Application of Differential Transform Method in ...

INTERNATIONAL JOURNAL OF MATHEMATICAL ANALYSIS AND OPTIMIZATION: THEORY AND APPLICATIONS VOL. 2017, PP. 250 - 260

Application of Differential Transform Method in Solving a Typhoid Fever Model Peter O. J1* and Ibrahim M. O 2 1,2. Department of Mathematics, University of Ilorin, Ilorin, Kwara State, Nigeria . *Corresponding author: [email protected]

Article Info Received: 27 November 2017 Accepted: 27 February 2018

Revised: 17 January 2018 Available online: 12 March 2018

Abstract In this paper, we present a deterministic model on the transmission dynamics of typhoid fever disease. Differential Transform Method (DTM) is employed to attempt the series solution of the model. The validity of the DTM in solving the model is established by classical fourth-order Runge-Kutta method implemented in Maple 18. The comparism between DTM solution and Runge-Kutta(RK4) were performed. The results obtained confirm the accuracy and potential of the DTM to cope with the analysis of modern epidemics.

Keywords: Typhoid Fever , Differential Transform Method, Runge-Kutta Method. MSC2010: 91A40

1

Introduction

Typhoid fever is one of the infectious diseases which is endemic in most part of the world. Typhoid fever is caused by the bacteria Salmonella Typhi. Typhoid fever infects 21 million people and kills 200,000 worldwide every year. Asymptomatic carriers are believed to play an essential role in the evolution and global transmission of Typhoid fever, and their presence greatly hinders the eradication of Typhoid fever using treatment and vaccination.[1]. Typhoid germs are passed in the faces and urine of infected people, people become infected after eating food or drinking beverages that have been handled by a person who is infected or by drinking water that has been contaminated by sewage containing the bacteria. Once the bacteria enters the body they travel in the human intestines, and then enter to the bloodstream.They enter to the blood through lymph nodes, gallbladder, spleen, liver etc. Abdominal pain, fever and general ill feeling are the symptoms of this disease. High fever (103 F, or 39.5 C) or higher and severe diarrhea occur as the disease gets worse. The incubation period is about 10-14 days, sometimes 3 days . It is endemic in Central America. [2], The Indian subcontinent.,[3] . Southeast Asia [4] and some parts of Africa. [5]. Treatment of typhoid is based on antibiotic susceptibility of the patient blood culture. The oral chloramphenicol, amoxicillin may be used if the strain is sensitive. The chronic carrier state may be eradicated using oral therapy using ciprooxacin or noroxacin. Multi-drug resistant strains of S.Typhi are increasingly common worldwide which makes treatment by antibiotics more difficult and costly. Typhoid symptoms vary widely and are very much similar to the symptoms of other microbial infections. Here are some of the common typhoid fever symptoms: Variable degrees of high grade fever in about 75% of cases, Muscle pains and body aches, Chills, Decreased appetite, Headaches, Nose bleeds, Pain in the abdomen in 20 to 40% of cases,

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INTERNATIONAL JOURNAL OF MATHEMATICAL ANALYSIS AND OPTIMIZATION: THEORY AND APPLICATIONS VOL. 2017, PP. 250 - 260 Dizziness, Rose spots (rashes) over the skin, Weakness and fatigue, Constipation or diarrhoea, sore throat and a cough. Lifshitz, [6]. In year 2000, it is estimated that the disease caused illness is 21.6 million and 216,500 deaths globally. [7]. Modelling the transmission dynamics of typhoid is an important and interesting topic for a lot of Computational mathematical researchers. The study of infectious diseases in the past has been focused mainly on their impact on the human population. Although infectious diseases are present in human populations at all times to some degree, the effects of epidemics are the most noticeable and spectacular [8]. . When dealing with large populations, as in the case of Typhoid fever, compartmental mathematical models are used. In the deterministic model, individuals in the population are assigned to different subgroups, each representing a specific stage of the epidemic. Several mathematical models have been developed on this disease [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] and [22]. In this paper, we will extend previous efforts by introducing a model that includes these effects. Education, Vaccination and Treatment as control strategies. The aim of this paper is to present the application of Differential Transform Method to the proposed model and to verify the validity of the Differential Transform Method in solving the model using Maple 18’s classical fourth-order Runge-Kutta method as a basis for comparisone. Differential Transform Method was first introduced by [24] to study the initial value problems in electrical circuit for a computation in linear and non-linear system. Differential transform method (DTM) is proved to be an excellent tool to investigate analytical and numerical solutions of nonlinear ordinary differential equations. The concept of differential transformation is briefly presented and some well-known properties of this DTM are rewritten in a more generalized forms. The Differential Transformation Method is one of the semi- analytical method commonly used for solving ordinary and partial differential equations in the forms of polynomials as approximations of exact solution [25]. Other semi- analytical methods includes: Homotopy Analysis Method (HAM), Homotopy Pertubation Method (HPM), Reduced Differential Transform Method (RDTM), Viarational Iterational Method (VIM), Parameter Expansion Method (PEM) .The methods mentioned above have been used as a tools to approximate linear and non-linear problems in Physics and Engineering . The main advantages of DTM over other techniques are, its provides desired accuracy and reliability to the series solution with remarkable convergence rate, it requires less size computational work and yield high accuracy [26] . In this study, we employ the (DTM) to the system of differential equations which describe our model and approximating the solutions in a sequence of time intervals. In other to illustrate the accuracy of the DTM, the obtained results are compared with fourth-order RungeKutta Method. 2 Model Formulation In this section, a deterministic, compartmental mathematical model to describe the transmission dynamics of typhoid fever is formulated to extend and complement the ones existing in literature. The model captures vaccination, education campaign and treatment as control parameters in order to study the impact of these control strategies on the dynamics of typhoid fever. The model consists of four compartments: The Susceptible class consists individuals that are susceptible to the disesase . The Infected class consists of individuals who have been infected with the disease are aware of their infection,and are capable of spreading the disease to those in the susceptible class. The Carrier class consists of individuals who are infected and capable of infecting others, but who do not shows any sign of infection.The Recovered class consists of individuals who have been infected and have subsequently recovered then recovered from the disease: those in this categorycannot be reinfected or transmit the infection to others. The number of individuals in the Susceptible, Infected, Carrier, and Recovered classes are functions of time denoted by S(t), I(t),Ic (t), R(t) respectively.

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INTERNATIONAL JOURNAL OF MATHEMATICAL ANALYSIS AND OPTIMIZATION: THEORY AND APPLICATIONS VOL. 2017, PP. 250 - 260 Individuals are recruited into the susceptible population by either immigration or birth at the rate  . We assume that proportion  of susceptible class progress to carrier class when infected, while the compliment 1   progress to infectious compartment. We assumed that the rate of transmission  for carriers is higher than the rate of transmission  of symptomatically infected individuals due to the fact that they are more likely to be unaware of their condition, and therefore continue with their regular activities. Carriers may become symptomatic at a rate  . Infectious individuals can receive treatment and recover at the rate  . Susceptible individuals receive vaccination to protect themself against infection at the rate  . 1 -  is an educational parameter that caters for limiting both carriers and symptomatic individuals from spreading typhoid. This parameter lies in the interval When  = 0 it means that no  education campaigns are in place so susceptible population are ignorant of typhoid fever and when  = 1 , then it means that all susceptible individuals are fully aware of typhoid fever, that is to say they know what causes the diseases, how it is spread and how to avoid contracting the disease. Detailed description of parameters is shown in Table 1 while the compartmental flow diagram of the model is shown by Figure 1. The system of equations governing the model is given as:

dS dt dIc dt dI dt dR dt  = I c  I

    =  S (1   )   2 I c   (1   ) I c   = (1   )S (1   )   (1   ) I c  ( 3   ) I    = S  I   4 R  =   1S  S (1   )  S

(1)

Substituting the value of force of infection in (1)

dS dt dIc dt dI dt dR dt

    =  S (  I c  I )(1   )   2 I c   (1   ) I c   = (1   )(1   ) S (  I c  I )   (1   ) I c  ( 3   ) I    = S  I   4 R  =   1S  S (  I c  I )(1   )  S

252

(2)

INTERNATIONAL JOURNAL OF MATHEMATICAL ANALYSIS AND OPTIMIZATION: THEORY AND APPLICATIONS VOL. 2017, PP. 250 - 260

Fig1: Pictorial Representation of the Model Table 1: Description of Variables and Parameters for Model Variables

S (t )

Description susceptible individuals at time

Ic(t )

carrier infectious individuals at time

I (t )

infectious individuals at time

R(t ) Parameters

t t

t recovered individuals at time t



Interpretation recruitment rate into susceptible class

1

natural death rate

2

natural rate for

3

natural death rate for I and disease induced death rate

4

natural death rate



     



Ic class and disease induced death rate

rate at which carriers develop symptom education parameter vaccination rate probability that newly infected individuals are asymptomatic or carrier transmission rate for carrier individual transmission rate for infectious individual force of infection recovery rate for infectious class

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INTERNATIONAL JOURNAL OF MATHEMATICAL ANALYSIS AND OPTIMIZATION: THEORY AND APPLICATIONS VOL. 2017, PP. 250 - 260

3

Differential Transform Method The process involved in DTM is as follows: Given an arbitrary function of

x,

suppose

y(x) is a non-linear funtion of x , then y(x) can be expanded in a Taylor series about a point x = 0 as 

y ( x ) = x k k =0

 1  dk  k y ( x) k!  dx  x=0

Thus,the differential Transform of y(x) is given as:

Y (k ) =

 1  dk  k y ( x) k!  dx  x=0

and the inverse differential Transform is given as 

y ( x ) = Y ( k ) x k k =0

gives some operational properties of DTM. In the table, c(x) and d(x) are arbitary functions with transforms C (k ) and D(k ) are the transformed functions respectively. Table 3

Table 2: Basic operation properties of the DTM S/No

Original Function

Transformed Function

1

y( x) = c( x)  d ( x)

Y (k ) = C(k )  D(k )

2

y( x) = c( x)

Y (k ) = C (k ),  is a constant

3 4

5

y ( x) =

dc( x) dx

Y (k ) = (k  1)C(k  1)

y ( x) =

d 2 c( x) dx 2

Y (k ) = (k  1)(k  2)C (k  2)

y ( x) =

d n c( x) dx n

Y (k ) = (k  1)(k  2)(k  n)C (k  n)

6 7

y(x) = 1

Y (k ) =  (k )

y ( x) = x

Y (k ) =  (k  1) ,  is the Kronecker delta

8

y( x) = e(x )

9

y( x) = c( x)d ( x)

10

y( x) = (1  x) n

Y (k ) =

k k!

Y (k ) = n =0D(n)C (k  n) k

Y (k ) =

254

n(n  1)(n  2)  (n  k  1) k!

INTERNATIONAL JOURNAL OF MATHEMATICAL ANALYSIS AND OPTIMIZATION: THEORY AND APPLICATIONS VOL. 2017, PP. 250 - 260

3.1 Solution of Typhoid Carrier Model In this section, we apply the steps involved in differential transform method to model (1) as follows: Using the operational properties (1), (2), (3) and (6) in Table 1 and applying them to the systyem of differential equations in (1) we obtain the following system of transformed equations below,

S (k  1) =

k 1 [ bk ,0  1   S l  I c k  l    I k  l   1   S k ] k 1 l =0

I c (k  1) = I (k  1) =

k 1 [  1   S l  I c k  l    I k  l   K1 I c k ] k 1 l =0

k 1 [1   1   S l  I c k  l    I k  l   K 2 I c k   K 3 M k ] k 1 l =0

R(k  1) = where

1 [ S k    I k    4 Rk ] k 1

K 2 =  (1   ), K1 = K 2  2 , K3 = 3  

Subject to the initial conditions S (0) = 60 , I c (0) = 40 , I (0) = 20 , R(0) = 10 . Using the initial conditions and the parameter values in the table and with the help of Maple 18, obtain the following expansion up to 5th order.

S (1) = 9.99931480  105 , S (2) = 5.709649910  105 , S (3) = 98122.20010, S (4) = 9.186204735  108 , S (5) = 6.704742246  108 

I c (1) = 4.60, I c (2) = 1.749891240  10 5 , I c (3) = 30915.19266, I c (4) = 4.593079842  10 8 , I c (5) = 3.322973420  10 8  I (1) = 10.40, I (2) = 1.749856100  105 , I (3) = 50162.88353, I (4) = 4.59315106222  108 , I (5) = 3.626130222 R(1) = 31.580, R(2) = 1.499913798  105 , R(3) = 20449.68858, R(4) = 1320.411710, R = 1.378007502  10 7  Hence, k

S (t ) = S (k )t k = 60  9.99931480  10 5 t  5.7096499  10 5 t 2  98122.20010t 3 n=0

 9.186204735  108 t 4  6704742246  108 t 5   k

I c (t ) = I (k )t k = 40  4.60t  1.749891240  10 5 t 2  30915.19266t 3 n=0

 4.593079842  108 t 4  3.626130222  108 t 5   k

I (t ) = R (k )t k = 20  10.40t  1.749856100  10 5 t 2  50162.88353t 3 n=0

 4.5931510622  108 t 4  3.626130222  108 t 5  

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5

R(t ) = R(k )t k = 10  31.580t  1.499913798t 2  20449.68858t 3 n=0

 1320.411710t 4  1.378007502  10 7 t 5  

4 Numerical Simulation and Graphical Illustration of the Model In this section,.we present the numerical simulation which demonstrate the analytical results for the proposed typhoid carrier individual in model (1). This is achieved by using the set of parameter values given in the Table which are derived from literature and well as assumptions . We considered the following initial conditions for the different compartments. S (0) = 60 ,

I c 0) = 40 , I (0) = 20 , R(0) = 10 . The DTM is demonstrated against Maple 18’s fourth order Runge-Kutta procedure for the solution of typhoid carrier individuals in Model (1). Fig (2) to (5) shows the combined plots of the solutions of S (t ) , I c (t ) I (t ) and R(t ) by DTM and RK4

Table 3: Parameters values for model

Parameter

Initial Value

Source Assumed

2

0.2

Assumed

1

0.3 0.142

3

0.2

Assumed

4

0.142

Mushayabasa, (2011)

0.3 0.5 0.02 0.01 0.75 0.3

Assumed







 

 



106

Mushayabasa, (2011)

Assumed Assumed Mushayabasa, (2011) Assumed Estimated Lauria et al.,(2009)

256

INTERNATIONAL JOURNAL OF MATHEMATICAL ANALYSIS AND OPTIMIZATION: THEORY AND APPLICATIONS VOL. 2017, PP. 250 - 260

Fig2: Solution of Susceptible Population by DTM and RK4

Fig3: Solution of Carrier Population by DTM and RK4

Fig4: Solution of Infected Population by DTM and RK4

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INTERNATIONAL JOURNAL OF MATHEMATICAL ANALYSIS AND OPTIMIZATION: THEORY AND APPLICATIONS VOL. 2017, PP. 250 - 260

Fig5: Solution of Recovered Population by DTM and RK4

5 Discussion Method

of

Results

for

Differential

Transform

The solutions obtained by using Differential Transform Method with given initial conditions compared favourably with the solution obtained by using classical fouth-other Runge-Kuta method. The solutions of the two methods follows the same pattern and behaviour. This shows that Differential Transform Method is suitable and efficient to conduct the analysis of typhoid models.

6

Conclusion

We present three deterministic models on the transmission dynamics of typhoid fever disease. We tested for the existence and uniqueness of solution for the model using the Lipchitz condition to ascertain the efficacy of the models and proceeded to determine both the disease free equilibrium (DFE) and the endemic equilibrium (EE) for the system of the equations. Differential Transform Method (DTM) is employed to attempt the series solution of the model. Numerical simulations were carried out to compare the results obtained by Differential Transform Method with the result of classical fourth-order Runge-Kutta method. The results of the simulations were displayed graphically.The results obtained confirm the accuracy and potential of the DTM to cope with the analysis of modern epidemics.

Conflict of Interest The authors declare that there are no conflicts of interest regarding the publication of the paper.

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