Modelling and Simulation of the Dynamics of the ... - Semantic Scholar

International Journal of Computer Trends and Technology- volume3Issue1- 2012

Modelling and Simulation of the Dynamics of the Transmission of Measles E.A. Bakare 1,2,Y.A. Adekunle3,K.O Kadiri4 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

ISSN: 2231-2803

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 lifelong 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

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International Journal of Computer Trends and Technology- volume3Issue1- 2012 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 finitedimensional 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.

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 β is the average number of adequate contacts (i.e., contacts sufficient for transmission) of a person per unit time, then is the average number of contacts with infective per unit time of one susceptible, and is the number of new cases per unit time due to the S(t) susceptible. The parameters are defined as following : β = Contact rate σ= rate of progression from exposed state to Infectious state γ =natural recovery rate = Birth rate µ =Mortality rate

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

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

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=

−( + )

=

−( + )

= =

− −

(N→ )

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International Journal of Computer Trends and Technology- volume3Issue1- 2012 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, = ( , , , )∈ℜ : → Lemma 1: The closed set is positively invariant and attracting Proof: Adding the four equations the model (1) gives the rate of change of the total human population: = − . Thus, the total human population (N) is bounded above by , So that N(t) =

. Therefore,

= 0 whenever

a standard comparison

Theorem[18,p.31] can be used to show that ( ) = +( − ) . In particular, N(t) = . If N(0) = , hence , the region attracts all solutions in ℜ . Since the region is positively invariant and attracting (Lemma 1), it is sufficient to consider the dynamics of the flow generated by the model (1) in where the usual existence, uniqueness, continuation results hold for the system(that is, the system(1) is mathematically and epidemiologically well posed in ). 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 . A. Stability of the Disease-Free Equilibrium (DFE) The model equation (1.0) has a DFE given by = ( ∗ , ∗ , ∗ , ∗ ) = ( , 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 V = 0 + 0 0 ∗

=

(

)(

∗

)

(

0

)

0

It follows that the basic reproduction number of the model (1.0) denoted by is given by

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∗

=

(

=

)(

(

)

)(

)

The Jacobian of (1.0) at the equilibrium point =( , 0,0,0) is J( ∗ ,

∗

, ∗,

− ⎛ ⎜

∗

∗ ∗

)

=

− ∗

0 −( + )

∗

0 0

⎝

−

0 0

− ⎛ J( , 0,0,0) = ⎜ 0 0 ⎝0

∗

0 0⎞ ∗ 0⎟ −( + ) − ⎠ ∗ ∗

0

0 0⎞ −( + ) 0⎟ 0 −( + ) − ⎠ 0

In the absence of infection ∗= ∗ = 0 and the absence of recovery ∗ = 0, the Jacobian of (1.0) at the =( , 0,0,0) is

disease-free equilibrium Its eigen values are

− − | −

0

|= 0 0

=− ,

0 −( + ) − 0 −( + ) − 0

= −( + ),

0 0 0 − −

= −( + ),

= −

Theorem 1 : The Disease-Free Equilibrium(DFE) of the model equation (1.0), given by , is locally asymptotically stable(LAS) if < 1, and unstable if > 1.Thus this theorem 1 implies that measles can be eliminated for the human population (when < 1) if the initial sizes of the sub-populations of the model equation are in the basin of attraction of the DFE, . Proof: As and are negative, we also see that are both negative too. We realized that, using the Routh-Hurwitz theorem, it is the case when + < 0 and >0 i.e. + = −( + 2 ) < 0 , we also have = ( + ) > 0 is also true.

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International Journal of Computer Trends and Technology- volume3Issue1- 2012 B. Global Stability of the DFE We define, first of all the region Ω∗ = {( , , , ) ∈ ℜ : ≤ ∗ } Lemma 2: The region Ω∗ is positively invariant and attracting Proof: It should be noted that the region is shown to be positively invariant and attracting (Lemma 1).We simplify the equation(1.0) = − − ≤ − = ( ∗− ) Hence, S(t) ≤ ∗ − ( ∗ − (0)) (note that ∗ < 0 if S(t)> ). Thus, it follows that either S(t) approaches ∗ asymptotically, or there is some finite time which S(t)≤ ∗ . Thus the set Ω∗ is attracting and positively invariant. We ascertain the following result Theorem 2: The DFE of the model equation(1.0) given by is GAS (Globally Asymptotically Stable) in Ω∗ , whenever ≤ 1.If > 1 then the DFE is unstable and EE is LAS. Proof: Infection free equilibrium is LAS if < 1, infection or Disease free equilibrium is an unstable saddle with vector into when > 1. We consider the Liapunov function V = +( + ) ̇ = +