Open access peer-reviewed chapter

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

Written By

Qiming Liu, Meici Sun and Shihua Zhang

Submitted: April 25th, 2017 Reviewed: September 20th, 2017 Published: December 20th, 2017

DOI: 10.5772/intechopen.71087

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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.

Keywords

  • epidemic spreading
  • scale-free network
  • basic reproductive number
  • time delay
  • stability

1. Introduction

Following the seminal work on scale-free network, in which the probability of p(k) for any node with klinks to other nodes is distributed according to the power law p(k) = Ckγ  (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 [327].

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, 1517]. 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 SEIRmodel with time delay and a suitable SEIRSmodel 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 SEIRSmodel with time delay on scale-free network is discussed. The SEIRSmodel with time delay on scale-free network is discussed in Section 3. Finally, the main conclusions of this work are summarized in Section 4.

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2. Analysis of the SEIRmodel with time delay

2.1. The SEIRmodel

Suppose that the size of the network is a constant Nand 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 Sk(t), Ek(t), Ik(t) and Rk(t) be the relative density of susceptible nodes, exposed nodes, infected nodes, and recovered nodes of connectivity kat time t, respectively, where k = m, m + 1, ⋯, n(mand nare the minimum and maximum degree in network topology) and nis related to the network age, measured as the number of nodes N[3]:

n=mN1/γ1.E1

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

Ṡkt=λkSktΘt,Ėkt=λkSktΘtλkSktτΘtτ,İkt=λkSktτΘtτμIkt,Ṙkt=μIktE2

with the normalization condition

Skt+Ekt+Ikt+Rkt=1E3

holds due to the fact that the number of total nodes with degree kis a constant p(k)Nduring the period of epidemic spreading. Where λ(k) is the degree-dependent infection rate such as λ(k) = λk[3] and λC(k) [4], μis 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

Θt=1kk=mnϕkpkIktE4

in which ⟨k⟩ = ∑kp(k)kstands 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) = Ain [5], φ(k) = akα/(1 + bkα), 0 < α < 1 in [7], and so on. Here, we point out that the delay τin the model (2) in this paper is different from one in the model (2)–(4) in [15]. The incubation period τin 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

Ekt=λktτtSksΘsdsE5

and the normalization condition becomes the following mathematical form

Skt+λktτtSksΘsds+Ikt+Rkt=1.E6

The initial conditions of system (2) are

Skθ=φkθ,Ikθ=Ψkθ,Rkt=ζkθ,θτ0E7

and satisfy Sk0+λkτ0SksΘsds+Ik0+Rk0=1which guarantees the normalization condition holds. And Φk = (ϕk(θ), Ψk(θ), ζ(θ), k = m, m + 1, ⋯, n − m + 1) ∈ Care nonnegative continuous on [−τ, 0], ϕk(0) > 0, Ψk(0) > 0, and ζ(θ) = 0 for θ = 0. Cdenotes the Banach space C([−τ, 0], R3(n − m + 1)) with the norm, where ∣f(θ)∣τ = supτ ≤ θ ≤ 0 ∣ f(θ)∣. ω=(i=mnΨiθτ2+ϕiθτ2+ζiθτ21/2.

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

M0=ŜÊÎR̂Ek=Ik=0Sk+Rk=1RkSk0k=mm+1nE8

in which Ŝ=SmSm+1Sn,Ê=EmEm+1En,Î=ImIm+1In,R̂=RmRm+1Rn.

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

f+=limt+ftE9

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

Rk+Rk0=μ0+Iksds.E10

According to the last equation of system (2), Rk(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 Rk(+∞), and Rk(0) exits, i.e., 0<0+Ikudu<+. Since Ik(t) is smooth nonnegative function, we know Ik(+∞) = 0, i.e., limt →  + ∞Ik(t) = 0.

Furthermore, we have from (5) that

0Ekt=λktτtSksΘsdsλktτtΘsds,E11

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

tτtΘsds=Θξτ,tτξt.E12

We obtain from Ik(+∞) = 0 that limξ →  + ∞Θ(ξ)τ = 0 and then limt →  + ∞Ek(t) = 0. Hence, M0 is globally attractive [27].

In addition, it follows from (2) that

Ṡkt+Ėkt+İkt=μIkt.E13

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

Sk0Sk++Ek0Ek++Ik0Ik+=μ0+Iksds.E14

Noting that Ik(+ ∞) = 0, Ek(0) = Ek(+ ∞) = 0, and Sk(+ ∞) exists due to existence of 0+Iksds, we have from (14) that

0+Iksds=1μSk0+Ik0Sk+.E15

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

lnSk0Sk=λkkkϕkpk0+Iksds.E16

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

lnSk0Sk+=λkkkϕkpk1μSk0+Ik0Sk+.E17

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

lnSk+=λkμkkϕkpkSk+1.E18

Consequently,

Rk=1Sk+.E19

Hence, we have the following result.

Theorem 2.1.The equilibrium setM0=ŜÊÎR̂Ek=Ik=0Sk+Rk=1k=12nof system(2) is globally attractive, i.e.,limt →  + ∞Ik(t) = 0, limt → ∞Ek(t) = 0. And Rk(+∞), Sk(+∞) are given byformulas (18) and(19).

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

+=λkμkkϕkpkE20

Obviously, Eq. (20) does not hold, i.e., Sk(+∞) = 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),

R0=λkϕkμkE21

is the basic reproductive number for system(2).

Proof.Note that k=mnφkpkIktmay 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

tdt=1kkϕkpkİkt=1kkϕkpkλkSktτΘtτμIkt=Θtτ1kkλkϕkpkSktτμΘt.E22

We have

dΘtdtt=0=Θτ1kkλkϕkpkSkτμΘ0).E23

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

dΘtdtt=0=μ1μkkλkϕkpk1Θ0=μR01Θ0.E24

If R0 > 1, tdtt=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 tdtt=00. 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

Skt=Sk0eλkΨtE25

in which Ψt=1kk0tφkpkIkudu.Hence, it follows from Eqs. (22) and (25) that

dΘtdt=Θtτ1kkλkϕkpkSk0eλkΨtτμΘt.E26

By way of contradiction, supposing that T <  + ∞, then we have ddtΘt=0, and there exists a t1 ∈ (t, t + τ] such that ddtΘt1>0. It follows that there is a t2 ∈ [t, t1) such that ddtΘt2=0and Θ(t2) < Θ(t1). Note that Θ(t2 − τ) ≥ Θ(t1 − τ). It follows from (25) that

0<ddtΘt1ddtΘt2=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, limt →  + ∞Ik(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 = (λk2⟩)/(⟨k⟩), which is identical with the results that the epidemic threshold λc = (λk⟩)/(⟨k2⟩) in [9]. And, R0 is always more than unity when Nis 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) = Ckγ, and Csatisfies k=1npk=1. Here, we set the maximum degree n = 100 and the minimum degree m = 1. Consider system (2), let φ(k) = akα/(1 + bα) in which a = 0.5, α = 0.75, b = 0.02 and λ(k) = λk, and let γ = 2.5. Figures 14 show the dynamic behaviors of system (2) with the initial functions satisfying condition (7).

Figure 1.

(a) The time evolutions for system(2)withλ = 0.5, μ = 0.4, τ = 5 andIk(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the otherkandR0 = 0.4006. (b) The time evolutions for system(2)with different initial values,λ = 0.4, μ = 0.4, τ = 5 andR0 = 0.4006.

Denote that

St=k=mnpkSkt,It=k=mnpkIkt,Rt=k=mnpkRkt.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)(Sk(t) + Rk(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)withλ = 1, μ = 0.1, τ = 5 andIk(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the otherkandR0 = 3.2051. (b) The time evolutions for system(2)with different initial values,λ = 1, μ = 0.1, τ = 5 andR0 = 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 andR0 = 0.4006. (b) Phase trajectories on SR-plane of system(2)with different initial values,λ = 1, μ = 0.1, τ = 5 andR0 = 3.2051.

Second, time delay τhas 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 τgets, 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].

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3. Analysis of the SEIRSmodel with time delays

3.1. The SEIRSmodel

Since some diseases confer temporary immunity, the recovered nodes cycle back into the susceptive class after an immune period. Let ωbe 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 Sk(t), Ek(t), Ik(t) and Rk(t), at the mean-field level, satisfy the following set of coupled different equations when t > 0:

Ṡkt=λkSktΘt+μIktω,Ėkt=λkSktΘtλkSktτΘtτ,İkt=λkSktτΘtτμIkt,Ṙkt=μIktμIktωE29

with the normalization condition (3).

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

Ekt=λktτtSksΘsds,Rkt=μtωtIksds.E30

Hence, the normalization condition becomes the following mathematical form:

Skt+λktτtSksΘsds+Ikt+μtωtIksds=1.E31

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

Ṡkt=λkSktΘt+μIktω,İkt=λkSktτΘtτμIktE32

with the normalization condition (31).

The initial conditions of system (32) are

Skθ=φkθ,Ikθ=Ψkθ,θσ0,E33

which satisfy Sk0+λkτ0SksΘsds+Ik0+μω0Iksds=1. Hence, the normalization condition (31) holds. And, Φk = (ϕk(θ), Ψk(θ), k = m, m + 1, ⋯, n − m + 1) ∈ Care nonnegative continuous on [−σ, 0], ϕk(0) > 0, Ψk(0) > 0, and ζ(θ) = 0 for θ = 0. Cdenotes the Banach space C([−σ, 0], R2(n − m + 1)) with the norm ω=(i=mnΨiθσ2+φiθσ21/2, where |f(θ)|σ = supτ ≤ θ ≤ 0 ∣ f(θ)∣.

3.2. The main results for the model

Denote that

R0=1μλkφkk,E34

where fk=k=mnfkpkin 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 equilibriumE=SmSm+1SnImIm+1Inwhen 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:

λkSkΘ+μIk=0,Sk+λkτSkΘ+Ik+ωμIk=1,E35

where

Θ=1kkϕkpkIk.E36

We obtain from (35) that

Ik=λkΘμ+λk1+μτ+ωμΘ.E37

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

Θ=λkk=mnϕkpkλkΘμ+λk1+μτ+ωμΘ=fΘ.E38

Note that

fΘΘ=0=λkk=mnϕkpkλkμμ+λk1+μτ+ωμΘ2Θ=0=λkμk=R0E39

and

f′′Θ=2λkk=mnϕkpkλ2k2μ1+μτ+ωμΘμ+λk1+μτ+ωμΘ3<0.E40

Hence, if R0 > 1, Eq. (38) has a unique positive solution. Consequently, system (32) has a unique endemic equilibrium ES1S2SnI1I2Insince (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

Vt=12Θ2t+γtτtΘ2μ,E41

where γis a constant to be determined. Let G=ϕ:V̇ϕ=0, and Mis the largest set in Gwhich is invariant with respect to system (32). Clearly, Mis not empty since E0 ∈ M. Calculating the derivative of V(t) along the solution of (32), we get

V̇t3.3=Θt1λkkφkpkλkSktτΘtτμΘt+γΘ2tγΘ2tτΘt1kλkϕkΘtτμΘt+γΘ2tγΘ2tτ12kλkϕkΘ2t+12kλkϕkΘ2tτμΘ2t+γΘ2tγΘ2tτ=12kλkϕkμ+γΘ2t+12kλkϕkγΘ2tτ.E42

Note that R0 ≤ 1 implies 1kλkφkμ<0; if we let γ=12kλkφk, we have from (42) that

V̇t3.31kλkϕkμΘ2t0.E43

It follows from Sk(t) + Ek(t) + Ik(t) + Rk(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:

ẋt=a1xtτa2xt,E44

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

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

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

Lemma 3.2.([29], p 273–280) Let X be a complete metric space, X = X0 ∪ ∂X0, whereX0, 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. A˜is isolated and has an acyclic covering M.

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

WsMiX0=,E45

where A˜=xAωx, 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 thatlimt →  + ∞ inf I(t) > ε, whereIt=k=mnϕkpkIkt.

Proof.Denote that

X=S¯Ψ¯:Ψkθ0forallθζ0k=mm+1n,E46
X0=S¯Ψ¯:Ψkθ>0forsomeθζ0k=mm+1n,E47

and, consequently,

X0=X/X0=S¯Ψ¯:Ψiθ=0forallθσ0imm+1n,E48

where S¯Ψ¯=SmSm+1SnΨmΨm+1Ψn.

Let (Sm(t), Im(t), ⋯, Sn, In(t)) = (Sm(t, ω), Im(t, ω), ⋯, Sn(t, ω), In(t, ω)) be the solution of (32) with initial function ω = (ζm(θ), Ψm(θ), ⋯, Ψn(θ), ϕn(θ)) and

T(t)(ω)(θ) = (Sm(t + θ, ω), Im(t + θ, ω), ⋯, Sn(t + θ, ω), In(t + θ, ω)), θ ∈ [−σ, 0]. Obviously, Xand X0 are positively invariant sets for T(t). T(t) is completely continuous for t > 0. Also, it follows from 0 < Sk(t), Ik(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, A˜=E0, while 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

limt+infSkt=1,limt+infIkt=0,k=1,2,,n.E49

Since R0 > 1, we may choose 0 < η < 1 such that α = η(λ(k)τφ(k)⟩ + 1 + μω) satisfies (1 − α)R0 > 1. At the same time, there exists a t1 > τsuch that Ik(t) < ηfor t > t1 due to limt →  + ∞ inf Ik = 0.

When t > t1, we obtain from (32) that

Skt=1Ikt+λktτtSksΘsds+μtωtIksds1η+λkτηϕk+μωη=1α.E50

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

Θ̇t=1kk=mnϕkpkİkt=1kk=mnϕkpkλkSktτΘtτμIkt1αλkϕkkΘtτ)μΘtE51

Note that (1 − α)R0 > 1, and it follows with 1αλkφkk>μ. Hence, we obtain from (51) that limt →  + ∞Θ(t) =  + ∞ according to Lemma 3.1 contradicts limt →  + ∞Θ(t) = 0 due to limt →  + ∞Ik(t) = 0. Then, Ws(E0) ∩ X0 = ∅.

Hence, the infection is uniformly persistent according to Lemma 3.2, i.e., there exists a positive constant εsuch that limt →  + ∞ inf Ik > εand, consequently, limt+infIt>k=mnpkε=ε. This completes the proof.

In addition, Liu and Zhang discussed a simple SEIRSmodel 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) = Ain 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) = Ckγ, and Csatisfies k=1npk=1. Here, we set still the maximum degree n = 100 and the minimum degree m = 1.

Consider system (32). Let φ(k = akα/(1 + bα) 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 14 show that the dynamic behaviors of system (32) with the initial functions satisfy condition (33) in which Ik(s) = 0.45, k = 2, 3, 4, 5 for s ∈ [−σ, 0] and Ik = 0, k ≠ 2, 3, 4, 5.

Figure 4.

(a) The time evolutions of the average relative densityI(t) for system(2)with differentτas well asIk(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the otherk,λ = 0.5, μ = 0.4 andR0 = 0.4006. (b) The time evolutions of the average relativeI(t) for system(2)with differentτas well asIk(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the otherk,λ = 1, μ = 0.1 andR0 = 3.2051.

Figure 5.

(a) The time evolutions of the average force of infectionΘ(t) for system(2)with differentτas well as withIk(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the otherk,λ = 0.5, μ = 0.4 andR0 = 0.4006. (b) The time evolutions of the average force of infectionΘ(t) for system(2)with differentτas well asIk(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the otherk,λ = 1, μ = 0.1 andR0 = 3.2051.

Figure 6.

(a) Evolutions ofI(t) for system(32)withτ = 3,λ(k) = 0.03k, andR0 = 0.8037. (b) Evolutions ofI15 andI85 for system(32)withτ = 3,λ(k) = 0.03k, andR0 = 0.8037.

Figure 7.

(a) Evolutions ofI(t) for system(32)withω = 6,λ(k) = 0.03k, andR0 = 0.8037. (b) Evolutions ofI15 andI85 for system(32)withω = 6,λ(k) = 0.03k, andR0 = 0.8037.

Figure 8.

(a) Evolutions ofI(t) for system(32)withτ = 3,λ(k) = 0.14k, andR0 = 3.7505. (b) Evolutions ofI15 andI85 for system(32)withτ = 3,λ(k) = 0.14k, andR0 = 3.7505.

Figure 9.

(a) Evolutions ofI(t) for system(32)withω = 6,λ(k) = 0.14k, andR0 = 3.7505. (b) Evolutions ofI15 andI85 for system(32)withω = 6,λ(k) = 0.14k, andR0 = 3.7505.

Figure 10.

(a) Evolutions ofI(t), for system (52) withω = 3, 8, 14, 23, respectively, whenR0 = 1.3881 > 1. (b) Evolutions ofI15, I45, I85, for system (1) withω = 23, respectively, whenR0 = 1.3881 > 1.

Although R0 is irrelative to τand ω. Figures 6 and 7 show that both the delay τand ωhave 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 ωgets or the smaller τgets. 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 kalways 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 τand ωhave much impact on the steady state of density of the infected nodes when R0 > 1, the density of infected decreases as the delay τand ωincrease, 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 SIRSmodel [26]:

Ṡkt=λkSktΘt+μIktω,İkt=λkSktΘtμIkt,Ṙkt=μIktμIktω.E52

with the normalization condition

Skt+Ikt+Rkt=1.E53

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

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4. Conclusion and discussion

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

R0=1μλkϕkk.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 limt →  + ∞Ik(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 SEIRSmodel with two time delays on the scale-free network, which formulated a disease transmission with constant latent and immune periods. For SEIRSmodel, 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 SEIRSmodel, 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.

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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.

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Written By

Qiming Liu, Meici Sun and Shihua Zhang

Submitted: April 25th, 2017 Reviewed: September 20th, 2017 Published: December 20th, 2017