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Dengue is one of the most rapidly spreading diseases in the world. It is transmitted to humans by the bite of infected aedes mosquitoes. In the present paper, the transmission dynamics of dengue disease in the presence and absence of awareness in host population is discussed. It is assumed that some hosts do not interact with the infected mosquitoes as they take different kinds of precautions due to their awareness towards the disease. Some susceptible population is supposed to avoid mosquito bites and some infected hosts are isolated so that they do not transmit the disease. A system of differential equations that models the population dynamics and the associated basic reproduction number are discussed in the paper. Stability analysis is made to determine the dynamical behavior of the system.
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International Journal of Advanced Engineering Research and Applications (IJAERA) ISSN: 2454-2377
Vol. 1, Issue 6, October 2015
www.ijaera.org
2015, IJAERA - All Rights Reserved
239
Mathematical Model of Dengue Fever with and
without awareness in Host Population
Ganga Ram Phaijoo1* & Dil Bahadur Gurung2
1Department of Natural Sciences (Mathematics), School of Science
Kathmandu University, Dhulikhel, NEPAL
*E-mail: gangaram@ku.edu.np
2Department of Natural Sciences (Mathematics), School of Science
Kathmandu University, Dhulikhel, NEPAL
E-mail: db_gurung@ku.edu.np
Abstract: Dengue is one of the most rapidly spreading diseases in the world. It is transmitted to humans
by the bite of infected aedes mosquitoes. In the present paper, the transmission dynamics of dengue
disease in the presence and absence of awareness in host population is discussed. It is assumed that
some hosts do not interact with the infected mosquitoes as they take different kinds of precautions due
to their awareness towards the disease. Some susceptible population is supposed to avoid mosquito
bites and some infected hosts are isolated so that they do not transmit the disease. A system of
differential equations that models the population dynamics and the associated basic reproduction
number are discussed in the paper. Stability analysis is made to determine the dynamical behavior of
the system.
Keywords: Dengue; Stability; Awareness; Basic reproduction number
I. INTRODUCTION
Dengue fever is an infectious vector borne disease spreading in tropical and subtropical
countries. Dengue is transmitted by aedes mosquitoes. Four serotypes of the dengue viruses DEN 1,
DEN 2, DEN 3 and DEN 4 cause the dengue fever. Nowadays, dengue fever is endemic in more than
hundred countries. In recent years, the number of dengue cases has been increasing dramatically.
Awareness towards the disease can change the whole dynamics of the transmission of the
disease. Due to awareness, people can take different kinds of precautions towards the disease. Avoiding
mosquito’s bite is the major precaution against dengue fever. Some of the precautions that can be taken
are: to keep home, environment and surrounding hygiene, to remove all stagnant water and containers,
to cover all containers properly to prevent dengue mosquito breeding there, to wrap all unused plastic
tyres, to use mosquito repellents to avoid mosquito bite, to use aerosols and mosquito coils to kill
mosquitoes, to wear long sleeve and fully covered clothes, to use mosquitoes net around bed while
sleeping etc.
To control the disease effectively, one should understand the dynamics of the disease
transmission and take all of the corresponding details into account. Kermack and McKendric
contributed on the development of the mathematical theory of epidemics [1]. In the paper, the authors
considered three compartments- Susceptible, Infectious and Removal for mathematical formulation of
the model. Esteva and Vargas made a study on the transmission of dengue fever with constant human
population [2] and variable vector population [3]. Edy and Supriatna developed a transmission model
for dengue fever restricting the dynamics for the constant host and vector populations, and reducing the
model to two-dimensional planar equations [4]. Different studies have been made to investigate the
dengue disease transmission [5 9]. The present paper considers SIR model with a fraction of
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susceptible host and infected host population aware of dengue disease transmission. It is supposed that
the populations do not interact with the mosquitoes.
II. FORMULATION OF THE MODEL
To study transmission process of the dengue fever, host population is divided into three
compartments susceptible, infected and recovered. People who are healthy and may potentially get
infected with dengue virus are considered to be in susceptible compartment, people who are infected
with dengue and are able to transmit the disease are considered to be in infected compartment and
people who have recovered from dengue disease are considered to be in recovered compartment.
The population of mosquitoes is divided into two compartments only, susceptible and infected
compartment: mosquitoes that may potentially become infected with dengue virus are considered to
be in susceptible compartment and mosquitoes that are infected with dengue and can transmit the
disease are considered to be in infected compartment. The recovered class in the mosquito population
does not exist as their infection period ends with their death.
Table 1. Description of state variables
:
Constant host (human) population size
s
h
:
Number of susceptibles in host population
i
h
:
Number of infectives in host population
r
h
:
Number of removals (recovered) in host population
n
m
:
Vector (mosquito) population size
s
m
:
Number of susceptibles in vector population
i
m
:
Number of infectives in the vector population
In the present model, the fraction p of susceptible host population aware of the disease
transmission who use mosquito repellent, mosquitoes net etc. to avoid the mosquito’s bite due to
awareness is considered. So, the population does not come in contact with the infected mosquitoes.
Also, a fraction q of infected hosts is isolated as a result of awareness. Thus, the present paper includes
the effect of awareness in the model proposed by Esteva and Vargas [2].
Table 2. The parameters used in the model and their dimensions
p
:
Fraction of susceptible host population aware of dengue transmission, Dimensionless
q
:
Fraction of infected host population aware of dengue transmission, Dimensionless
h
:
Birth/death rate in the host population, Time -1
m
:
Death rate in the vector population, Time -1
h
:
Transmission probability from vector to host, Dimensionless
m
:
Transmission probability from host to vector, Dimensionless
h
:
Recovery rate in the host population, Time -1
b
:
Biting rate of vector, Time -1
A
:
Recruitment rate, Mosquitoes
Time -1
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The systems of differential equations which describe the present model are;
For human population:
(1 )
(1 ) ( )
sn s i s
n
is i i
n
rir
hb
dh h h p h m h h
dt h
hb
dh p h m h h h
dt h
dh h h h h
dt



 
 

(1)
For vector population:
(1 )
(1 )
ss i s
n
is i i
n
mb
dm A q m h m m
dt h
mb
dm q m h m m
dt h
  
 
(2)
Also,
s i r n
si
h h h h
A
mmm
 

(3)
Above equations can be reduced to three equations
(1 )
(1 ) ( )
(1 )
sn s i s
n
is i i
n
ii i i
n
hb
dh h h p h m h h
dt h
hb
dh p h m h h h
dt h
q m b
dm Am h m m
dt h m


 
 

 



(4)
Introducing the proportions
,,
/
s i i
nn
h h m
x y z
h h A m
 
(5)
We obtain,
(1 )
(1 )
dx h x xz
dt
dy xz y
dt
dz z y z
dt


 

 
(6)
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where,
(1 ) , , (1 ) ,
n
bh A
p h h q bm m
mh
 
 
   
III. EQUILIBRIUM POINTS AND STABILITY ANALYSIS
A. Disease free equilibrium point
In a disease free situation,
0, 0yz
. Hence, from (6),
1.x
So, the disease free equilibrium
point is (1, 0, 0).
B. Basic Reproduction Number
The basic reproduction number is the expected number of secondary infections produced by an
index case in a completely susceptible population. According to the values of the basic reproduction
number, disease can persist with
01R
and the disease can die out when
01.R
Using the next generation method, we compute the basic reproduction number
0
R
, associated
with the disease free equilibrium (1, 0, 0). Using last two equations of the system of Equations (6), the
non-negative matrix
,F
of the infection terms and the non-singular matrix
V
, of the transition terms
are given, respectively [10] by:
0
0
F



and
0
0
V



And, therefore
10
0
FV






Eigenvalues of
1
FV
are


Therefore, the basic reproduction number,
0
R
= spectral radius of the matrix,
1
FV
=


.
Proposition 1 (Stability of disease free equilibrium point)
The disease free equilibrium point is asymptotically stable if
01R
and unstable if
01R
.
Proof: Jacobian matrix of system of equations (6) at the disease free equilibrium point (1, 0, 0) is
0
0
0
h
J









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Using the matrix
,J
we find the following characteristic equation:
22
0
( )[ ( ) (1 )] 0hR
  
 
(7)
From characteristic equation (7), it is observed that first eigenvalue is
h
which is negative.
Next,
22
0
( ) (1 ) 0R
  
 
(8)
The two conditions of Routh Hurwitz Criteria, for local asymptotical stability of second order
characteristic polynomial
212
0aa

 
are
i.
10a
ii.
20a
We have,
1
a h h m
 

 
which is always positive. Also,
2
20
(1 )aR


.
Here,

is positive. And
2
0
10R
if
01R
. So,
2
20
(1 ) 0aR

 
if
01R
.
Hence, if
01R
the eigenvalues will have negative real parts and the disease free equilibrium point
becomes asymptotically stable. If
01R
, the two eigenvalues of equation (8) are one negative and
one positive real number. So, the disease free equilibrium point becomes unstable if
01R
.
Proposition 2 (Existence of the Endemic equilibrium point)
The Endemic equilibrium point of the system of equations (6) exists if
01R
.
Proof: Solving the system of equations (6), the equilibrium points found are (1, 0, 0) and
( , , )
e e e
x y z
where
( ) ( )
( , , ) , ,
( ) ( ) ( )
e e e h h h
x y z h h h
 
 
     
    

 


  

The first point is disease free equilibrium point. The second point becomes endemic equilibrium
point and exists if
0
 

. Which implies that
1


and then
01R
.
Hence, the endemic equilibrium point exists if
01R
.
IV. NUMERICAL RESULTS AND DISCUSSION
In the present paper we explored the effect of awareness in the transmission of dengue disease taking
different values of awareness parameters p and q. For the value p = 0, q = 0 present model changes to
the model proposed by Esteva and Vargas [2].
To illustrate the dynamics of the dengue disease, the different values of the awareness fraction p and
q used are given in Table 3.
Table 3. Parameters and their values used
Parameters
h
m
h
m
b
n
h
A
,pq
values
0.0000457/day
0.14286/day
0.75
0.75
0.14286/day
1/3
10000
5000
Variable
references
[2]
[7]
[6]
[6]
[2]
[7]
[2]
[2]
---
Figure 1 is drawn for p = 0 = q (no awareness). Figures 2 and 3 (with awareness) are drawn for p =
0.5, q = 0.7 and p = 0.8, q = 0.9 and compared with the Figure 1. Figures 2 and 3 illustrate the effect
of awareness in the transmission of the dengue epidemic showing that there is small number of people
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infected of disease and large number of people remained susceptible when compared with Figure 1.
The size of infected mosquito population is seen smaller in Figure 2 and 3 than in Figure 1.
Figure 1. Dynamics of host and vector population with p
= 0 = q.
Figure 2. Dynamics of host and vector population with p
= 0.50, q = 0.70.
The infection rate is observed higher when there is no awareness (Figure 1) than when there is
awareness in people (Figure 2 and Figure 3). Also, the rate of infection is seen decreasing with the
increasing values of awareness parameters (Figure 2 and Figure 3). Figure 3 shows that when there is
sufficient number of people aware of the disease transmission, there is very less number of host and
vector infected of the disease. Thus, increasing awareness in people can decrease the infection rate and
consequently can decrease the number of infected population.
Figure 3. Dynamics of host and vector population with p =
0.80, q = 0.90.
Figure 4. Basic reproduction number with different
values of awareness terms p and q.
For p = 0 = q, the value of basic reproduction number,
03.27R
and for p = 0.8, q = 0.9, the value of
basic reproduction number,
00.46R
. Thus, the value of basic reproduction number is less when there
is no awareness and more in the presence of awareness in host population. Also, Figure 4 shows that
the value of basic reproduction number approaches unity and becomes less than unity for sufficiently
higher values of awareness parameters p and q.
V. CONCLUSION
A dengue model was studied by including the impact of the awareness parameters in the
transmission dynamics of dengue fever. An analysis of the disease transmission was made with and
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without the presence of awareness in the host population. A fraction of susceptible host population is
supposed not to come in contact with mosquitoes. The population is supposed to be aware of the
disease transmission and they are supposed to take all possible precautions such as using mosquito
repellent, mosquito net etc so that they can avoid the bite of mosquitoes. Also, a fraction of infected
host population is supposed not to transmit the disease as the population is isolated due to awareness.
Small values of basic reproduction number were appeared in the presence of higher levels of
awareness and large value of basic reproduction number was appeared in the absence of awareness in
host population. Hence, the spread of the disease comes under control with the increase in awareness
in host population.
For higher level of awareness, the disease is seen to affect less number of people and
mosquitoes. Large number of people is seen to be affected from the disease when there is no awareness
in the host population. So, the present study suggests that with the increase in the awareness in host
population, remarkable susceptible host population size can be saved from being infected. So, spread
of awareness about the disease transmission plays an important role in controlling the dengue disease
transmission.
VI. REFERENCES
[1] W. O. Kermack and A. G. McKendrick (1927). A contribution to the mathematical theory of epidemics ,
Proceedings of the Royal Society of London, vol. 115, pp. 700 - 721.
[2] L. Esteva and C. Vargas (1998). Analysis of a dengue disease transmission model, Math. Bio-sciences,
vol. 150, pp. 131 151.
[3] L. Esteva and C. Vargas (1999). A model for dengue disease with variable human population, J. Math.
Biol., vol. 38, pp. 220 240.
[4] E. Soewono and A. K. Supriatna (2001). A two dimensional model for the transmission of dengue fever
disease, Bull. Malaysian Math. Sc. Soc., vol. 24, pp. 49 57.
[5] P. Pongsumpun (2008). Mathematical model of dengue disease with the incubation period of virus, World
Academy of Sc. Engg. and Tech., vol. 44, pp. 328 332.
[6] S. T. R. Pinho, C. P. Ferreira , L. Esteva, F. R. Barreto, V. C. Morato e Silva and M. G. L. Teixeira
(2010). Modelling the dynamics of dengue real epidemics, Phil Trans. R. Soci., vol. 368, pp. 5679-5693.
[7] R. Kongnuy and P. Pongsumpun (2011). Mathematical Modeling for Dengue Transmission with the
Effect of Season, International journal of Mathematical, Computational, Physical, Electrical and
Computer Engineering, vol. 5, pp. 9 -13.
[8] S. Side and S. M. Noorani (2013). A SIR model for spread of dengue fever disease (simulation for south
Sulawesi, Indonesia and Selangor, Malaysia, world Journal of Modelling and Simulation, vol. 9, pp. 96
105.
[9] S. Gakkhar and N. C. Chavda (2013). Impact of awareness on the spread of dengue infection in human
population, Applied Mathematics, vol. 4, pp. 142 147.
[10] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz (1990). On the definition and computation of the
basic reproduction ratio
0
R
in models for infectious diseases in heterogeneous populations, Journal of
Mathematical Biology, vol. 28, pp. 365 382.
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