Global Dynamics of an SIRS Epidemic Model with Distributed Delay on Heterogeneous Network

Copyright © 2017 Qiming Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A novel epidemic SIRS model with distributed delay on complex network is discussed in this paper. The formula of the basis reproductive number R0 for the model is given, and it is proved that the disease dies out when R0 < 1 and the disease is uniformly persistent when R0 > 1. In addition, a unique endemic equilibrium for the SIRS model exists when R0 > 1, and a set of sufficient conditions on the global attractiveness of the endemic equilibrium for the system is given.


Introduction
Following the seminal work on small-world network by Watts and Strogatz [1], and the scale-free network, in which the probability of () for any node with  links to other nodes is distributed according to the power law () =  − (2 <  ≤ 3), suggested by Barabási and Albert [2], the spreading of epidemic disease on heterogeneous network, that is, scalefree network, has been studied by many researchers .
Compared with the ordinary differential equation (ODE) models (see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein), more realistic models should be retarded functional differential equation (RFDE) models which can include some of the past states of these systems.Time delay plays an important role in the process of the epidemic spreading; for instance, the incubation period of the infectious diseases, the infection period of infective members, and the immunity period of the recovered individuals can be represented by time delays [24].However, less attention has been paid to the epidemic models with time delays on heterogeneous network [19][20][21][22].
Zou et al. constructed a delayed SIR model without birth rate and death rate on scale-free network [19].In the model, the discrete delay in model represents the incubation period in 2011.In 2014, Liu et al. also presented a delayed SIR model with birth rate and death rate on scale-free network.In this model, the discrete delay also represents the incubation period during which the infectious agents develop in the vector [21].However, the assumption that the incubation period of an infective vector is determinate is somewhat idealized.And it is interesting to discuss the spreading of disease by using functional differential equation model with distributed delay [25].Motivated by the work of Zou et al. [19] and Wang et al. [22], considering the fact the immune individual may become the susceptible individual [15], we will present a novel functional differential equation  model with distributed delay on heterogeneous network in this paper to investigate the epidemic spreading, where the distributed delay represents the incubation period of an infective vector.
We consider the whole population and their contacts on network in which every individual is considered as a node in the network.Suppose the size of the network is a constant  during the period of epidemic spreading; we also suppose that the degree of each node is time invariant; let   (),   (), and   () be the relative density of susceptible nodes, infected nodes, and recovered nodes of connectivity  at time , respectively, where  = , +1, . . .,  in which  and  are the minimum and maximum number of contact each node, respectively.
In the process of the epidemic propagation via vector (such as mosquito), when a susceptible vector is infected by an infected nodes, there is a delay  during which the 2 Mathematical Problems in Engineering infectious agents develop in the vector, and infected vector becomes itself infectious after the delay.At the same time, the vector's usual activities are in a limited range; that is, if a vector is infected by an infected node, its usual activities are in the vicinity of the infected node.Furthermore, if the vector population size is large enough, we can suppose that the number of the infectious vector population in the vicinity of the infected nodes with degree  ( = 1, 2, . . ., ) at any time  is simply proportional to the number of the infected nodes with degree  at time  −  [25,26].Let the kernel function () denote the probability that an susceptible vector who is infected at time  −  and becomes infective at time .Meanwhile, let () be the correlated (-dependent) infection rate such as  and () [11].The susceptible nodes may acquire temporary immunity and the removal rate from the susceptible nodes to the recovered nodes is given by .And  is removal rate from the recovered nodes to the susceptible nodes because the recovered nodes lose the temporary immunity.In addition, the infected nodes are cured with rate .The dynamical equations for the density   (),   (), and   (), at the mean-field level, satisfy the following set of functional differential equations when  > 0: with   () +   () +   () = 1,  = ,  + 1, . . ., , (2) due to the fact that the number of total nodes with degree  is a constant () during the period of epidemic spreading.The dynamics of  groups of  subsystems are coupled through the function Θ(), which represents the probability that any given link points to an infected site.Assuming that the network has no degree correlations [3,11], we have where ⟨⟩ = ∑  () stands for the average node degree and () =   /(1 +   ) [7] (0 ≤  < 1,  > 0,  ≥ 0) denotes an infected node with degree  occupied edges which can transmit the disease.If  ̸ = 0, () gradually become saturated with the increase of degree , that is, lim →+∞ () = /.
It can be verified that solutions of system (1) in  with initial conditions above remain positive for all  ≥ 0.
The rest of this paper is organized as follows.The dynamical behaviors of the  model with distributed delay are discussed in Section 2. Numerical simulations and discussions are offered to demonstrate the main results in Section 3.

Dynamical Behaviors of the Model
Since   () +   () +   () = 1, system (1) is equivalent to the following system (8): Thus we only discuss system (8) if we want to discuss the dynamical behaviors of system (1). Denote where ⟨()⟩ = ∑  ()() in which () is a function.Note that we can obtain from the first equation of system (8) that By the standard comparison theorem in the theory of differential equations, we have lim Hence we know is positively invariant with respect to system (8), and every forward orbit in  Proof.Obviously, the disease-free equilibrium  0 of system (8) always exists.Now we discuss the existence of the endemic equilibrium of system (8).Combined with where From ( 13), we obtain that Substituting it into (14), we obtain the self-consistency equality (16) and it can be verified that (15) has a unique positive solution when  0 > 1 using the same proof as for Theorem 1 in [21]; consequently, system (8) has a unique endemic equilibrium  * since ( 13) and ( 15) hold.
Proof.Obviously, we need only discuss global attractiveness of system (8) in  0 .
Hence, the infection is uniformly persistent according to Lemma 3; that is, there exists a  is a positive constant such that lim →∞ inf   > , and the disease-free equilibrium  0 is unstable accordingly.This completes the proof.
At last, let us discuss the global stability of the endemic equilibrium of system (8) by constructing suitable Lyapunov function.
Note that the endemic equilibrium of system (1) satisfies We have from ( 29) and (30) that Let us consider (32) Calculating the derivative of   () along solution of (31), we get In addition, the matrix is irreducible, so the following matrix is irreducible: Hence there exists a positive vector  = ( )) = 0.

Numerical Simulation and Discussion
The basic reproductive number for system (8) (or (1)) is The equilibrium  0 is globally attractive and the infection eventually disappears when  0 < 1, and the infection will always exist when  0 > 1.Note that  0 is irrelevant to the distributed delay.Extensive numerical simulations are carried out on scalefree model to demonstrate the mentioned theorems above.The simulations are based on system (8) and a scale-free networks in which the degree distribution is () =  − , and  satisfies ∑  = () = 1.Supposing the network is finite one, the maximum connectivity  of any node is related to the network age, measured as the number of nodes  [3,7]: (47) Obviously, () is the relative average density of the infected nodes.