This paper addresses a time-delayed SIQRS model with a linear incidence rate. Immunity gained by experiencing the disease is temporary; whenever infected, the disease individuals will return to the susceptible class after a fixed period of time. First, the local and global stabilities of the infection-free equilibrium are analyzed, respectively. Second, the endemic equilibrium is formulated in terms of the incidence rate, and locally asymptotic stability. Finally we use the adomian decomposition method is applied to the system epidemiologic. This method yields an analytical solution in terms of convergent infinite power series.
Adomian Method; Epidemiology; Mathematical Model; The Epidemic Model; The Equilibrium Points1. Introduction
In the past, the epidemiology is restricted to the study of morbid phenomena resulting in an increase in sudden sharp and localized in space, the number of cases and time. It focuses primarily on infectious diseases. Epidemiology is a set of methods research by conducting investigations and a decision tool. Infectious diseases are one area where the theoretical were more developed in epidemiology. The mathematical theory of epidemics provides a framework for reconstruction history of past pandemics, contributing to a better understanding transmission mechanisms, a warning earlier vis-à-vis emergent phenomena, and now the prediction of epidemic spread in time and space. Generally, a model contains a disease-free equilibrium and one or multiple equilibria are endemic. The stability of a disease-free status equilibrium and the existence of other nontrivial equilibria can be determined by the ratio called the basic reproductive number, which quantifies the number of secondary infections arise from a simply put infected in a population of sensitive. When the basic reproduction number is less than unity, disease-free equilibrium is locally asymptotically stable, and therefore the disease off after a certain period of time. Similarly, when endemic equilibrium is global attractor, epidemiologically, it means that the disease will prevail and persist in a population and overall stability of these models is relatively low, especially models with delays.
In this paper, we discuss the equilibrium and stability of the model SIQR epidemic with constant infectious period which is made of a delay time. A particular assumption is made that the time which individuals remain infectious can be described by an exponential distribution. This distribution corresponds to the assumption that the chances of recovery in a given time interval are independent of time since infection. To solve the system is the autile decomposition method and for this we use the references of [1-14]. In order to describe the effects of disease immunity temporal delays are often incorporated in such models [15-19].
2. Model Equations
The system is described by equations which are defined as follows:
D’oùdenotes the population size at time; and denote the sizes of the population susceptible to disease, and infectious members, quarantine members and those who were removed from the possibility of infection through temporary immunity, respectively. It is assumed that all new borns are susceptible.
The positive constants and represent the death rates of susceptible, infectious and those who are being quarantiane and recovered, respectively. Biologically, it is natural to assume that. The positive constants and represent the birth rate (from insidence) of the population and the recovery rate of infection, respectively. The positive constant is the average number of contacts per infective. The positive constants are the numbers of transfers or conversions of infected people quarantined and quarantined at recovered. The term indicates that an individual has survived to natural death in a pool recovery before becoming susceptible again, where is the length of immunity period.
The initial condition of (1) is given as:
where such that
We have denote the Banach space of continuous functions mapping the interval into. By a biological meaning, we further assume that for
Since does not appear explicitly in the first three equations of (1), instead of (1) we consider the system:
With the initial condition
where, we obtain and
.
The region
is positively invariant set of (3).
3. The Disease Free Equilibrium and Its Stability
An equilibrium point of system (3) satisfies:
We calculate the points of equilibria in the absence and presence of infection.
In the absence of infection, substituting in the system we obtain the first equilibrium point:
We calculate the Jacobian matrix according to the system (3) with.
The epidemic is locally asymptotically stable if and only if all eigenvalues of the Jacobian matrix J have negative real part. The eigenvalues can be determined by solving the characteristic equation.
So the three eigenvalues are:
In order for and to be negative, it is required that
Then we define the basic reproduction number of the infection as follows
3.1. Stability of Equilibrium Point <img src="2-5300166\0e0b625b-4f45-49b6-94a2-bc577acbd6cc.jpg" />
The basic reproduction number is defined as the total number of infected population in the resulting subinfected population where almost all of the uninfected.
Theorem 1. The disease-free equilibrium is locally asymptotically stable if and unstable if.
3.2. Existence of Endemic Equilibrium and Its Locally Asymptotical Stability
From the previous section it is follows that when the trivial equilibrium of system (3) is locally asymptotically stable, then endemic equilibrium does not exist. When, system (3) has a unique non-trivial equilibrium other than the disease-free equilibrium.
In the presence of infection, substituting in the system we obtain the second equilibrium point:
Using the simplified and intuitive point theorem, Lyapunov square near the fixed point, and the solutions of the nonlinear system associated with the linear system by applying the Taylor formula of order 1.
Let
we note h, k, and m are the small perturbations. The formula for the Taylor series expansion.
We calculate the Jacobian matrix according to the system (3) with The epidemic is locally asymptotically stable if and only if all eigenvaluesof the Jacobian matrix J have negative real part. The eigenvalues can be determined by solving the characteristic equation.
Since, we have
Then (10) is then written:
The caracteristic equation is as follow:
with the notations:
When we have and The real parts of eigenvalues are négative, then the equilibrium is locally asymptotically stable.
With, system (3) has a unique non-trivial equilibrium is locally asymptotically stable.
4. The Adomian Decomposition Method
The Adomian decomposition method has been applied to broad classes of problems in many fields such as mathematics, physics, biology. This method solves the functional equations of different types, and the advantage of this method is that it solves a problem at the direct scheme, the solution is obtained as a series sounds fast converging. We consider the operator equation, when is the operator represents a general nonlinear ordinary differential and is a given function. The linear part of can be decomposed into, is easily invertible and R is the remainder of F. It is therefore assumed that the nonlinear problem can be written as
where represents the nonlinear terms (is a nonlinear operator).: is invertible (L is the derivative highest for what is supposed to be invertible). is a linear differential operator (of the order of less than L) and is the source term.
It can be written
Since L is invertible we also have
where is the solution of the homogeneous equation, with initial conditions. The decomposition of the nonlinear term, and to do so, Adomian developed a very elegant technique as follows. We define the parameter decomposition, then is a function of next expansion Maclurian series from.
We have in the form of a series,
We decompose the nonlinear term, as a series of special polynomials called Adomian polynomials,
These polynomials are obtained by introducing a parameter and writing,
we deduce that
we have:
To determine the we can use the following method,
Then
Finally a solution is given as
The exact solution is
4.1. The Resolution to the System with Adomian Decomposition Method
It has a direct application of the Adomian decomposition method system. We note that the system is a more general homogeneous system of ordinary differential equations where the nonlinear term is the product of two variables. We consider the general form of a system of differential equations given as follows.
We can write the system of equations above as operator with is the first nonlinear term and the second term is linear, differential operator
With applying the differential operator inverse we have
The solution is given as
The first nonlinear term is
With applying the differential operator inverse we have
The first linear term is
With applying the differential operator inverse about the Equation (18) on obtient:
For we have
So if we write the solution for each as follows
With applying the Adomian polynomial and then the general solution, is defined as follows
Solutions are given as follows:
4.2. Solve the System Using the Method (MADM)
The solution explicite
The coefficients are given with relations reccurence as follows
5. Conclusion
This paper examined the asymptotic stability of the disease-free equilibrium and endemic equilibrium equation. If, we proved that the disease-free equilibrium is globally asymptotically stable for any delay time, and if, it was proved that the endemic equilibrium is zero, and disease-free equilibrium becomes unstable. We also derived two sufficient conditions for local asymptotic stability of the endemic equilibrium and sufficient condition for asymptotic stability. We resolve the system with the the adomian decompostion method.
REFERENCESReferencesR. M. Anderson, et al., “A Preliminary Study of the Transmission Dynamics of the Human Immunodeficiency Virus (HIV), the Causative Agent of AIDS,” Mathematical Medicine and Biology, Vol. 3, No. 4, 1986, p. 229-263. doi:10.1093/imammb/3.4.229Y. Asif and K. Dogan, “A Numerical Comparison for Coupled Boussines Equations by Using the ADM,” Proceedings of Dynamical Systems and Applications, 5-10 July 2004, Antalya, pp. 730-736.N. T. J. Bailley, “Some Stochastic Models for Small Epidemics in Large Population,” Applied Statistics, Vol. 13, No. 1, 1964, pp. 9-19. doi:10.2307/2985218N. T. J. Bailley, “The Mathematical Theory of Infection Diseases and Its Application,” Applied Statistics, Vol. 26, No. 1, 1977, pp. 85-87. doi:10.2307/2346882M. S. Bartlett, “An Introduction to Stochastic Processes,” 3rd Edition, Cambridge University Press, Cambridge, 1978.B. Batiha, M. S. M. Noorani and I. Hashim, “Numerical Solutions of the Nonlinear Integro-Differential Equations,” International Journal of Open Problems in Computer Science, Vol. 1, No. 1, 2008, pp. 34-42.D. J. Evansa and K. R. Raslan, “The Adomian Decompositio Methode for Solving Delay Differential Equation,” International Journal of Computer Mathematics, Vol. 00, No. 0, 2004, pp.1-6.H. A. Zedan and Al-A. Eman, “Numerical Solutions for a Generalized Ito System by Using Adomian Decomposition Method,” International Journal of Mathematics and Computation, Vol. 4, No. S09. 2009, pp. 9-19.D. Kaya and Inc, “On the Solution of the Nonlinear Wave Equation by the Decomposition Method,” Bull. Malaysian Math. Soc. (Second Series) 22. 1999, p. 151-155.K. R. Raslan, “The Decomposition Methode for a Hirota-Satsuma Coupled KdV Equation and a Coupled MKdV Equation,” International Journal of Computer Mathematics, Vol. 81, No. 12, 2004, pp. 1497-1505.
doi:10.1080/0020716042000261405S. Pamuk, “An Application for Linear and Nonlinear Heat Equations by Adomian’s Decomposition Method,” Applied Mathematics and Computation, Vol. 163, No. 1, 2005, pp. 89-96. doi:10.1016/j.amc.2003.10.051T. M.-D. Syed, “On Numerical Solutions of Two-Dimensional Boussinesq Equations by Using Adomian Decomposition and He’s Homotopy Perturbation Method,” Applications and Applied Mathematics. An International Journal, No. 1, 2010, pp. 1-11.V. Makarov and D. Dragunov, “A Numeric-Analytical Method for Solving the Cauchy Problem for Ordinary Diferential Equations,” Applied Mathematics and Computation, 2010, pp. 1-26.L. Wu, F.-D. Zong and J.-F. Zhang, “Adomian Decomposition Method for Nonlinear Differential-Difference Equation,” Communications in Theoretical Physics, Vol. 48, No. 6, 2007, pp. 983-986.
doi:10.1088/0253-6102/48/6/004K. Wang, W. Wang, X. Liu, “Viral Infection Model with Periodic Lytic Immune Response,” Chaos, Solitons & Fractals, Vol. 28, No. 1, 2006, pp. 90-99.
doi:10.1016/j.chaos.2005.05.003W. Wang, “Global Behavior of an SEIRS Epidemic Model with Time Delays,” Applied Mathematics Letters, Vol. 15, No. 4, 2002, pp. 423-428.
doi:10.1016/S0893-9659(01)00153-7D. Greenhalgh, Q. J. A. Khanand and F. I. Lewis, “Recurrent Epidemic Cycles in an Infectious Disease Model with a Time Delay in Loss of Vaccine Immunity,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 63, No. 5-7, 2005, pp. 779-788. doi:10.1016/j.na.2004.12.018G. Li and J. Zhen, “Global Stability of a SEIR Epidemic Model with Infectious Force in Latent, Infected and Immune Period,” Chaos, Solitons & Fractals, Vol. 25, No. 5, 2005, pp. 1177-1184. doi:10.1016/j.chaos.2004.11.062Y. N. Kyrychko and K. B. Nlyuss, “Global Properties of a Delayed SIR Model with Temporary Immunity and Nonlinear Incidence Rate,” Nonlinear Analysis: Real World Applications, Vol. 6, No. 3, 2005, pp. 495-507.
doi:10.1016/j.nonrwa.2004.10.001
denotes the population size at time
;
and 
denote the sizes of the population susceptible to disease, and infectious members, quarantine members and those who were removed from the possibility of infection through temporary immunity, respectively. It is assumed that all new borns are susceptible.
and 
represent the death rates of susceptible, infectious and those who are being quarantiane and recovered, respectively. Biologically, it is natural to assume that
. The positive constants 
and 
represent the birth rate (from insidence) of the population and the recovery rate of infection, respectively. The positive constant 
is the average number of contacts per infective. The positive constants 
are the numbers of transfers or conversions of infected people quarantined and quarantined at recovered. The term 
indicates that an individual has survived to natural death in a pool recovery before becoming susceptible again, where 
is the length of immunity period.
such that
denote the Banach space 
of continuous functions mapping the interval 
into
. By a biological meaning, we further assume that 
for 
does not appear explicitly in the first three equations of (1), instead of (1) we consider the system:
we obtain 
and
.
, substituting in the system we obtain the first equilibrium point:
.
and 
to be negative, it is required that
as follows
is defined as the total number of infected population in the resulting subinfected population where almost all of the uninfected.
is locally asymptotically stable if 
and unstable if
.
of system (3) is locally asymptotically stable, then endemic equilibrium does not exist. When
, system (3) has a unique non-trivial equilibrium 
other than the disease-free equilibrium.
, substituting in the system we obtain the second equilibrium point:
The epidemic is locally asymptotically stable if and only if all eigenvaluesof the Jacobian matrix J have negative real part. The eigenvalues can be determined by solving the characteristic equation.
, we have
we have 
and 
The real parts of eigenvalues are négative, then the equilibrium 
is locally asymptotically stable.
, system (3) has a unique non-trivial equilibrium 
is locally asymptotically stable.
, when 
is the operator represents a general nonlinear ordinary differential and 
is a given function. The linear part of 
can be decomposed into
, 
is easily invertible and R is the remainder of F. It is therefore assumed that the nonlinear problem can be written as
represents the nonlinear terms (
is a nonlinear operator).
: is invertible (L is the derivative highest for what is supposed to be invertible). 
is a linear differential operator (of the order of less than L) and 
is the source term.
is the solution of the homogeneous equation
, with initial conditions. The decomposition of the nonlinear term
, and to do so, Adomian developed a very elegant technique as follows. We define the parameter 
decomposition, then 
is a function of 
next expansion 
Maclurian series from
.
in the form of a series,
as a series of special polynomials called Adomian polynomials,
and writing,
we can use the following method,
is the first nonlinear term and 
the second term is linear, 
differential operator 
we have
is given as
we have
about the Equation (18) on obtient:
we have
as follows
are given as follows:
, we proved that the disease-free equilibrium is globally asymptotically stable for any delay time, and if
, it was proved that the endemic equilibrium is zero, and disease-free equilibrium becomes unstable. We also derived two sufficient conditions for local asymptotic stability of the endemic equilibrium and sufficient condition for asymptotic stability. We resolve the system with the the adomian decompostion method.