J. Nig. Soc. Phys. Sci. 3 (2021) 82–88

Journal of the
Nigerian Society

of Physical
Sciences

Comments on “The Solution of a Mathematical Model for
Dengue Fever Transmission Using Differential Transformation

Method: J. Nig. Soc. Phys. Sci. 1 (2019) 82-87”

Gurpreet Singh Tutejaa,∗, Tapshi Lalb

a Zakir Husain Delhi College, University of Delhi
b Satyawati College, University of Delhi

Abstract

The mathematical model for dengue fever transmission studied by [1], has been re-investigated. The differential transformation method
(DTM) is used to compute the semi-analytical solutions of the non-linear differential equations of the compartment (SIR) model of dengue
fever. This epidemiology problem is well-posed. The effect of treatment as a control measure is studied through the growth equations
of exposed and infected humans. The inadvertent errors in the recurrence relations (DTM) of equations for dengue disease transmission
including initial conditions have been removed. Furthermore, the semi-analytic solutions of the model are obtained and verified with the
built-in function AsymptoticDSolveValue of Wolfram Mathematica. It has been found that results obtained from the DTM are valid only for
small-time t (t < 1.5), as t becomes large, the human population (exposed and recovered) and infected vector population become negative.

DOI:10.46481/jnsps.2021.170

Keywords: SIR model, Differential Transformation Method (DTM), Dengue Fever, Treatment

Article History :
Received: 01 March 2021
Received in revised form: 2 April 2021
Accepted for publication: 01 April 2021
Published: 18 May 2021

©2021 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: B. J. Falaye

1. Introduction

Dengue fever is a mosquito-borne flavivirus that is mostly
found in tropical and sub-tropical regions of the world. The
disease is spread by Aedes mosquito due to day-biting [2].
Dengue fever is the fastest-spreading vector-borne viral dis-
ease affecting 40 per cent of the world’s population, and now
endemic in over 100 countries. Over the last two decades,
the number of dengue cases registered to WHO has increased
from 505,430 cases in 2000 to over 2.4 million in 2010, and

∗Corresponding author tel. no:
Email addresses: gstuteja@gmail.com (Gurpreet Singh Tuteja ),

tapshisingh@ymail.com (Tapshi Lal)

4.2 million in 2019. Between 2000 and 2015, the number of
registered deaths increased from 960 to 4032 [3]. A second
potential vector, Aedes Albopictus, resides in temperate re-
gions (North America and Europe), where it may give rise to
occasional dengue outbreaks [4, 5].

The spread of infectious diseases is studied through vari-
ous epidemiological models, including observational studies,
interventional studies apart from mathematical modelling us-
ing the compartment model [6]. The pioneering work using
the SIR model for contagious diseases is done by [7, 8, 9]. In
the compartment model, the population is primarily divided
into three distinct mutually exclusive compartments: suscep-
tible S(t ), infected/infectious I (t ) and recovered R (t ) at any

82



Tuteja & Lal / J. Nig. Soc. Phys. Sci. 3 (2021) 82–88 83

time t , based on the epidemiological status of the population
[10].
For the first time, the DTM was used for solving electrical cir-
cuit problems [11]. The application of DTM in finding so-
lutions for the set of non-linear ordinary differential equa-
tions obtained using the SIR model for various epidemiologi-
cal diseases including Typhoid [12, 13], Malaria [14] and sea-
sonal diseases [15] has been studied. In this SIR model for
dengue disease, we consider two separate but dependent sets
of non-linear ordinary differential equations related to hu-
man and vector population [16, 17]. The purpose of this pa-
per is to find semi-analytical solutions to the dengue fever
(SIR) model including the effect of treatment as a measure
of control. The semi-analytic solutions are obtained by using
the differential transform method (DTM) and are confirmed
by using a built-in function: AsymptoticDSolveValue of
Wolfram Mathematica and are discussed graphically also.
The paper is organized in the following sections: In section
2, the SIR model for dengue fever with model parameters in-
cluding treatment is briefly described. The existence, unique-
ness and positivity of the solution to the epidemiology prob-
lem are discussed in Section 3. Section 4 presents a theoret-
ical concept and implementation of DTM. In Section 5, nu-
merical solutions are obtained with a graphical discussion.
Finally, concluding remarks in section 6.

2. Formulation of the Problem

Two sets of populations consisting of human and vector
are considered in Figure 1. The population is divided into
some mutually exclusive compartments as given below. The
total human population is divided into the following mutu-
ally exclusive epidemiological classes: susceptible humans
Sh (t ), humans with dengue in latent stage Eh (t ), humans in-
fected with dengue Ih (t ), humans treated for dengue Rh (t )
(recovered), while the vector population is divided into three
classes: susceptible vectors S v (t ), vectors with dengue in la-
tent stage E v (t ), vectors with dengue I v (t ). The class of treated
vectors (Rv (t )) is not taken into consideration. Let Nh (t ) and
Nv (t ) denote the total number of humans and vectors at time
t , respectively. Hence, we have that, Nh (t ) = Sh (t ) + Eh (t ) +
Ih (t ) + Rh (t ) and Nv (t ) = S v (t ) + E v (t ) + I v (t ), where Nh (t ) >
0, Nv (t ) > 0, Sh (t ) > 0, S v (t ) > 0, Eh (t ) ≥ 0, Ih (t ) ≥ 0, Rh (t ) ≥
0, E v (t ) ≥ 0, I v (t ) ≥ 0.
The susceptible humans are recruited at a rate Λh , while the
susceptible vectors are recruited at a rate Λv . The susceptible
humans’ contract to dengue at a rate:

λDV =
βv h (ηv E v + I v )

Nh
, (1)

where ηv < 1, this accounts for the relative infectiousness of
vectors with latent dengue E v compared to vectors in the I v
class. Susceptible vectors acquire dengue infection from in-
fected humans (infected blood is passed to a vector through
a bite) at a rate:

λD H =
βh v (ηA Eh +ηB Ih )

Nh
, (2)

where ηA < ηB this accounts for the relative infectiousness of
humans with latent dengue Eh compared to humans in the
Ih class [1].
The model equations for dengue disease transmission includ-
ing treatment as a control measure are:

d Sh
d t

=Λh −µh Sh −λDV Sh , (3)
d Eh
d t

= λDV Sh − (γh +µh )Eh , (4)
d Ih
d t

= γh Eh − (τh +µh +δDh )Ih , (5)
d Rh
d t

= τh Ih −µh Rh , (6)
d S v
d t

=Λv −µv S v −λD H S v , (7)
d E v
d t

= λD H S v − (γv +µv )E v , (8)
d I v
d t

= γv E v − (µv +δD v )I v , (9)

where Λh ,Λv are the recruitment rates and µh , µv are natu-
ral death rates of susceptible human and vector population,
respectively. βv h and βh v are the effective contact rate for
dengue from vectors to humans and humans to vectors, re-
spectively. The treatment rate for infected humans is τh . γh
and γv are the progression rate of the human and vector pop-
ulation from the latent class (exposed) to the active dengue
class, respectively. The disease induced deaths in human and
vector are denoted by δDh and δD v . ηv , ηA , ηB are the modi-
fication parameters of E v , Eh and Ih , respectively.

3. Existence, Uniqueness and Positivity of Solution

We will use the Lipchitz condition to verify the existence
and uniqueness of solution [18] for the model equations (3)-
(9):

E1 =Λh −µh Sh −λDV Sh ,
E2 = λDV Sh − (γh +µh )Eh ,
E3 = γh Eh − (τh +µh +δDh )Ih ,
E4 = τh Ih −µh Rh ,
E5 =Λv −µv S v −λD H S v ,
E6 = λD H S v − (γv +µv )E v ,
E7 = γv E v − (µv +δD v )I v .

Let B denote the region, |t −t0| ≤ δ, ||x−x0|| ≤ α, where x =
(x1, x2, . . . xn ), x0 = (x10, x20, . . . xn0) also suppose that a(t , x )
satisfies the Lipschitz condition:

||a(t , x1) − a(t , x2)|| ≤ k||x1 − x2||
whenever the pairs (t , x1), (t , x2) belong to B where k is a posi-
tive constant, then there is a positive constant δ ≥ 0, such that
there exists a unique and continuous vector solution x (t ) of

83



Tuteja & Lal / J. Nig. Soc. Phys. Sci. 3 (2021) 82–88 84

Figure 1: Dengue Fever Model

the system in the interval |t − t0| < δ. The condition is sat-
isfied by the requirement that

∂ai
∂x j

,i , j = 1, 2, 3, ..n, be continu-
ous and bounded in B. Considering the model equation (3)-
(9), we are interested in the region 0 ≤ α ≤ R [13].
Let B denote the region 0 ≤ α ≤ R , then equations (3) – (9) will
have a unique solution if

∂ai
∂x j

,i , j = 1, 2, 3, ..7 are continuous
and bounded in B.
For E1:∣∣∣∣ ∂E1∂Sh

∣∣∣∣ = |− (µh +λDV )| < ∞,
∣∣∣∣ ∂E1∂Eh

∣∣∣∣ = 0 < ∞,
∣∣∣∣∂E1∂Ih

∣∣∣∣ = 0 < ∞,∣∣∣∣ ∂E1∂Rh
∣∣∣∣ = 0 < ∞,

∣∣∣∣∂E1∂S v
∣∣∣∣ = 0 < ∞,

∣∣∣∣ ∂E1∂E v
∣∣∣∣ = 0 < ∞,

∣∣∣∣∂E1∂I v
∣∣∣∣ = 0 < ∞.

For E2:∣∣∣∣ ∂E2∂Sh
∣∣∣∣ = |λDV | < ∞,

∣∣∣∣ ∂E2∂Eh
∣∣∣∣ = |− (γh +µh )| < ∞,

∣∣∣∣∂E2∂Ih
∣∣∣∣ = 0 < ∞,∣∣∣∣ ∂E2∂Rh

∣∣∣∣ = 0 < ∞,
∣∣∣∣∂E2∂S v

∣∣∣∣ = 0 < ∞,
∣∣∣∣ ∂E2∂E v

∣∣∣∣ = 0 < ∞,
∣∣∣∣∂E2∂I v

∣∣∣∣ = 0 < ∞.
For E3:∣∣∣∣ ∂E3∂Sh

∣∣∣∣ = 0 < ∞,
∣∣∣∣ ∂E3∂Eh

∣∣∣∣ = |γh| < ∞,∣∣∣∣∂E3∂Ih
∣∣∣∣ = |− (τh +µh +δDh )| < ∞,∣∣∣∣ ∂E3∂Rh
∣∣∣∣ = 0 < ∞,

∣∣∣∣∂E3∂S v
∣∣∣∣ = 0 < ∞,

∣∣∣∣ ∂E3∂E v
∣∣∣∣ = 0 < ∞,

∣∣∣∣∂E3∂I v
∣∣∣∣ = 0 < ∞.

For E4:∣∣∣∣ ∂E4∂Sh
∣∣∣∣ = 0 < ∞,

∣∣∣∣ ∂E4∂Eh
∣∣∣∣ = 0 < ∞,

∣∣∣∣∂E4∂Ih
∣∣∣∣ = |τh| < ∞,∣∣∣∣ ∂E4∂Rh

∣∣∣∣ = |−µh| < ∞,
∣∣∣∣∂E4∂S v

∣∣∣∣ = 0 < ∞,
∣∣∣∣ ∂E4∂E v

∣∣∣∣ = 0 < ∞,∣∣∣∣∂E4∂I v
∣∣∣∣ = 0 < ∞.

These partial derivatives exist, are continuous and bounded,
similarly for E5, E6, E7. Hence the model has a unique solu-
tion. The positivity of the solution is presented in the follow-
ing theorems:

Positivity of the Solution: We show that the model equa-
tions (3)–(9) are biologically and epidemiologically meaning-
ful and well-posed as the solutions of all the stated variables
are non-negative [16].
If Sh (0) > 0, Eh (0) ≥ 0, Ih (0) ≥ 0, Rh (0) ≥ 0, S v (0) > 0, E v (0) ≥
0 and I v (0) ≥ 0, then the solution region Sh (t ), Eh (t ), Ih (t ),
Rh (t ), S v (t ), E v (t ) and I v (t ) of the system of equations (3)–(9)
is always non-negative.
We consider each differential equation separately and show
that its solution is positive.

Theorem 1: Positivity of susceptible human population: Con-
sider the differential equation (3):

d Sh
d t

=Λh − (µh +λDV )Sh (t ) ≥ −(µh +λDV )Sh (t ),
Λh > 0 being recruitment rate of humans, we can write as:

d Sh
Sh

= −(µh +λDV )d t

On integrating, the solution is Sh = Sh0e−
∫ t

0 (µh +λDV )d t . It is
clear from the solution that Sh (t ) is positive since Sh0 = Sh (0) >
0 and the exponential function is always positive.

Theorem 2: Positivity of latent human population: Con-
sider the differential equation (4):

d Eh
d t

= λD H Sh (t ) − (γh +µh )Eh (t ) ≥ −(γh +µh )Eh (t ),
Sh (t ) is positive in time t and λD H > 0 being humans’ contract
rate to dengue, we can write as:

d Eh
Eh

= −(γh +µh )d t .
84



Tuteja & Lal / J. Nig. Soc. Phys. Sci. 3 (2021) 82–88 85

On integrating, the solution is Eh = Eh0e−
∫ t

0 (γh +µh )d t . It is
clear from the solution that Eh (t ) is positive since Eh0 = Eh (0) ≥
0 and the exponential function is always positive.

Theorem 3: Positivity of infected human population: Con-
sider the differential equation (5):

d Ih
d t

= γh Eh (t ) − (τh +µh +δDh )Ih (t ) ≥ −(τh +µh +δDh )Ih (t ),

γh being the progression rate of humans from latent class to
active dengue class and Eh (t ) ≥ 0, we can write as:

d Ih
Ih

= −(τh +µh +δDh )d t .

On integrating, the solution is Ih = Ih0e−
∫ t

0 (τh +µh +δDh )d t . So,
it is clear from the solution that Ih (t ) is positive since Ih0 =
Ih (0) ≥ 0 and exponential function is always positive.

Theorem 4: Positivity of recovered human population: Con-
sider the differential equation (6):

d Rh
d t

= τh Ih (t ) −µh Rh (t ) ≥ −µh Rh (t ),

τh > 0 being the treatment rate for infected humans and Ih (t )
is positive in time t , we can write as:

d Rh
Rh

= −µh d t .

On integrating, the solution is Rh = Rh0e−
∫ t

0 µh d t . It is clear
from the solution that Rh (t ) is positive since Rh0 = Rh (0) ≥ 0
and the exponential function is always positive.

Theorem 5: Positivity of susceptible vector population: Con-
sider the differential equation (7):

d S v
d t

=Λv − (µv +λD H )S v (t ) ≥ −(µv +λD H )S v (t ),

Λv > 0 being recruitment rate of vectors, we can write as:
d S v
S v

= −(µv +λD H )d t .

On integrating, the solution is S v = S v 0e−
∫ t

0 (µv +λD H )d t . It is
clear from the solution that S v (t ) is positive since S v 0 = S v (0) >
0 and the exponential function is always positive.

Theorem 6: Positivity of latent vector population: Consider
the differential equation (8):

d E v
d t

= λD H S v (t ) − (γv +µv )E v (t ) ≥ −(γv +µv )E v (t ),

S v (t ) is positive in time t and λD H > 0 being vectors’ contract
rate to dengue due to infected humans, we can write as:

d E v
E v

= −(γv +µv )d t .

On integrating, the solution is E v = E v 0e−
∫ t

0 (γv +µv )d t . It is clear
from the solution that E v (t ) is positive since E v 0 = E v (0) ≥ 0
and the exponential function is always positive.

Theorem 7: Positivity of infected vector population: Con-
sider the differential equation (9):

d I v
d t

= γv E v (t ) − (µv +δD v )I v (t ) ≥ −(µv +δD v )I v (t ),

γv > 0 being the progression rate of vectors from latent class
to active dengue class and E v (t ) ≥ 0, we can write as:

d I v
I v

= −(µv +δD v )d t .

On integrating, the solution is I v = I v 0e−
∫ t

0 (µv +δD v )d t . So, it is
clear from the solution that I v (t ) is positive since I v 0 = I v (0) ≥
0 and exponential function being positive always.
Hence, the stated problem is epidemiologically meaningful,
well-posed and has a unique solution.

4. Differential Transform Method (DTM)

The differential transformation of the k t h derivative of f (x )
is defined as:

F (k ) = 1
k !

[
d k f (x )

d x k

]
x0

. (10)

We obtain,

f (x ) =
∞∑

k=0
F (k )(x − x0)k , (11)

is called the inverse differential transformation of F(k). In real
applications, the function f(x) can be expressed as a finite se-
ries and equation (11) can be expressed as:

f (x ) =
n∑

k=0
F (k )(x − x0)k . (12)

So, we have

f (x ) =
n∑

k=0
(x − x0)k

1

k !

[
d k f (x )

d x k

]
x0

. (13)

From equations (10) and (11), the following properties are ob-
tained:

1. If z (x ) = f (x ) ± g (x ), then Z (k ) = F (k ) ±G (k ).
2. If z (x ) = αF (x ), then Z (k ) = αF (k ).
3. If z (x ) = f ′(x ), then Z (k ) = (k + 1)F (k + 1).
4. If z (x ) = f ′′(x ), then Z (k ) = (k + 1)(k + 2)F (k + 2).
5. If z (x ) = f (l )(x ), then Z (k ) = (k+1)(k+2)...(k+l )F (k+l ).
6. If z (x ) = u(x )v (x ), then Z (k ) = ∑k

l =0 F (l )G (k − l ).
7. If z (x ) = αx l , then Z (k ) = αδ(k − l ), where Kronecker

delta, δ(k − l ) =
{

1,k=l
0,k,l

Using the fundamental operations of differential transforma-
tion method, let Sh (k ),Eh (k ), Ih (k ), Rh (k ), S v (k ), E v (k ) and
I v (k ) denote the differential transformations of Sh (t ), Eh (t ),

85



Tuteja & Lal / J. Nig. Soc. Phys. Sci. 3 (2021) 82–88 86

Ih (t ), Rh (t ), S v (t ),E v (t ) and I v (t ) respectively, the recurrence
relation to each equation of the system (3)–(9) is:

Sh (k + 1) =
1

k + 1
{
Λhδ(k ) −µh Sh (k )

−βv hηv
Nh

k∑
m=0

Sh (m)E v (k − m) −
βv h

Nh

k∑
m=0

Sh (m)I v (k − m)
}

(14)

Eh (k + 1) =
1

k + 1

{
βv hηv

Nh

k∑
m=0

Sh (m)E v (k − m)

+ βv h
Nh

k∑
m=0

Sh (m)I v (k − m) − (γh +µh )Eh (k )
}

(15)

Ih (k + 1) =
1

k + 1 {γh Eh (k ) − (τh +µh +δDh )Ih (k )} (16)

Rh (k + 1) =
1

k + 1 {τh Ih (k ) −µh Rh (k )} (17)

S v (k + 1) =
1

k + 1
{
Λv δ(k ) −µv S v (k )

− βh v ηA
Nh

k∑
m=0

S v (m)Eh (k − m)

− βh v ηB
Nh

k∑
m=0

S v (m)Ih (k − m)
}

(18)

E v (k + 1) =
1

k + 1

{
βh v ηA

Nh

k∑
m=0

S v (m)Eh (k − m)

+ βh v ηB
Nh

k∑
m=0

S v (m)Ih (k − m) − (γv +µv )E v (k )
}
(19)

I v (k + 1) =
1

k + 1
{
γv E v (k ) − (µv +δD v )I v (k )

}
(20)

The recurrence relations (14), (15), (18) and (19) are the rec-
tified forms of (8), (9), (12) and (13) of the study [1]. The
semi-analytical solutions and numerical solutions have sig-
nificantly changed due to these corrections.

5. Numerical and Graphical Simulation of the Model Equa-
tions

With the initial conditions Sh (0) = 3503, Eh (0) = 490, Ih (0) =
390, Rh (0) = 87, S v (0) = 390,E v (0) = 100, I v (0) = 130, we com-
pute the semi-analytical solutions for k = 4 using following
values of the parameters: Nh = 4470, Nv = 620, Λh = 500,
Λv = 1, 000, 000, µh = 0.02041, µv = 0.5, βv h = 0.5, βh v = 0.4,
γh = 0.3254, γv = 0.03, δDh = 0.365, δD v = 0, ηv = 0.4, ηA =
0.2, ηB = 0.5 [5].

5.1. Low Dengue Treatment(τh = 0.25)

sh (t ) =
4∑

k=0
Sh (k )t

k = 3503 + 361.8919131767338t

+ 8.365058543813413t 2 − 686.2825200034454t 3
+ 197.4722799192922t 4,

eh (t ) =
4∑

k=0
Eh (k )t

k = 490 − 102.83504317673376t

+ 5.722527622691168t 2 + 685.5659739627513t 3
− 253.23941572498939t 4,

i h (t ) =
4∑

k=0
Ih (k )t

k = 390 − 88.3639t + 11.342391324645417t 2

− 1.7816527943897462t 3 + 56.05381198239061t 4,

rh (t ) =
4∑

k=0
Rh (k )t

k = 87 + 95.72433t − 12.02235428765t 2

+ 1.026991360724097t 3 − 0.11659352306745382t 4,

sv (t ) =
4∑

k=0
S v (k )t

k = 390 + 999794.7744966443t

− 263054.4930350626t 2 + 48072.332846142934t 3
− 6858.820737023293t 4,

e v (t ) =
4∑

k=0
E v (k )t

k = 100 − 42.774496644295304t

+ 13117.134652512259t 2 − 6547.277795576331t 3
+ 1717.2934391692909t 4,

i v (t ) =
4∑

k=0
I v (k )t

k = 130 − 62.t + 14.85838255033557t 2

+ 128.69494943339998t 3 − 65.19145214599749t 4

These solutions are the same as obtained from the in-built
function AsymptoticDSolveValue of Wolfram Mathematica
13. Following is the program:

A s y m p t o t i c D Sol v eV al ue [

{S′h [t ] −Λh +µh Sh [t ] + (βv hηv /Nh )Sh [t ]
E v [t ] + (βv h /Nh )Sh [t ]I v [t ] == 0,

E ′h [t ] − (βv hηv /Nh )Sh [t ]
E v [t ] − (βv h /Nh )Sh [t ]I v [t ] + (γh +µh )Eh [t ] == 0,

I ′h [t ] −γh Eh [t ] + (τh +µh +δDh )Ih [t ] == 0,
R′h [t ] −τh Ih [t ] +µh Rh [t ] == 0,

S′v [t ] −Λv +µv S v [t ] + (βh v ηA /Nh )S v [t ]
Eh [t ] + (βh v ηB /Nh )S v [t ]Ih [t ] == 0,

E ′v [t ] − (βh v ηA /Nh )S v [t ]Eh [t ] − (βh v ηB /Nh )
S v [t ]Ih [t ] + (γv +µv )E v [t ] == 0,

I ′v [t ] −γv E v [t ] + (µv +δD v )
86



Tuteja & Lal / J. Nig. Soc. Phys. Sci. 3 (2021) 82–88 87

I v [t ] == 0, Sh [0] == Sh[0], Eh [0] == Eh [0],
Ih [0] == I h[0], Rh [0] == R h[0], S v [0] == Sv [0], E v [0] == E v [0],

I v [0] == I v [0]},
{Sh [t ], Eh [t ], Ih [t ], Rh [t ], S v [t ], E v [t ], I v [t ]}, {t , 0, 4}]

5.2. Moderate Dengue Treatment(τh = 0.50)

sh (t ) =
4∑

k=0
Sh (k )t

k = 3503 + 361.8919131767338t

+ 8.365058543813413t 2 − 686.2380770354108t 3
+ 254.42405449756384t 4

eh (t ) =
4∑

k=0
Eh (k )t

k = 490 − 102.83504317673376t

+ 5.722527622691168t 2 + 685.5215309947168t 3
− 310.1875748678114t 4,

i h (t ) =
4∑

k=0
Ih (k )t

k = 390 − 185.8639t + 65.5516163246454t 2

− 18.725982040526862t 3 + 59.912219486045935t 4,

rh (t ) =
4∑

k=0
Rh (k )t

k = 87 + 193.22433t − 48.43782928765t 2

+ 11.25480808602788t 3 − 2.3981754133248154t 4,

sv (t ) =
4∑

k=0
S v (k )t

k = 390 + 999794.7744966443t

− 263053.6423639217t 2 + 49525.708598154626t 3
− 7943.074169380729t 4,

e v (t ) =
4∑

k=0
E v (k )t

k = 100 − 42.774496644295304t

+ 13116.28398137132t 2 − 8000.645040876618t 3
+ 2812.4460625275524t 4,

i v (t ) =
4∑

k=0
I v (k )t

k = 130 − 62.t + 14.85838255033557t 2

+ 128.6864427219906t 3 − 76.09064314682345t 4

5.3. High Dengue Treatment(τh = 0.75)

sh (t ) =
4∑

k=0
Sh (k )t

k = 3503 + 361.8919131767338t

+ 8.365058543813413t 2 − 686.1936340673763t 3
+ 311.3702737048312t 4,

eh (t ) =
4∑

k=0
Eh (k )t

k = 490 − 102.83504317673376t

+ 5.722527622691168t 2 + 685.4770880266824t 3
− 367.13017863962915t 4,

Figure 2: Susceptible, Exposed, Infected Human (τ = 0.5)

i h (t ) =
4∑

k=0
Ih (k )t

k = 390 − 283.3639t + 144.1358413246454t 2

− 53.93038836999731t 3 + 71.07183667576527t 4,

rh (t ) =
4∑

k=0
Rh (k )t

k = 87 + 290.72433t − 109.22830428765t 2

+ 36.777076894665t 3 − 10.299602854229525t 4,

sv (t ) =
4∑

k=0
S v (k )t

k = 390 + 999794.7744966443t

− 263053.6423639217t 2 + 50978.94257164284t 3
− 9299.822488317648t 4,

e v (t ) =
4∑

k=0
E v (k )t

k = 100 − 42.774496644295304t

+ 13115.43331023038t 2 − 9453.870507653413t 3
+ 4180.0925091263725t 4,

i v (t ) =
4∑

k=0
I v (k )t

k = 130 − 62.t + 14.85838255033557t 2

+ 128.6779360105812t 3 − 86.98877080872326t 4.

The graph of susceptible, exposed, infected human pop-
ulation with moderate treatment and exposed, infected vec-
tors is plotted against time t Figures 2 and 3. It is found that
the susceptible and infected human population grows with
time t while the exposed human population becomes nega-
tive after t = 2.25, being population graph, it can’t be negative
(limitation of DTM). Similarly, the graph of the infected vec-
tors becomes negative.
The effect of the treatment as a control measure can be stud-
ied from Figures 4 and 5, where the effect of better treatment
on infected population is found positive initially, the recov-
ered population also increases initially and then consequently
decreases (after t = 1.70, for high treatment rate) due to fast
growth of infected population.

6. Conclusions

The compartmental model for dengue fever with treat-
ment control measure [1] has been re-investigated and the

87



Tuteja & Lal / J. Nig. Soc. Phys. Sci. 3 (2021) 82–88 88

Figure 3: Exposed and Infected Vectors

Figure 4: Infected Human with different Treatment (τ)

Figure 5: Recovered Human with different Treatment (τ)

inadvertent errors have been removed from the recurrence
relations of the model equations due to DTM. The existence,
uniqueness and positivity of the solutions have been estab-
lished. The semi-analytical solutions of the model equations
are re-computed using the DTM and built-in function Asymp-
toticDSolveValue of Wolfram Mathematica and are found to
be the same. It has been found that results obtained from
the DTM are valid only for a small interval of time t (t < 1.5),
as t becomes large, the exposed, recovered human popula-
tion and infected vector population becomes negative. For
smaller t, the better the treatment is, recovery will be faster

Figure 5.

Acknowledgments

We thank the anonymous referees for the positive enlight-
ening comments and suggestions, which have greatly helped
us in making improvements to this paper.

References

[1] F. Y. Eguda, A. James & S. Babuba, “The Solution of a Mathematical
Model for Dengue Fever Transmission Using Differential Transforma-
tion Method”, Journal of Nigerian Society of Physical Sciences 1 (2019)
82.

[2] Dengue. Retrieved from World Health Organization:
https://www.who.int/immunization/diseases/dengue/en/(2021)

[3] Dengue-and-Severe-Dengue. Retrieved from World Health Organiza-
tion:
https://www.who.int/news-room/fact-sheets/detail/dengue-and-
severe-dengue (2021).

[4] A. Mathieu, N Hens, C Marais & B P Beutels, “Dynamic Epidemiological
Models for Dengue” PLOSONE 7 (2012) 1.

[5] Messina, J. P., Brady, O. J., Golding, N., Kraemer .et.al. The current and
future global distribution and the population at risk of dengue. Nature
Microbiology 4 (2019), 1508–1515.

[6] V Capasso & G Serio, “A generalization of the Kermack-McKendrick de-
terministic epidemic model”, Mathematical Biosciences 42 (1978) 43.

[7] W. O. Kermack & MA. G. McKendrick, “Contribution to the Mathemati-
cal Theory of Epidemics” (Part 1). Proceedings of Royal Society London
138A (1932) 55.

[8] W. O. Kermack & MA. G. McKendrick, “Contribution to the Mathemati-
cal Theory of Epidemics (Part II)”, Proceedings of Royal Society London
141 (1932) 94.

[9] W. O. Kermack & MA. G. McKendrick, “A contribution to the mathemat-
ical theory of epidemics”, Proceedings of the Royal Society of London A:
Mathematical, Physical and Engineering Science 115 (1927), 700.

[10] H Hethcote, “The Mathematics of Infectious Diseases”, SIAM Review,
42 (2000) 599.

[11] J K Zhou, “Differential Transformation and Its Applications for Electri-
cal Circuits”, Huazhong University Press, China 1986.

[12] P Olumuyiwa et. al. “Direct and Indirect Transmission Dynamics of Ty-
phoid Fever Model by Differential Transform Method”, ATBU Journal of
Science, Technology and Education, 6 (2018) 167.

[13] P. Olumuyiwa & M O. Ibrahim, “Application of Differential Transform
Method in Solving a Typhoid Fever Model”, International Journal Of
Mathematical Analysis And Optimization: Theory And Applications 1
(2017) 250.

[14] A. Abioye, M. O. Ibrahim, O. Peter, A. Sylvanus & F. Abiodun, “Differen-
tial Transform Method for Solving Mathematical Model of SEIR and SEI
Spread of Malaria”, International Journal of Sciences: Basic and Applied
Research 40 (2018) 197.

[15] J. A. Abraham, G. P. Gilberto & Benito M. Chen-Charpentier, “Dynami-
cal analysis of the transmission of seasonal diseases using the differen-
tial transformation method”, Mathematical and Computer Modelling
50 (2009), 765.

[16] M. Derouich & A. Boutayeb, “Dengue fever: Mathematical modelling
and computer simulation”, Applied Mathematics and Computation 177
(2006) 528.

[17] Z. N. Meksianis, N Anggriani & A. K. Supriatna, “Application of differ-
ential transformation method for solving dengue transmission mathe-
matical model”, AIP Conference Proceedings 2018.

[18] N. R. Derrick & S. L. Grossman, “Differential Equation with applica-
tions", Addison Wesley Publishing Company Inc., Philippines 1976.

88