. Using a compartmental approach,
one may assume that a susceptible individual first goes through a latent
period after infection before becoming infected. The resulting models are of
SE!R types depending on whether the acquired immunity is permanent or
otherwise. In this article, the SE!R model with varying total population size
will be discussed and this model is in the form of nonlinear.
population size followed by comparison of results between the approximate
solutions and the solutions obtained by numerical method. The last section is
conclusion.
We know that except a limited number of these problems, most of them
'do not have analytical solution. Therefore, these nonlinear equations should
2. Mathematical Formulation
be solved using other methods. Some of them arc solved using numerical
considered so as to avoid divergence or inappropriate result. In the analytical
Formulation of SE!R model with varying total population size using by a
compartmental approach is given by Li and Fang [7]. It is assumed that the
perturbation method, we should exert the small parameter in the equation.
local density of the total population is a constant. A population of size N(t)
Therefore, finding the small parameter and exerting it into the equation are
difficulties of this method. Many different mathematical methods have been
is partitioned into subclasses of individuals who are susceptible, exposed
(infected but not yet infectious), infectious and recovered, with sizes denoted
recently introduced to eliminate the small parameter. In 1992, Liao [9] has
by S(t), E(t), l(t), and R(t), respectively. The sum E(t) + l(t) is the total
proposed a new analytical method called the homotopy analysis method,
infected population. Our assumptions on the dynamical transfer of the
population are then demonstrated in the form of SEIR model as depicted by
Figure I.
techniques. In the numerical method, stability and convergence should be
which introduce an embedding parameter to construct a homotopy and then
analyzes it by means of Taylor formula. Therefore, unlike the perturbation
190
Jaharuddin
Homotopy Perturbation Method for a SEIR Model ...
subject to the initial conditions
;t {
€E
191
S(O) ｾ＠
0, E(O) ｾ＠ 0, 1(0) ｾ＠ 0, /1(0) ｾ＠
0.
The total population size N(t) can be determined by N(t) = S(t) + E(t)
Figure I. A schematic representation of the flow of individuals between
+ l(t) + R(t) or from the differential equation
epidemiological classes.
dN = (b _ d)N - a.I,
dt
The parameter h > 0 is the constant rate for natural birth and d > 0 is
that of natural death. It is assumed that all newborns arc susceptible and
(2)
which is derived by adding the equations in ( l ).
vertical transmission can be neglected. l'hc parameter a is the constant rate
3. Analysis of Method
for disease-related death and y is the rate for recovery. The rate of removal e
of individuals from the exposed class is assumed lo be a constant so that
l/t
can be regarded as the mean latent period. In the limiting case, when
E
-> oo, or equivalently, when the mean latent period lfe -> 0, the SEIR
In this section, we illustrate the basic idea of the homotopy perturbation
method for solving nonlinear differential equation in which we consider the
following general nonlinear problem:
model becomes a SIR model. The recovered individuals are assumed to
A(u(r)) = 0, r
ac.quire permanent immunity; there is no transfer from R class back to the S
E Q,
(3)
class. The per capita contact rate A., which is the average number of effective
where A is a nonlinear operator, u(r) is an unknown function, r is an
co_ntacts with other individual hosts per unit time, is then a constant. A
independent variable, and
fraction l(t)/ N(t) of these contacts is with infectious individuals and thus
v(r, p) : n x [O, l] -> IR which satisfies
the average number of relevant contacts of each individual with the
n
is the domain. W c construct a homotopy
H(v, p) = (1 - p)(L(v)- L(v0 )) + pA(v),
'infectious class is A.i(t)/ N(r). The total number of new infections at a time t
l]
is an embedding parameter, L denotes an auxiliary linear
is given by A.l(t)S(t)/ N(t). The following differential equations are derived
where p e [0,
based on the basic assumption and using the transfer diagram:
operator and v0 is an initial approximation of the exact solution. By equating
dS
dt
to zero the homotopy function, the zero-order deformation equation is
=
bN - dS _ A.IS
constructed as
N'
(1 - p)(L(v)- L(vo)) + pA(v) = 0.
(4)
dE _ A.IS _ (t + d)E,
dt -
N
Setting p = 0, the zero-order deformation equation (4) becomes
di
dt = EE - (y +CJ.+ d)I,
L(v)-L(vo) = 0.
Using (5), by linearity
dR
dt
= yl -
dR,
(I)
v(r, 0) = vo(r).
(5)
,,,,
Jahamddin
Homotopy Perturbation Method for a SEIR Model ...
When p = 1, the zero-order deformation equation (4) is reduced to
dr
d1 = yi - br + air,
193
(8)
A(v(r, I))= 0,
subject to the restriction s + e + i + r = I. Note that the total population size
which is exactly the same as the nonlinear equation (3 ), provided
N(I) does not appear in (8); this is a direct of the homogeneity of the system
v(r, I)= 11(r).
(8). Also observe that the variable r does not appear in the first three
According to the homotopy perturbation method, the solution of the
equation (4) can be expressed as a series inp in the form:
= VQ
V
+
fJVI
+p
2
V2
+ · · ·.
equations of (8). This allows us to attack (8) by studying the subsystem:
ds =b-bs-f.is+ais,
di
(6)
When p-> I, equation (4) corresponds to the original equations (3) and (6)
ｾ［＠
becomes the approximate solution of equation (3), i.e.,
ll
lim v =
=
Vo
+
V1
+ V2 + ....
(7)
= f.is - (E + h )e + aie,
di
.2
di =Ee- (y+a.+ I)'
Jl+fJ.I,
ｰｾｉ＠
The convergence of the series in equation (7) is discussed by He in [5].
(9)
and determining r from r = I - s - e - r or
4. Application of Method
dr
di = yi - hr + a.ir.
In this section, the homotopy perturbation method described in the
previous section for solving a SEIR model with varying total population size
Subsystem (9) will be solved by generalizing the described homotopy
is applied. Then comparison is made with the numerical method to assess the
accuracy and the effectiveness of the homotopy perturbation method. The
perturbation method. The linear operators L1, l2, and L3 can be defined as
below:
first step is to transform the variables.
Let s = S/N, e
= E/N, i = I/N,
and r
= R/N
.) d• '-() de'-(')
di
l(
I S = di ' ｾｌ＠
e = di ' ｾｊ＠ 1 = di .
denote the fractions of
the classes S, E. I. and R in the population, respectively. It is easy to verify
thats, e, i, and r satisfy the system of differential equations:
From (9) nonlinear operators A1, A2 , and A3 can be defined as
d1· = h _ hs _;\.is + a.is,
A1(s) =
de
, . - (E + b) e + a1e,
.
dt = A.ls
Ai(e) =
d1
di
( y+a.+ b)'l+fJ.I,
.2
-,=Eeal
l
A3(i)
］ｾ［Ｍｅ･Ｋ＠
ｾ＠
ｾ［＠
- b + hs + f.is - ais,
- f.is + (E + b)e - a.ie,
(y +a+ b)i - ai2.
• 194
Homotopy Perturbation Method for a SEIR Model ...
Jaharuddin
ei = [ai0 eo + 'J...i0 s0
According to equation (4 ). we can obtain
dt
di
Coefficient of
de -deo)
(1-p) ( - +p(de
--'J...is+(r.+h)e-aie ) =0
dtdl
ＨＱＭｰＩｾ＠
dt
ＭＺｾＩ＠
'
+ 1{:;-r.e+(y+a +h)i-ai2) = 0,
(10)
so(t) = s(O), e0(1) = e(O), i0(t) = i(O).
,\' = SQ + fJSJ + fl 2Sz + p .1 S3 + "·,
s2 =
Ｍｾｨ｛｢＠
2.
";7
= -bs1 - ('J... - a) (ios1 + i1so ),
d;7
= 'J...(i1s 0 + i0s1) + a(i0 e1 + i1eo)-(r. + b)ei,
- bs0
e -- eo + pe1 + p 2ez + p 3e3 + ... ,
.
gives the following initial value problem:
with initial conditions: s 2(0) = 0, e2(0) = 0, i2(0) = 0. Solving the above
equations, we obtain the following approximations:
In the following we assume the solution for system (I 0) in the form:
.
p2
diz
(
/)'1 + 2··
dt=r.e1+y+a+>1
at0 11,
and the initial approximations arc as follows:
.
+ b)eo]t,
it = [r.eo -(y +a+ b)i0 + ai5]t.
(I_ p)(ds - di·o) + p(d' - h + hs +/...is - ais) = 0,
di
- (r.
195
J.
1 = 10 + P11 + P 12 + P 13 + .. "
- ('J... - a)i0s 0 ]1
Ｐ Ｈ｢＠
｡Ｉ｛ｾｩ
-('J... (I I)
+
ｾ｛ｲＮ･ｯ＠
- bs 0
2
- ('J... -
a)i0 s0 )1
2
-(y +a+ b)io + ai6Js 0 t
J.
2
Substituting equation ( 11) into (I 0) and equating the term with identical
powers of p, we obtain the set of initial value problems. Coefficient of p
e2 =
ｾＧｊＮ｛ｲ･
Ｐ＠
gives the following initial value problem:
+
.
, .
(
= atoeo + "''o"o - r. +
h) eo,
-
dii = i:e0 - (y +a+ h)io + ai6,
d1
with initial conditions: s 1(0) = 0, e1(0) = 0, i1(0) = 0. Solving the above
equations, we obtain the following approximations:
s 1 = [h - hs0
- ('J... -
a)i0so]t,
2
(y +a+ b)i0 + ai5Js0 t +
+ -'-ai0 [ai0e0 + 'J...i0 s0 - (r. + b)e0 ]t
2
d'1 = ;, _ h.i·o - ('J... - a)ioso.
di
de1
dr
-
i2 =
ｾ｡｛ｲＮ･
Ｐ＠
-
- (r. +
-'-r.[ai0 e0 + 'J...i0 s0 -(r. + b)e0 ]t
Ｐ ｛｢＠
- bs0 - ('J... - a)i0so]t
2
(y +a+ b)i0 + ai5Je0 t
ｾ＠ (r. + b)[ai0e0 + 'J...i0 s0
ｾＧｊＮｩ
2
2
b)e0 ]t ,
2
2
+ ｾＨｹ＠
+a+ b)[r.eo - (y +a+ b)i0 + ai5]t
2
+ aio[r.eo -(y +a+ b)io + ai5Jt ,
2
2
• 196
Jaharuddin
Homotopy Perturbation Method for a SEIR Model ...
and so on, in the same manner the rest of the components can be obtained
Table 2. Comparison between absolute error of homotopy perturbation
ethod (HPM) and numerical method (NM)
using the symbolic package. According to the homotopy perturbation
method, we can obtain the solution in a series formed as follows:
I
S ::::
so
+ SI + .'lz + ,.. '
197
Is(t)HPM -s(t)NM I Ie(t)HPM -e(t)NM I Ii(t)HPM -i(l)NM I
0
0
0
0
0.1
3.0000 x 10-IO
3.6000 x 10- 10
2.2769 x 10-1
{12)
0.2
2. 7000 x 10-9
2.7600 x 10-9
9.1061x10-1
Suppose given the following data: the initial number of susceptible
0.3
8.9000 x 10-9
9.3200 x 10-9
2.0485 x 10-6
0.4
2.1200 x 10-8
2.2050 x 10-8
3.6413 x 10-6
0.5
4.1500 x 10-8
4.30 I 0 x 10-8
5.6887 x 10-6
0.6
7.1600 x 10-8
7.4700 x 10-8
8.1902x10-6
0.7
1.1400 x 10-1
l.1922x 10-7
1.1146 x 10-5
0.8
1.7040 x 10-1
1.7842 x 10-1
1.4554 x 10-5
0.9
2.4260 x 10-1
2.5412 x J0- 7
1.8418 x 10-5
1
3.3290 x 10-1
3.4813 x 10-1
2.2734 x 10-5
e = e0 + e1 + ez + ···,
i=io+ii+iz+···.
individual in the location S(O) = JO thousand: the initial number of expose
people 1:"(0) = I thousand: initial number of actively infected people
/(0) = O: The parameter values used for numerical simulation is given in
Table I.
Table I. Values used in the numerical simulation [2]
A.
The parameter controlling how ollen a susceptibleinfected contact results in a new exposure
y
The rate an infected recovers and moves into the
resistant phase
"
The rate at which an exposed person becomes
infective
b
The natural birth rate
a
The rate for diseases related death
0.05
0.003
0.05
5. Conclusions
In this paper, homotopy perturbation method has been successfully
0.00001
applied to find the solution of the SEIR model with varying total population
0.002
size. The method is reliable and easy to use. The main advantage of the
Then following approximate solution is obtained as result of first 6
method is the fact that it provides its user with an analytical approximation,
terms of the series decomposition {12). Results calculated by homotopy
in many cases an exact solution, in a rapidly convergent sequence with
perturbation method arc compared with numerical solution in Table 2. The
elegantly computed term.
explicit Rungc-Kutta method in symbolic computation package has been
References
used to find numerical solution of s(t), e(l), and i(t). As it can be seen in
Table 2, there exists a very good agreement between homotopy perturbation
method result and numerical solutions.
[I]
J. E. Cohen, Infectious diseases of hu1nans: dynamics and control, The Journal of
the American Medical Association 268 (1992), 3381-3392.
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Jaharuddin
[21
E. Dcmirci, A. Unal and N. Ozalp, A fractional order SEIR model with density
dependent dca!h rate, Hacct J. Ma!h. Slat 40 (2011), 287-295.
{3)
A. Fercidoon, H. Yaghoobi and M. Davoudabadi, Application of the homotopy
perturbation method for solving the foam drainage equation, Int. J. Differ. Equ.
2011, Art. ID 864023, 13 pp. doi:I0.1155/2011/864023.
[4]
J. 1-1. lie, A coupling method of a homotopy technique and a perturbation
technique ICJr non-linear problerns, Internal. J. Non-Linear Mech. 35 (2000),
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[5]
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(7]
X. Li and B. Fang, Stability of an age-structured SEIR epidemic model with
infcc1ivi1y in la!en! period, Appl. Appl. Math. 4 (2009), 218-236.
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M. Y. Li and J. S. Muldowney, A geometric approach to global-stability
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[91
S. J. Liao, The proposed homotopy analysis technique for rhc solutions of
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[10]
M. Y. Li, J. R. Graef, L. C. Wang and J. Karsai, Global dynamics of a SEIR
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FAR EAST JOURNAL OF MATHEMATICAL SCIENCES (FJMS;
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...
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Far East Journal of Mathematical Sciences (FJMS)
© 2014 Pushpa Publishing House, Allnhnbad, India
Alms and Sc-: The Far Easl Journal of MathemaYcal Sciences (FJMS) Is devoted to publishing
original resean:h pepers and critical survey articles In the field of Pure and Applied Mathematics,
Computer Applicllllons and Statistics. The FJMS is a fortnightly journal published in twelve volumes
annuslly and each volume comprises of two Issues.
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Volume 84, Number 2, 2014, Pages 187-198
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HOMOTOPY PERTURBATION METHOD FOR A SEIR
MODEL WITH VARYING TOTAL POPULATION SIZE
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Bogor Agricultural University
Jalan Meranti, Kampus !PB Dramaga
Bogor 16680, Indonesia
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e-mail: jaharipb@yahoo.com
Abstract
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We consider a SElR model with varying total population size. We find
the analytical solution of the proposed model
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solution of the nonlinear problem. By using this 1ncthod, we solve the
problem analytically and then compare the nu1nerical result with other
standard 1nethods. We also justify the numerical simulation and their
results. The comparison reveals that our approximate solutions are in
very good agreement with those by numerical 1nethod. Moreover, the
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I. Introduction
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Mathematical modelling has become important tools in analyzing the
spread and control of infectious diseases. The model helps us to understand
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perturbation method which is one of the best 1nethods for finding the
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different factors such as the transmission and recovery rates and to predict
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Keywords and phrases: epidemic models, homotopy perturbation method, system of nonlinear
differential equation.
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188
Homotopy Perturbation Method for a SEIR Model ...
Jaharuddin
189
.: how the diseases will spread over a period of time I 11. In recent years, many
method, this method is independent of small parameters and can overcome
: aitcmpts have been made to develop realistic mathematical models for
the restrictions of the perturbation methods. The homotopy perturbation
' investigating the transmission dynamic of infectious diseases. One of them is
method provides a universal technique to introduce a perturbative parameter.
: a ·model of epidemic which is usually given as a system of differential
This method has been introduced in 1999 by He [6] and was further
equations. To understand the behavior of epidemic model, we need to know
developed and improved by him [4]. The method has been used by many
the analysis of steady state and their stability 18 I. Studies of epidemic models
authors [3] in a wide variety of scientific and engineering applications to
that incorporate disease caused death and varying total population have
solve different types of governing differential equations. In this paper, the
become one of the in1portant areas in the mathematical theory of
basic idea of the homotopy perturbation method is introduced and then, the
epidemiology. Most of the research literatures on these types of models
nonlinear equation of SE!R model is solved through the homotopy
assume that the disease incubation is negligible so that once infected, each
perturbation method. The purpose of this paper is to extend the homotopy
susceptible individual instantaneously becomes infected and later recovered
perturbation method for computing the approximate analytical solution of a
with a permanent or tcrnporary acquired immunity. A compartmental model
SEIR model with varying total population size and then see how these
based on these assumptions is customarily called a SIR model. Models that
solutions compare with the solutions by numerical method.
arc more general than the SIR types need to be studied to investigate the role
This paper is organized as follows: Sections 2 and 3 are devoted to a
short description of the SEIR models with varying total population size and
the analysis of homotopy perturbation method, respectively. In Section 4, we
present the analytical approximate solutions obtained by implementing the
homotopy perturbation method to the SEIR model with varying total
of incubation in disease transmission I 101. Using a compartmental approach,
one may assume that a susceptible individual first goes through a latent
period after infection before becoming infected. The resulting models are of
SE!R types depending on whether the acquired immunity is permanent or
otherwise. In this article, the SE!R model with varying total population size
will be discussed and this model is in the form of nonlinear.
population size followed by comparison of results between the approximate
solutions and the solutions obtained by numerical method. The last section is
conclusion.
We know that except a limited number of these problems, most of them
'do not have analytical solution. Therefore, these nonlinear equations should
2. Mathematical Formulation
be solved using other methods. Some of them arc solved using numerical
considered so as to avoid divergence or inappropriate result. In the analytical
Formulation of SE!R model with varying total population size using by a
compartmental approach is given by Li and Fang [7]. It is assumed that the
perturbation method, we should exert the small parameter in the equation.
local density of the total population is a constant. A population of size N(t)
Therefore, finding the small parameter and exerting it into the equation are
difficulties of this method. Many different mathematical methods have been
is partitioned into subclasses of individuals who are susceptible, exposed
(infected but not yet infectious), infectious and recovered, with sizes denoted
recently introduced to eliminate the small parameter. In 1992, Liao [9] has
by S(t), E(t), l(t), and R(t), respectively. The sum E(t) + l(t) is the total
proposed a new analytical method called the homotopy analysis method,
infected population. Our assumptions on the dynamical transfer of the
population are then demonstrated in the form of SEIR model as depicted by
Figure I.
techniques. In the numerical method, stability and convergence should be
which introduce an embedding parameter to construct a homotopy and then
analyzes it by means of Taylor formula. Therefore, unlike the perturbation
190
Jaharuddin
Homotopy Perturbation Method for a SEIR Model ...
subject to the initial conditions
;t {
€E
191
S(O) ｾ＠
0, E(O) ｾ＠ 0, 1(0) ｾ＠ 0, /1(0) ｾ＠
0.
The total population size N(t) can be determined by N(t) = S(t) + E(t)
Figure I. A schematic representation of the flow of individuals between
+ l(t) + R(t) or from the differential equation
epidemiological classes.
dN = (b _ d)N - a.I,
dt
The parameter h > 0 is the constant rate for natural birth and d > 0 is
that of natural death. It is assumed that all newborns arc susceptible and
(2)
which is derived by adding the equations in ( l ).
vertical transmission can be neglected. l'hc parameter a is the constant rate
3. Analysis of Method
for disease-related death and y is the rate for recovery. The rate of removal e
of individuals from the exposed class is assumed lo be a constant so that
l/t
can be regarded as the mean latent period. In the limiting case, when
E
-> oo, or equivalently, when the mean latent period lfe -> 0, the SEIR
In this section, we illustrate the basic idea of the homotopy perturbation
method for solving nonlinear differential equation in which we consider the
following general nonlinear problem:
model becomes a SIR model. The recovered individuals are assumed to
A(u(r)) = 0, r
ac.quire permanent immunity; there is no transfer from R class back to the S
E Q,
(3)
class. The per capita contact rate A., which is the average number of effective
where A is a nonlinear operator, u(r) is an unknown function, r is an
co_ntacts with other individual hosts per unit time, is then a constant. A
independent variable, and
fraction l(t)/ N(t) of these contacts is with infectious individuals and thus
v(r, p) : n x [O, l] -> IR which satisfies
the average number of relevant contacts of each individual with the
n
is the domain. W c construct a homotopy
H(v, p) = (1 - p)(L(v)- L(v0 )) + pA(v),
'infectious class is A.i(t)/ N(r). The total number of new infections at a time t
l]
is an embedding parameter, L denotes an auxiliary linear
is given by A.l(t)S(t)/ N(t). The following differential equations are derived
where p e [0,
based on the basic assumption and using the transfer diagram:
operator and v0 is an initial approximation of the exact solution. By equating
dS
dt
to zero the homotopy function, the zero-order deformation equation is
=
bN - dS _ A.IS
constructed as
N'
(1 - p)(L(v)- L(vo)) + pA(v) = 0.
(4)
dE _ A.IS _ (t + d)E,
dt -
N
Setting p = 0, the zero-order deformation equation (4) becomes
di
dt = EE - (y +CJ.+ d)I,
L(v)-L(vo) = 0.
Using (5), by linearity
dR
dt
= yl -
dR,
(I)
v(r, 0) = vo(r).
(5)
,,,,
Jahamddin
Homotopy Perturbation Method for a SEIR Model ...
When p = 1, the zero-order deformation equation (4) is reduced to
dr
d1 = yi - br + air,
193
(8)
A(v(r, I))= 0,
subject to the restriction s + e + i + r = I. Note that the total population size
which is exactly the same as the nonlinear equation (3 ), provided
N(I) does not appear in (8); this is a direct of the homogeneity of the system
v(r, I)= 11(r).
(8). Also observe that the variable r does not appear in the first three
According to the homotopy perturbation method, the solution of the
equation (4) can be expressed as a series inp in the form:
= VQ
V
+
fJVI
+p
2
V2
+ · · ·.
equations of (8). This allows us to attack (8) by studying the subsystem:
ds =b-bs-f.is+ais,
di
(6)
When p-> I, equation (4) corresponds to the original equations (3) and (6)
ｾ［＠
becomes the approximate solution of equation (3), i.e.,
ll
lim v =
=
Vo
+
V1
+ V2 + ....
(7)
= f.is - (E + h )e + aie,
di
.2
di =Ee- (y+a.+ I)'
Jl+fJ.I,
ｰｾｉ＠
The convergence of the series in equation (7) is discussed by He in [5].
(9)
and determining r from r = I - s - e - r or
4. Application of Method
dr
di = yi - hr + a.ir.
In this section, the homotopy perturbation method described in the
previous section for solving a SEIR model with varying total population size
Subsystem (9) will be solved by generalizing the described homotopy
is applied. Then comparison is made with the numerical method to assess the
accuracy and the effectiveness of the homotopy perturbation method. The
perturbation method. The linear operators L1, l2, and L3 can be defined as
below:
first step is to transform the variables.
Let s = S/N, e
= E/N, i = I/N,
and r
= R/N
.) d• '-() de'-(')
di
l(
I S = di ' ｾｌ＠
e = di ' ｾｊ＠ 1 = di .
denote the fractions of
the classes S, E. I. and R in the population, respectively. It is easy to verify
thats, e, i, and r satisfy the system of differential equations:
From (9) nonlinear operators A1, A2 , and A3 can be defined as
d1· = h _ hs _;\.is + a.is,
A1(s) =
de
, . - (E + b) e + a1e,
.
dt = A.ls
Ai(e) =
d1
di
( y+a.+ b)'l+fJ.I,
.2
-,=Eeal
l
A3(i)
］ｾ［Ｍｅ･Ｋ＠
ｾ＠
ｾ［＠
- b + hs + f.is - ais,
- f.is + (E + b)e - a.ie,
(y +a+ b)i - ai2.
• 194
Homotopy Perturbation Method for a SEIR Model ...
Jaharuddin
ei = [ai0 eo + 'J...i0 s0
According to equation (4 ). we can obtain
dt
di
Coefficient of
de -deo)
(1-p) ( - +p(de
--'J...is+(r.+h)e-aie ) =0
dtdl
ＨＱＭｰＩｾ＠
dt
ＭＺｾＩ＠
'
+ 1{:;-r.e+(y+a +h)i-ai2) = 0,
(10)
so(t) = s(O), e0(1) = e(O), i0(t) = i(O).
,\' = SQ + fJSJ + fl 2Sz + p .1 S3 + "·,
s2 =
Ｍｾｨ｛｢＠
2.
";7
= -bs1 - ('J... - a) (ios1 + i1so ),
d;7
= 'J...(i1s 0 + i0s1) + a(i0 e1 + i1eo)-(r. + b)ei,
- bs0
e -- eo + pe1 + p 2ez + p 3e3 + ... ,
.
gives the following initial value problem:
with initial conditions: s 2(0) = 0, e2(0) = 0, i2(0) = 0. Solving the above
equations, we obtain the following approximations:
In the following we assume the solution for system (I 0) in the form:
.
p2
diz
(
/)'1 + 2··
dt=r.e1+y+a+>1
at0 11,
and the initial approximations arc as follows:
.
+ b)eo]t,
it = [r.eo -(y +a+ b)i0 + ai5]t.
(I_ p)(ds - di·o) + p(d' - h + hs +/...is - ais) = 0,
di
- (r.
195
J.
1 = 10 + P11 + P 12 + P 13 + .. "
- ('J... - a)i0s 0 ]1
Ｐ Ｈ｢＠
｡Ｉ｛ｾｩ
-('J... (I I)
+
ｾ｛ｲＮ･ｯ＠
- bs 0
2
- ('J... -
a)i0 s0 )1
2
-(y +a+ b)io + ai6Js 0 t
J.
2
Substituting equation ( 11) into (I 0) and equating the term with identical
powers of p, we obtain the set of initial value problems. Coefficient of p
e2 =
ｾＧｊＮ｛ｲ･
Ｐ＠
gives the following initial value problem:
+
.
, .
(
= atoeo + "''o"o - r. +
h) eo,
-
dii = i:e0 - (y +a+ h)io + ai6,
d1
with initial conditions: s 1(0) = 0, e1(0) = 0, i1(0) = 0. Solving the above
equations, we obtain the following approximations:
s 1 = [h - hs0
- ('J... -
a)i0so]t,
2
(y +a+ b)i0 + ai5Js0 t +
+ -'-ai0 [ai0e0 + 'J...i0 s0 - (r. + b)e0 ]t
2
d'1 = ;, _ h.i·o - ('J... - a)ioso.
di
de1
dr
-
i2 =
ｾ｡｛ｲＮ･
Ｐ＠
-
- (r. +
-'-r.[ai0 e0 + 'J...i0 s0 -(r. + b)e0 ]t
Ｐ ｛｢＠
- bs0 - ('J... - a)i0so]t
2
(y +a+ b)i0 + ai5Je0 t
ｾ＠ (r. + b)[ai0e0 + 'J...i0 s0
ｾＧｊＮｩ
2
2
b)e0 ]t ,
2
2
+ ｾＨｹ＠
+a+ b)[r.eo - (y +a+ b)i0 + ai5]t
2
+ aio[r.eo -(y +a+ b)io + ai5Jt ,
2
2
• 196
Jaharuddin
Homotopy Perturbation Method for a SEIR Model ...
and so on, in the same manner the rest of the components can be obtained
Table 2. Comparison between absolute error of homotopy perturbation
ethod (HPM) and numerical method (NM)
using the symbolic package. According to the homotopy perturbation
method, we can obtain the solution in a series formed as follows:
I
S ::::
so
+ SI + .'lz + ,.. '
197
Is(t)HPM -s(t)NM I Ie(t)HPM -e(t)NM I Ii(t)HPM -i(l)NM I
0
0
0
0
0.1
3.0000 x 10-IO
3.6000 x 10- 10
2.2769 x 10-1
{12)
0.2
2. 7000 x 10-9
2.7600 x 10-9
9.1061x10-1
Suppose given the following data: the initial number of susceptible
0.3
8.9000 x 10-9
9.3200 x 10-9
2.0485 x 10-6
0.4
2.1200 x 10-8
2.2050 x 10-8
3.6413 x 10-6
0.5
4.1500 x 10-8
4.30 I 0 x 10-8
5.6887 x 10-6
0.6
7.1600 x 10-8
7.4700 x 10-8
8.1902x10-6
0.7
1.1400 x 10-1
l.1922x 10-7
1.1146 x 10-5
0.8
1.7040 x 10-1
1.7842 x 10-1
1.4554 x 10-5
0.9
2.4260 x 10-1
2.5412 x J0- 7
1.8418 x 10-5
1
3.3290 x 10-1
3.4813 x 10-1
2.2734 x 10-5
e = e0 + e1 + ez + ···,
i=io+ii+iz+···.
individual in the location S(O) = JO thousand: the initial number of expose
people 1:"(0) = I thousand: initial number of actively infected people
/(0) = O: The parameter values used for numerical simulation is given in
Table I.
Table I. Values used in the numerical simulation [2]
A.
The parameter controlling how ollen a susceptibleinfected contact results in a new exposure
y
The rate an infected recovers and moves into the
resistant phase
"
The rate at which an exposed person becomes
infective
b
The natural birth rate
a
The rate for diseases related death
0.05
0.003
0.05
5. Conclusions
In this paper, homotopy perturbation method has been successfully
0.00001
applied to find the solution of the SEIR model with varying total population
0.002
size. The method is reliable and easy to use. The main advantage of the
Then following approximate solution is obtained as result of first 6
method is the fact that it provides its user with an analytical approximation,
terms of the series decomposition {12). Results calculated by homotopy
in many cases an exact solution, in a rapidly convergent sequence with
perturbation method arc compared with numerical solution in Table 2. The
elegantly computed term.
explicit Rungc-Kutta method in symbolic computation package has been
References
used to find numerical solution of s(t), e(l), and i(t). As it can be seen in
Table 2, there exists a very good agreement between homotopy perturbation
method result and numerical solutions.
[I]
J. E. Cohen, Infectious diseases of hu1nans: dynamics and control, The Journal of
the American Medical Association 268 (1992), 3381-3392.
198
Jaharuddin
[21
E. Dcmirci, A. Unal and N. Ozalp, A fractional order SEIR model with density
dependent dca!h rate, Hacct J. Ma!h. Slat 40 (2011), 287-295.
{3)
A. Fercidoon, H. Yaghoobi and M. Davoudabadi, Application of the homotopy
perturbation method for solving the foam drainage equation, Int. J. Differ. Equ.
2011, Art. ID 864023, 13 pp. doi:I0.1155/2011/864023.
[4]
J. 1-1. lie, A coupling method of a homotopy technique and a perturbation
technique ICJr non-linear problerns, Internal. J. Non-Linear Mech. 35 (2000),
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[5]
J. II. He, Homotopy perturbation method: a ne\v nonlinear analytical technique,
Appl. Ma!h. Compu!. 135 (2003), 73-79.
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Engrg. 178(3-4) ( 1999), 257-262.
(7]
X. Li and B. Fang, Stability of an age-structured SEIR epidemic model with
infcc1ivi1y in la!en! period, Appl. Appl. Math. 4 (2009), 218-236.
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M. Y. Li and J. S. Muldowney, A geometric approach to global-stability
problems. SIAM J. Math. Anal. 27 (1996), 1070-1083.
[91
S. J. Liao, The proposed homotopy analysis technique for rhc solutions of
nonlinear problems, Ph.D. dissertation, Shanghai Jiao Tong University, China,
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[10]
M. Y. Li, J. R. Graef, L. C. Wang and J. Karsai, Global dynamics of a SEIR
model with varying total population size, Math. Biosci. 160 ( 1999), 191-213.
FAR EAST JOURNAL OF MATHEMATICAL SCIENCES (FJMS;
Editorial Board
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Principal Editor
Azad, K. K. (India)
...
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