The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network

Abstract

Two novel delayed epidemic spreading models with latent period on scale-free network are presented. The formula of the basic reproductive number and the analysis of dynamical behaviors for the models are presented. Meanwhile, numerical simulations are given to verify the main results.

1. Introduction

Following the seminal work on 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 [1], the researches of complex network have attracted more and more interests. It was found that many relevant networks, for instance, the internet, the World Wide Web (WWW), the patterns of human sexual contacts, biology network, transportation infrastructure, etc., exhibit power-law or “scale-free” degree distributions.

The dynamical behaviors of epidemic diseases have been studied for a long time. The epidemic spreading process on network is primarily dominated by two factors: one is the macroscopic topology of the underlying network, and the other is the microscopic infection scheme, which includes properties of disease, infection pattern, individual differences, infectivity of individuals, etc. The traditional epidemic dynamics is based on homogeneous network, and the infectivity rate is equally likely over all links [2]. However, the real disease transmission network exhibits scale-free properties, and the spreading of epidemic disease (e.g., computer virus spreading, epidemic disease between human beings) on heterogeneous network, i.e., scale-free network, has been studied by many researchers [3–27].

A handful of existing works address the complex behavior of epidemic spreading using compartmental differential equations [2, 15]. Comparing with the ordinary differential equation models, more realistic models should be retarded functional differential equation 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 latent period of the infectious diseases or computer virus, the infection period of infective members, and the immunity period of the recovered individuals can be represented by time delays [2]. Recently, some researchers discussed the epidemic spreading model with time delays [7, 15–17]. Susceptible-Infected-Removed (SIR) model is a basic and important epidemic model, Zou and Wu discussed a delayed SIR model without birth and death [15], and Wang and Wang et al. discussed a delayed SIR model with birth rate and death rate [16]. However, it is suitable to divide the nodes being considered into disjoint classes of susceptible, exposed, infective, and recovered nodes in modeling disease transmission [2, 10], i.e., a susceptive node first through an incubation period (and it is said to become exposed) after infection and before becoming infectious. For example, the latent period of epidemic cholera is about 1–3 days, hepatitis B virus 100 days, measles 10–11 days, chincough 7–10 days, diphtheria 2–4 days, scarlatina 2–5 days, poliomyelitis 7–14 days, and so on [25]; the resulting model is Susceptible-Exposed-Infected-Removed (SEIR) model. In addition, some diseases confer temporary immunity, and the recovered nodes cycle back into the susceptive class after an immune period; the resulting model is Susceptible-Exposed-Infected-Removed-susceptible (SEIRS) model.

In this paper, we will present a suitable SEIR model with time delay and a suitable SEIRS model with time delay on heterogeneous network by using functional differential equation to investigate the dynamical behaviors of epidemic spreading.

The rest of this paper is organized as follows: In Section 2, the SEIRS model with time delay on scale-free network is discussed. The SEIRS model with time delay on scale-free network is discussed in Section 3. Finally, the main conclusions of this work are summarized in Section 4.

2. Analysis of the SEIR model with time delay

2.1. The SEIR model

Suppose that the size of the network is a constant N and the degree of each node is time invariant during the period of epidemic spreading, p(k) denotes the degree distribution of the network. We classify all the nodes in the network into n groups such that the nodes in the same group have the same degree. That is, each node in the kth group has the same connectivity (k = m, m + 1, ⋯, n). Let S_k(t), E_k(t), I_k(t) and R_k(t) be the relative density of susceptible nodes, exposed nodes, infected nodes, and recovered nodes of connectivity k at time t, respectively, where k = m, m + 1, ⋯, n (m and n are the minimum and maximum degree in network topology) and n is related to the network age, measured as the number of nodes N [3]:

$n={\mathit{mN}}^{1/\left(\gamma -1\right)}.$E1

Let τ be the latent period of the disease, i.e., each exposed node becomes an infected node after τ. The relative density S_k(t), E_k(t), I_k(t) and R_k(t), at the mean-field level, satisfy the following set of coupled different equations when t > 0 [15, 16]:

$\left\{\begin{array}{rrr}\hfill {\dot{S}}_k\left(t\right)& \hfill =& \hfill -\lambda \left(k\right)S_k\left(t\right)\Theta \left(t\right),\\ \hfill {\dot{E}}_k\left(t\right)& \hfill =& \hfill \lambda \left(k\right)S_k\left(t\right)\Theta \left(t\right)-\lambda \left(k\right)S_k\left(t-\tau \right)\Theta \left(t-\tau \right),\\ \hfill {\dot{I}}_k\left(t\right)& \hfill =& \hfill \lambda \left(k\right)S_k\left(t-\tau \right)\Theta \left(t-\tau \right)-\mu I_k\left(t\right),\\ \hfill {\dot{R}}_k\left(t\right)& \hfill =& \hfill \mu I_k\left(t\right)\end{array}\right.$E2

with the normalization condition

$S_k\left(t\right)+E_k\left(t\right)+I_k\left(t\right)+R_k\left(t\right)=1$E3

holds 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. Where λ(k) is the degree-dependent infection rate such as λ(k) = λk [3] and λC(k) [4], μ is the recovery rate of the infected nodes; Θ(t) represents the probability that any given link points to an infected node. Assuming that the network has no degree correlations [7, 16, 22], we have

$\Theta \left(t\right)=\frac{1}{⟨k⟩}\sum _{k=m}^n\varphi \left(k\right)p\left(k\right)I_k\left(t\right)$E4

in which ⟨k⟩ = ∑_kp(k)k stands for the average node degree and ϕ(k) means the occupied edges which can transmit the disease (i.e., represents the infectivity of infected nodes) [22]; they have many different forms, such as φ(k) = A in [5], φ(k) = ak^α/(1 + bk^α), 0 < α < 1 in [7], and so on. Here, we point out that the delay τ in the model (2) in this paper is different from one in the model (2)–(4) in [15]. The incubation period τ in the model in [15] is another kind of time period, during which the infectious agents develop in the vector and the infected vector becomes infectious after that time.

Note that we obtain from the third equation of system (2) that

$E_k\left(t\right)=\lambda \left(k\right){\int }_{t-\tau }^t{S}_k\left(s\right)\Theta \left(s\right)\mathit{ds}$E5

and the normalization condition becomes the following mathematical form

$S_k\left(t\right)+\lambda \left(k\right){\int }_{t-\tau }^t{S}_k\left(s\right)\Theta \left(s\right)\mathit{ds}+I_k\left(t\right)+R_k\left(t\right)=1.$E6

The initial conditions of system (2) are

$S_k\left(\theta \right)={\phi }_k\left(\theta \right),I_k\left(\theta \right)={\Psi }_k\left(\theta \right),R_k\left(t\right)={\zeta }_k\left(\theta \right),\theta \in \left[-\tau ,0\right]$E7

and satisfy $S_k\left(0\right)+\lambda \left(k\right){\int }_{-\tau }^0{S}_k\left(s\right)\Theta \left(s\right)\mathit{ds}+I_k\left(0\right)+R_k\left(0\right)=1$ which guarantees the normalization condition holds. And Φ_k = (ϕ_k(θ), Ψ_k(θ), ζ(θ), k = m, m + 1, ⋯, n − m + 1) ∈ C are nonnegative continuous on [−τ, 0], ϕ_k(0) > 0, Ψ_k(0) > 0, and ζ(θ) = 0 for θ = 0. C denotes the Banach space C([−τ, 0], R^{3(n − m + 1)}) with the norm, where ∣f(θ)∣_τ = sup_{−τ ≤ θ ≤ 0} ∣ f(θ)∣. $‖\omega ‖=({\sum }_{i=m}^n{\left({\left|{\Psi }_i\left(\theta \right)\right|}_{\tau }^2+{\left|{\varphi }_i\left(\theta \right)\right|}_{\tau }^2+{\left|{\zeta }_i\left(\theta \right)\right|}_{\tau }^2\right)}^{1/2}.$

2.2. The main results for the model

In this section, we first discuss the final size relation of solutions for system (2).

It is easy to know that system (2) only has a disease-free equilibrium set

$M_0=\left\{\left(\widehat{S},\widehat{E},\widehat{I},\widehat{R}\right)|E_k=I_k=0,S_k+R_k=1,R_k,S_k\ge 0,k=m,m+1,\cdots ,n\right\}$E8

in which $\widehat{S}=\left(S_m,S_{m+1},\cdots ,S_n\right),$$\widehat{E}=\left(E_m,E_{m+1},\cdots ,E_n\right),$$\widehat{I}=\left(I_m,I_{m+1},\cdots ,I_n\right),$$\widehat{R}=\left(R_m,R_{m+1},\cdots ,R_n\right).$

Supposing f(t) is an arbitrary nonnegative continuous function f(t), we adopt the following convention:

$f\left(+\infty \right)=\underset{t\to +\infty }{lim}f\left(t\right)$E9

and we obtain from the last equation of system (2) that

$R_k\left(+\infty \right)-R_k\left(0\right)=\mu {\int }_0^{+\infty }{I}_k\left(s\right)\mathit{ds}.$E10

According to the last equation of system (2), R_k(t) is increasing and bounded above by 1, and it has a limit as t →  + ∞. Thus, the left-hand side of (10) is finite due to boundedness of R_k(+∞), and R_k(0) exits, i.e., $0<{\int }_0^{+\infty }{I}_k\left(u\right)\mathit{du}<+\infty$. Since I_k(t) is smooth nonnegative function, we know I_k(+∞) = 0, i.e., lim_{t →  + ∞}I_k(t) = 0.

Furthermore, we have from (5) that

$0\le E_k\left(t\right)=\lambda \left(k\right){\int }_{t-\tau }^t{S}_k\left(s\right)\Theta \left(s\right)\mathit{ds}\le \lambda \left(k\right){\int }_{t-\tau }^t\Theta \left(s\right)\mathit{ds},$E11

In addition, by using mean value theorem for integrals, we have

${\int }_{t-\tau }^t\Theta \left(s\right)\mathit{ds}=\Theta \left(\xi \right)\tau ,\left(t-\tau \le \xi \le t\right).$E12

We obtain from I_k(+∞) = 0 that lim_{ξ →  + ∞}Θ(ξ)τ = 0 and then lim_{t →  + ∞}E_k(t) = 0. Hence, M₀ is globally attractive [27].

In addition, it follows from (2) that

${\dot{S}}_k\left(t\right)+{\dot{E}}_k\left(t\right)+{\dot{I}}_k\left(t\right)=-\mu I_k\left(t\right).$E13

Integrating (13) from 0 to +∞, we obtain that

$S_k\left(0\right)-S_k\left(+\infty \right)+E_k\left(0\right)-E_k\left(+\infty \right)+I_k\left(0\right)-I_k\left(+\infty \right)=\mu {\int }_0^{+\infty }{I}_k\left(s\right)\mathit{ds}.$E14

Noting that I_k(+ ∞) = 0, E_k(0) = E_k(+ ∞) = 0, and S_k(+ ∞) exists due to existence of ${\int }_0^{+\infty }{I}_k\left(s\right)\mathit{ds}$, we have from (14) that

${\int }_0^{+\infty }{I}_k\left(s\right)\mathit{ds}=\frac{1}{\mu }\left(S_k\left(0\right)+I_k\left(0\right)-S_k\left(+\infty \right)\right).$E15

Additionally, integrating the first equation of system (2) from 0 to + ∞, we have

$ln\frac{S_k\left(0\right)}{S_k\left(\infty \right)}=\frac{\lambda \left(k\right)}{⟨k⟩}\sum _k\varphi \left(k\right)p\left(k\right){\int }_0^{+\infty }{I}_k\left(s\right)\mathit{ds}.$E16

Substituting (15) into (16), we obtain that

$ln\frac{S_k\left(0\right)}{S_k\left(+\infty \right)}=\frac{\lambda \left(k\right)}{⟨k⟩}\sum _k\varphi \left(k\right)p\left(k\right)\frac{1}{\mu }\left(S_k\left(0\right)+I_k\left(0\right)-S_k\left(+\infty \right)\right).$E17

Because there are only several infective nodes at the beginning of disease spreading, we take S_k(0) ≈ 1 and obtain from (17) that

$ln{S}_k\left(+\infty \right)=\frac{\lambda \left(k\right)}{\mu ⟨k⟩}\sum _k\varphi \left(k\right)p\left(k\right)\left(S_k\left(+\infty \right)-1\right).$E18

Consequently,

$R_k\left(\infty \right)=1-S_k\left(+\infty \right).$E19

Hence, we have the following result.

Theorem 2.1. The equilibrium set $M_0=\left\{\left(\widehat{S},\widehat{E},\widehat{I},\widehat{R}\right)|E_k=I_k=0,S_k+R_k=1,k=1,2,\cdots ,n\right\}$ of system (2) is globally attractive, i.e., lim_{t →  + ∞}I_k(t) = 0, lim_{t → ∞}E_k(t) = 0. And R_k(+∞), S_k(+∞) are given by formulas (18) and (19).

Note that it is impossible for every susceptible to be infected. Supposing S_k(+∞) = 0, we know from (16) that

$+\infty =\frac{\lambda \left(k\right)}{\mu ⟨k⟩}\sum _k\varphi \left(k\right)p\left(k\right)$E20

Obviously, Eq. (20) does not hold, i.e., S_k(+∞) = 0. Similar results were obtained in the early literature [19].

Secondly, we discuss the basic reproductive number of model (2). The basic reproductive number is an important conception; it represents the average number of secondary infections infected by an individual of infective during the whole course of disease in the case that all the members of the population are susceptible [2].

Theorem 2.2. For system (2),

$R_0=\frac{⟨\lambda \left(k\right)\varphi \left(k\right)⟩}{\mu ⟨k⟩}$E21

is the basic reproductive number for system (2).

Proof. Note that ${\sum }_{k=m}^n\phi \left(k\right)p\left(k\right)I_k\left(t\right)$ may be considered as the force of infection [15] and Θ(t) may be considered as the average force of infection. Letting Θ(t) be an auxiliary function and computing its time derivative along the solution of (2), we get

$\begin{array}{rr}\hfill \frac{\mathit{d\Theta }\left(t\right)}{\mathit{dt}}& =\frac{1}{⟨k⟩}\sum _k\varphi \left(k\right)p\left(k\right){\dot{I}}_k\left(t\right)\\ & \hfill =\frac{1}{⟨k⟩}\sum _k\varphi \left(k\right)p\left(k\right)\left(\lambda \left(k\right)S_k\left(t-\tau \right)\Theta \left(t-\tau \right)-\mu I_k\left(t\right)\right)\\ & \hfill =\Theta \left(t-\tau \right)\frac{1}{⟨k⟩}\sum _k\lambda \left(k\right)\varphi \left(k\right)p\left(k\right)S_k\left(t-\tau \right)-\mathit{\mu }\Theta \left(t\right).\end{array}$E22

We have

${\left.\frac{\mathit{d}\Theta \left(t\right)}{\mathit{dt}}\right|}_{t=0}=\Theta \left(-\tau \right)\frac{1}{⟨k⟩}\sum _k\lambda \left(k\right)\varphi \left(k\right)p\left(k\right)S_k\left(-\tau \right)-\mathit{\mu }\Theta \left(0\right)).$E23

Since each exposed node becomes infected node after τ, I_k(−τ) = I_k(0). It follows that Θ(−τ) = Θ(0). Meanwhile, S_k(−τ) ≈ 1. Hence, we have from (23) that

${\left.\frac{\mathit{d}\Theta \left(t\right)}{\mathit{dt}}\right|}_{t=0}=\mu \left(\frac{1}{\mu ⟨k⟩}\sum _k\lambda \left(k\right)\varphi \left(k\right)p\left(k\right)-1\right)\Theta \left(0\right)=\mu \left(R_0-1\right)\Theta \left(0\right).$E24

If R₀ > 1, ${\left.\frac{\mathit{d\Theta }\left(t\right)}{\mathit{dt}}\right|}_{t=0}>0$, which means that Θ(t) increases at the beginning of the epidemic and there exists at least one outbreak.

Meanwhile, if R₀ ≤ 1, we obtain from (24) that ${\left.\frac{\mathit{d\Theta }\left(t\right)}{\mathit{dt}}\right|}_{t=0}\le 0$. Let t^∗ = sup {T ≥ 0 : Θ(t) decreases on [0, T]}. Then, it follows from the above discussion that T ≥ 0. We will prove that T =  +  ∞ . Note that we obtain from the first equation of system (2) that

$S_k\left(t\right)=S_k\left(0\right){\mathrm{e}}^{-\lambda \left(k\right)\Psi \left(t\right)}$E25

in which $\Psi \left(t\right)=\frac{1}{⟨k⟩}\sum _k{\int }_0^t\phi \left(k\right)p\left(k\right)I_k\left(u\right)\mathit{du}.$ Hence, it follows from Eqs. (22) and (25) that

$\frac{\mathit{d}\Theta \left(t\right)}{\mathit{dt}}=\Theta \left(t-\tau \right)\frac{1}{⟨k⟩}\sum _k\lambda \left(k\right)\varphi \left(k\right)p\left(k\right)S_k\left(0\right){\mathrm{e}}^{-\lambda \left(k\right)\Psi \left(t-\tau \right)}-\mathit{\mu \Theta }\left(t\right).$E26

By way of contradiction, supposing that T <  + ∞, then we have $\frac{d}{\mathit{dt}}\Theta \left(t^{\ast }\right)=0$, and there exists a t₁ ∈ (t^∗, t^∗ + τ] such that $\frac{d}{\mathit{dt}}\Theta \left(t_1\right)>0$. It follows that there is a t₂ ∈ [t^∗, t₁) such that $\frac{d}{\mathit{dt}}\Theta \left(t_2\right)=0$ and Θ(t₂) < Θ(t₁). Note that Θ(t₂ − τ) ≥ Θ(t₁ − τ). It follows from (25) that

$0<\frac{d}{\mathit{dt}}\Theta \left(t_1\right)\le \frac{d}{\mathit{dt}}\Theta \left(t_2\right)=0,$E27

which is a contradiction. Hence, Θ(t) decreases on [0, +∞), and there is no one outbreak when R₀ ≤ 1. Hence, R₀ is the basic reproductive number for system (2).

It follows from Theorems 2.1 and 2.2 that R₀ is the basic reproductive number for system (2), which is irrelative to τ. There exists at least one outbreak for the spreading of epidemic if R₀ > 1, and there is no outbreak if R₀ ≤ 1. Whether or not there exists one outbreak for the spreading of epidemic, lim_{t →  + ∞}I_k(t) = 0 due to global attractivity of M₀.

Besides, if we let τ = 0, φ(k) = k, λ(k) = λk, μ = 1, the model (2) reduces to the model in [9]. Furthermore, the basic reproductive number for system (2) is R₀ = (λ⟨k²⟩)/(⟨k⟩), which is identical with the results that the epidemic threshold λ_c = (λ⟨k⟩)/(⟨k²⟩) in [9]. And, R₀ is always more than unity when N is large enough [3, 7], and it means the lack of any basic reproductive number. This result is consistent with the results in epidemic dynamics on heterogeneous network [3, 10].

2.3. Numerical simulation for the model

Now, we present numerical simulations to support the results obtained in previous sections and analyze the effect of time delay on behaviors of disease spreading.

The degree distribution of scale-free network is p(k) = Ck^−γ, and C satisfies ${\sum }_{k=1}^{n}p\left(k\right)=1$. Here, we set the maximum degree n = 100 and the minimum degree m = 1. Consider system (2), let φ(k) = ak^α/(1 + b^α) in which a = 0.5, α = 0.75, b = 0.02 and λ(k) = λk, and let γ = 2.5. Figures 1–4 show the dynamic behaviors of system (2) with the initial functions satisfying condition (7).

Figure 1.

(a) The time evolutions for system (2) with λ = 0.5, μ = 0.4, τ = 5 and I_k(0) = 0.2, k = 2, 3, 4, 5, I_k(0) = 0 for the other k and R₀ = 0.4006. (b) The time evolutions for system (2) with different initial values, λ = 0.4, μ = 0.4, τ = 5 and R₀ = 0.4006.

Denote that

$S\left(t\right)=\sum _{k=m}^{n}p\left(k\right)S_k\left(t\right),I\left(t\right)=\sum _{k=m}^{n}p\left(k\right)I_k\left(t\right),R\left(t\right)=\sum _{k=m}^{n}p\left(k\right)R_k\left(t\right).$E28

They are the relative average density of susceptible nodes, exposed nodes, infected nodes, and recovered nodes at time t, respectively.

First, Figures 1 and 2 show that the infection eventually disappears, whatever R₀ < 1 or not, and the outbreak of disease spreading appears when R₀ > 1 and the outbreak of disease spreading does not appear when R₀ ≤ 1. Meanwhile, Figure 3 shows that phase trajectories on SR-plane of system (2) with different initial values tend to be S(t) + R(t) = 1, i.e., ∑_kp(k)(S_k(t) + R_k(t)) = 1, which is consistent with the fact that the equilibrium M₀ is globally attractive. The numerical simulation results are identical with Theorems 2.1–2.2.

Figure 2.

(a) The time evolutions for system (2) with λ = 1, μ = 0.1, τ = 5 and I_k(0) = 0.2, k = 2, 3, 4, 5, I_k(0) = 0 for the other k and R₀ = 3.2051. (b) The time evolutions for system (2) with different initial values, λ = 1, μ = 0.1, τ = 5 and R₀ = 3.2051.

Figure 3.

(a) Phase trajectories on SR-plane of system (2) with different initial values, λ = 0.5, μ = 0.4, τ = 5 and R₀ = 0.4006. (b) Phase trajectories on SR-plane of system (2) with different initial values, λ = 1, μ = 0.1, τ = 5 and R₀ = 3.2051.

Second, time delay τ has no effects on the basic reproductive number R₀ according to (21), but it has much impact on the of process of the disease; the slower the relative density of infected nodes converges to zero, the larger τ gets, i.e., time delay may slow down the speed of disappearing the disease spreading on network. Meanwhile, time delay may effectively reduce the peak value of the relative density of infected nodes when R₀ > 1. Thus, the delay cannot be ignored.

At last, we know from Figure 5 that time evolutions of the average force of infection for system (2) is consistent with time evolutions of the average relative density I(t). However, there is only one outbreak, which is different from the phenomenon in [15].

3. Analysis of the SEIRS model with time delays

3.1. The SEIRS model

Since some diseases confer temporary immunity, the recovered nodes cycle back into the susceptive class after an immune period. Let ω be furthermore the immune period of the recovered node, and the recovered node cycles back into the susceptive class after an immune period ω. Denote that σ = max {τ, ω}. Based on the model (2), the relative densities S_k(t), E_k(t), I_k(t) and R_k(t), at the mean-field level, satisfy the following set of coupled different equations when t > 0:

$\left\{\begin{array}{l}{\dot{S}}_k\left(t\right)=-\lambda \left(k\right)S_k\left(t\right)\Theta \left(t\right)+\mu I_k\left(t-\omega \right),\hfill \\ {\dot{E}}_k\left(t\right)=\lambda \left(k\right)S_k\left(t\right)\Theta \left(t\right)-\lambda \left(k\right)S_k\left(t-\tau \right)\Theta \left(t-\tau \right),\hfill \\ {\dot{I}}_k\left(t\right)=\lambda \left(k\right)S_k\left(t-\tau \right)\Theta \left(t-\tau \right)-\mu I_k\left(t\right),\hfill \\ {\dot{R}}_k\left(t\right)=\mu I_k\left(t\right)-\mu I_k\left(t-\omega \right)\hfill \end{array}\right.$E29

with the normalization condition (3).

Furthermore, we obtain from the third equation and the fourth equation of system (29) that

$E_k\left(t\right)=\lambda \left(k\right){\int }_{t-\tau }^t{S}_k\left(s\right)\Theta \left(s\right)\mathit{ds},R_k\left(t\right)=\mu {\int }_{t-\omega }^t{I}_k\left(s\right)\mathit{ds}.$E30

Hence, the normalization condition becomes the following mathematical form:

$S_k\left(t\right)+\lambda \left(k\right){\int }_{t-\tau }^t{S}_k\left(s\right)\Theta \left(s\right)\mathit{ds}+I_k\left(t\right)+\mu {\int }_{t-\omega }^t{I}_k\left(s\right)\mathit{ds}=1.$E31

Obviously, if we discuss the dynamical behaviors of system (29), we just need to discuss the following system:

$\left\{\begin{array}{l}{\dot{S}}_k\left(t\right)=-\lambda \left(k\right)S_k\left(t\right)\Theta \left(t\right)+\mu I_k\left(t-\omega \right),\hfill \\ {\dot{I}}_k\left(t\right)=\lambda \left(k\right)S_k\left(t-\tau \right)\Theta \left(t-\tau \right)-\mu I_k\left(t\right)\hfill \end{array}\right.$E32

with the normalization condition (31).

The initial conditions of system (32) are

$S_k\left(\theta \right)={\phi }_k\left(\theta \right),I_k\left(\theta \right)={\Psi }_k\left(\theta \right),\theta \in \left[-\sigma ,0\right],$E33

which satisfy $S_k\left(0\right)+\lambda \left(k\right){\int }_{-\tau }^0{S}_k\left(s\right)\Theta \left(s\right)\mathit{ds}+I_k\left(0\right)+\mu {\int }_{-\omega }^0{I}_k\left(s\right)\mathit{ds}=1$. Hence, the normalization condition (31) holds. And, Φ_k = (ϕ_k(θ), Ψ_k(θ), k = m, m + 1, ⋯, n − m + 1) ∈ C are nonnegative continuous on [−σ, 0], ϕ_k(0) > 0, Ψ_k(0) > 0, and ζ(θ) = 0 for θ = 0. C denotes the Banach space C([−σ, 0], R^{2(n − m + 1)}) with the norm $‖\omega ‖=({\sum }_{i=m}^n{\left({\left|{\Psi }_i\left(\theta \right)\right|}_{\sigma }^2+{\left|{\phi }_i\left(\theta \right)\right|}_{\sigma }^2\right)}^{1/2}$, where |f(θ)|_σ = sup_{−τ ≤ θ ≤ 0} ∣ f(θ)∣.

3.2. The main results for the model

Denote that

$R_0=\frac{1}{\mu }\frac{⟨\lambda \left(k\right)\phi \left(k\right)⟩}{⟨k⟩},$E34

where $⟨f\left(k\right)⟩={\sum }_{k=m}^{n}f\left(k\right)p\left(k\right)$ in which f(k) is a function.

Theorem 3.1. System (32) always has a disease-free equilibrium E₀ (1, 1, ⋯, 1, 0, 0, ⋯, 0), and it has a unique endemic equilibrium $E_{\ast }=\left(S_m^{\ast },S_{m+1}^{\ast },\cdots ,S_n^{\ast },I_m^{\ast },I_{m+1}^{\ast },\cdots ,I_n^{\ast }\right)$ when R₀ > 1.

Proof. Obviously, the disease-free equilibrium E₀ of system (32) always exists. Now, we discuss the existence of the endemic equilibrium of system (32). Note that the equilibrium E_∗ should satisfy:

$\begin{array}{l}-\lambda {\mathit{kS}}_k^{\ast }{\Theta }^{\ast }+\mu I_k^{\ast }=0,\hfill \\ S_k^{\ast }+\lambda \left(k\right)\tau S_k^{\ast }{\Theta }^{\ast }+I_k^{\ast }+\mathit{\omega \mu }{I}_k^{\ast }=1,\hfill \end{array}$E35

where

${\Theta }^{\ast }=\frac{1}{⟨k⟩}\sum _{k^{\prime }}\varphi \left(k^{\prime }\right)p\left(k^{\prime }\right)I_k^{\ast }.$E36

We obtain from (35) that

$I_k^{\ast }=\frac{\mathit{\lambda k}{\Theta }^{\ast }}{\mu +\mathit{\lambda k}\left(1+\mathit{\mu \tau }+\mathit{\omega \mu }\right){\Theta }^{\ast }}.$E37

Substituting it into Eq. (4), we obtain the self-consistency equality:

${\Theta }^{\ast }=\frac{\lambda }{⟨k⟩}\sum _{k=m}^n\varphi \left(k\right)p\left(k\right)\frac{\mathit{\lambda k}{\Theta }^{\ast }}{\mu +\mathit{\lambda k}\left(1+\mathit{\mu \tau }+\mathit{\omega \mu }\right){\Theta }^{\ast }}=f\left({\Theta }^{\ast }\right).$E38

Note that

${\left.f^{\prime }\left({\Theta }^{\ast }\right)\right|}_{{\Theta }^{\ast }=0}={\left.\frac{\lambda }{⟨k⟩}\sum _{k=m}^n\varphi \left(k\right)p\left(k\right)\frac{\mathit{\lambda k\mu }}{{\left(\mu +\mathit{\lambda k}\left(1+\mathit{\mu \tau }+\mathit{\omega \mu }\right){\Theta }^{\ast }\right)}^2}\right|}_{{\Theta }^{\ast }=0}=\frac{\lambda ⟨\mathit{k\varphi }\left(k\right)⟩}{\mu ⟨k⟩}=R_0$E39

and

${f^{\prime \prime }}^{}\left({\Theta }^{\ast }\right)=\frac{-2\lambda }{⟨k⟩}\sum _{k=m}^n\varphi \left(k\right)p\left(k\right)\frac{{\lambda }^2{k}^2\mu \left(1+\mathit{\mu \tau }+\mathit{\omega \mu }\right){\Theta }^{\ast }}{{\left(\mu +\mathit{\lambda k}\left(1+\mathit{\mu \tau }+\mathit{\omega \mu }\right){\Theta }^{\ast }\right)}^3}<0.$E40

Hence, if R₀ > 1, Eq. (38) has a unique positive solution. Consequently, system (32) has a unique endemic equilibrium $E_{\ast }\left(S_1^{\ast },S_2^{\ast },\cdots ,S_n^{\ast },I_1^{\ast },I_2^{\ast },\cdots ,I_n^{\ast }\right)$ since (35) and (37) hold.

Theorem 3.2. If R₀ ≤ 1, the disease-free equilibrium E₀ of system (32) is globally attractive.

Proof. We define a Lyapunov function V(t) as

$V\left(t\right)=\frac{1}{2}{\Theta }^2\left(t\right)+\gamma {\int }_{t-\tau }^t{\Theta }^2\left(\mu \right)\mathit{d\mu },$E41

where γ is a constant to be determined. Let $G=\left\{\varphi :\dot{V}\left(\varphi \right)=0\right\}$, and M is the largest set in G which is invariant with respect to system (32). Clearly, M is not empty since E₀ ∈ M. Calculating the derivative of V(t) along the solution of (32), we get

$\begin{array}{l}{\left.\dot{V}\left(t\right)\right|}_{\left(3.3\right)}=\Theta \left(t\right)\left[\frac{1}{\lambda \left(k\right)⟩}\sum _k\phi \left(k\right)p\left(k\right)\left(-\lambda {\mathit{kS}}_k\left(t-\tau \right)\Theta \left(t-\tau \right)-\mathit{\mu \Theta }\left(t\right)\right)\right]+\gamma {\Theta }^2\left(t\right)-\gamma {\Theta }^2\left(t-\tau \right)\hfill \\ \le \Theta \left(t\right)\left[\frac{1}{⟨k⟩}⟨\lambda \left(k\right)\varphi \left(k\right)⟩\Theta \left(t-\tau \right)-\mathit{\mu \Theta }\left(t\right)\right]+\gamma {\Theta }^2\left(t\right)-\gamma {\Theta }^2\left(t-\tau \right)\hfill \\ \le \frac{1}{2⟨k⟩}⟨\lambda \left(k\right)\varphi \left(k\right)⟩{\Theta }^2\left(t\right)+\frac{1}{2⟨k⟩}⟨\lambda \left(k\right)\varphi \left(k\right)⟩{\Theta }^2\left(t-\tau \right)-\mu {\Theta }^2\left(t\right)+\gamma {\Theta }^2\left(t\right)-\gamma {\Theta }^2\left(t-\tau \right)\hfill \\ =\left(\frac{1}{2⟨k⟩}⟨\lambda \left(k\right)\varphi \left(k\right)⟩-\mu +\gamma \right){\Theta }^2\left(t\right)+\left(\frac{1}{2⟨k⟩}⟨\lambda \left(k\right)\varphi \left(k\right)⟩-\gamma \right){\Theta }^2\left(t-\tau \right).\hfill \end{array}$E42

Note that R₀ ≤ 1 implies $\frac{1}{⟨k⟩}⟨\lambda \left(k\right)\phi \left(k\right)⟩\le \mu <0$; if we let $\gamma =\frac{1}{2⟨k⟩}⟨\lambda \left(k\right)\phi \left(k\right)⟩$, we have from (42) that

${\left.\dot{V}\left(t\right)\right|}_{\left(3.3\right)}\le \left(\frac{1}{⟨k⟩}⟨\lambda \left(k\right)\varphi \left(k\right)⟩-\mu \right){\Theta }^2\left(t\right)\le 0.$E43

It follows from S_k(t) + E_k(t) + I_k(t) + R_k(t) = 1 that M = E₀. Therefore, by the LaSalle invariance principle [24], the disease-free equilibrium E₀ is globally attractive.

Lemma 3.1. [28] Consider the following equation:

$\dot{x}\left(t\right)=a_{1}x\left(t-\tau \right)-a_{2}x\left(t\right),$E44

where a₁, a₂, τ > 0; x(t) > 0 for −τ ≤ t ≤ 0. We have

1. if a₁ < a₂, then lim_{t →  + ∞}x(t) = 0,

2. if a₁ > a₂, then lim_{t →  + ∞}x(t) =  + ∞.

Lemma 3.2. ([29], p 273–280) 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

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

2. T(t) is point dissipative in X.

3. ${\tilde{A}}_{\text{∂}}$ is isolated and has an acyclic covering M.

Then, T(t) is uniformly persistent if and only if, for each M_i ∈ M,

$W^s\left(M_i\right)\cap X^0=\varnothing ,$E45

where ${\tilde{A}}_{\text{∂}}={\displaystyle \bigcup _{x\in A_{\text{∂}}}}\omega \left(x\right)$, 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) = T(t)|_∂X⁰.

Theorem 3.3. For system (32), if R₀ > 1, the disease is uniformly persistent, i.e., there exists a positive constant ε such that lim_{t →  + ∞} inf I(t) > ε, where $I\left(t\right)={\sum }_{k=m}^n\varphi \left(k\right)p\left(k\right)I_k\left(t\right)$.

Proof. Denote that

$X=\left\{\left(\overline{S},\overline{\Psi }\right):{\Psi }_k\left(\theta \right)\ge 0,\mathit{\text{for}}\mathit{all}\theta \in \left[-\zeta ,0\right],k=m,m+1,\cdots ,n\right\},$E46

$X^0=\left\{\left(\overline{S},\overline{\Psi }\right):{\Psi }_k\left(\theta \right)>0,\mathit{\text{for}}\mathit{\text{some}}\theta \in \left[-\zeta ,0\right],k=m,m+1,\cdots ,n\right\},$E47

and, consequently,

$\partial X^0=X/X^0=\left\{\left(\overline{S},\overline{\Psi }\right):{\Psi }_i\left(\theta \right)=0,\mathit{\text{for}}\mathit{all}\theta \in \left[-\sigma ,0\right],i\in \left\{m,m+1,\cdots ,n\right\}\right\},$E48

where $\left(\overline{S},\overline{\Psi }\right)=\left(S_m,S_{m+1},\cdots ,S_n,{\Psi }_m,{\Psi }_{m+1},\cdots ,{\Psi }_n\right)$.

Let (S_m(t), I_m(t), ⋯, S_n, I_n(t)) = (S_m(t, ω), I_m(t, ω), ⋯, S_n(t, ω), I_n(t, ω)) be the solution of (32) with initial function ω = (ζ_m(θ), Ψ_m(θ), ⋯, Ψ_n(θ), ϕ_n(θ)) and

T(t)(ω)(θ) = (S_m(t + θ, ω), I_m(t + θ, ω), ⋯, S_n(t + θ, ω), I_n(t + θ, ω)), θ ∈ [−σ, 0]. Obviously, X and X⁰ are positively invariant sets for T(t). T(t) is completely continuous for t > 0. Also, it follows from 0 < S_k(t), I_k(t) ≤ 1 for t > 0 that T(t) is point dissipative. E₀ is the unique equilibrium of system (32) on ∂X⁰, and it is globally stable on ∂X⁰, ${\tilde{A}}_{\text{∂}}=\left\{E_0\right\}$, while E₀ is isolated and acyclic.

Finally, the proof will be done if we prove W^s(E₀) ∩ X⁰ = ∅, where W^s(E₀) is the stable manifold of E₀. Suppose it is not true, then there exists a solution $\left(\overline{S},\overline{I}\right)$ in X⁰ such that

$\underset{t\to +\infty }{lim}inf{S}_k\left(t\right)=1,\underset{t\to +\infty }{lim}inf{I}_k\left(t\right)=0,k=1,2,\cdots ,n.$E49

Since R₀ > 1, we may choose 0 < η < 1 such that α = η(λ(k)τ⟨φ(k)⟩ + 1 + μω) satisfies (1 − α)R₀ > 1. At the same time, there exists a t₁ > τ such that I_k(t) < η for t > t₁ due to lim_{t →  + ∞} inf I_k = 0.

When t > t₁, we obtain from (32) that

$S_k\left(t\right)=1-\left(I_k\left(t\right)+\lambda \left(k\right){\int }_{t-\tau }^t{S}_k\left(s\right)\Theta \left(s\right)\mathit{ds}+\mu {\int }_{t-\omega }^t{I}_k\left(s\right)\mathit{ds}\right)\ge 1-\left(\eta +\lambda \left(k\right)\mathit{\tau \eta }⟨\varphi \left(k\right)⟩+\mathit{\mu \omega \eta }\right)=1-\alpha .$E50

On the other hand, for t > t₁ we have from (4) and (50) that

$\begin{array}{ll}\dot{\Theta }\left(t\right)\hfill & =\frac{1}{⟨k⟩}\sum _{k=m}^n\varphi \left(k\right)p\left(k\right){\dot{I}}_k\left(t\right)\hfill \\ \hfill & =\frac{1}{⟨k⟩}\sum _{k=m}^n\varphi \left(k\right)p\left(k\right)\left[\lambda \left(k\right)S_k\left(t-\tau \right)\Theta \left(t-\tau \right)-\mu I_k\left(t\right)\right]\hfill \\ \hfill & \ge \left(1-\alpha \right)\frac{⟨\lambda \left(k\right)\varphi \left(k\right)⟩}{⟨k⟩}\Theta \left(t-\tau \right))-\mathit{\mu \Theta }\left(t\right)\hfill \end{array}$E51

Note that (1 − α)R₀ > 1, and it follows with $\left(1-\alpha \right)\frac{⟨\lambda \left(k\right)\phi \left(k\right)⟩}{⟨k⟩}>\mu$. Hence, we obtain from (51) that lim_{t →  + ∞}Θ(t) =  + ∞ according to Lemma 3.1 contradicts lim_{t →  + ∞}Θ(t) = 0 due to lim_{t →  + ∞}I_k(t) = 0. Then, W^s(E₀) ∩ X⁰ = ∅.

Hence, the infection is uniformly persistent according to Lemma 3.2, i.e., there exists a positive constant ε such that lim_{t →  + ∞} inf I_k > ε and, consequently, ${lim}_{t\to +\infty }infI\left(t\right)>{\sum }_{k=m}^{n}p\left(k\right)\epsilon =\epsilon$. This completes the proof.

In addition, Liu and Zhang discussed a simple SEIRS model without delay in [25], and the basic productive number for the model in [25] is λA/γ, which is consistent with R₀ for the model (32) in which ϕ(k) = A in this paper.

3.3. Numerical simulations for the model

Now, we present the results of numerical simulations. The degree distribution of the scale-free network is p(k) = Ck^−γ, and C satisfies ${\sum }_{k=1}^{n}p\left(k\right)=1$. Here, we set still the maximum degree n = 100 and the minimum degree m = 1.

Consider system (32). Let φ(k = ak^α/(1 + b^α) in which a = 0.5, α = 0.75, b = 0.02 and λ(k) = λk, and let γ = 2.5 and μ = 0.06. Figures 1–4 show that the dynamic behaviors of system (32) with the initial functions satisfy condition (33) in which I_k(s) = 0.45, k = 2, 3, 4, 5 for s ∈ [−σ, 0] and I_k = 0, k ≠ 2, 3, 4, 5.

Figure 4.

(a) The time evolutions of the average relative density I(t) for system (2) with different τ as well as I_k(0) = 0.2, k = 2, 3, 4, 5, I_k(0) = 0 for the other k, λ = 0.5, μ = 0.4 and R₀ = 0.4006. (b) The time evolutions of the average relative I(t) for system (2) with different τ as well as I_k(0) = 0.2, k = 2, 3, 4, 5, I_k(0) = 0 for the other k, λ = 1, μ = 0.1 and R₀ = 3.2051.

Figure 5.

(a) The time evolutions of the average force of infection Θ(t) for system (2) with different τ as well as with I_k(0) = 0.2, k = 2, 3, 4, 5, I_k(0) = 0 for the other k, λ = 0.5, μ = 0.4 and R₀ = 0.4006. (b) The time evolutions of the average force of infection Θ(t) for system (2) with different τ as well as I_k(0) = 0.2, k = 2, 3, 4, 5, I_k(0) = 0 for the other k, λ = 1, μ = 0.1 and R₀ = 3.2051.

Figure 6.

(a) Evolutions of I(t) for system (32) with τ = 3, λ(k) = 0.03k, and R₀ = 0.8037. (b) Evolutions of I₁₅ and I₈₅ for system (32) with τ = 3, λ(k) = 0.03k, and R₀ = 0.8037.

Figure 7.

(a) Evolutions of I(t) for system (32) with ω = 6, λ(k) = 0.03k, and R₀ = 0.8037. (b) Evolutions of I₁₅ and I₈₅ for system (32) with ω = 6, λ(k) = 0.03k, and R₀ = 0.8037.

Figure 8.

(a) Evolutions of I(t) for system (32) with τ = 3, λ(k) = 0.14k, and R₀ = 3.7505. (b) Evolutions of I₁₅ and I₈₅ for system (32) with τ = 3, λ(k) = 0.14k, and R₀ = 3.7505.

Figure 9.

(a) Evolutions of I(t) for system (32) with ω = 6, λ(k) = 0.14k, and R₀ = 3.7505. (b) Evolutions of I₁₅ and I₈₅ for system (32) with ω = 6, λ(k) = 0.14k, and R₀ = 3.7505.

Figure 10.

(a) Evolutions of I(t), for system (52) with ω = 3, 8, 14, 23, respectively, when R₀ = 1.3881 > 1. (b) Evolutions of I₁₅, I₄₅, I₈₅, for system (1) with ω = 23, respectively, when R₀ = 1.3881 > 1.

Although R₀ is irrelative to τ and ω. Figures 6 and 7 show that both the delay τ and ω have certain influence on the relative density of the infected nodes when R₀ < 1, for example, the faster the relative density of infected nodes converges to zero, the larger ω gets or the smaller τ gets. In addition, Figures 6 and 7 show that the average relative density of the infected nodes I(t) monotonically decreases to zero, whereas the relative density of infected nodes of connectivity k always breaks out first and then decreases to zero; the reason of the phenomenon appears that the spreading network is a scale-free one. Note that Figures 8 and 9 show that the delay τ and ω have much impact on the steady state of density of the infected nodes when R₀ > 1, the density of infected decreases as the delay τ and ω increase, which is consistent with the formula (37). We also know from (37) that I(t) → 0 as ω →  + ∞ or τ →  + ∞.

Especially, system (29) reduces to the following SIRS model [26]:

$\left\{\begin{array}{l}{\dot{S}}_k\left(t\right)=-\lambda \left(k\right)S_k\left(t\right)\Theta \left(t\right)+\mu I_k\left(t-\omega \right),\hfill \\ {\dot{I}}_k\left(t\right)=\lambda \left(k\right)S_k\left(t\right)\Theta \left(t\right)-\mu I_k\left(t\right),\hfill \\ {\dot{R}}_k\left(t\right)=\mu I_k\left(t\right)-\mu I_k\left(t-\omega \right).\hfill \end{array}\right.$E52

with the normalization condition

$S_k\left(t\right)+I_k\left(t\right)+R_k\left(t\right)=1.$E53

Figure 10 shows that when R₀ > 1, the quarantine delay ω can impact the density of infected nodes at the stationary state, and raising the quarantine period will suppress the viruses when ω is not large enough, which coincides with formula (37). Moreover, there exists periodic oscillation near the endemic equilibrium when ω is large enough. This is an interesting phenomenon which means that a bifurcation may appear.

4. Conclusion and discussion

An SEIR model with time delay on the scale-free network, which formulated a disease or computer virus transmission with constant latent period, is presented. For SEIR model, the basic reproduction number is

$R_0=\frac{1}{\mu }\frac{⟨\lambda \left(k\right)\varphi \left(k\right)⟩}{⟨k⟩}.$E54

When R₀ ≤ 1, there is no outbreak of the disease spreading, and the infection eventually disappears. When R₀ > 1, there exists at least one outbreak for the spreading of epidemic, and then lim_{t →  + ∞}I_k(t) = 0 due to global attractivity of M₀. If the recovered nodes cycle back into the susceptive class after an immune period, we obtain a SEIRS model with two time delays on the scale-free network, which formulated a disease transmission with constant latent and immune periods. For SEIRS model, the basic reproduction number is still R₀ shown in (54). If R₀ ≤ 1, although the equilibrium E₀ is globally stable and the infection eventually disappears, the equilibrium E₀ may lose its stability when R₀ > 1 and the infection will always exists.

Although R₀ is irrelevant to time delays, they influence the dynamical behaviors of the model such as slowing down the speed disappear of disease spreading on network, depressing the density of infected nodes at the stationary state.

In addition, for SEIRS model, numerical simulations show that the endemic equilibrium E_∗ may be globally asymptotically stable under some conditions when R₀ > 1 (shown in Figures 8 and 9). We would like to mention here that it is interesting but challenging to discuss the stability of equilibrium E_∗ when R₀ > 1.

Furthermore, more and more researchers realize the fundamental role of the stochastic nature of diseases on their dynamics. In order to gain analytical insight into the behavior of the epidemic spreading, we also may extend the models (2) and (29) to the ones with random perturbations, i.e., stochastic differential equation models.

Acknowledgments

This research was supported by the Hebei Provincial Natural Science Foundation of China under Grant No. A2016506002. Qiming Liu would like to thank Professor Zhen Jin of Shanxi University in China for his helpful suggestions that led to truly significant improvements of this chapter.