_{1}

^{*}

In this paper, we present an SEIQRS epidemic model with non-linear incidence function. The proposed model exhibits two equilibrium points, the virus free equilibrium and viral equilibrium. The model stability is connected with the basic reproduction number
*R _{0}*. If

*R*< 1 then the virus free equilibrium point is stable locally and globally. In the opposite case

_{0}*R*> 1, then the model is locally and globally stable at viral equilibrium point. Numerical methods are used for supporting the analytical work.

_{0}The Malicious objects are harmful codes that reproduce and spread by way of internet [_{j} RS E-Epidemic model for various groups of infection in computer networking [

In this paper, we present a Propagation Model with non-linear incidence function, (susceptible, exposed, infected, quarantined, and recovered) which exhibits two equilibria the virus free and viral equilibrium point. The stability of both equilibrium points is connected with the threshold quantity. On the off chance that its value is less then unity then the virus free equilibrium point is locally and additionally globally asymptotically stable and in the opposite case of the threshold quantity, the same condition of stability is satisfied for viral equilibrium point.

The rest of the paper is set as follows, in Section 2 we shall formulate the new model, in Section 3 the basic propertied is discussed like reproduction number, virus free and viral equilibrium. In Section 4, we shall study the local and global stability of the virus free equilibrium point. In Section 5, we shall examine the local and global stability at viral equilibrium point. Finally, we support our analytical work with numerical simulations.

As usual the computer it either internal or external (i.e. connected with the internet or not). In this model we divide the internal computers into five states. Susceptible, exposed, infected, quarantined and recovered. Where the variable S(t), E(t), I(t), Q(t) and R(t) denote the sizes of nodes at time t in the states susceptible, exposed, infectious, quarantined and recovered respectively. N(t) the total number of computer at time t, where N(t) = S(t) + E(t) + I(t) + Q(t) + R(t). The governed model is given below:

With Initial conditions

where

We suppose that the like transmission rate in the form of

In this subsection, we shall investigate the feasible region and the threshold quantity for the proposed model (1). Where the size of total population is represented by N satisfy the following equation.

and then

Therefore, for the system (1) the positively invariant feasible region is given below.

Thus, the solution with initial condition will be analyzes inside the feasible region

For viral free equilibrium point we take

To investigate the reproduction Number for system (1) we use the next generation method [

Assume that

where

The spectral radius of the matrix

Now we examine the virus equilibrium point for the given system (1) and denoted by

where

Now by substituting the above mention values of

we get

When

Next to take the derivative of

Since

Therefore,

Proposition 3.1. Suppose that

In this section, we shall study the local and global stability of the given system (1) at virus free equilibrium point

Theorem 4.1. The given system (1) is locally asymptotically stable if

Proof:

To examine the local stability of the given system (1) we construct the jacobian matrix at

With row operation we can get the characteristic equation of the above jacobian matrix is

The first three roots of the above equation is

After a little algebraic calculation we can get the following equation

where

Theorem 4.2. The system (1) at

Proof: To examine the global stability of the proposed model consider a Lyapunov function as below:

Taking the derivative i.e.

Since for

In this section we will discuss the local as well as the global stability of the system (1) for the viral equilibrium point.

Theorem 5.1. The given system (1) at

Proof: Toinvestigate the local stability of the given system (1) we find the jacobian matrix at

Trace of the above jacobian Matrix

And after some row operation we get the following matrix

Thus, the system (1) at

In this segment we should look at the global stability of the given system (1) at viral equilibrium point. We use the method presented in [

Consider

where

・

・

We know that equilibrium point

If a neighborhood

Lemma 5.1. Assume that conditions

The accompanying Bandixson criterion is displayed in [

where

The matrix

And

It is shown in [

Lemma 5.2. Let the simply connected set

Now for the analysis of global stability at viral equilibrium we follow the method presented by Li and Muldowney in [

Theorem 5.2. The viral equilibrium

Proof: For the global stability of the system (1) we find the second additive compound matrix

Let us choose a function

Then,

So,

Therefore,

Let

where,

Suppose the norm in

where

where,

Therefore,

using second equation of system (1)

Again,

where,

And

using third equation of system (1),

With a compact absorbing set

Thus the viral equilibrium

Next we talk about the given subsystem of system (1)

And it limit system is

Based on (6), we get

This implies that as

Then according to [

The aim of this work is to study and analyze the dynamic behavior of an epidemic model SEIQRS with a nonlinear incidence function. We consider a mathematical model of the type SEIQRS and obtained the basic reproduction number, to determine its dynamical behavior. In epidemiology, the reproduction ratio is very important, because the stability of the proposed model is associated with reproduction ratio. For virus free equilibrium point, the model is stable locally and globally if

The purpose of this section is to support the analytic results mentioned in above work are supported through numerical results. Numerical results we choose different values of parameters, which we discuss below with the help of graph. For

Thus we can write

If

Thereby,

Nothing that when

age

The author is grateful to the anonymous reviewers for their constructive suggestions that greatly improve the quality of this paper.

QaisarBadshah, (2015) Global Stability of SEIQRS Computer Virus Propagation Model with Non-Linear Incidence Function. Applied Mathematics,06,1926-1938. doi: 10.4236/am.2015.611170