Abstract
A novel epidemic model with distributed delay on complex network is discussed in this paper. The formula of the basis reproductive number for the model is given, and it is proved that the disease dies out when and the disease is uniformly persistent when . In addition, a unique endemic equilibrium for the model exists when , and a set of sufficient conditions on the global attractiveness of the endemic equilibrium for the system is given.
1. 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 , suggested by Barabási and Albert [2], the spreading of epidemic disease on heterogeneous network, that is, scale-free network, has been studied by many researchers [3–23].
Compared with the ordinary differential equation (ODE) models (see [3–18] 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 [19–22].
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 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 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 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 :with 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 , 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 [7] denotes an infected node with degree occupied edges which can transmit the disease. If , gradually become saturated with the increase of degree , that is, .
The kernel function is nonnegative and continuous on and satisfies where is a positive number. And there are many types of kernel functions such as(1)the gamma distribution , where is a real number and is an integer, especially when , , and then ,(2)the uniform distribution where is real number,(3)the Delta-distribution , where is real constant.
Define the following Banach space of fading memory type (see [27] and references therein): with norm , and let be such that .
Consider system (1) in phase space . Standard theory of functional differential equation implies system (1) has a unique solution satisfying the initial conditions where .
It can be verified that solutions of system (1) in with initial conditions above remain positive for all .
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.
2. Dynamical Behaviors of the Model
Since , 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 Hence we know is positively invariant with respect to system (8), and every forward orbit in eventually enters .
Theorem 1. System (8) has always a disease-free equilibrium . System (8) has a unique endemic equilibrium when .
Proof. Obviously, the disease-free equilibrium of system (8) always exists. Now we discuss the existence of the endemic equilibrium of system (8). Combined with , it is easy to know that the equilibrium satisfies where From (13), we obtain that Substituting it into (14), we obtain the self-consistency equality and it can be verified that (15) has a unique positive solution when using the same proof as for Theorem 1 in [21]; consequently, system (8) has a unique endemic equilibrium since (13) and (15) hold.
Theorem 2. If , the disease-free equilibrium of system (8) is globally attractive.
Proof. Obviously, we need only discuss global attractiveness of system (8) in .
Consider the following Lyapunov function where Calculating the derivative of along solution of (8), for , we get Thus when , and if and only if . Note that the fact means ; moreover, ; the largest invariant set of is a singleton . Hence the disease-free equilibrium is globally attractive when according to the LaSalle Invariance Principle [28, Chapter 2, Theorem 5.3].
Lemma 3 (see [28, p273–280]). Let be a complete metric space, , where , assumed to be nonempty, is the boundary of . Assume the semigroup on satisfies , and(i)there is a such that is compact for ;(ii) is point dissipative in ;(iii) is isolated and has an acyclic covering .Then is uniformly persistent if and only if, for each , where , and is the omega limit set of through , and is global attractor of in in which .
Theorem 4. For system (8), if , the disease-free equilibrium is unstable, and the disease is uniformly persistent; that is, there exists a positive constant such that .
Proof. Denote and consequently, where .
Let be the solution of (8) with initial function and Obviously, and are positively invariant set for . is completely continuous for . Also, it follows from , for that is point dissipative. is the unique equilibrium of system (8) on and it is globally stable on , , and is isolated and acyclic. Finally, the proof will be done if we prove , where is the stable manifold of . Suppose it is not true; then there exists a solution in such that Since , we may choose such that . At the same time, there exists a and such that for due to .
When , we obtain from the first equation of system (8) that Hence there exist a such that the following equality holds when :For , we have from (3) and (26) that By and the comparison principle furthermore, it is easy to see that , contradicting as . Hence ; moreover, there exists such that , contradicting , .
Hence, the infection is uniformly persistent according to Lemma 3; that is, there exists a is a positive constant such that , and the disease-free equilibrium 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.
Theorem 5. If , and , the endemic equilibrium of system (8) is globally asymptotically attractive.
Proof. For convenience, we still discuss system (1). According to (13) and , we know is positively invariant with respect to system (8), and every forward orbit in eventually enters .
Thus we just need to discuss the global attractiveness of system (1) in .
Denote , and then is the endemic equilibrium of system (1). System (1) may be rewritten as follows:where .
Note that the endemic equilibrium of system (1) satisfiesWe have from (29) and (30) that Let us consider Calculating the derivative of along solution of (31), we get Since hold, where and , we can obtain from (30), (33), and (34) that Furthermore, by and , we have from the last equation of system (31) that Since , we can take , and it follows from (42) that there exists a such that when . Hence when , that is, In addition, the matrix is irreducible, so the following matrix is irreducible: Hence there exists a positive vector such that in which is the cofactor of the th diagonal of , [29, Lemma 2.1]. It follows from that which leads to that is,Define a Lyapunov function where is defined by (32), and we have from (35), (37), and (42) that Moreover, if and only if , . Therefore, LaSalle Invariance Principle [28, Chapter 2, Theorem 5.3] implies that the endemic equilibrium of system (1) is globally attractive when , and . The proof is completed.
3. Numerical Simulation and Discussion
The basic reproductive number for system (8) (or (1)) is The equilibrium is globally attractive and the infection eventually disappears when , and the infection will always exist when . Note that is irrelevant to the distributed delay.
Extensive numerical simulations are carried out on scale-free 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 . 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]: Let and is a suitable assumption. Meanwhile, let in which , , , and . The initial functions are and for .
Denote Obviously, is the relative average density of the infected nodes.
Case 1. Let , , , , , and ; we can obtain from (45) that . Figure 1 shows the dynamical behaviors of system (8). The numerical simulation shows , it follows that , and the infection eventually disappears. The numerical result is consistent with Theorem 2.
Case 2. Let , , , , and , and . We can obtain from (45) that and . Figure 2 shows the dynamic behaviors of system (8). The relative density and the relative average density converge to positive constant as , respectively, and the infection is uniformly persistent. The numerical result is consistent with Theorems 4 and 5.
Moreover, let , , , , and , and . Figure 3 shows the dynamic behaviors of system (8). The parameters of system (8) do not satisfy Theorem 5, but the relative density and the relative average density still converge to positive constant as , respectively. Therefore, Theorem 5 has room for improvement.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
The authors thank Professor Rui Xu with Shijiazhuang Mechanical Engineering College for his helpful suggestions. This research was supported by the Hebei Provincial Natural Science Foundation of China under Grant no. A2016506002 and the Innovation Foundation of Shijiazhuang Mechanical Engineering College under Grant no. YSCX1201.