The role of vaccine success is introduced into an epidemic spreading model consisting of three states: susceptible, infectious, and vaccinated. Moreover, the effect of three types, namely, contact, local, and global, of infection awareness and immunization awareness is also taken into consideration. The model generalizes those considered in Pastor-Satorras and Vespignani [Phys. Rev. E 63, 066117 (2001)], Pastor-Satorras and Vespignani [Phys. Rev. E 65, 036104 (2002)], Moreno et al. [Eur. Phys. J. B 26, 521–529 (2002)], Wu et al. [Chaos 22, 013101 (2012)], and Wu et al. [Chaos 24, 023108 (2014)]. Our main results contain the following. First, the epidemic threshold is explicitly obtained. In particular, we show that, for any initial conditions, the epidemic eventually dies out regardless of what other factors are whenever some type of immunization awareness is considered, and vaccination has a perfect success. Moreover, the threshold is independent of the global type of awareness. Second, we compare the effect of contact and local types of awareness on the epidemic thresholds between heterogeneous networks and homogeneous networks. Specifically, we find that the epidemic threshold for the homogeneous network can be lower than that of the heterogeneous network in an intermediate regime for intensity of contact infection awareness while it is higher otherwise. In summary, our results highlight the important and crucial roles of both vaccine success and contact infection awareness on epidemic dynamics.

In this work, we study the effect of vaccine success and the impact of awareness on a general epidemic spreading model consisting of three states: susceptible, infectious, and vaccinated. We obtain two unexpected results. First, if people are aware of the importance of immunization and vaccination is always successful, then, for any initial conditions, the epidemic can be under control even if the spreading rate of the disease is extremely high. Second, it is shown that, although the epidemic threshold is smaller for the heterogeneous network in low and high intensity of contact infection awareness, there is an intermediate regime where the epidemic threshold is in contrast higher for the homogeneous network.

The vaccination is proved to be an effective method to control the spread of epidemic diseases. However, not many theoretical results are done to discuss the impact of vaccine failure, which may go as high as 50%6 on epidemic spreading. It is reported6 that, although vaccine success has usually been about 85%, success as low as 44% has also been observed. To the best of our knowledge, the first theoretical work to discuss the role of vaccine failure was given in Ref. 7.

In the study of epidemiology, finding ways to prevent the outbreak of an epidemic disease is always an important issue. In the past few decades, several effective immunization strategies have been proposed to minimize the risk of the outbreak of epidemic diseases on complex networks.2,5,8–16 Recently, the study of aspects of human responses towards the spread of epidemic diseases has drawn much attention4,7,17–28 due to the fact that the change of individual behaviors has an effect on the epidemic dynamics. In the literature,4,18,29,30 according to the source of information, awareness is classified into three types, which are termed contact awareness, local awareness, and global awareness. For the first type, it is assumed4,29 that individuals with larger contact number in a network are more willing to change their behavior in order to reduce the risk of being infected. The second one is based on personal local information or local infection density.4,18 The third one is based on the global infection density in a whole community.4,31 The information relating to that may come from national media, e.g., public health authorities. In this work, we shall further differentiate each of the three types of awareness into two cases: infection awareness and immunization awareness. People with infection awareness are more likely to take extra steps to avoid infection, while people with immunization awareness increase likelihood for getting vaccinated. It should also be noted that in heterogeneous networks such as scale-free (SF) ones, the effect of these three types of awareness cannot be separated completely. Such combined effect is also addressed in Ref. 4.

In this work, we consider an epidemic spreading model consisting of three states: susceptible (S), infectious (I), and vaccinated (V), for which the changes of states between S and I or S and V take into account the impact of individual awareness and the role of vaccine success. Our main results contain the following. First, the epidemic threshold is explicitly obtained. In particular, if immunization awareness is considered and vaccination has a perfect success, then the epidemic dies out eventually regardless of what other factors are or what initial conditions are. Furthermore, such threshold depends on the contact type of awareness and local infection awareness, and is independent of all the other types of awareness. Such result indeed suggests that the effect of these three types of awareness against the outbreak of an epidemic disease in decreasing order in terms of their importance is contact awareness, local awareness, and global awareness. Second, we compare the epidemic threshold for the homogeneous network, where individuals have roughly the same degree (contact number), and the heterogeneous network, where individuals' degree distribution has a heavy tail. We show that the epidemic threshold for the homogeneous network can be lower than that of the heterogeneous network in an interval J of intensity of contact infection awareness, while it is higher otherwise. In certain extreme cases, the interval J is empty or infinite.

The organization of the paper is as follows. We introduce our discrete-time epidemic spreading model and its continuous-time version in Section II. The study of the epidemic threshold and the stability of the disease free equilibrium (DFE) on the continuous-time epidemic spreading model are given in Section III. In Section IV, we discuss the effect of heterogeneity of networks on the epidemic threshold. Numerical simulations to support our theoretical results and their other implications are given in Section V. In Section VI, we discuss some of the works related to ours. The summary of our obtained results and the future work is provided in Section VII. In  Appendix A, we give the detailed derivation of the continuous-time epidemic spreading model from the discrete-time one. All the proofs of our results are recorded in  Appendix B.

In this section, we shall depict the epidemic spreading model under consideration. We begin with the formulation based on probabilistic discrete-time Markov chains. It is assumed that each individual, at each time t, is in one of the following three discrete states: infectious (I), vaccinated (V), and susceptible (S), and each could transfer its state at time t to a new one at time t + 1 through one of the following four different ways: (i) SI, (ii) SV, (iii) VS, and (iv) IS. Letting pi(t) (respectively, qi(t)), i=1,,N, be the probability that a node i is infectious (respectively, vaccinated) at time t, our model under consideration reads as follows:

(1)
pi(t+1)=(1γ)pi(t)+[1pi(t)qi(t)]Probi(SI),qi(t+1)=(1δ)qi(t)+[1pi(t)qi(t)]Probi(SV),
(1a)
where

Probi(SI)=1[1λψ(ki)(1βp(t))(1αsiki)]si,
(1b)
Probi(SV)=1ψ̃(ki)(1β̃p(t))(1α̃siki).
(1c)

Here ki denotes the degree of a node i (i.e., the number of neighbors of a node i); si, ranging from 0 to ki, denotes the number of infectious neighbors of a node i; and p(t) denotes the fraction of infectious nodes in the network. For other parameters and terms in (1a)(1c), their epidemic meanings are explained in the following list.

  • The parameter γ[0,1] in (1a) denotes the recovering probability for infectious individuals for the whole time period.

  • The parameter δ[0,1] in (1a) denotes the probability of vaccine failure for vaccinated individuals.

  • The term Probi(SI), defined in (1b), gives the probability that a node i changes its state from S to I. The quantity λ[0,1] denotes the spreading rate for which an infectious node would actually transmit a disease through an edge connecting to a susceptible node.32,33

    • The term ψ(ki) in (1b) describes the contact awareness to avoid infection for a node i,4,29 which is to be termed as the contact infection awareness in short. Similar usage of the term is to be followed. We assume that a node with a higher degree is aware of the higher risk to be infected and, consequently, it increases its protection and hence reduces the probability of getting infected. Thus, it is assumed that function ψ(k) is decreasing in the degree k. In Ref. 4, ψ(k) is chosen to be kb for some nonnegative constant b. The larger the quantity b is, the smaller the ψ(k) and the term defined in (1b) is, and hence the less likely a susceptible individual would get infected. The constant b is to be termed as the intensity of contact infection awareness. Note that, if ψ(k) is independent of k (i.e., b = 0), then there is no contact infection awareness in the epidemic spreading.

    • The term (1βp(t)) in (1b) is a decreasing function of p(t) and β. Thus, this term indicates that, as the infectious density p(t) in the population increases, one increases its protection and hence reduces the probability of getting infected. And so, this term represents the global infection awareness.30,34 Clearly, such setup indicates that people are made aware of the epidemic disease through national media. Moreover, β[0,1] describes the strength of the average risk assessment from global awareness and hence β is to be termed as the intensity of global infection awareness. For β = 0, this means that global infection awareness is not considered in the epidemic spreading.

    • The term (1α(si/ki)) in (1b) is a decreasing function of (si/ki) and α. Thus, this term indicates that as the local infectious density (si/ki) of a node i increases, one increases its protection and hence reduces the probability of getting infected. And so, this term represents the local infection awareness.5,30,34 The parameter α[0,1] indicates the strength of the average risk assessment from local awareness, and hence α is to be termed as the intensity of local infection awareness. Note that α = 0 means that there is no local infection awareness in the epidemic spreading.

    • The term
      [1λψ(ki)(1βp(t))(1αsiki)],
      represents the probability that a susceptible node i will not get infected when it makes a contact with exactly one infectious individual. Thus
      [1λψ(ki)(1βp(t))(1αsiki)]si,

      is the probability that a node i will not change its state from susceptible to infectious when it makes contacts with si infectious individuals. Thus, Probi(SI), defined in (1b), gives the probability that a node i changes its state from susceptible to infectious.

  • The term Probi(SV), defined in (1c), gives the probability that a node i changes its state from S to V when the epidemic disease is spreading.5 

    • Similarly, the term ψ̃(ki) in (1c) describes the contact awareness to get immunized, which is to be termed as the contact immunization awareness in short. Similar usage of the term is to be followed. We assume that a node with a higher degree is aware of the higher risk to be infected, and it is more likely to get vaccinated to reduce the probability of getting infected. Thus, it is assumed that function ψ̃(k) is decreasing in k. We also set ψ̃(k)=kb̃ for some nonnegative constant b̃, where the parameter b̃ is to be termed as the intensity of contact immunization awareness. That is, for higher b̃, a susceptible individual tends to have a higher probability of getting vaccinated.

    • The terms (1β̃p(t)) and β̃[0,1] in (1c) describe the global immunization awareness and the intensity of global immunization awareness, respectively. Note that Probi(SV) is increasing in β̃. Consequently, for higher β̃, a susceptible individual is more likely to get vaccinated.

    • The term (1α̃(si/ki)) in (1c) represents the probability that a node i changes its state from susceptible to vaccinated due to its local awareness. Note that Probi(SV) is increasing in α̃. Likewise, (1α̃(si/ki)) and the constant α̃[0,1] are called local immunization awareness and the intensity of local immunization awareness, respectively.

We summarize the epidemic meaning of each parameter and function in Eq. (1) in Table I.

TABLE I.

The depiction of each parameter and function in epidemic spreading model (1). Note that a node is less (respectively, more) likely to get infected (respectively, immunized) with the increase of intensity α, β, or b (respectively, α̃,β̃, or b̃).

DescriptionParameters
Spreading rate λ  
Recovering probability γ  
Probability of vaccine failure δ  
Awareness Infection Immunization 
Intensity of local awareness α α̃ 
Intensity of global awareness β β̃ 
Contact awareness ψ(k) ψ̃(k) 
Intensity of contact awareness b b̃ 
DescriptionParameters
Spreading rate λ  
Recovering probability γ  
Probability of vaccine failure δ  
Awareness Infection Immunization 
Intensity of local awareness α α̃ 
Intensity of global awareness β β̃ 
Contact awareness ψ(k) ψ̃(k) 
Intensity of contact awareness b b̃ 

To investigate the effect of the heterogeneity of networks on the epidemic dynamics, we next make a coarse-graining approximation on (1) to derive a continuous-time degree-based mean-field model1–3,35,36 by assuming that (i) individuals with the same degree have the same property of dynamical behaviors, (ii) the variable si could be approximated by its expected value, (iii) the underlying network is uncorrelated, and (iv) the high order terms in (1b)(1c) are negligible. To begin with, we divide individuals into several distinct groups depending on their degrees k. The fraction of the number of individuals with degree k is denoted by P(k) and the corresponding infectious and vaccinated densities among nodes with degree k are denoted by pk(t) and qk(t), respectively. The model then reads as follows:

(2)
ṗk(t)=γpk(t)+[1pk(t)qk(t)]{λψ(k)(1βp(t))Θ(t)[k(1αΘ(t))α(1Θ(t))]},
(2a)
q̇k(t)=δqk(t)+[1pk(t)qk(t)]{1ψ̃(k)(1β̃p(t))(1α̃Θ(t))},
(2b)
where

p(t)=kP(k)pk(t),
(2c)
Θ(t)=kkP(k)pk(t)kkP(k):=kkP(k)pk(t)k.
(2d)

The detailed derivation of the above formula is provided in  Appendix A due to its similarity to those in Refs. 4 and 5 and the tediousness.

We end the section by claiming that the epidemic spreading model (2) is well-defined in the sense that if initial conditions pk(0) and qk(0) satisfy 0<pk(0),qk(0), and pk(0)+qk(0)<1, for all k, then pk(t) and qk(t) also satisfy 0<pk(t),qk(t), and pk(t)+qk(t)<1, for all k and t > 0. The proof of Proposition 1 is given in  Appendix B.

Proposition 1. Define

Δ2n={(pk,qk)1kn2n:pk,qk0andpk+qk1}.

Then Δ2n is positively an invariant for (2).

In this section, we study the epidemic spreading model (2) and compute the threshold for effective spreading rate λ̂(:=λ/γ).1,37 Our derived results are summarized in the following theorem and the proof is to be provided in  Appendix B.

Theorem 1 (see, e.g., Table II and Fig. 1). Consider the epidemic spreading model (2). Then the following two assertions hold. (i) In the case that δ = 0 and either α̃2+β̃2>0 or ψ̃(k)1, the epidemic dies out. In particular, every solution (pk(t),qk(t))1kn of (2) converges to a disease free equilibrium (DFE) (0,q¯k*)1kn for some q¯k* in [0,1]. (ii) In other cases, there exists an epidemic thresholdλ̂c, as given in the following:

λ̂c:=kk(1qk*)ψ(k)(k2αk)P(k),
(3)

where

qk*={(1ψ̃(k))δ+(1ψ̃(k))ifδ>0,qk(0)ifδ=0,α̃=β̃=0,ψ̃(k)1,
(4)

such that the epidemic dies out when the effective spreading rateλ̂(:=λγ) is smaller than λ̂c; otherwise, the disease breaks out.

TABLE II.

The table lists the parameters chosen for simulation in Fig. 1. We also record the corresponding epidemic thresholds λ̂c and final epidemic sizes p(:=limtp(t)). The simulation is evaluated for (2) under the degree distribution P(k)=kr/c, k=m,,M, where γ=2.85, m = 2, M = 1000, and c=k=mMkr, and with the recovering probability γ=0.15. In the table, the dashed line means that the values of corresponding parameters are chosen to be the same as those given in No. 1.

No.λ̂αbb̃δββ̃α̃λ̂cp
1.93 0.6 0.8 0.25 0.25 0.1 0.1 0.1 1.90 0.71% 
… … … … … 0.6 … … … 0.56% 
… … … … … … 0.6 … … 0.53% 
… … … … … … … 0.6 … 0.52% 
1.87 … … … … … … … … 
… 0.7 … … … … … … 1.98 
… … 0.85 … … … … … 2.07 
… … … 0.3 … … … … 2.05 
… … … … 0.2 … … … 2.15 
10 … 0.1 Nonexistence 
No.λ̂αbb̃δββ̃α̃λ̂cp
1.93 0.6 0.8 0.25 0.25 0.1 0.1 0.1 1.90 0.71% 
… … … … … 0.6 … … … 0.56% 
… … … … … … 0.6 … … 0.53% 
… … … … … … … 0.6 … 0.52% 
1.87 … … … … … … … … 
… 0.7 … … … … … … 1.98 
… … 0.85 … … … … … 2.07 
… … … 0.3 … … … … 2.05 
… … … … 0.2 … … … 2.15 
10 … 0.1 Nonexistence 
FIG. 1.

Simulation results in model (2) for the ten cases given in Table II. For case Nos. 1–4, their corresponding time series (t,p(t)),0t104, are plotted with the bold lines and the values of p(t) are labeled on the right vertical axis of the log scale ranging from 103 to 102. For other cases Nos. 5–10, their corresponding time series (t,p(t)),0t104, are plotted with the fine lines and the values of p(t) are labeled on the left vertical axis of the log scale, where p(t) are less than 1010 when t7000.

FIG. 1.

Simulation results in model (2) for the ten cases given in Table II. For case Nos. 1–4, their corresponding time series (t,p(t)),0t104, are plotted with the bold lines and the values of p(t) are labeled on the right vertical axis of the log scale ranging from 103 to 102. For other cases Nos. 5–10, their corresponding time series (t,p(t)),0t104, are plotted with the fine lines and the values of p(t) are labeled on the left vertical axis of the log scale, where p(t) are less than 1010 when t7000.

Close modal

Remark 1. (i) The implication of the first assertion of the theorem is that if people are aware of the importance of immunization and vaccination has a perfect success, then the epidemic is to die out eventually regardless of what other factors are. (ii) We see clearly, via (3) and (4), that, when δ>0

λ̂c=kkδδ+(1ψ̃(k))ψ(k)(k2αk)P(k),
(5)

which depends on the contact type of awareness (ψ and ψ̃), local infection awareness (α) and is independent of local immunization awareness (α̃) and global type of awareness (β and β̃). It should also be noted, though, that α̃, β, and β̃ play the role on decreasing the final epidemic size (limtp(t)) when the disease breaks out (see, e.g., Table II and Fig. 1 in Section V).

In this section, we consider the impact of awareness on the epidemic threshold λ̂c as given in (5). Specifically, we assume the contact type of awareness to be given as follows:4 

ψ(k)=kbandψ̃(k)=kb̃.
(6)

It means that individuals with quite a large degree have negligible probability of being infected. Such assumption is also consistent with the strategy of the targeted immunization for the susceptible individuals in the epidemic spreading model,2 where individuals with a degree larger than some constant k0 are much more inclined to take immunization.

Complex networks describe a wide range of systems in nature and society such as the World Wide Web links, biological networks, and the contact network of individuals.1 SF networks,38 which are heterogeneous, have frequently been introduced into the epidemic spreading models. Their degree distributions follow a power law P(k)kr and, typically, the exponent r(2,3] and k[m,M] where m and M are, respectively, the minimum and maximum of degrees among nodes in SF networks. In this setting, the average of connections k is finite but the variance k2k2 is infinite as M approaches to infinity. Here k2:=kk2P(k). Remark that, throughout the paper, we denote by f(k)=kf(k)P(k) for any f(k).

For a SF network, if neither type of awareness is considered, that is, α=α̃=β=β̃=0 and ψ(k)=ψ̃(k)1, then we have, via (5), that λ̂c(SF)=kk2 and hence limMλ̂c(SF)=0. It implies that the epidemic disease is bound to spread in a SF network with large M. However, by increasing the intensity b of contact infection awareness, we shall prove in the following theorem that the epidemic can be under control.

Theorem 2. Suppose that the probability δ of vaccine failure is positive. Then the epidemic threshold λ̂c(SF) for a SF network with the exponent r(2,3] and k[m,M] tends to be 0 as M approaches to if and only if b3r.

The above theorem indicates that the intensity b of contact infection awareness plays a critical role in determining the outbreak of the epidemic disease in SF networks. When the vaccine success is not perfect and the intensity b of contact infection awareness is low, the epidemic threshold λ̂c becomes zero in the limit. As a result, even for extremely low effective spreading rates, the disease would be able to diffuse through the population and prevail in the SF networks. However, by sufficiently increasing the intensity b of contact infection awareness, the epidemic threshold λ̂c then in the limit becomes a positive finite value.

In this subsection, we consider the epidemic disease spreading in the homogeneous network and compare its epidemic threshold with that of the heterogeneous network (specifically, the SF network) under the assumption that two networks have the same average degree number k. Contrary to the heterogeneous network that owns a long-tail degree distribution, another wide class of networks has exponentially bounded degree fluctuations and each node in the network has roughly the same number of links, kk. Networks of this property are called the homogeneous networks. Paradigmatic homogenous networks are the Erdös-Rényi random graphs and the Watts-Strogatz (WS) small-world models. For simplicity, we assume herein that all nodes in the homogeneous network have exactly the same degree. Then, via (5), we have that the corresponding epidemic threshold is

λ̂c=kδδ+[1kb̃]kb[k2αk]=:λ̂c(Homo).
(7)

Note that, when neither type of awareness is considered, that is, b̃=b=α=0, we have, via (7), that λ̂c(Homo)=k/k2. Consequently

λ̂c(SF)(=k/k2)<λ̂c(Homo)(=k/k2),
(8)

which implies that the epidemic disease is easier to break out in the heterogeneous network than in the homogeneous network. However, when awareness is taken into account, some more complicated and interesting results concerning the epidemic threshold λ̂c can be observed. We will state our finding in the following theorems. The detailed proofs of these results, making use of Jesen's inequality and some properties of log-convex functions, are recorded in  Appendix B.

Theorem 3. Suppose b̃=0,δ>0 and k[m,M]. Then, for any α[0,1), there exists some b2(=b2(α,m,M))[2,21α) such that

{λ̂c(SF)<λ̂c(Homo)ifb[0,1)(b2,),λ̂c(SF)=λ̂c(Homo)ifb=1orb2,λ̂c(SF)>λ̂c(Homo)ifb(1,b2)=:J0,α,δ,m.

Moreover, b2(α,m,M) is strictly increasing in α and satisfies k2b2k2b2k1b2k1b2=α.

The theorem is essentially amount to saying that, for b̃=0, there exists a nonempty and finite interval J=(1,b2) for which the epidemic disease is easier to break out in the homogeneous network than in the SF network provided that the intensity b of contact infection awareness lies in J. On the other hand, if b[1,b2], then the epidemic disease is easier to break out in the SF network.

In the next theorem, we show that, if the intensity b̃ of contact immunization awareness is nonzero, then the corresponding interval J could be empty, nonempty and finite, or infinite.

Theorem 4. Suppose b̃>0,δ>0 and k[m,M]. Then, for any α[0,1], there exists at most one open and connected interval Jb̃,α,δ,m such that

λ̂c(SF)>λ̂c(Homo)ifandonlyifbJb̃,α,δ,m.

Moreover, λ̂c(SF)=λ̂c(Homo) if and only if b satisfies

δk2bδ+1kb̃δδ+1kb̃k2bδk1bδ+1kb̃δδ+1kb̃k1b=α.

Note that Jb̃,α,δ,m could be either of the form (a, b), 0<a<b<, or of the form (a,), or the empty set.

For m, the minimum degree among nodes in the SF network, sufficiently large and δ0, we are able to show that Jb̃,α,δ,m must be an open, connected, nonempty, and finite interval. In particular, we prove that, given any set of parameters (b,b̃,α)+×+×[0,1], its corresponding sign of λ̂c(SF)λ̂c(Homo) can be determined as long as m is sufficiently large and δ0. The above mentioned result is recorded in the following theorem.

Theorem 5. Consider b̃,δ>0 and let A be the union of the sets Ai, i=1,2,,5, where A1=([0,1)(2,))×+×[0,1], A2={1}×(1,)×[0,1], A3={1,2}×{1}×[0,1δ+1), A4={2}×(1,)×{0}, A5={2}×(0,1)×[0,1], and B={1,2}×{1}×{1δ+1}. Here +=[0,). Then the following holds.

  • For any set of parameters (b,b̃,α)A, there exists some positive integer m0 such that λ̂c(SF)<λ̂c(Homo) whenever the minimum degree m in the SF network is greater than or equal to m0. In particular, when (b,b̃,α)A3A4, we have that m0=1.

  • For any set of parameters (b,b̃,α)B, λ̂c(SF)=λ̂c(Homo).

  • For any set of parameters (b,b̃,α)/AB, there exists some positive integer m0 such that λ̂c(SF)>λ̂c(Homo) whenever the minimum degree m in the SF network is greater than or equal to m0. In particular, when b = 1 or 2, b̃=1 and α>1δ+1, we have that m0=1.

Above theorem implies that, for any given b̃0,0α1 and δ>0, the interval Jb̃,α,δ,m is open, connected, nonempty, and finite whenever m is sufficiently large. Indeed, for any b(0,1)(2,) (respectively, b(1,2)), there exists some integer m0 such that bJb̃,α,δ,m (respectively, bJb̃,α,δ,m) whenever mm0 since (b,b̃,α)A1 (respectively, (b,b̃,α)AB).

In the first part of this section, we illustrate some numerical simulations for (coarse-graining) epidemic spreading model (2) to verify that the observations made in Remark 1 (ii) are indeed true. To see this, we set, in (2), P(k)=kr/c,k=m,,M, where r = 2.85, m = 2, M = 1000, and c=k=mMkr, while all other parameters are treated as testing variables. The simulation results in model (2) are provided in Table II and Fig. 1. In Table II, we record the parameters used for simulation, epidemic thresholds λ̂c computed by (5) and the finial epidemic sizes p(:=limtp(t)) from the simulation. It can be observed that, for epidemic spreading model (2), if the parameters are chosen as those in No. 1 (respectively, No. 5) in Table II, then λ̂(=1.93)>λ̂c(1.90) (respectively, λ̂(=1.87)<λ̂c(1.90)). Consequently, we have that the epidemic disease breaks out (respectively, dies out) and its corresponding p0.71%. Meanwhile, if we increase the intensity b̃, b, or α of awareness or decrease the probability δ of vaccine failure, then the outbreak of the epidemic disease can be prevented (see Nos. 6–9 in Table II). However, if we increase the intensity β, β̃, or α̃ of awareness, then it cannot prevent the epidemic outbreak but helps to reduce the final epidemic size (see Nos. 2–4 in Table II). We also point out that, if the vaccine success is perfect and immunization awareness is introduced, then the epidemic would eventually die out (see No. 10 in Table II). The time series simulation for the cases in Table II are given in Fig. 1.

We next aim to compare the size of λ̂c(SF) and λ̂c(Homo) claimed in Theorems 4 and 5. To this end, we fix α=0.6 and b̃=δ=0.25. In addition, to compute λ̂c(SF), we set P(k)=kr/c,k=m,,M, where r = 2.85, M = 1000, and c=k=mMkr in (5), and, to compute λ̂c(Homo), we set k=k=mMk1r/c in (7). Here, the parameters m and b are treated as the testing ones, and their choices and their corresponding epidemic thresholds λ̂c are given in Table III. In Table III, we see that the sign of λ̂c(SF)λ̂c(Homo) changes twice from case 1 to 3 as b varies from b = 0.1 to b = 0.8 to b = 2.8. This indicates the existence of the interval Jb̃,α,δ,m as claimed in Theorem 4. On the other hand, the computed epidemic thresholds λ̂c for cases 2, 4, and 5 seem to suggest that λ̂c(SF)<λ̂c(Homo) whenever m2(=m0) as claimed in Theorem 5(i).

TABLE III.

The table gives the values of epidemic thresholds λ̂c(SF) and λ̂c(Homo) with fixed parameters α=0.6 and b̃=δ=0.25 and varying parameters m and b.

Casembλ̂c
0.1 λ̂c(SF)(0.65)<λ̂c(Homo)(1.64) 
0.8 λ̂c(SF)(2.19)>λ̂c(Homo)(2.14) 
2.8 λ̂c(SF)(4.05)<λ̂c(Homo)(4.59) 
0.8 λ̂c(SF)(1.90)<λ̂c(Homo)(1.95) 
0.8 λ̂c(SF)(1.84)<λ̂c(Homo)(1.91) 
Casembλ̂c
0.1 λ̂c(SF)(0.65)<λ̂c(Homo)(1.64) 
0.8 λ̂c(SF)(2.19)>λ̂c(Homo)(2.14) 
2.8 λ̂c(SF)(4.05)<λ̂c(Homo)(4.59) 
0.8 λ̂c(SF)(1.90)<λ̂c(Homo)(1.95) 
0.8 λ̂c(SF)(1.84)<λ̂c(Homo)(1.91) 

For clarity, we summarize the results of Theorems 3–5 and our observations from the simulations in the following Table IV. Let Jb̃,α,δ,m be the set of the parameters b so that λ̂c(SF)>λ̂c(Homo). The table gives the range of the parameters (b̃,α,δ,m) for which their corresponding Jb̃,α,δ,m is finite, infinite, or empty. The assertions made in the second row in Table IV is rigorous. The assertions in the third and fourth rows are based on numerical simulations. It is clear then that Jb̃,α,δ,m being nonempty and finite is generic. Consequently, the epidemic disease is easier (respectively, harder) to break out in the homogeneous network than in the heterogeneous network whenever the intensity of contact infection awareness is neither too low nor too high (respectively, intermediate).

TABLE IV.

Three possible cases for interval Jb̃,α,δ,m and their corresponding range of parameters. Here Jb̃,α,δ,m is the collection of intensity b of contact infection awareness so that λ̂c(SF)>λ̂c(Homo).

Jb̃,α,δ,mRange of parameters
0<|Jb̃,α,δ,m|< (b̃=0,0α<1,0<δ1,m) 
 (b̃>0,0α1,0<δ1,mm0) 
|Jb̃,α,δ,m|= (b̃0,α=1,0<δ1,m=1) 
|Jb̃,α,δ,m|=Ø (b̃>0,α1,0<δ1,mm1) 
Jb̃,α,δ,mRange of parameters
0<|Jb̃,α,δ,m|< (b̃=0,0α<1,0<δ1,m) 
 (b̃>0,0α1,0<δ1,mm0) 
|Jb̃,α,δ,m|= (b̃0,α=1,0<δ1,m=1) 
|Jb̃,α,δ,m|=Ø (b̃>0,α1,0<δ1,mm1) 

In this section, we demonstrate that our model (2) is a generalized model for those considered in Refs. 1–5. Indeed, (i) if there is no immunization awareness, i.e., α̃=β̃=0,ψ̃(k)1 and the vaccine success is perfect, i.e., δ = 0, and if, in addition, we let qk(0)=0, then the model becomes the one considered in Ref. 4. Moreover, λ̂c in (3) is reduced to the following:

λ̂c=kkψ(k)(k2αk)P(k)=kk2ψ(k)αkψ(k),

which is the one obtained in [Eq. (11), Ref. 4]. (ii) For model without considering infection awareness, i.e., α=β=0,ψ(k)1, we have that

ṗk(t)=γpk(t)+λkΘ(t)[1pk(t)],

which is the model considered in Refs. 1–3. The corresponding epidemic thresholds λ̂c(SF) and λ̂c(Homo) are computed as in (8). They are in agreement with those given in Refs. 1–3 and 39. (iii) If only local immunization awareness is considered, i.e., α̃>0,α=β=β̃=0 and ψ(k)=ψ̃(k)1, then our discrete-time version of the epidemic spreading model given in (A6) with h = 1 agrees with that considered in Ref. 5.

We proposed an epidemic spreading model including the element of vaccine failure and three types of infection awareness and immunization awareness. Our results generalize the established results on reduced forms of the model presented here. We also find that the epidemic threshold for the homogeneous network can be lower than that of the heterogeneous network provided that the intensity of contact infection awareness lies in an intermediate regime. It is of interest to study the effect of vaccine failure and awareness on cooperative40–42 or competitive43,44 disease dynamics.

The authors would like to thank the referees for their valuable comments which help to improve the quality of the paper. This research was supported by the Ministry of Science and Technology of Taiwan under Grant No. Most 105-2115-M-009-002-MY2.

In this section, we give the detailed derivation of the epidemic spreading model (2) from (1). We first make some coarse-graining approximations on (1) to get the discrete-time heterogeneous mean-field model of it. To this end, we divide the individuals into several distinct groups depending on their degrees k. Denote by P(k), the fraction of the number of individuals with degree k, and pk(t) and qk(t) by the corresponding infectious and vaccinated densities among nodes with degree k, respectively. Then it is assumed that each individual with the same degree has the same property of dynamical behaviors. To be more precise, we assume that, for each node i in the subgroup with degree k

kik,sis,pipkandqiqk.

It follows that (1b) and (1c) become, respectively:

Probi(SI)=1[1λψ(k)(1αsk)(1βp(t))]s=:1ω(s,k),Probi(SV)=1ψ̃(k)(1α̃sk)(1β̃p(t))=:ω̃(s,k),

where p(t)=j=1Npj(t)=kP(k)pk(t). Then we use the expected value of variable s to get an approximation of it. That is, we assume that

Probi(SI)E[1ω(s,k)]=s=1k[1ω(s,k)]B(s,k),Probi(SV)E[ω̃(s,k)]=s=1kω̃(s,k)B(s,k).
(A1)

Here B(s, k) is the probability that a node with degree k has exactly s infectious neighbors, and it is assumed to satisfy the binomial distribution.45 That is

B(s,k)=(ks)[Θ(t)]s[1Θ(t)]ks,
(A2)

where Θ(t) is the probability of a randomly selected link pointing to an infectious individual, and it is assumed to be1,37

Θ(t)=kkP(k)kpk(t)=kkP(k)pk(t)k,
(A3)

as defined in (2d). Herein, we assume that the underlying network is uncorrelated.

Following by the assumptions of (A1), (A2), and (A3), we compute that:

Probi(SI)s=0k{1[1λψ(k)(1αsk)(1βp(t))]s}B(s,k)=1s=0k[1λψ(k)(1αsk)(1βp(t))]sB(s,k)1s=0k[1sλψ(k)(1αsk)(1βp(t))]B(s,k)=λψ(k)(1βp(t))s=0ks(1αsk)B(s,k)=λψ(k)(1βp(t))(E[s]αkE[s2])=λψ(k)(1βp(t))Θ(t)[k(1αΘ(t))α(1Θ(t))],
(A4)

since E[s]=kΘ and E[s2]=k2Θ2+kΘkΘ2. Similarly

Probi(SV)s=0k[1ψ̃(k)(1α̃sk)(1β̃p(t))]B(s,k)=[1ψ̃(k)(1β̃p(t))(1α̃Θ(t))].
(A5)

Next, to derive the continuous version of the epidemic spreading model (2), we first shorten the time interval for the iterations from [t,t+1) to [t,t+h). In the time interval [t,t+h), we assume that the probability of recovering and vaccine failure to be γh and δh, respectively. Similarly, we assume that Probi,h(SI)=hProbi(SI) and Probi,h(SV)=hProbi(SV), which means that the probability that an individual changes its state depends linearly on the length of exposure. Then, by (1a) and letting pipk and qiqk, we have that the discrete-time heterogeneous mean-field model of (1) reads as follows:

pk(t+h)=(1γh)pk(t)     +h[1pk(t)qk(t)]Probi(SI),qk(t+h)=(1δh)qk(t)     +h[1pk(t)qk(t)]Probi(SV).
(A6)

Using (A6), (A4), and (A5) and letting h tends to be 0, we arrive at the continuous version of the epidemic spreading model as given in (2).

In this section, we give detailed proofs of Proposition 1, and Theorems 1–5.

Proof of Proposition 1. To show that Δ2n is positively invariant for (2), it suffices to claim that the vector field defined by (2) is tangent or points into Δ2n on the boundary Δ2n of Δ2n. Clearly, (pk,qk)Δ2n if (pk,qk)Δ2n and either pk0=0,qk0=0 or pk0+qk0=1 for some k0. Note, via (2), that we have ṗk00 (respectively, q̇k00) whenever pk0=0 (respectively, qk0=0). Similarly, ṗk0+q̇k0=γpk0δqk00 provided that pk0+qk0=1. The proof of the proposition is just completed. □

Proof of Theorem 1. Let (p*,q*) be a DFE of (2), where p*:=(p1*,p2*,,pn*)=(0,0,,0)(=:0) and q*:=(q1*,q2*,,qn*)[0,1]n. Then some direct computation from (2) yields that qk* satisfies [δ+(1ψ̃(k))]qk*+(1ψ̃(k))=0. Hence, (i) if δ = 0 and ψ̃(k)=1, then qk* can be any arbitrary value in [0,1], while (ii) if δ>0 or ψ̃(k)1, then qk*=(1ψ̃(k))δ+(1ψ̃(k)), as defined in the first equation of (4).

To see the first assertion of the theorem, let p=(p1,p2,,pn) and q=(q1,q2,,qn). Consider the Lyapunov candidate function V(p,q)=12k(qk1)2. Then V̇(p,q)=k(qk1)q̇k0 since, as can been seen in Eq. (2), q̇k(t)0 when δ = 0, and qk1 by Proposition 1. By LaSalle's invariant principle, all solutions of (2) approach the largest invariant S of {(p,q)Δ2n:V̇(p,q)=0}. Clearly, when δ = 0, V̇=0 if and only if pk+qk=1 or p=0 by (2) and the definitions of p and Θ since either α̃2+β̃2>0 or ψ̃(k)1. However, when pk+qk=1,ṗk+q̇k=γpk0 and the equality holds if and only if pk = 0. It implies that pk = 0 for points in S. Thus, for every solution (pk(t),qk(t))1kn of (2), pk(t) converges to 0 for each k. Moreover, since qk(t) is increasing and bounded above, qk(t) also converges to some fixed point q¯k* in [0,1].

We next prove the second assertion of the theorem. From (2), it can be easily seen that

ṗk(t)=γpk(t)+λψ(k)(kα)[1qk(t)]Θ(t)+o(p(t)),q̇k(t)=δqk(t)+[1pk(t)qk(t)]×[1ψ̃(k)(1α̃Θ(t)β̃p(t))]+o(p(t)).
(B1)

Hence the Jacobian matrix of (2) at the DFE (p*,q*) is

J=[P0RQ],
(B2)

where P=(pkk)n×n with

pkk=γδkk+λψ(k)(kα)(1qk*)kP(k)k,
(B3)

δkk=1 if k=k, otherwise δkk=0, and

Q=[δ+(1ψ̃(k))]I,

and I, 0, are the identity and zero matrices of size n × n, respectively. Clearly, λmax(J)=max{λmax(P),λmax(Q)} where λmax(·) takes the maximum real parts of the eigenvalues of a matrix. Define u:=(u1,u2,,un)T and v:=(v1,v2,,vn)T with uk:=ψ(k)(kα)(1qk*) and vk:=kP(k)k. Then, by (B3), we have that P=γI+λuvT. And so, λmax(P)=γ+λuTv=γ+λkψ(k)(kα)(1qk*)kP(k)k. On the other hand, since δ,ψ̃(k)[0,1], we have that λmax(Q)=mink[δ+(1ψ̃(k))]<0. Hence the DFE (p*,q*) of model (2) is stable if and only if λmax(P)<0, or equivalently, λγ<λ̂c, where λ̂c is defined as in (3). Moreover, since the epidemic disease breaks out when the stability of DFE of model (2) is lost, λ̂c is indeed the threshold for the outbreak of epidemic disease.

To complete the proof, we note that, in the case of δ = 0, α̃=β̃=0 and ψ̃(k)1, we have that, by (2)

ṗk(t)=γpk(t)+[1pk(t)qk(0)]{λψ(k)(1βp(t))Θ(t)[k(1αΘ(t))α(1Θ(t))]},qk(t)qk(0).

Hence, q*=q(0). The proof is complete. □

Proof of Theorem 2. In the case that P(k)=kr/c,k=m,,M, where c=k=mMkr, and functions ψ and ψ̃ are defined as in (6), we compute, via (5), that

λ̂c=kk=mMδδ+(1kb̃)kb(k2αk)(kr/c)=k=mMk1rk=mMδ(k2αk)(δ+1)kb+rkb+rb̃=:λ̂c(SF).
(B4)

Since the numerator of the above equation is finite for any r(2,3] and the denominator goes to as M tends to if and only if b+r3, the proof of theorem is complete. □

In Theorems 3–5, we aim to compare the size of λ̂c(SF) and λ̂c(Homo). To this end, we need to apply some well-known results, which are stated in the following two lemmas for ease of the references.

Lemma B.1 (Jesen's inequality). Let P(k) be a discrete probability distribution and f be a continuous function defined on [m,M]. If f is convex (respectively, concave) in [m,M], then f(kxkP(k)) (respectively, ) kf(xk)P(k). Moreover, both of the above equalities hold if and only if xk are all equal or f is linear.

Lemma B.2 (Theorem F in p. 19 of Ref. 46). Let f be a positive function defined in an interval I. Then f is said to be a log-convex function in I if ln(f) is convex in I. The class of log-convex functions in I is closed under addition, multiplication, and taking the limits, provided that the limits exist and are positive.

Proposition B.2. Define

ψ̂c(x)=δδ+(1xb̃)xb(x2αx).
(B5)

Then

  • λ̂c(SF)λ̂c(Homo), whenever ψ̂c(x) is convex in [m,M],

  • λ̂c(SF)λ̂c(Homo), whenever ψ̂c(x) is concave in [m,M].

Moreover, both of the above equalities hold if and only if (b,b̃,α) equals to (1,0,a),(1,1,1δ+1), (2, 0, 0) or (2,1,1δ+1) where a[0,1].

Proof. From (B4) and (7), it is clear that

λ̂c(SF)=kkψ̂c(k)P(k),

where P(k)=kr/c, and

λ̂c(Homo)=kψ̂c(k),

respectively. Then by applying Lemma B.1 to function ψ̂c and letting xk = k, the proof of the theorem is complete. □

Lemma B.3. Suppose b̃=0. Then the following assertions hold.

  • If 0b<1, then λ̂c(SF)<λ̂c(Homo);

  • If b = 1, then λ̂c(SF)=λ̂c(Homo)(=kkα);

  • If 1<b<2, then λ̂c(SF)>λ̂c(Homo);

  • If b = 2, then λ̂c(SF)λ̂c(Homo) and the equality holds only when α = 0.

  • If b > 2 and α<b2b, then λ̂c(SF)<λ̂c(Homo).

Proof. In the case of b̃=0, we compute directly by (B5) that

ψ̂c(x)=(1b)[(2b)x+αb]x1b.

It follows that ψ̂c(x)>0 (respectively, < 0) in [1,) if b < 1 (respectively, 1<b<2). Hence, statements (i) and (iii) in the lemma hold true by Proposition B.2.

For b > 2, since limxψ̂c(x)=+ and sgn(ψ̂c(x))=sgn((b2)xαb) is increasing in x, ψ̂c(x) has the same sign in [1,) if and only if sgn(ψ̂c(1))=sgn((b2)αb)=1, or equivalently, α<b2b. Hence, statement (v) holds true. The remaining statements (ii), (iv), and (v) in the lemma can be shown similarly and thus are omitted. □

We are now in the position to prove Theorems 3–5.

Proof of Theorem 3. In the following, we only give the proof for the case that M=. When M<, the proof can be obtained similarly.

For any α[0,1],b̃0 and r(2,3], define function g(b) in (3r,) by

g(b):=ln([λ̂c(SF)(b,α)]1)ln([λ̂c(Homo)(b,α)]1).
(B6)

Then by Theorem 2 and Eq. (7), g is well defined for all r(2,3]. Moreover, when b=3r, since λ̂c(SF)=0 and λ̂c(Homo)>0, we have that limb(3r)+g(b)=+. We now show that g is convex. Since, by (B4)

[λ̂c(SF)(b,α)]1=1kkδ(k2αk)(kr/c)δ+(1kb̃)kb,

we compute hat 2b2ln[δ(k2αk)ckrδ+(1kb̃)kb]0. It implies that [δ(k2αk)ckrδ+(1kb̃)kb] is a log-convex function. Consequently, [λ̂c(SF)(b,α)]1 is also a log-convex function of b by Lemma B.2. On the other hand, since, by (7)

[λ̂c(Homo)(b,α)]1=1kδ[k2αk]δ+[1kb̃]kb,

we compute that 2b2ln[[λ̂c(Homo)(b,α)]1]0. From the above, we conclude that g(b)0 on (3r,). That is, g is convex on (3r,).

We next show the existence of b2(α,m,M) for α[0,1) and m in the case that b̃=0. From Lemma B.3, it is easy to see that, for any α[0,1), g(b) > 0 when b[3r,1)(21α,),g(1)=0, while g(b) < 0 when b(1,2). Then the convex property of g implies that it has exactly two zeros at b = 1 and some b2(=b2(α,m,M))[2,21α). In fact, g(b) > 0 when b[0,1)(b2,) and g(b) < 0 when b(1,b2). Then by the definition of g, the inequalities stated in the theorem hold true.

Now, we show that b2(α,m,M) is strictly increasing in α[0,1). Suppose not, then there exist α1α2[0,1) such that b2(α1,m,M)=b2(α2,m,M)(=:b*). By the definition of function b2(α,m,M), λ̂c(SF)(b*,α1)=λ̂c(Homo)(b*,α2) and b*2 as claimed above. For each fixed b>3r, define hb(α) in + by

hb(α):=[λ̂c(SF)(b,α)]1[λ̂c(Homo)(b,α)]1=[δk2bδ+1kb̃δδ+1kb̃k2bk][δk1bδ+1kb̃δδ+1kb̃k1bk]α.
(B7)

Then clearly, λ̂c(SF)(b,α)=λ̂c(Homo)(b,α) if and only if hb(α)=0. Hence, hb*(α1)=hb*(α2)=0. Note that in the case that b̃=0, function hb becomes

hb,b̃=0(α)=[k2bk2bk][k1bk1bk]α.

Since, by Lemma B.1, k2bk2b for b2 and the equality holds if and only if b = 2, we have that k1b*k1b*>0. It follows that hb* has exactly one zero, which is a contradiction to the fact that hb*(α1)=hb*(α2)=0. Moreover, it is clear to see that, since hb*(α)=0, we have that:

k2b2*k2b2*k1b2*k1b2*=α.

Hence, the proof is complete. □

Proof of Theorem 4. Note that the facts that the function g(b), defined in (B6), is convex on (3r,) and limb(3r)+g(b)=+ are proved for all b̃0 and α[0,1]. Moreover, by Theorem 2 and Eq. (7), g(b) > 0 for b on (3r,) or there exists some nonempty open interval J such that g(b) < 0 if and only if bJ. These two scenarios correspond to λ̂c(SF)<λ̂c(Homo) and λ̂c(SF)>λ̂c(Homo), respectively.

On the other hand, consider the function hb(α) defined in (B7). Then λ̂c(SF)(b,α)=λ̂c(Homo)(b,α) if and only if hb(α)=0 or equivalently, b satisfies

δk2bδ+1kb̃δδ+1kb̃k2bδk1bδ+1kb̃δδ+1kb̃k1b=α.

Hence, the proof is complete. □

Proof of Theorem 5. For b̃>0, we compute directly that

ψ̂c(x)=δx2b̃b1[δ+(1xb̃)]3ψˇ(x),

where

ψˇ(x)=(b1)(b2)(δ+1)2x2b̃+1αb(b1)(δ+1)2x2b̃(2b22bb̃b̃26b+3b̃+4)(δ+1)xb̃+1+α(2b22bb̃b̃22b+b̃)(δ+1)xb̃+(bb̃1)(bb̃2)xα(bb̃)(bb̃1).

Then sgn(ψ̂c(x))=sgn(ψˇ(x)). Clearly, when b[0,1)(2,), we have that limxψˇ(x)=+, which implies that there exists some positive integer m0 such that ψˇ(x)>0 for xm0. Hence, by Proposition B.2, λ̂c(SF)<λ̂c(Homo) whenever the minimum degree in the SF network is greater than m0. On the other hand, when b(1,2), since limxψˇ(x)=, λ̂c(SF)>λ̂c(Homo) whenever the minimum degree in the SF network is greater than some positive integer m0. The remaining cases stated in the Theorem can be proved similarly. □

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