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An Analytical Approaches to Solve a Lung Infection Model 

  

J. Jothibasu 𝟏,  C. Murugesan 𝟐, G. Tamil Preethi 𝟑,∗ And N. Magesh 𝟒. 

Post-Graduate and Research Department of Mathematics,  Government Arts College for Men, Krishnagiri 635001, 

Tamilnadu, India. 

e-mail:  1jbvnb22@gmail.com,  2 cmurugesan27@gmail.com, 

 3,∗ tamilpreethig@gmail.com and   4 nmagi_2000@yahoo.co.in 

 3,∗ORCID Address: https://orcid.org/0000-0003-3453-0029. 

 4ORCID Address: https://orcid.org/0000-0002-0764-8390. 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  

Acute respiratory infections cause morality in young childhood. We assessed the effects 

of water(𝑊), sanitation(𝑆), hand hygiene(𝐻) and nutritional(𝑁) interventions on 

childhood ARI. Geographic cluster of kids from rural and urban places are taken a 

random survey of WSHN. Acute respiratory illness was defined as caregiver-reported 

persistent cough, panting, wheezing or difficult in breathing. The main limitations of this 

study include low adherence to some of the targeted behaviours, reliance on caregiver-

reported outcomes (excluding visible runny nose), subjected to bias, the data are 

collected from World Health Organization- defined pneumonia. The idea of our study is 

to make the people understand the importance of cleanliness by the dynamical behaviour 

of infections. Media plays a vital role in improving the urgency of habitual change in 

day-to-day life. 

2020 Mathematics Subject Classifications: 65L06, 92D25, 92D30. 

Keywords: Lung infection, Differential transformation method, Pade approximation. 

 

1. Introduction 

Many people associate pneumonia with the elderly but it is actually the biggest it claims the lives of 

more than 6 million humans, under the age of 10 including new born and adolescence who are 

particularly vulnerable to infection. Pneumonia may be caused by bacteria, viruses or fungus. One or 

both of the lungs might become infected with pneumonia. It makes the lungs air sacs or alveoli swell 

with fluid or pus. An infection of the lungs called pneumonia. An infection of the lower respiratory 

tract is another name for it. In children three years of age and below, viruses are the main cause of 

pneumonia. Most pneumonia instances in older kids and teens are brought on by bacterial infections. 

A youngster may get a viral pneumonia at first, which might later develop into a bacterial pneumonia. 

Numerous methods exist for spreading pneumonia. When breathed, the common viruses and bacteria 

in a child’s nose or throat can infect the lungs. Additionally, airborne droplets from a cough or sneeze 

might transmit them. Additionally, particularly during and right after childbirth, pneumonia can spread 

through blood. The many pneumonia-causing bacteria and the modes of transmission require more 

study since they are crucial for both therapy and prevention. Every day at least one child dies every 45 



Communications on Applied Nonlinear Analysis 

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seconds from pneumonia where almost all of these deaths are preventable. Lung infection caused by 

bacteria is easily preventable with vaccine. However, more than 50% of children around the world are 

not fully protected with primary vaccine to prevent pneumonia - the Pneumococcal (PCV) vaccine 

[20]. A new vaccine for one of the main viral causes of pneumonia is under development. In adults, 

Pneumonia can show signs and symptoms include a cough that produces green, yellow or even red 

mucus, fever, perspiration chills that shake respiration difficulty and quick breathing. Chest discomfort 

that is piercing or stabbing and that worsens when you cough or breathe deeply poor energy, weariness 

and loss of appetite [5]. The existing model has significant drawbacks, despite the fact that our Inf-Net 

segmented infected areas with promising results. The Inf-Net first concentrates on COVID-19 patient 

lung infection segmentation. On the other hand, in clinical practice, it frequently necessitates 

classifying COVID-19 patients before segmenting the infection zones for further therapy. The 

quantitative response caused by the deposition of a test substance could be indicated in the various 

parts of the lung model surface if test bacteria with a suitable bio luminescent reporter system were 

used to seed the agar coating the lung model surface, this would allow local deposition levels to be 

precisely determined [9]. 

Now-a-days media takes vital part in creating awareness to public through health care units. Treating 

pneumonia requires health workers to be with easy reach of families and to have the right training in 

medicine handling and diagnostic tools. Here, media helps to reach each and every family in all part 

of the world by giving precautions about the virus or ongoing pandemic, preventing measures and most 

importantly about vaccination.  

2. Mathematical Modeling  

This chapter examines the SIVMR (Susceptible-Infected-Vaccinated-Media Awareness-Recovery) 

lung infection using DTM, CFDTM and LPDTM methods. Utilizing a variety of examples [4, 7, 8, 

10–15, 17–19], the mathematical model is framed with five compartments of susceptible, infected, 

media awareness, vaccinated and recovery, whereas the solutions has been provided. 

𝑠′(𝑡) = 𝛼 − 𝛽 𝑠(𝑡) 𝑖(𝑡) − 𝑚 𝑠(𝑡)𝑚(𝑡) − 𝜇 𝑠(𝑡) − 𝜙 𝑠(𝑡)                                

𝑣′(𝑡) = 𝜙 𝑠(𝑡) − (𝑑 + 𝜇) 𝑣(𝑡)                                                                                 

                 𝑖′(𝑡) = 𝛽 𝑠(𝑡) 𝑖(𝑡) − (𝜇 + 𝜈 + 𝜎) 𝑖(𝑡)                                                                           (1) 

𝑟′(𝑡) =  𝑖(𝑡) − 𝜇 𝑟(𝑡)                                                                                             

𝑚′(𝑡) = 𝑚 𝑠(𝑡) 𝑚(𝑡) − 𝜇 𝑖(𝑡).                                                                                  

To the series solution with LPDTM, we employed DTM for equation (1) with the initial conditions 

𝑆(0) =  𝑆0  ≥  0, 𝑉(0) =  𝑉0 ≥  0,   𝐼(0) = 𝐼0  ≥  0,   𝑅(0)  = 𝑅0  ≥  0 and 𝑀(0) = 𝑀0 ≥ 0, to 

obtain the recursion relation 

𝑆(𝑝 + 1) =
1

(𝑝 + 1)
[𝛼 𝛿(𝑝) − 𝛽 ∑ 𝑆(𝑝 − 𝑟) 𝐼(𝑟) − 𝜇 𝑆(𝑝) − 𝜙 𝑆(𝑝) − 𝑚 ∑ 𝑆(𝑝 − 𝑟)𝑀(𝑟)

𝑝

𝑟=0

𝑝

𝑟=0

] 

𝑉(𝑝 + 1) =
1

(𝑝 + 1)
[𝜙 𝑆(𝑝) − (𝑑 + 𝜇) 𝑉(𝑝)]                                                                  



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            𝐼(𝑝 + 1) =
1

(𝑝 + 1)
[𝛽 ∑ 𝑆(𝑝 − 𝑟) 𝐼(𝑟) − (𝜇 + 𝜈 + 𝜎) 𝐼(𝑝)

𝑝

𝑟=0

]                                      (2) 

𝑅(𝑝 + 1) =
1

(𝑝 + 1)
[𝜈 𝐼(𝑝) − 𝜇 𝑅(𝑝)]                                                                              

𝑀(𝑝 + 1) =
1

(𝑝 + 1)
[𝑚 ∑ 𝑆(𝑝 − 𝑟) 𝑀(𝑟) − 𝜇 𝐼(𝑝)

𝑝

𝑟=0

]                                                   

Initial Conditions: 

                                 {
𝑆(0) = 10000,       𝑉(0) = 200,       𝐼(0) = 200
𝑅(0) = 20,              𝑀(0) = 200.                             

                                            (3) 

 

Table 1. Parameter values and its descriptions 

𝛼 Recruitment rate 271.23 

𝛽 Force of infection 0.10 

𝑚 Rate of publicity od awareness program 0.04 

𝜙 Rate at which susceptible are vaccinated 0.9 

𝑑 Disease induced death rate 0.004 

𝜇 Natural death rate 0.18 

𝜎 Rate of news collection from the infectives 0.025 

 

3. Differential Transform Method and its Properties 

The fundamental concepts and operational characteristics of the differential transformation technique 

were covered in this section. A predetermined system of differential equations and associated 

beginning conditions make up this methodology. These are converted into a set of recurrence 

equations, which in turn become a set of algebraic equations whose solutions are the coefficients of a 

power series solution. 

The following is the definition of a function’s differential transformation: 

                                                 𝑍(𝑙) =
1

𝑙 !
[
𝑑 𝑙𝑧(𝑡)

𝑑𝑡  𝑙
]

𝑡=𝑡0

                                                                        (4) 

In equation (4), 𝑍(𝑙) is the transformed function, sometimes known as the 𝑇 −function, while 

𝑧(𝑡) is the original function. The definition of a differential inverse transform for 𝑍(𝑙) is: 



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                                                  𝑧(𝑡) = ∑ 𝑍(𝑙)(𝑡 − 𝑡0)𝑙

∞

𝑙=0

                                                                    (5) 

From equation (4) and (5), we obtain 

                                                   𝑧(𝑡) = ∑
(𝑡 − 𝑡0)𝑙

𝑙!

∞

𝑙=0

[
𝑑 𝑙𝑧(𝑡)

𝑑𝑡  𝑙
]

𝑡=𝑡0

                                                   (6) 

Equation (6) suggests that the Taylor series expansion is where the idea of a differential transform 

comes from and that relative derivatives are determined in an iterative manner as indicated by the 

altered equations of the original functions. Finite series are used to express the function 𝑧(𝑡), then (5) 

can be written 

                                                   𝑧(𝑡) ≈ ∑ 𝑍(𝑙) (𝑡 − 𝑡0)𝑙

∞

𝑙=0

                                                                  (7) 

with 𝑁 is a convergence of natural frequency. For the properties and its applications one can refer [1, 

3, 6, 16, 23, 24]. 

4. Pade Approximation 

A Maclaurin’s expansion-based analytical function 𝑧(𝑡) is given 

                                       𝑧(𝑡) = ∑ 𝑧𝑛 𝑡𝑛,   

∞

𝑛=0

             0 ≤ 𝑡 ≤ 𝑇.                                                         (8) 

The Pade approximation to 𝑧(𝑡) of order [𝐿, 𝑀] which we denote by [
𝐿

𝑀
]

𝑧
(𝑡) is defined by [2]  

                                     [
𝐿

𝑀
]

𝑧

(𝑡) =
𝑝0 + 𝑝1 + ⋯ + 𝑝𝐿𝑡𝐿

1 + 𝑞1𝑡 + ⋯ + 𝑞𝑀𝑡𝑀
 ,                                                                (9) 

when 𝑞0 = 1 was taken into account and there are no common factors between the numerator and 

denominator. 

In the equation (9), the numerator and denominator are designed so that 𝑧(𝑡) and [𝐿 /𝑀]𝑧(𝑡) and the 

derivatives of them coincide at 𝑡 = 0 up to 𝐿 + 𝑀. Which is 

                        𝑧(𝑡) − [
𝐿

𝑀
]

𝑧
(𝑡) = 𝑂(𝑡𝐿+𝑀+1).                                                                      (10) 

From (10), we have 

                                      𝑧(𝑡) ∑ 𝑞𝑛 𝑡𝑛 − ∑ 𝑝𝑛 𝑡𝑛

𝐿

𝑛=0

𝑀

𝑛=0

= 𝑂(𝑡𝐿+𝑀+1)                                                 (11) 

From (11), the following algebraic linear systems are obtained: 

𝑧𝐿 𝑞1 + ⋯ + 𝑧𝐿−𝑀+1𝑞𝑀 = −𝑧𝐿+1     

𝑧𝐿+1 𝑞1 + ⋯ + 𝑧𝐿−𝑀+2 𝑞𝑀 = −𝑧𝐿+2          



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                                                                                              ⋮                                                                  (12) 

𝑧𝐿+𝑀−1 𝑞1 + ⋯ + 𝑧𝐿 𝑞𝑀 = −𝑧𝐿+𝑀     

 

𝑝0 = 𝑧0                                       

𝑝1 = 𝑧1 + 𝑧0 𝑞1                       

                                                                   ⋮                                                                                             (13) 

                              𝑝𝐿 = 𝑧𝐿 + 𝑧𝐿−1 𝑞1 + ⋯ + 𝑧0 𝑞𝐿.                        

From (12), we calculate first all the coefficients 𝑞𝑛 , 1 ≤ 𝑛 ≤ 𝑀. Then, we determine the coefficients 

𝑝𝑛 ,   0 ≤ 𝑛 ≤ 𝐿 from (13). 

It should be noted that error (10) occurs for a fixed value of 𝐿 + 𝑀 + 1 when the denominator and 

numerator of (9) have the identical grade or where the value of the numerator has a degree that is one 

more than the denominator [2]. 

5. Method of Laplace-Pade Re-summation 

Solutions in the form of polynomial power series are provided via many approximations. However, 

this kind of solutions may lack substantial areas of convergence. As a result, the Laplace-Pade re-

summation approach is employed in literature to broaden the range of solutions that may be found by 

convergent analysis or to identify all possible solutions. 

You may sum up the Laplace-Pade approach as follows: 

1) Initially, power sequences are subjected to the Laplace transformation (5). 

2) Following that, in the resultant equation, 1/𝑡 is used to replace 𝑠. 

3) Later, by generating its Pade approximation of order [𝑁 /𝑀], we change the altered series into 

a metamorphic function. Although 𝑁 and 𝑀 were picked at random, they should be less than the order 

of the power series, order to improve accuracy and convergence, the Pade approximation increases the 

domain of the truncated series solution at this step. 

4) Now 1/𝑠 is used to replace 𝑡. 

5) At last, we arrive to the precise or approximate solution via the inverse Laplace 𝑠 

transformation. 

Applying the parameter values and initial conditions in (2), we obtain up to certain order solution as 

follows: 

𝑠(𝑡) = ∑ 𝑆(𝑝) × 𝑡𝑝

∞

𝑝=0

                                                                                                             

  = 10000 − 290528.7700 t − 111721211.6 𝑡2 −  30978513960.0 𝑡3 

−5924327228000.0 𝑡4 − 609173762600000.0 𝑡5  +  ⋯ 



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𝑣(𝑡) = ∑ 𝑉(𝑝) × 𝑡𝑝

∞

𝑝=0

                                                                                                             

  = 200 +  8814.0 t − 134836.4565 𝑡2 − 33474564.17 𝑡3                       

−6962382805.0 𝑡4 −  1065083898000.0 𝑡5  + ⋯                   

𝑖(𝑡) = ∑ 𝐼(𝑝) × 𝑡𝑝

∞

𝑝=0

                                                                                                             

                    = 200 + 199893.0 t +  96987740.90 𝑡2 +  29631320500.0 𝑡3                   (14) 

            + 5986223935000.0 𝑡4 + 660172409800000.0 𝑡5  + ⋯                   

 𝑟(𝑡) = ∑ 𝑅(𝑝) × 𝑡𝑝

∞

𝑝=0

                                                                                                             

              = 20 +  62.40000000 t + 32976.72900 𝑡2 + 10666672.90 𝑡3                       

     + 2444103941.0 𝑡4 + 395002792000.0 𝑡5  +  ⋯                   

𝑚(𝑡) = ∑ 𝑀(𝑝) × 𝑡𝑝

∞

𝑝=0

                                                                                                             

              = 200 +  79995.0 t + 14834386.26 𝑡2 +  1369308719.0 𝑡3                           

          −57680894920.0 𝑡4 −  50389449540000.0 𝑡5  + ⋯                   

To enhance the exactness of DTM equation (14) solution series, we applied a 𝑡 − Laplace transform, 

later replace 𝑠 by 1/𝑡 and implement Pade approximation to the transformed series. Finally, 𝑡 is 

replaced by 1/𝑠 and applied inverse Laplace 𝑠 −transform for the generated interpretation to acquire 

the estimate. 

Put in Laplace transform to equation (14) yields 

ℒ[𝑠(𝑡)] = 10000 𝑠−1 − 290528.7700 𝑠−2 − 223442423.2 𝑠−3 −  185871083800.0 𝑠−4 

    −142183853500000.0 𝑠−5 −  73100851510000000.0 𝑠−6  + ⋯   

ℒ[𝑣(𝑡)] = 200 𝑠−1 + 8814.0 𝑠−2 − 269672.9130 𝑠−3 −  200847385.0 𝑠−4                         

    −167097187300.0 𝑠−5 −  127810067800000.0 𝑠−6  +  ⋯               

ℒ[𝑖(𝑡)] = 200 𝑠−1 + 199893.0 𝑠−2 + 193975481.8 𝑠−3 +  177787923000.0 𝑠−4            

                        +143669374400000.0 𝑠−5 + 79220689180000000.0 𝑠−6  + ⋯              (15) 

ℒ[𝑟(𝑡)] = 20 𝑠−1 + 62.40000000 𝑠−2 + 65953.45800 𝑠−3 +  64000037.40 𝑠−4            

   +58658494580.0 𝑠−5 +  47400335040000.0 𝑠−6  +  ⋯                   

ℒ[𝑚(𝑡)] = 200 𝑠−1 + 8814.0 𝑠−2 − 269672.9130𝑠−3 −  200847385.0 𝑠−4                         



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    −167097187300.0 𝑠−5 −  127810067800000.0 𝑠−6  +  ⋯               

For the purpose of clarity, we substitute 𝑠 = 1/𝑡 in (15) to acquire: 

ℒ[𝑠(𝑡)] = 10000 𝑡 − 290528.7700 𝑡2 − 223442423.2 𝑡3 −  185871083800.0 𝑡4          

    −142183853500000.0 𝑡5 −  73100851510000000.0 𝑡6  +  ⋯   

ℒ[𝑣(𝑡)] = 200 𝑡 + 8814.0 𝑡2 − 269672.9130 𝑡3 −  200847385.0 𝑡4                                    

    −167097187300.0 𝑡5 −  127810067800000.0 𝑡6  + ⋯               

ℒ[𝑖(𝑡)] = 200 𝑡 + 199893.0 𝑡2 + 193975481.8 𝑡3 +  177787923000.0 𝑡4                       

                        +143669374400000.0 𝑡5 + 79220689180000000.0 𝑡6  +  ⋯              (15) 

ℒ[𝑟(𝑡)] = 20 𝑡 + 62.40000000 𝑡2 + 65953.45800 𝑡3 +  64000037.40 𝑡4                          

   +58658494580.0 𝑡5 +  47400335040000.0 𝑡6  + ⋯                   

ℒ[𝑚(𝑡)] = 200 𝑡 + 8814.0 𝑡2 − 269672.9130𝑡3 −  200847385.0 𝑡4                                      

    −167097187300.0 𝑡5 −  127810067800000.0 𝑡6  + ⋯               

 

Following (16); we workout Pade approximates of order [5/4] to each equation in (16) of transformed 

series, we acquire: 

[
5

4
]

𝑠
=

10000.0 t +  3159069 𝑡2 −  766424340 𝑡3 + 11966339 𝑡4 − 63343485 𝑡5

1.0 + 3188.12 t − 7549275 𝑡2  +  118368350 𝑡3 −  60856609310 𝑡4
              

[
5

4
]

𝑣
=

200.0 t +  8157.77 𝑡2  +  85211.1 𝑡3 −  38241998 𝑡4 +  24576395900 𝑡5

1.0 −  3.28112 t + 1919.01 𝑡2 −  1820575 𝑡3 +  139253041 𝑡4
                

 

[
5

4
]

𝑖
=

200.0 t − 601369 𝑡2 +  245732 𝑡3 −  2856058 𝑡4  +  1995223 𝑡5

1.0 −  4006.31 t + 1532090 𝑡2 −  265963 𝑡3 + 14665463 𝑡4
                       (17) 

[
5

4
]

𝑟
=

20.0 t + 12005.9 𝑡2 + 3931762 𝑡3 −  2675397980 𝑡4  +  215764814500 𝑡5

1.0 +  597.178 t +  1960720 𝑡2 −  133882766 𝑡3 + 10818702450 𝑡4
             

 

[
5

4
]

𝑚
=

200.0 t +  145496.3 𝑡2  +  7750203 𝑡3  +  23075868 𝑡4 + 1844297578 𝑡5

1.0 +  327.5068 t +  108171.7 𝑡2 −  17549540.31 𝑡3 + 76655610 𝑡4
               

 

Applying 𝑡 = 1/𝑠, we obtain [5/4]𝑠 , [5/4]𝑣 , [5/4]𝑖 , [5/4]𝑟   and  [5/4]𝑚 in terms of 𝑠 and taking 

inverse Laplace the following series obtained. 

 

𝑠(𝑡) = 10408.64526 +  0.5820268 (𝑥) −  4.455814 × 10−12𝑒−0.0001943056 𝑥                       



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+ (0.00000344015 +  0.00000238646 i)𝑒(0.000377555−0.000684391 𝑖)𝑥 

+ (0.0000034401 −  0.00000238646 i)𝑒(0.000377555+0.000684391 𝑖)𝑥    

+0.00080717 𝑒0.0013842 𝑥                                                                                

𝑣(𝑡) = 176.4873186 −  0.0438851 (x)                                                                                             

−(0.000025997 −  0.000001192 i)𝑒(−0.0017612−0.0020394 𝑖) 𝑥               

−(0.000025997 +  0.000001192 i)𝑒(−0.0017612+0.0020394 𝑖) 𝑥               

−(0.000011790 −  0.00012290 i)𝑒(0.0024149−0.0020143 𝑖)𝑥                   

−(0.000011790 + 0.00012290 i)𝑒(0.0024149+0.0020143 𝑖)𝑥                   

𝑖(𝑡) = 1.360491042 −  0.192279 (x)                                                                                          

−(0.000000001931 + 0.0000000003505 i)𝑒(0.000079009 − 0.00030585 𝑖) 𝑥 

                  −(0.000000001931 − 0.0000000003505 i)𝑒(0.000079009+ 0.00030585 𝑖) 𝑥         (18) 

+ 0.00013612 𝑒0.00078436  𝑥 − 0.00031868 𝑒0.0008711 𝑥                              

𝑟(𝑡) = 19.9436869 −  0.00004882 (x)                                                                                     

−(2.43212167 × 10−15 − 2.864294 × 10−12 𝑖)𝑒(−0.0001973−0.000307 𝑖) 𝑥 

−(2.43212167 × 10−15 + 2.864294 × 10−12 𝑖)𝑒(−0.0001973+0.000307 𝑖) 𝑥 

− (0.00000002033 −  0.0000000018 i)𝑒(0.0008161−0.0001611 𝑖) 𝑥               

− (0.0000000203 +  0.0000000018 i)𝑒(0.0008161+0.0001611 𝑖) 𝑥                

𝑚(𝑡) = 240.5952508 +  0.35611 (x)                                                                                           

−(0.00002962 −  0.00002697 i) 𝑒(−0.001179474 − 0.0011528 𝑖) 𝑥                  

−(0.000029621 +  0.00002697 i) 𝑒(−0.001179474+ 0.0011528 𝑖) 𝑥                

+ (0.00040615 −  0.00018201 i)  𝑒(0.001293 −  0.0017666  𝑖) 𝑥                    

+ (0.00040615 +  0.00018201 i)  𝑒(0.001293+  0.0017666  𝑖) 𝑥                    

 

7. Result and Discussion 

Although most types of lung infection are treatable and most people recover, their infections are very 

dangerous too. Especially in case of infants, older adults and people with lung infections or weak 

immune system. In this chapter, the lung infection model is taken into account as deterministic 

epidemic model. There are five compartments framed including vaccination and media awareness. In 

past few years the world-wide pandemic has shown us the importance of vaccine and also through 

media the awareness is created about vaccine and self-discipline. The five compartments that we 

considered are susceptible 𝑠(𝑡), vaccination 𝑣(𝑡), infected class 𝑖(𝑡), media awareness 𝑚(𝑡) and 



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recovery class 𝑟(𝑡). In the above calculations, we employed a hybrid technique called Laplace-Pade 

Differential Transform Method (LPDTM) for the considered deterministic compartment model. With 

initial values of each compartment (3) and parameter values from Table 1, we generated a recurrence 

relation of each equation in (1) up to certain order. In particular, we discussed the two important 

compartments that is infected class and recovery class. In infected class that includes susceptible, 

vaccination and death rate of infected, whereas in recovery class, media and vaccination plays a vital 

role. Equations (15) - (18) describes the LPDTM procedure and Figure 1 - 2 gives us the picture view 

of infected class and recovery class. 

  

 

 

 

 

 

 

 

 

FIGURE 1. : Infected individuals 𝒊(𝒕) by LPDTM 

FIGURE 2. : Recovered individuals 𝒓(𝒕) by LPDTM 

 

 

       



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8. Conclusion 

The present study gives the better understanding of the Laplace Pade Differential Transform Method 

and its applications which results the convergence of analytical solution. DTM is effectively applied 

to the system of nonlinear differential equations and analysed that this method can be implemented to 

various real life problem models for predicting the causes and prevention. 

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