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A new epidemic SIRS model with discrete delay on scale-free network is presented. We give the formula of the basic reproductive number for the model and prove that the disease dies out when the basic reproductive number is less than unity, while the disease is uniformly persistent when the basic reproductive number is more than unity. Numerical simulations are given to demonstrate the main results.

Since the modelling of the seminal works on the scale-free network, in which the probability of

by Barabá and Albert [

Continuous time deterministic epidemic models are traditionally formulated as systems of ordinary differential equations. More realistic models should include some peat states of these systems, and ideally, a real system should be modeled by delay differential equation. Time delay plays an important role in propagation process of the epidemic, we can simulate the latent period of infectious diseases, the infections period of patients and the immunity period of recovery of the disease with time delay. Much attention has been given to the dynamical behaviors of the epidemic spreading model with time delay on homogeneous network [

The rest of this paper is organized as follows: The SIRS model on scale-free network with discrete delay is presented in Section 2. The basic reproductive number is given and dynamical behaviour of the system is analyzed in Section 3. Numerical simulations are given to demonstrate the main results in Section 4. Conclusion is finally drawn in Section 5.

Suppose that the size of the network is a constant N during the period of epidemic spreading, we also suppose that the degree of each degree is time invariant. Let

The dynamical equation for the density

where

may become susceptible nodes because they are recovered and are not immunized, where

where

The initial condition of system (1) is

where

Denote

where

Obviously, the second equation of system (1) can be furthermore transformed into the following integral equation:

Note that

Theorem 1. The system (5) has always a disease-free equilibrium

Proof. Denote

where

It yields that

Substituting it into (7), we obtain the self-consistency equality

Obviously,

and

Hence, if

Theorem 2. Consider the system (5), the following assertions hold.

(1) If

(2) If

Proof. First, According to the Equation (4), similar to the proof of Theorem 1 in [

Second, motivated by the work in [

Step 1. We will prove that for any

Since

in which

Suppose

for

for

for

It follows from (2) and (12) that

Set

which leads to

Choose a positive constant

If the claim is not valid, there exists a

Thus

which leads to

for

Step 2. We will prove that there exists a positive constant

Since it is impossible that

Case 1:

Case 2:

Suppose

Let us define

Hence,

At last, since

The basic reproductive number for system (4) is

Remark. When

The basic reproductive number for system (1) is

Note that an epidemic always occurs on a finite networks in the real world, the maximum connectivity n of any node is related to the network age, which is measured as the number of nodes N [

where m is the minimum connectivity of the network. It follows from (14) and (15) that

Now we present the results of numerical simulations by using MATLAB 7.0 to support the results obtained in previous sections. Since the equilibria were obtained from system (5), the simulations are based on system (5)

and a scale-free network in which the degree distribution is

From the dynamical behaviors of the SIRS model (5) shown in

According to

An SIRS model with discrete delay has been proposed for investigating the dynamical behaviors of the epidemics on scale-free networks. Through mathematical analysis, we obtained the basic reproduction number

TaoLi,QimingLiu,BaochenLi, (2015) The Analysis of an SIRS Epidemic Model with Discrete Delay on Scale-Free Network. Applied Mathematics,06,1939-1946. doi: 10.4236/am.2015.611171