International J ournal of Computer Trends and Technology- volume3Issue1- 2012

ISSN: 2231-2803 http://www.internationaljournalssrg.org Page 174

Modelling and Simulation of the Dynamics of
the Transmission of Measles
E.A. Bakare
1,2
,Y.A. Adekunle
3
,K.O Kadiri
4

1
Department of Computer and Info Sc
Lead City University, Ibadan, Oyo State, Nigeria.
2
Department of Mathematics
University of Ibadan, Ibadan, Oyo State, Nigeria.
3
Department of Computer and Mathematics
Babcock University, Ilishan-Remo, Ogun State, Nigeria.
4
Department of Electrical/Electronics Engr
Federal Polythecnic, offa, Kwara State, Nigeria.

Abstract: We derive a compartmental
mathematical model of the dynamics of measles
within a particular population with variable size.
We used the compartmental model which we
expressed as a set of differential equations to see
the dynamics of measles infection. The stability of
the disease-free and endemic equilibrium is
addressed. Numerical Simulation are carried out.
We discussed in details the implications of our
analytical and numerical findings.

Keywords: measles, compartmental, differential
equations, Simulations, endemic equilibrium,
disease-free equilibrium, stability.

I.INTRODUCTION
Over the past one hundred years, mathematics has
been used to understand and predict the spread of
diseases, relating important public-health questions to
basic transmission parameters. From prehistory to the
present day, diseases have been a source of fear and
superstition. A comprehensive picture of disease
dynamics requires a variety of mathematical tools,
from model creation to solving differential equations
to statistical analysis. Although mathematics has been
so far done quite well in dealing with epidemiology
but there is no denying that there are certain factors
which still lack proper mathematization.
Infectious diseases pose a great challenge to both
humans and animals world-wide. Control and
prevention are therefore important tasks both from a
humane and economic point of views. Efficient
intervention hinges on complete understanding of
disease transmission and persistence (Finkenstadt et
al., 2002). Dynamic modeling of diseases has
contributed greatly to this (Anderson & May, 1991).
In this work we focus on measles, a childhood
disease. The Measles virus is a paramyxovirus, genus
Morbillivirus. Measles is an infectious disease highly
contagious through person-to-person transmission
mode, with > 90% secondary attack rates among
susceptible persons. It is the first and worst eruptive
fever occurring during childhood. It produces also a
characteristic red rash and can lead to serious and
fatal complications including pneumonia, diarrhea
and encephalitis. Many infected children
subsequently suffer blindness, deafness or impaired
vision. Measles confer life long immunity from
further attacks .
Measles is a viral respiratory infection that attacks
the immune system and is so contagious that any
person not immunized will suffer from the disease
when exposed. Measles virus causes rash, cough,
running nose, eye irritation and fever. It can lead to
ear infection, pneumonia, seizures, brain damage and
death (WHO,2005). Children under five years are
most at risk. Measles infects about 30 to 40 million
children each year and causing a mortality of over
500,000, often from complications related to
pneumonia, diarrhea and malnutrition
(WHO/UNICEF, 2001). Survivors are left with life-
long disabilities that include blindness, deafness or
brain damage. Available records revealed that in
2003 alone, 530,000 deaths were recorded in the
world as a result of measles (WER, 2005). Despite
the availability of measles vaccine for more than 40
years, many regions of the world are still being
plagued by the disease. In 1989, the World Health
Assembly set specific goals for the reduction in
measles morbidity and mortality (WHO, 1990),
resulting in the WHO/UNICEF measles mortality
reduction and regional elimination strategic plan
(WHO, 2005). Majority of measles deaths occur in
14 countries where immunization coverage for
children was reported to be less than 50 %. In 2005,
measles killed more than 500 children in Nigeria. Of
International J ournal of Computer Trends and Technology- volume3Issue1- 2012
ISSN: 2231-2803 http://www.internationaljournalssrg.org Page 175

the 23,575 cases recorded in 2005, more than 90%
were in Northern Nigeria, where people are wary of
vaccinations largely for religious reasons (WHO,
2005). Because measles is both an epidemic and
endemic disease, it is difficult to accurately estimate
its incidence on the global level, particularly in the
absence of reliable surveillance systems. Although
many counties reported the number of incident cases
directly to WHO, the heterogeneity of these systems
with differential underreporting does not permit an
accurate assessment of the global measles incidence.
In view of these difficulties, models have been used
to estimate the burden of measles. We realized that
finding the threshold conditions that determine
whether an infectious disease will spread or will die
out in a population remains one of the fundamental
questions of epidemiological modeling. For this
reason, there exists a key epidemiological quantity
R0, the basic reproductive number. R0 is the number
of secondary cases that result from a single infectious
individual in an entirely susceptible population.
Introduced by Ross in 1909, the current usage of R0
is the following : if R0 < 1, the modeled disease dies
out, and if R0 > 1, the disease spreads in the
population. Reproductive numbers turned out to be
an important factor in determining targets for
vaccination coverage. In mathematical models, the
reproductive number R0 is determined by the
dominant eigenvalue of the Jacobian matrix at the
infection-free equilibrium for models in a finite-
dimensional space.
In this paper we organized the section as follows :
Section II introduces the formulation of model using
study the dynamics of measles in the absence of any
intervention strategy. The reproductive numbers are
computed in Section III and the qualitative behavior
of the disease-free steady state is also studied. In
Section IV, we discuss the analysis of the model
Section V is devoted to Numerical simulations,
discussions about our results and next future work.

II. MODEL EQUATIONS
Following the classical assumption, we formulate a
deterministic, compartmental, mathematical model to
describe the transmission dynamics of measles. The
population is homogeneously mixing and reflects the
demography of a typical developing country, as it
experiments an exponential increasing dynamics.
Compartments with labels such as S,E, I, and R are
often used for the epidemiological classes. As most
mothers has been infected, IgG antibodies transferred
across the placenta, to newborn infants give them
temporary passive immunity to measles infection.
After the maternal antibodies remains in the body up
to nine months, we consider that the infant enters
directly in the susceptible class S at birth. So, all the
newborns were assumed to be susceptible. When
there is an adequate contact of a susceptible with an
infective so that transmission occurs, then the
susceptible enters the exposed class E of those in the
latent period, who are infected but not yet infectious.
After the latent period ends, the individual enters the
class I of infectives, who are infectious in the sense
that they are capable of transmitting the infection.
When the infectious period ends, the individual
enters the recovered class R consisting of those with!
permanent infection-acquired immunity, otherwise
passes away. We exclude vertical incidence in our
model, which means that the infection rate of
newborns by their mothers. Our model belongs to a
more general SEIR transmission model.
Therefore, we divide the population into five
compartments : S(t), E(t), I(t) and R(t) as susceptible,
exposed, infectious and the immune individuals,
where t represents the time. So, at time t an
homogeneous population of size N(t) is categorized
to disease status :
S(t) =Susceptible individuals
E(t) =Exposed individuals, but not yet infectious
I(t) =Infectious individuals; they can spread the
disease
R(t) =Recovered from disease or removed
If is the average number of adequate contacts (i.e.,
contacts sufficient for transmission) of a person per
unit time, then
[I
N
is the average number of contacts
with infective per unit time of one susceptible, and
[SI
N
is the number of new cases per unit time due to
the S(t) susceptible.
The parameters are defined as following :
= Contact rate
= rate of progression from exposed state to
Infectious state
=natural recovery rate
z = Birth rate
=Mortality rate

Thus, the differential equations for the deterministic
model are as follows:

dS
dt
=z
[SI
N
pS

dL
dt
=
[SI
N
(o +p)E

dI
dt
=oE (y +p)I

dR
dt
=yI pR

dN
dt
= z pN (N
x

)
International J ournal of Computer Trends and Technology- volume3Issue1- 2012
ISSN: 2231-2803 http://www.internationaljournalssrg.org Page 176

III. BASIC PROPERTIES
Since the model (1) monitors human populations,
all the associated parameters and state variables are
non-negative t 0(it is easy to show that the state
variables of the model remain non-negative for all
non-negative initial conditions). Consider the
biologically feasible region,
u=](S,E,I,R)
+
4
:N
x

Lemma 1: The closed set u is positively invariant
and attracting
Proof: Adding the four equations the model (1)
gives the rate of change of the total human
population:

dN
dt
= z pN. Thus, the total human population
(N) is bounded above by
x

, So that
dN
dt
=0 whenever
N(t) =
x

. Therefore, a standard comparison

Theorem[18,p.31] can be used to show that N(t) =
x

+(N
0

x

)c
-t
. In particular, N(t) =
x

. If N(0) =
x

,
hence , the region u attracts all solutions in
+
4
.
Since the region u is positively invariant and
attracting (Lemma 1), it is sufficient to consider the
dynamics of the flow generated by the model (1) in u
where the usual existence, uniqueness, continuation
results hold for the system(that is, the system(1) is
mathematically and epidemiologically well posed in
u).
IV.ANALYSIS OF THE MODEL
We consider the human population model, given
by the four systems of equations. Hence, it is
sufficient to consider the dynamics of the human
system in u.
A. Stability of the Disease-Free Equilibrium
(DFE)
The model equation (1.0) has a DFE given by n
0
=
(S

,E

,I

,R

) =(
x

,0,0,0) The local stability of will

be investigated using the next generation matrix
method. We calculate the next generation matrix for
the systems of equation(1.0) by enumerating the
number of ways that
1. new infections arise 2.The number of ways that
individuals can move but only one way to create an
infections
F = _
0
[S

N
0 0
_ V = [
y +p o
0 o +p

FI
-1
=_
[S

c
N(c+)(y+)
[S

N(y+)
0 0
_
It follows that the basic reproduction number of the
model (1.0) denoted by R
0
is given by
R
0
=
[S

c
N(c+)(y+)
=
[xc
N(c+)(y+)

The Jacobian of (1.0) at the equilibrium point
n
0
=(
x

,0,0,0) is
J(S

,E

,I

,R

) =

[I

p 0
[S

[I

(o +p)
[S

0 0 (y +p)
0 0 y

0
0
0
p

J(
x

,0,0,0) =

p 0
-[x
N
0 (o +p)
[x
N
0 0 (y +p)
0 0 y

0
0
0
p

In the absence of infection E

=I

=0 and the absence

of recovery R

= 0, the Jacobian of (1.0) at the

disease-free equilibrium n
0
=(
x

,0,0,0) is
Its eigen values are

|[ zI|=
_
_
p z 0
-[x
N
0 (o +p) z
[x
N
0 0 (y +p) z
0 0 y

0
0
0
p z
_
_

z
1
=p,z
2
= (o +p),z
3
= (y +p),z
4
= p
Theorem 1 : The Disease-Free Equilibrium(DFE)
of the model equation (1.0), given by n
0
, is locally
asymptotically stable(LAS) if R
0
<1, and unstable
if R
0
>1.Thus this theorem 1 implies that measles
can be eliminated for the human population (when
R
0
<1) if the initial sizes of the sub-populations of
the model equation are in the basin of attraction of
the DFE, n
0
.
Proof: As z
1
and z
2
are negative, we also see that
z
3
onJ z
4
are both negative too. We realized that,
using the Routh-Hurwitz theorem, it is the case when
z
3
+z
4
<0 and z
3
z
4
>0
i.e. z
3
+z
4
= (y +2p) <0 is truc, we also
have z
3
z
4
=(y +p)p >0 is also true.
International J ournal of Computer Trends and Technology- volume3Issue1- 2012
ISSN: 2231-2803 http://www.internationaljournalssrg.org Page 177

B. Global Stability of the DFE
We define, first of all the region

={(S,E,I,R)
+
4
:S S

}
Lemma 2: The region

is positively invariant
and attracting
Proof: It should be noted that the region u is
shown to be positively invariant and attracting
(Lemma 1).We simplify the equation(1.0)

dS
dt
=z
[SI
N
pS
z pS
=p(S

S)
Hence, S(t) S

(S

S(0))c
-t
(note that
dS
dt
<0 if S(t)>S

). Thus, it follows that either S(t)

approaches S

asymptotically, or there is some finite

time which S(t) S

. Thus the set

is attracting and
positively invariant.
We ascertain the following result
Theorem2: The DFE of the model equation(1.0)
given by n
0
is GAS (Globally Asymptotically Stable)
in

, whenever R
0
1.If R
0
>1 then the DFE is
unstable and EE is LAS.
Proof: Infection free equilibrium is LAS if R
0
<1,
infection or Disease free equilibrium is an unstable
saddle with vector into u when R
0
>1. We consider
the Liapunov function V =oE +(o +p)I
I

=
v
L
L
t
+
v
I
I
t
<0
I

=(o _
[SI
N
(o +p)] E +((o +p)oE
(y +p))I 0
I

=(o _
[SI
N
(o +p)] E +((o +p)oE
(y +p))I 0
I

=
[cSI
N
(y +p)I 0
Since
[SI
N
y +p
Where I

=0 is the face of u with I=0

dI
dt
=oE
I mo:cs o tis occ unlcss E =0
E=I=0
dR
dt
=pR R 0
E=I=R=0
dS
dt
=pS S 0 origin is the only
positive invariant subset of S
Implies that all paths in u approach the origin.
V. NUMERICAL SOLUTIONS AND
RESULTS
In this section we present a computer
simulation of some solution of the system(1.0). From
practical point of view, we realized that numerical
solutions are very important beside analytical study.
We take the parameters as follows:
= 18
= 2.6
= 3.2
=6.2
=4.52
(S(0), E(0), I(0), R(0)) =(4, 1, 1, 1) at time
t=0, over the time interval of [0, 10].
Mathematically, our result rely upon local
and global stability of the disease-free equilibrium
point. We have studied the local and global stability
of the disease-free equilibrium again by linearization,
Jacobian matrix and Routh Hurwitz condition. We
also looked at the global stability for SEIR
epidemiological models by Lyapunov functions. We
also determine the epidemiological threshold
conditions R
0
as an important public health interest.
Nevertheless, this approach has limits due to the
pattern of the transmission dynamics.

Fig 1.0 Shows the trends of the disease in S, E, I, R population as
times goes on.

REFERENCES
[1] Grais R.F., M.J . Ferrari, C. Dubray, O.N. Bjrnstad,B.T.
Grenfell, A. Djibo, F. Fermon, P.J . Guerin Estimating
transmission intensity for a measles epidemic in Niamey,
Niger: lessons for intervention. Transactions of the Royal
Society of Tropical Medicine and Hygiene (2006) (In
Press).
[2] Hethcote H.W. and Waltman P. Optimal vaccination
schedules in a deterministic epidemic model.Math. Biosci.
18 (1973), 365-382.
[3] Hethcote H.W. Optimal ages for vaccination for measles.
Math. Biosci. 89 (1989), 29 -52
[4] Hethcote H.W., 2000. The mathematics of infectious
diseases. Society for Industrial and Applied Mathematics
SiamReview, Vol. 42, No. 4, pp. 599653.
[5] Kaninda A.V., D. Legros, I.M. J ataou, P. Malfait, M.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
10
20
30
40
50
60
70
80
90
100
time
P
o
p
u
l
a
t
i
o
n

o
f

S
,

E
,

I
,

R


Susceptible
Exposed
Infected
Recovered
International J ournal of Computer Trends and Technology- volume3Issue1- 2012
ISSN: 2231-2803 http://www.internationaljournalssrg.org Page 178

Maisonneuve,C. Paquet and A. Moren Measles
vaccine effectiveness in standard and early immunization
strategies, Niger. Pediatr. Infect. Dis. J . 17 (1998), 1034
1039.
[6] Malfait, I.M. J ataou, M.C. J ollet, A. Margot, A.C.
DeBenoist and A. Moren. Measles epidemic in
the urban community of Niamey: transmission patterns,
vaccine efficacy and immunization strategies, Niger, 1990
to 1991. Pediatr. Infect. Dis. J . 13 (1994), 38 - 45.
[7] Mazer A., Sankale M., Guide de medecine en Afrique et
Ocean indien. EDICEF, Paris, 1988.
[8] Ousmane M.F., Mathematical model for control of
measles by vaccination(2006).
[9] World Health Organization, Department of vaccinces and
biologicals, 2001. Measles Technical Working Group:
strategies for measles control and elimination. Report of a
meeting, Geneva, 11-12 May 2000. Geneva, Switzerland :
World Health Organization.
[10] World Health Organization. Measles vaccines: WHO
position paper. Wkly. Epidemiology. Rec. 79, 130-140.
[11] World Health Organization, Department of Immunization
Vaccines and Biologicals, 2004. Vaccine Assessment and
Monitoring TeamImmunization Profile - Niger. Vaccines,
immunizations andbiologicalsp://www.who.int/immunization
monitoring/data/en/[accessed November 2005].