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.

Natural Science Foundation of Hebei ProvinceA2016506002Innovation Foundation of Shijiazhuang Mechanical Engineering CollegeYSCX12011. Introduction

Following the seminal work on small-world network by Watts and Strogatz [1], and the scale-free network, in which the probability of p(k) for any node with k links to other nodes is distributed according to the power law p(k)=Ck-γ(2<γ≤3), 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 SIRS 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 N during the period of epidemic spreading; we also suppose that the degree of each node is time invariant; let Sk(t), Ik(t), and Rk(t) be the relative density of susceptible nodes, infected nodes, and recovered nodes of connectivity k at time t, respectively, where k=m,m+1,…,n in which m and n 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 μ 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 k(k=1,2,…,n) at any time t is simply proportional to the number of the infected nodes with degree k at time t-μ [25, 26]. Let the kernel function f(u) denote the probability that an susceptible vector who is infected at time t-μ and becomes infective at time t. Meanwhile, let λ(k) be the correlated (k-dependent) infection rate such as λk and λc(k) [11]. The susceptible nodes may acquire temporary immunity and the removal rate from the susceptible nodes to the recovered nodes is given by δ. And μ 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 r. The dynamical equations for the density Sk(t), Ik(t), and Rk(t), at the mean-field level, satisfy the following set of functional differential equations when t>0:(1)S˙kt=-λkSkt∫0+∞Θt-ufudu-δSkt+μRkt,I˙kt=λkSkt∫0+∞Θt-ufudu-rIkt,R˙kt=rIkt-μRkt+δSktwith (2)Skt+Ikt+Rkt=1,k=m,m+1,…,n, due to the fact that the number of total nodes with degree k is a constant p(k)N during the period of epidemic spreading. The dynamics of n groups of SIRS subsystems are coupled through the function Θ(t), 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 (3)Θt=1k∑k=mnφkpkIkt,where k=∑kp(k)k stands for the average node degree and φ(k)=akα/1+bkα [7] 0≤α<1,a>0,b≥0 denotes an infected node with degree k occupied edges which can transmit the disease. If b≠0, φ(k) gradually become saturated with the increase of degree k, that is, limt→+∞φ(k)=b/a.

The kernel function f(u) is nonnegative and continuous on [0,+∞) and satisfies (4)∫0+∞fudu=1,∫0+∞fueαμdu<+∞, where α is a positive number. And there are many types of kernel functions such as

the gamma distribution f(u)=un-1/(n-1)!bne-u/b, where b>0 is a real number and n>1 is an integer, especially when n=1, b=1, and then f(u)=e-u,

the uniform distribution (5)fu=1/h,0≤u≤h0,u>h,

where h>0 is real number,

the Delta-distribution f(u)=δ(u-τ), where τ>0 is real constant.

Define the following Banach space of fading memory type (see [27] and references therein): (6)C¯=ϕ∈C-∞,0,R:ϕseαsisuniformlycontinuousfors∈-∞,0,sups≤0ϕseαs<+∞ with norm ϕ=sups≤0ϕseαs, and let ϕt∈C¯ be such that ϕt(s)=ϕ(t+s),s∈(-∞,0].

Consider system (1) in phase space X=C¯×C¯×C¯n-m+1. Standard theory of functional differential equation implies system (1) has a unique solution satisfying the initial conditions (7)Skθ=ϕ1ks,Ikθ=ϕ2ks,Rkθ=ϕ3ks,ϕiks≥0,s∈-∞,0,ϕik0>0,i=1,2,3,k=m,m+1,…,n,where ϕ1ms,ϕ2ms,ϕ3ms,…,ϕ1ns,ϕ2ns,ϕ3ns∈X.

It can be verified that solutions of system (1) in X with initial conditions above remain positive for all t≥0.

The rest of this paper is organized as follows. The dynamical behaviors of the SIRS 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 Sk(t)+Ik(t)+Rk(t)=1, system (1) is equivalent to the following system (8):(8)S˙kt=-λkSkt∫0+∞Θt-ufudu-δSkt+μ1-Skt-Ikt,I˙kt=λkSkt∫0+∞Θt-ufudu-rIkt.Thus we only discuss system (8) if we want to discuss the dynamical behaviors of system (1).

Denote (9)R0=μμ+δrλkφkk,where f(k)=∑kf(k)p(k) in which f(k) is a function.

Note that we can obtain from the first equation of system (8) that (10)S˙kt≤μ-μ+δSkt. By the standard comparison theorem in the theory of differential equations, we have (11)limt→+∞supSkt≤μμ+δ. Hence we know(12)D0=Sm,Im,…,Sn,In∈R+2n-m+1∣0<Sk,Ik,Sk+Ik≤1,0≤Sk≤μμ+δ is positively invariant with respect to system (8), and every forward orbit in R+2(n-m+1) eventually enters D0.

Theorem 1.

System (8) has always a disease-free equilibrium E0(μ/μ+δ,…,μ/μ+δ,0,…,0). System (8) has a unique endemic equilibrium E∗(Sm∗,Sm+1∗,…,Sn∗,Im∗,Im+1∗,…,In∗) when R0>1.

Proof.

Obviously, the disease-free equilibrium E0 of system (8) always exists. Now we discuss the existence of the endemic equilibrium of system (8). Combined with ∫0+∞f(u)du=1, it is easy to know that the equilibrium E∗ satisfies (13)-λkSk∗Θ∗-δSk∗+μ1-Sk∗-Ik∗=0,λkSk∗Θ∗-rIk∗=0,where (14)Θ∗=1k∑kφkpkIk∗.From (13), we obtain that (15)Ik∗=λkμΘ∗μ+δr+λkr+μΘ∗.Substituting it into (14), we obtain the self-consistency equality (16)Θ∗=1k∑kφkpkλkμΘ∗μ+δr+λkr+μΘ∗,and it can be verified that (15) has a unique positive solution when R0>1 using the same proof as for Theorem 1 in [21]; consequently, system (8) has a unique endemic equilibrium E∗ since (13) and (15) hold.

Theorem 2.

If R0<1, the disease-free equilibrium E0 of system (8) is globally attractive.

Proof.

Obviously, we need only discuss global attractiveness of system (8) in D0.

Consider the following Lyapunov function (17)Vt=12V12t+R0r2V2t, where (18)V1t=∫0+∞Θ2t-ufudu,V2t=∫0+∞fu∫t-2ut-uΘ2sdsdu.

Calculating the derivative of V(t) along solution of (8), for t>T1, we get (19)V˙t9=∫0+∞Θt-u1k∑kϕkpkλkSkt∫0+∞Θt-2ufudu-rIkt-udu+R0r2V1t-∫0+∞fuΘ2t-2udu≤∫0+∞Θt-u1k∑kϕkpkλkμμ+δ∫0+∞Θt-2ufudu-rIkt-udu+R0r2V1t-∫0+∞fuΘ2t-2udu=R0r∫0+∞Θt-uΘt-2ufudu-rV1t+R0r2∫0+∞fuΘ2t-udu-∫0+∞fuΘ2t-2udu≤R0r2∫0+∞Θ2t-u+Θ2t-2ufudu-rV1t+R0r2∫0+∞fuΘ2t-udu-∫0+∞Θ2t-2udu=rR0-1V1t. Thus V˙t(9)≤0 when R0<1, and V˙t(9)=0 if and only if V1(t)=0. Note that the fact V˙(t) means Ik=0; moreover, limt→+∞Sk(t)=μ/(μ+δ); the largest invariant set of V˙t(9)=0 is a singleton E0. Hence the disease-free equilibrium E0 is globally attractive when R0<1 according to the LaSalle Invariance Principle [28, Chapter 2, Theorem 5.3].

Lemma 3 (see [<xref ref-type="bibr" rid="B28">28</xref>, p273–280]).

Let X be a complete metric space, X=X0∪∂X0, where ∂X0, assumed to be nonempty, is the boundary of X0. Assume the C0-semigroup T(t) on X satisfies T(x):X0→X0, T(x):∂X0→∂X0 and

there is a t0 such that T(t) is compact for t>t0;

T(t) is point dissipative in X;

A~∂ is isolated and has an acyclic covering M.

Then T(t) is uniformly persistent if and only if, for each Mi∈M, (20)WsMi∩X0=∅, where A~∂=⋃x∈A∂ω(x), and ω(x) is the omega limit set of T(x) through x, and A∂ is global attractor of T∂(t) in ∂X0 in which T∂(t)=Tt∂X0.Theorem 4.

For system (8), if R0>1, the disease-free equilibrium E0 is unstable, and the disease is uniformly persistent; that is, there exists a positive constant ϵ such that limt→+∞infIk(t)>ϵ,k=m,m+1,…,n.

Proof.

Denote (21)X=S¯,ψ¯:ψkθ≥0,∀θ∈-∞,0,k=m,m+1,…,n,X0=S¯,ψ¯:ψkθ>0, for some θ∈-∞,0,k=m,m+1,…,n,and consequently, (22)∂X0=XX0=S¯,ψ¯:ψiθ=0,∀θ∈-∞,0,i∈m,m+1,…,n, where (S¯,ψ¯)=Sm,Sm+1,…,Sn,ψm,ψm+1,…,ψn.

Let (Sm(t),…,Sn(t),Im(t),…,In(t))=Smt,ω,…,Snt,ω,Imt,ω,…,Int,ω be the solution of (8) with initial function ω=ϕ1ms,…,ϕ1ns,ϕ2ms,…,ϕ2ns and (23)Ttωθ=Smt+θ,ω,…,Snt+θ,ω,Imt+θ,ω,…,Int+θ,ω,θ∈-∞,0. Obviously, X and X0 are positively invariant set for T(t). T(t) is completely continuous for t>0. Also, it follows from 0<Sk, Ik≤1 for t>0 that T(t) is point dissipative. E0 is the unique equilibrium of system (8) on ∂X0=X/X0 and it is globally stable on ∂X0=X/X0, A~∂=E0, and E0 is isolated and acyclic. Finally, the proof will be done if we prove Ws(E0)∩X0=∅, where Ws(E0) is the stable manifold of E0. Suppose it is not true; then there exists a solution (S¯,I¯) in X0 such that (24)limt→+∞Skt=μμ+δ,Ikt=0,k=m,m+1,…,n. Since R0>1, we may choose 0<η<1 such that μ(1-η)/(λ(k)η+μ+δ)r〈λ(k)φ(k)〉/〈k〉>1. At the same time, there exists a T2>0 and 0<ϵ<η such that 0≤Ik(t)<η for t>T2 due to limt→+∞infIk=0.

When t>T2, we obtain from the first equation of system (8) that(25)S˙kt>-λkSkη+μ1-Skt-η-δSkt=μ1-η-λkη+μ+δSkt. Hence there exist a T3>T2 such that the following equality holds when t>T2:(26)Skt≥μ1-ηλkη+μ+δ.For t>T2, we have from (3) and (26) that (27)Θ˙t=1k∑kφkpkI˙kt=1k∑kφkpkλkSkt∫0+∞Θt-τfudu-rIkt≥μ1-ηλkη+μ+δλkφkk∫0+∞Θt-τfudu-rΘt. By μ(1-η)/(λ(k)η+μ+δ)r〈λ(k)φ(k)〉/〈k〉>1 and the comparison principle furthermore, it is easy to see that limt→+∞Θ(t)=+∞, contradicting limt→+∞Θ(t)=0 as limt→+∞Ik(t)=0. Hence limt→+∞infΘ(t)≠0; moreover, there exists k0∈{m,m+1,…,n} such that limt→+∞infIk0(t)≠0, contradicting limt→+∞infIk(t)=0, k=m,m+1,…,n.

Hence, the infection is uniformly persistent according to Lemma 3; that is, there exists a ϵ is a positive constant such that limt→∞infIk>ϵ, and the disease-free equilibrium E0 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 R0>1, δ<r and Ik∗<μ/(μ+δ)(δ/r),k=m,m+1,…,n, the endemic equilibrium E∗ of system (8) is globally asymptotically attractive.

Proof.

For convenience, we still discuss system (1). According to (13) and Sk(t)+Ik(t)+Rk(t)=1,k=m,m+1,…,n, we know(28)D~0=Sm,Im,Rm,…,Sn,In,Rn∈R+2n-m+1∣0<Sk,Ik,Rk,Sk+Ik+Rn=1,0≤Sk≤μμ+δ is positively invariant with respect to system (8), and every forward orbit in R+3(n-m+1) eventually enters D~0.

Thus we just need to discuss the global attractiveness of system (1) in D~0.

Denote Rk∗=1-Sk∗-Ik∗, and then E∗Sm∗,Im∗,Rm∗,Sm+1∗,Im+1∗,Rm+1∗,…,Sn∗,In∗,Rn∗, is the endemic equilibrium of system (1). System (1) may be rewritten as follows:(29)S˙kt=-∑l=mnβklSkt∫0+∞Ilt-ufudu-δSkt+μRkt,I˙kt=∑l=mnβklSkt∫0+∞Ilt-ufudu-rIkt,R˙kt=r-μ+rRkt-r-δSkt,where βkl=λ(l)φ(l)p(l)/k,l=m,m+1,…,n.

Note that the endemic equilibrium of system (1) satisfies(30)-∑l=mnβklSk∗Il∗-δSk∗+μRk∗=0,∑l=mnβklSk∗Il∗=rIk∗or ∑l=mnβklSk∗∫0+∞Il∗fudu=rIk∗,μ+rRk∗+δ-rSk∗=r.We have from (29) and (30) that (31)S˙kt=-∑l=mnβklSkt∫0+∞Ilt-ufudu-∑l=mnβklSk∗∫0+∞Il∗fudu-δSkt-Sk∗+μRkt-Rk∗,I˙kt=∑l=mnβklSkt∫0+∞Ilt-ufudu-1Ik∗∑l=mnβklSk∗∫0+∞Il∗fuduIkt,R˙kt=-μ+rRkt-Rk∗+δ-rSkt-Sk∗.Let us consider (32)Vkt=Skt-Sk∗-Sk∗lnSktSk∗+Ikt-Ik∗-Ik∗lnIktIk∗+∑l=mnβklSk∗∫0+∞fu∫t-utIls-Il∗-Il∗lnIlsIl∗dsdu+μSk∗r-δRkt-Rk∗22.Calculating the derivative of Vk(t) along solution of (31), we get (33)V˙kt32=-δ1-Sk∗SkSk-Sk∗+μ1-Sk∗SkRk-Rk∗+∑l=mnβklSk∗Il∗∫0+∞2-Sk∗Sk-IkIk∗-SkIlt-τIk∗Sk∗IkIl∗-lnIlIl∗+lnIlt-τIl∗+IlIl∗fudu+μRk-Rk∗1-SkSk∗-μr+μSk∗r-δRk-Rk∗2.Since (34)∑l=mnβklSk∗Il∗∫0+∞2-Sk∗Sk-IkIk∗-SkIlt-uIk∗Sk∗IkIl∗-lnIlIl∗+lnIlt-uIl∗+IlIl∗fudu=∑l=mnβklSk∗Il∗HIkIk∗-HIlIl∗-∑l=mnβklSk∗Il∗GSk∗Sk+GSkSk∗-∑l=mnβklSk∗Il∗∫0+∞GSkIlt-uIk∗Sk∗IkIl∗fudu+∑l=mnβklSk∗Il∗GSkSk∗,-δ1-Sk∗SkSk-Sk∗=-δSk∗GSkSk∗+GSk∗Skμ1-Sk∗SkRk-Rk∗+μRk-Rk∗1-SkSk∗=-μRk-Rk∗GSkSk∗+GSk∗Sk=-μRkGSkSk∗-μRkGSk∗Sk+μRk∗GSkSk∗+GSk∗Skhold, where H(a)=-a+lna and G(a)=a-1-lna≥0, we can obtain from (30), (33), and (34) that (35)V˙kt32≤∑l=mnβklSk∗Il∗HIkIk∗-HIlIl∗-μRkGSkSk∗+∑l=mnβklSk∗Il∗GSkSk∗≤∑l=mnβklSk∗Il∗HIkIk∗-HIlIl∗+-μRk+rIk∗GSkSk∗.

Furthermore, by r>δ and Sk(t)≤μ/μ+δ, we have from the last equation of system (31) that (36)R˙kt≥r-μ+rRkt-r-δμμ+δ=μ+rδμ+δ-Rkt.Since Ik∗<δ/(μ+δ)(μ/r), we can take ε=δ/(μ+δ)-Ik∗r/μ>0, and it follows from (42) that there exists a T4>T1 such that Rk(t)≥δ/(μ+δ)-ε when t>T4. Hence -μRk+rIk∗≤-μ(δ/(μ+δ)-ε)+rIk∗=0 when t>T4, that is, (37)-μRk+rIk∗GSkSk∗≤0.In addition, the matrix (38)βklSk∗Il∗n-m+1×n-m+1=λlφlplkSk∗Il∗n-m+1×n-m+1 is irreducible, so the following matrix is irreducible:(39)B=∑l≠mβmlSm∗Il∗-βm+1,mSm+1∗Im∗⋯-βnmSn∗Im∗-βm,m+1Sm∗Im+1∗∑l≠m+1βm+1,lSm+1∗Il∗⋯-βn,m+1Sn∗Im+1∗⋯⋯⋯⋯-βmnSm∗In∗-βm+1,nSm+1∗In∗⋯∑l≠nβnlSn∗Il∗. Hence there exists a positive vector C=(c1,c2,…,cn) such that BC=0 in which ck is the cofactor of the kth diagonal of B, m≤k≤n [29, Lemma 2.1]. It follows from BC=0 that (40)∑l=mnclβlkSl∗Ik∗=ck∑l=1nβklSk∗Il∗,k=m,m+1,…,n, which leads to (41)∑k=mnck∑l=mnβklSl∗Il∗HIkIk∗=∑k=mn∑l=mnclβlkSl∗Ik∗HIkIk∗=∑k=mn∑l=mnckβklSk∗Il∗HIlIl∗=∑k=mnck∑l=mnβklSk∗Il∗HIlIl∗,that is,(42)∑k=1nck∑l=1nβklSk∗Il∗HIkIk∗-HIlIl∗=0.Define a Lyapunov function (43)Vt=∑k=1nckVkt,where Vk(t) is defined by (32), and we have from (35), (37), and (42) that (44)V˙t32≤0. Moreover, V˙t(32)=0 if and only if Sk=Sk∗,Ik=Ik∗, Rk=Rk∗. Therefore, LaSalle Invariance Principle [28, Chapter 2, Theorem 5.3] implies that the endemic equilibrium E∗ of system (1) is globally attractive when R0>1, δ<r and Ik∗<μ/(μ+δ)(δ/r),k=m,m+1,…,n. The proof is completed.

3. Numerical Simulation and Discussion

The basic reproductive number for system (8) (or (1)) is (45)R0=μμ+δrλkφkk.The equilibrium E0 is globally attractive and the infection eventually disappears when R0<1, and the infection will always exist when R0>1. Note that R0 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 p(k)=Ck-γ, and C satisfies ∑k=mnp(k)=1. Supposing the network is finite one, the maximum connectivity n of any node is related to the network age, measured as the number of nodes N [3, 7]: (46)n=mN1/γ-1.Let n=100 and m=1 is a suitable assumption. Meanwhile, let ϕ(k)=akα/(1+bkα) in which a=0.5, α=0.75, b=0.02, and λ(k)=λk. The initial functions are Ik(s)=0.45,k=2,3,4,5 and Ik=0,k≠2,3,4,5 for s∈(-∞,0].

Denote (47)It=∑kpkIkt. Obviously, I(t) is the relative average density of the infected nodes.

Case 1.

Let λ=0.3, r=0.4, δ=0.7, μ=0.6, γ=2.5, and f(u)=e-u; we can obtain from (45) that R0=0.5064<1. Figure 1 shows the dynamical behaviors of system (8). The numerical simulation shows limt→+∞I(t)=0, it follows that limt→+∞Ik(t)=0, and the infection eventually disappears. The numerical result is consistent with Theorem 2.

Dynamical behaviors of system (8) with R0=0.5547.

Case 2.

Let λ=0.7, r=0.5, δ=0.4, μ=0.6, γ=2.5 and f(u)=e-u, and δ<r. We can obtain from (45) that R0=1.4596>1 and maxIk∗=0.3668<μ/(μ+δ)(δ/r)=0.4800. Figure 2 shows the dynamic behaviors of system (8). The relative density Ik(t) and the relative average density I(t) converge to positive constant as t→+∞, respectively, and the infection is uniformly persistent. The numerical result is consistent with Theorems 4 and 5.

Dynamical behaviors of system (8) with r=0.5>δ=0.4, maxIk∗=0.3668<μ/(μ+δ)(δ/r)=0.4800 and R0=1.2944.

Moreover, let λ=0.1, r=0.4, δ=0.7, μ=0.6, γ=2.5 and f(u)=e-u, and δ>r. Figure 3 shows the dynamic behaviors of system (8). The parameters of system (8) do not satisfy Theorem 5, but the relative density Ik(t) and the relative average density I(t) still converge to positive constant as t→+∞, respectively. Therefore, Theorem 5 has room for improvement.

Dynamical behaviors of system (8) with r=0.4<δ=0.7 and R0=1.3462.

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.

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