A New Modification Of The Differential Transform Method For A Sirc Influenza Model

A New Modification Of The Differential Transform Method For A Sirc Influenza Model

A New Modification Of The Differential Transform Method For A Sirc Influenza Model.

S. F. M. Ibrahim, Soad Moftah Ismail

1. Ain Shams University, Faculty of Education, Department of Mathematics, Cairo, Egypt.

2. Misr University For Science &Technology Faculty of Engineering, Dept. of Basic science, 6Th of October City, Egypt.

In this paper, approximate analytical solution of SIRC model associated with the evolution of influenza A disease in human population is acquired by the modified differential transform method (MDTM). The differential transform method (DTM) is mentioned in summary. MDTM can be obtained from DTM applied to Laplace, inverse Laplace transform and padé approximant. The MDTM is used to increase the accuracy and accelerate the convergence rate of truncated series solution getting by the DTM. We use this analytical-numerical technique in order to produce simulations with different initial conditions, parameter values for different values of the basic reproduction number.

Key words: SIRC model, Epidemic models, Modified Differential transformation method, padé approximants.

Influenza is caused by a virus that attacks mainly the upper respiratory tract, nose, throat and bronchi and rarely also the lungs. Most people recover within 1-2 weeks without requiring any medical treatment. In the very young, the elderly and people suffering from medical conditions such as lung diseases, diabetes, cancer, kidney or heart problems, influenza poses a serious risk. In these people, the infection may lead to severe complications of underlying diseases, pneumonia and death. In annual influenza epidemics 5-15% of the population are affected with upper respiratory tract infections. Hospitalization and deaths mainly occur in high risk groups (elderly, chronically ill). Annual epidemics are possibly between three and five million cases of severe illness and between 250,000 and 500,000 deaths every year around the world. Influenza is transmitted by a virus that can be of three different types, namely A, B and C [1]. Among these taypes, the virus A is epidemiologically the most important one for human beings, because it can recombine its genes with those of strains circulating in animal

populations such as birds, swine, horses etc . Unfortunately, within type A virus, there are several subtypes, H1N1, H3N2, H5N1, etc., each one of these has been pointed as the causal of recent pandemics. Much evidence shows that the antigenic distance between two different strains influences the degree of partial immunity, often called cross-immunity, conferred to a host already infected by one of the strains with respect to the other[2]. Mathematical models have proven to be useful tools to study the dynamics of viral infections, within these models, we encounter compartmental models of ordinary or partial differential equations. When we incorporate multiple strains into these models, the mathematical analysis becomes difficult. Over the last two decades, a number of epidemic models for predicting the spread of influenza through human population have been proposed based on either the classical susceptible-infected-removed (SIR) model developed by Kermack and McKendrick[3]. Casagrandi et al.[2] have introduced SIRC model by adding a new compartment C, which can be called cross-immune compartment, to the SIR model. This cross-immune compartment (C) describes an intermediate state between the fully susceptible (S) and the fully protected (R)one. They have studied the dynamical behaviors of this model numerically [4]. Jodar et al. [5] developed two nonstandard finite difference schemes to obtain numerical solutions of a influenza A disease model presented by Casagrandi et al.[2] . Very recently Samanta[4] considered a nonautonomous SIRC epidemic model for Influenza A with varying total population size and distributed time delay. This model assumes no immune interference between the different A virus subtypes, that is why they only considered one virus subtype. In this chapter, we apply The modified differential transform method (MDTM) will be employed in a straightforward manner without any need of linearization or smallness assumptions. DTM was first applied in the engineering domain by [6]. DTM provides an efficient explicit and numerical solution with high accuracy, minimal calculations, sparing of physically unrealistic assumptions. However, DTM

has some drawbacks. By using DTM, we obtain a series solution, in practice a truncated series solution. This series solution does not exhibit the periodic behavior which is characteristic of oscillator equations and gives a good approximation to the true solution in a very small region. In order to develop the accuracy of DTM, we use an alternative technique which modifies the series solution for non-linear oscillatory systems as follows: we first apply the Laplace transformation to the truncated series obtained by DTM, then convert the transformed series into a meromorphic function by forming its Padé approximants([7],[8],[9],[10],[11]), and finally accept an inverse Laplace transform to obtain an analytic solution, which may be periodic or a better approximation solution than the DTM truncated

some techniques exist to increase the convergence of a given series. Among them, the so- called padé technique is widely applied In this section we introduce the notion of rational approximations for functions. The function f(x) will be approximated over a small portion of its domain. For example, if f(x)=cos(x), it is sufficient to have a formula to generate approximations on the interval [0,/2]. Then trigonometric identities can be used to compute cos(x) for any value x that lies outside [0,/2]. A rational approximation to f(x) on [a,b] is the quotient of two

polynomials PN x and QM x of degrees N

and M, respectively. We use the notation [N/M] (x) to denote this quotient:

series solution.

N / M x

PN x

QM x

for a x

b.

(5)

Casagrandi et al. [2] considered the model

dS 1 S C SI ,

dt (1)

dI SI SI I ,

dt (2)

Our goal is to make the maximum error as small as possible. For a given amount of computational effort, one can usually construct a rational approximation that has a smaller overall error on [a,b] than a polynomial approximation. Our development is an introduction and will be limited to Padé approximations. The method of Padé

dR 1 CI I R ,

dt

dC R CI C ,

dt

(3)

(4)

requires that f(x) and its derivative be continuous at x=0. There are two reasons for the arbitrary choice of x=0. First, it makes the manipulations simpler. Second, a change of variable can be used to shift the calculations over to an interval that contains

With initial conditions

zero. The polynomials used in Eq. (5) are

2 N

2 N

S 0 M , I 0 M

, R 0 M

,C 0 M . N

(6)

PN x p0 p1x p2x … p x

PN x p0 p1x p2x … p x

where

1 2 3 4

And

Q x 1 q x q x 2 … q x M

(7)

is the mortality rate,

is the rate of progression from infective to recovered per year,

is the rate of progression from recovered to cross-immune per year,

is the rate of progression from recovered to susceptible per year,

o is the recruitment rate of cross-immune into the infective,

is the contct rate per year.

The disease free equilibrium is locally

M 1 2 M

The polynomials in (6) and (7) are constructed so that f(x) and [N/M] (x) agree at x=0 and their derivatives up to N+M agree at x=0. In the case Q(x) =1, the approximation is just the Maclaurin expansion for f(x). For a fixed value of N+M the error is smallest when PN x and QM x have

the same degree or when PN x has degree one higher than QM x . Notice that the constant

asymptotically stable if and only if

1

and

coefficient of QM is q =1. This is permissible,

because it cannot be 0 and [N/M] (x) is not changed

.

.

unstable if

1

when both PN x and QM x are divided by

There exists a unique and positive endemic equilibrium point if and only if (/(+))>1 which is locally asymptotically stable under some conditions on the coefficients[12]

the same constant. Hence the rational function [N/M](x) has N+M+1 unknown coefficients. Assume that f(x) is analytic and has the Maclaurin expansion

f x a a x a x 2 … a x k …,

(8)

introduced it in a study of electrical circuits.

0 1 2 k

And form the difference

f x QM x PN x z x :

Additionally, differential transformation has been applied to solve a variety of problems that are modeled with differential equations

M N

a x j q x j

p x j

c x j (9)

([14],[15],[16],[17])

j

j

j

j

j 0

j 0

j 0

j N M 1

The method consists of, given system of

The lower index j=M+N+1 in the summation on the right side of (9) is chosen because the first N+M derivatives of f(x) and [N/M](x) are to agree at x=0. When the left side of (9) is multiplied out and the coefficients of the powers of x j are set equal to zero for k= 0,1,…,N+M, the result is a system of N+M+1 linear equations:

a0 p0 0

q1a0 a1 p1 0

q a q a a p 0

differential equations and related initial conditions; these are transformed into a system of recurrence equations that finally leads to a system of algebraic equations whose solutions are the coefficients of a power series solution.

For the sake of clarity in the presentation of the DTM and in order to help to the reader we summarize the main issues of the method that may

be found in [6].

2 0 1 1 2 2

q a q a q a a p 0

10

Definition 4.1 A differential transformation Y(k)

3 0 2 1 1 2 3 3

qM aN M qM 1aN M 1 … aN pN 0

And

of function y (x) is defined as follows [18]

(12)

qM aN M 1 qM 1aN M 2 … q1aN aN 1 0

qM aN M 2 qM 1aN M 3 … q1aN 1 aN 2 0

qM aN qM 1aN 1 … q1aN M 1 aN M 0

11

In (12), y(x) is the Original function and Y(k) is the transformed function. Differential inverse transform of Y(k) is defined as follows

(13)

Notice that in each equation the sum of the subscripts on the factors of each product is the same, and this sum increases consecutively from 0 to N+M. The M equations in (11) involve only the unknowns q,q,…,qM and must be solved first.

In fact. From (12) and (13), we obtain

(14)

Then the equations in (10) are used successively to

find p,p,…,pN.

1. Pukhov [13] proposed the concept of differential transformation, where the image of a transformed function is computed by differential operations, which is different from the traditional integral transforms as are Laplace and Fourier. Thus, this method becomes a numerical-analytic technique that formalizes the Taylor series in a totally different manner. Differential transformation is a computational method that can be used to solve linear (or non-linear) ordinary (or partial) differential equations with their corresponding boundary conditions. A pioneer using this method to solve initial value problems is Zhou [6] , who

   Equation (14) implies that the concept of differential transformation is derived from the Taylor series expansion.

   From Equation (12) and (13), it is easy to obtain the following mathematical operations:

   1. If then

      .

   2. If then is a constant.

   3. If ,then

      .

   4. If If then

      .

   5. If then

      Y , is the

      Kronecker delta .

   6. If then

      .

      Author names and affiliations are to be centered beneath the title and printed in Times 12-point, non-boldface type. Multiple authors may be shown in a two- or three-column format, with their affiliations italicized and centered below their respective names. Include e-mail addresses if possible. Author information should be followed by two 12-point blank lines.

      bottom margin should be 1-1/8 inches (2.86 cm) from the bottom edge of the page for 8.5 x 11-inch paper; for A4 paper, approximately 1-5/8 inches (4.13 cm) from the bottom edge of the page.

2. In this section, the differential transformation technique is applied to solve several nonlinear differential equations system.

   By using the fundamental operations of differential transformation method. We obtained the following recurrence relation to the SIRC influenza model Eq.s (1)-(4) and initial conditions

   1 k

   (18)

   S k 1

   k S k S l I k l C k

   k 1

   l 0

   I k 1 1 k S l I k l k C l I k l I k (19)

   If x (t) and y (t) are two uncorrelated

   k 1

   l 0

   l 0

   functions with time t where X (k) and Y(K) are the

   R k 1 1 1 k C l I k l I k R k (20)

   transformed functions corresponding to x(t) and

   k 1

   l 0

   y(t) then we can easily proof the fundamental

   C k 1 1 R k k C l I k l C k (21)

   mathematics operations performed by differential

   k 1

   l 0

   Transformation and are listed as follows [16]:

   1. Linearity. If X(k) = D[x(t)] , Y(k) = D[y (t)] and

      and are independent of t and k then

      with S 0 M 1, I 0 M 2 , R 0 M 3 ,C 0 M 4.

      where

      1 d k S (t )

      (22)

      Sk

      k !

      dt k ,

      (15) t 0

      1 d k I (t )

      Thus, if c is a constant, Then

      I k

      k !

      dt k ,

      D[c ] c k , where is the kronecer delta function.

   2. Convolution. if

   t 0

   1 d k R (t )

   R k

   k !

   dt k

   t 0

   1 d k C (t )

   C k

   k !

   dt k

   and denote the convolution and Symbol D

   denoting the differential transformation process. Then

   t 0

   Are differential transform of

   S t , I t , R t ,C t

   (16)

   If then

   respectively.

   Thus, from a process of inverse differential transformation, it can be obtained the solutions in the power series

   n

   n

   S t S k t k ,

   k 0

   n

   (17)

   The proof of above properties is deduced from the definition of the differential trans

   The second and following pages should begin 1.0 inch (2.54 cm) from the top edge. On all pages, the

   I t I k t k ,

   k 0

   n

   n

   R t C k t k ,

   k 0

   n

   n

   C t C k t k ,

   k 0

   Therefore

   (23)

   S t M 1 M 1M 2 M 4 1 M 1 t

   M=0.8, M=0.1, M=0.04, M=0.06.

   These values correspond to table 2 in [12]. By differential transform method we have

   M 1M 2 M 4 1 M 1

   1 2/p>

   M 2M 4 M 3 M 4 t

   S 0 0.8, I 0 0.1, R 0 0.04, C0 0.06.

   2

   M 2 M 1M 2 M 4 1 M 1

   1 1 2 2 2 4

   1 1 2 2 2 4

   M M M M M M

   .And from equations (18)-(21). We have the closed form of the solution can be easily written as :

   I t M 2 M 1M 2 M 2 M 2M 4 t

   S t S k t k 0.8 3.966t 75.6219t 2 929.833t 3

   k 0

   7823.46t 4 34261.t 5 258727t 6 +8.4739 106 t 7

   (24)

   M 1M 2 M 2 M 2M 4

   1.22332 108t 8 1.19552 109 t 9 …

   M M M M 1 M k 2 3

   1

   2 1 2 4 1

   t 2

   I t I k t

   0.1 3.287t 44.0704t

   137.968t

   2 M M M M M M

   k 0

   1 1 2 2 2 4

   M 2 M 2M 4 M 3 M 4

   5355.53t 4 112877.t 5 1.1166 106 t 6 +3.15402 106 t 7

   9.4118107 t 8 1.96714 109t 9 …

   M M M M M M k 2

   4 1 2 2 2 4

   R t R k t

   k 0

   0.04 7.5442t 129.199t

   R t M M M M M 1 t

   1188.38t 3 3585.91t 4 71322.8t 5 1.3641106t 6

   3 2 3 2 4

   1.22489 107 t 7 4.09274 107 t 8 6.14701108t 9 …

   C t C k t k 0.06 0.2912t 9.50631t 2

   M 2 M 3 M 2M 4 1

   k 0

   1 2

   120.581t 3 1117.98t 4 7292.85t 5 11224.9t 6

   M 1M 2 M 2 M 2M 4 t

   2

   621014.t 7 1.27138107 t 8 1.56927 108t 9 …

   M M M M M

   2 2 4 3 4

   In this section, we apply Laplace transformation to

   1 M

   M M M M M

   (24), which yields

   4 1 2 2 2 4

   0.8 3.966 151.244 5579. 187763. 4.11132106

   L S t 2 3 4 5 6

   s s s s s s

   C t M 4 M 2 M 4 M 3 M 4 t

   186283440 4.270851010 4.932431012 4.33831014

   s

   s

   7

   s 8 s 9

   s 10

   …

   M 2 M 4 M 3 M 4

   0.1 3.287 88.1408 827.808 128533 1.35452107

   1 2

   L I t 2 3 4 5 6

   (25)

   M 2 M 3 M 2 M 4 1 t

   s s s s s s

   2

   8.03952 108

   1.589631010

   3.794841012

   7.138361014

   M 2 M 2 M 4

   M 3 M 4

   …

   M

   M M M M M

   s 7 s 8 s 9 s 10

   4 1 2 2 2 4

   0.04 7.5442 258.398 7130.28 86061.8 8.55874106

   L R t 2 3

   4 5 6

   s s s s s s

   9.82152108 6.173451010 1.650191012 2.230631014

s 7 s 8

s 9 s 10

…

In this section, we present the numerical results

L C t 0.06 0.2912 19.0126 723.486 26831.5 875142.

based on the application of the (MDTM) to SIRC

s s 2 s 3 s 4

s 5 s 6

8.08193106 3.12991109 5.12621011 5.69457 1013

influenza model. Since most of the non-linear differential equations do not have exact analytic

s 7 s 8 s 9 s 10

…

solutions, so approximation and numerical techniques must be used.

6.1 Disease free equilibrium

(R=(/(+))<1)

For numerical study, (for R<1) we use the following parameters:

1/ 50y 1, 73y 1, 1y 1

0.5y 1, 50, 0.05

This was done with the standard parameter values given above and initial values

For simplicity, replacing s= (1/t)

L S t 0.8t 3.966t 2 151.244t 3

5579.t 4 187763.t 5 4.11132 106t 6

186283440t 7 4.270851010 t 8

0.00035

0.00030

Infected population

Infected population

0.00025

4.932431012 t 9

4.33831014 t 10 …

0.00020

L I t 0.1t 3.287t 2 88.1408t 3

827.808t 4 128533t 5

1.35452 107 t 6 8.03952 108t 7

1.589631010 t 8 3.79484 1012t 9 4.33831014t 10 …

L R t 0.04t 7.5442t 2 258.398t 3

0.00015

0.00010

0.00005

0.00000

0.0 0.2 0.4 0.6 0.8 1.0

t time

7130.28t 4 86061.8t 5 8.55874 106t 6

9.82152 108t 7 6.173451010 t 8

1.65019 1012 t 9 2.230631014 t 10 …

L C t 0.06t 0.2912t 2 19.0126t 3

723.486t 4 26831.5t 5 875142.t 6

8.08193106 t 7 3.12991109 t 8

5.1262 1011t 9 5.69457 1013t 10 …

padé approximant [4/4] of (26) and substituting

1

(26)

Figure2:I(t)for =0.02,=50,=1,=0.5,=0.05,=73]

0.04

population Recovered

population Recovered

0.03

0.02

0.01

0.00

0 2 4 6 8 10 12 14

t , we obtain [4/4] in terms of S.

s

By using the inverse Laplace transformation, we obtain

t time

Figure3:R(t)for=0.02,=50,=1,=0.5,=0.05,=73]

0.14

Cross immunepopulation

Cross immunepopulation

0.12

38.0568t

0.0512041t

104.54730.8174i t 0.0000353126 0.000130581i

S t 0.10517e

0.694759e e

0.0000353126 0.000130581i e 61.6348it

0.10

I t 3.95371107 e 221.928t 0.00371616e 99.6621t 0.0172969e 80.5991t

0.113581e 37.9539t

0.08

0.06

37.6893t

1.03203t

86.93738.04011i t 0.00573314 0.00574379i

0.04

R t 0.23128e

0.259813e e

0.00573314 0.00574379i e16.0802it

0.02

C t 7.1784106e 132.238t 0.0119018e 39.554t 0.0483217e 3.83114t 0.000216349e 30.0015t

0.00

0 2 4 6 8 10 12 14

t time

1.00 Figure4:C(t)for=0.02,=50,=1,=0.5,=0.05,=73]

0.95

population susceptible

population susceptible

6.2 Endemic equilibrium R0 1

0.90

0.85

0.80

0.75

0.70

0 5 10 15

t time

For numerical study, (for R>1) we use the following parameters:

1/ 50y 1, 73y 1, 1y 1

0.5y 1, 100, 0.05

This was done with the standard parameter values given above and initial values

Figure1: S(t)for =0.02,=50,=1,=0.5,=0.05,=73]

M=0.8, M=0.1, M=0.04, M=0.06.

These values correspond to table 2 in [12]. For the four- component model. An approximation for S(t), I(t),R(t),C(t), the solution can be easily written as:

S t S k t k 0.8 7.966t 10.6419t 2

k 0

1153.98t 3 5541.77t 4 199026t 5

1.80856 106 t 6 3.30018107 t 7

4.88965108t 8 4.91921109t 9 …

I t I k t k 0.1 0.728t 37.3279t 2

k 0

248.328t 3 10102.4t 4 51998.1t 5 2.44897 106 t 6

6 7 8 8 8 9

(27)

L S t 0.8t 7.966t 2 21.2838t 3

6923.88t 4 133002t 5 23883120t 6

1.30216 109 t 7 1.66329 1011t 8

1.971511013t 9 1.785081015t 10 …

L I t 0.1t 0.728t 2 74.6558t 3

1489.97t 4 242458t 5 6.23977 106 t 6

1.76326 109 t 7 3.7946 1010 t 8

7.52897 10 t 5.9559 10 t 3.87404 10 t …

2.25614 1013t 9 1.405811014 t 10 …

R t R k t k 0.04 7.8292t 21.8457t 2

k 0

984.964t 3 3838.52t 4 158877.t 5 467477.t 6

L R t 0.04t 7.8292t 2 43.6914t 3

4 5 7 6

2.71389 107 t 7 3.00545107 t 8 4.71775109 t 9 …

5909.78t 92124.5t 1.90652 10 t (29)

C t C k t k 0.06 0.5912t 4.84031t 2

k 0

79.3108t 3 722.135t 4 11848.7t 5 172935.t 6

1.66611106 t 7 4.05398107 t 8 1.85948108t 9 …

In this section, we apply Laplace transformation to (27),which yields

L S t 0.8 7.966 21.2838 6923.88 133002

336583440t 7 1.36781011t 8

1.21181012 t 9 1.711981015t 10 …

L C t 0.06t 0.5912t 2 9.68062t 3

475.865t 4 17331.2t 5 1.42184 106t 6

1.24513108t 7 8.39719 109 t 8

1.63456 1012 t 9 6.747681013t 10 …

padé approximant [4/4] of (29) and substituting

1

s s 2

s 3 s 4 s 5

t , we obtain [4/4] in terms of S.

23883120 1.30216 109 1.66329 1011 s

s 6 s 7 s 8

By using the inverse Laplace transformation, we obtain

1.971511013 1.785081015

s 9 s 10

…

L I t 0.1 0.728 74.6558 1489.97 242458

s s 2 s 3

s 4 s 5

6.23977 106 1.76326 109 3.7946 1010

s 6 s 7 s 8

(28)

2.25614 1013 1.405811014

s 9 s 10

…

L R t 0.04 7.8292 43.6914 5909.78 92124.5

s s 2 s 3

s 4 s 5

1.90652 107 336583440 1.36781011 1.21181012

s 6

1.711981015

s 10

s 7 s 8 s 9

…

L C t 0.06 0.5912 9.68062 475.865 17331.2

s s 2 s 3

s 4 s 5

1.42184 106 1.24513108 8.39719 109

s 6 s 7 s 8

1.63456 1012 6.747681013

s 9 s 10

…

For simplicity, replacing s= (1/t)

S t 1.04173106e 1964.37t

0.777741e 8.17002t e 17.407659.9515i t

0.25

population Rcovered

population Rcovered

0.20

0.15

0.011129 0.010211i

0.10

0.011129 0.010211i e119.903it

I t e 2.0881688.2489i t

0.05

0.00

0 5 10 15 20 25 30 35

t time

0.00176682 0.000752284i

Figure7:R(t)for =0.02,=100,=1,=0.5,=0.05,=73]

0.00176682 0.000752284i e176.498it

0.22

Cross immunepopulation

Cross immunepopulation

0.20

0.0482332

0.00613219i

0.00613219i

e 0.22.6662i t

0.0482332

e 3.36506t

0.18

0.16

0.14

0.00613219i

e 3.3650645.3325i t

0.12

0 5 10 15 20 25 30 35

R t e 1.1909787.9418i t

0.000457939 0.00163777i

0.000457939 0.00163777i e175.884it

e 0.22.5497

0.0204579 0.163655i e 3.89398t

0.0204579 0.163655i e 3.8939845.0995i t

C t 0.0608182e 8.10172t 0.000185688e 70.5152t

e 2.5188564.6573i t

0.000501921 0.000882289i

0.000501921 0.000882289i e129.315it

t time

Figure8:C(t)for=0.02,=100,=1,=0.5,=0.05,=73]

In this chapter, differential transorm method was used for finding the solutions of nonlinear ordinary differential equation systems such as SIRC dynamical model. We demonstrated the accuracy and efficiency of these methods by solving some ordinary differential equation systems. We use Laplace transformation and padé approximant to obtain an analytic solution and to improve the accuracy of differential transorm method.

The computations and graphs associated with the example in this chapter were performed using Mathematica ver.8.

1. P. Palese, J. Young, Variation of influenza A,

   0.85

   population susceptible

   population susceptible

   0.80

   0.75

   0.70

   0.65

   0.60

   0 5 10 15 20 25 30 35

   t time

   B, and C viruses. Science 215 (1982)1468-1474.

2. R. Casagrandi, L. Bolzoni, S.A. Levin, V. Andreasen, The SIRC model and influenza A, Math. Biosci. 200 (2006) 152-169.

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   0.002

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   t time

   Figure6:I(t)for =0.02,=100,=1,=0.5,=0.05,=73]

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