Since the modelling of the seminal works on the scale-free network, in which the probability of for any node with links to other nodes is distributed according to the power law, suggested

by Barabá and Albert [1] , it is well known that the real disease transmission networks exhibit scale-free properties (see for example [2] [3] ), and the spreading of epidemic disease on scale-free network has been studied by many researchers [4] - [21] .

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 [18] . However, up to now, compared with studies of the dynamical behaviors of the epidemic models with time delays on hetergeneous network, only a few attentions have been paid to them on heterogeneous networks. Recently, Liu and Xu presented a delay differential equation SEIRS epidemic model with discrete time delays which represent the latent period and the immune period [19] . Liu and Deng et al. discussed epidemic SIS model with discrete time delay which represents the infectious period [20] , they obtained the basic reproduction number and discussed the persistence of the disease. Wang and Wang et al. discussed an epidemic SIR model with discrete time delay which represented the latent period [21] . Motivated by these, in this paper, we will present a suitable epidemic SIRS model with discrete delay which represents the infectious period on scale-free network using functional differential equations to investigate the epidemic spreading.

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, and be the relative density of susceptible nodes, infected nodes and recovered nodes of connectivity k at time t, respectively. Obviously, the following normalization condition holds due to the fact that the number of total nodes with degree k is a constant during the period of epidemic spreading.

The dynamical equation for the density, and, at the mean-field level, satisfy the following system when:

where is the correlated (k-dependent) infection rate such as [5] [8] [20] , [6] and so on, and represents the average infectious period. The term describes some infected nodes

may become susceptible nodes because they are recovered and are not immunized, where is called incomplete cure rate, and consequently, is cure rate. In reality, for example, some computer users find network virus and kill it after due to its destruction of the user, but they do not take further action to immunize computer from the network virus, but the other users can get the technique to protect their computer and their computer cannot be infected again. The dynamics of n groups of SIRS subsystems are coupled through the function, which represents the probability that any given link points to an infected site. Assume that the network has no degree correlations [4] , we have

where stands for the average node degree, and has many different forms, such as in [4] [5] , in [7] , , in [8] , and, in [9] and so on.

The initial condition of system (1) is

where are nonnegative continuous on, and and for. C denote the Banach space with the norm, and is Euclidean norm of.

Denote

where in which is a function.

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

Note that and can be replaced by in the first equation of system (1). Thus we obtain the following equivalent system of system (1):

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

Proof. Denote, and (some constants) and substitute them into (5), we have

where

It yields that

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

Obviously, always satisfies (9), it follows that from (8) that the disease-free equilibrium of system (5) always exists. Note that

and

Hence, if, the Equation (9) has a unique positive solution, consequently, system (5) has a unique positive equilibrium since (8) holds. is an unique endemic equilibrium.

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

(1) If, the equilibrium of system (5) is globally attractive.

(2) If, the disease is uniformly persistent, i.e., there exists a positive constant such that , and the equilibrium of system (5) is unstable.

Proof. First, According to the Equation (4), similar to the proof of Theorem 1 in [20] , we can obtain that the equilibrium of system (5) is globally attractive.

Second, motivated by the work in [22] , we will prove that conclusion (2) in Theorem 2 holds step by step, i.e., we prove that the disease is uniformly persistent when.

Step 1. We will prove that for any, it is impossible that for.

Since, there exists a small enough such that

in which

Suppose for, which implies

for. It follows from the first equation of system (5) that

for. Hence there exists a such that

for, and there a such that

It follows from (2) and (12) that

Set. We claim that for. If not, there exists a such that and for. It follow that

which leads to, contradicting. This proves the claim.

Choose a positive constant which satisfies. We claim now that for all. Note that

If the claim is not valid, there exists a such that and for.

Thus

which leads to, contradicting. This proves the claim. By induction method, we conclude that

for. It follows that if t is sufficiently large, contradicting. So, for any, it is impossible that for.

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

Since it is impossible that for. Hence, there are two cases to be considered for. Hence, there are two cases to be considered for.

Case 1: when t is sufficiently large.

Case 2: is oscillates about when t is sufficiently large.

Suppose and for, where is sufficiently large such that holds. Consequently, and for. is uniformly continuous since the positive solutions of system (4) are bounded. Hence, there is a (independent of the choice of) such that for. If, there is nothing to prove. If, for, we have from (15) that

Let us define, then for. If, by similar method in step 1, we can obtain that for. Thus, for case 2, when t is sufficiently large, and where.

Hence, when t is sufficiently large, where, consequently, , and the disease is uniformly persistent.

At last, since when, the equilibrium is unstable when. This completes the proof of Theorem 2.

The basic reproductive number for system (4) is. If, the disease will disappear due to the global attractivity of. If, the disease will always exists due to.

Remark. When system (1) reduce to system (4) in [20] , and the results still holds.

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 [8] [9] :

where m is the minimum connectivity of the network. It follows from (14) and (15) that, which depends both N and , can be approximately computed.

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, and C satisfies. Assume the network is a finite network with and are suitable assumptions. Let, , , , in which, , , and. Figures 1-3 show the dynamic behaviors of system (5).

From the dynamical behaviors of the SIRS model (5) shown in Figure l and Figure 2, it can be seen that if, the infection eventually disappears. If, the relative density of infected nodes will tend to a positive constant, and the infection will always exist. The numerical results are consistent with the theoretical results.

According to, if, the equilibrium is globally attractive and the disease eventually disappear. However, may hold as increases due to the fact is proportional to, that is to say, the equilibrium may lose its stability when the average infection period of disease is large enough and the infection will always exist (shown in Figure 3).

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. The main results reveal that when, the disease-free equilibrium is globally attractive, while, the disease-free equilibrium is unstable and the disease is uniformly persistent. In addition, numerical simulations show that the endemic equilibrium is globally asymptotically stable when (as shown in Figure 2). We would like to mention here that it is interesting but challenging to discuss the stability of equilibrium, we leave this for our future work.

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