The impact of vaccine success and awareness on epidemic dynamics

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.


I. INTRODUCTION
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 reported 6 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][9][10][11][12][13][14][15][16] Recently, the study of aspects of human responses towards the spread of epidemic diseases has drawn much attention 4,7,[17][18][19][20][21][22][23][24][25][26][27][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 assumed 4,29 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 a) 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.

II. EPIDEMIC SPREADING MODEL
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 þ 1 through one of the following four different ways: (i) S ! I, (ii) S ! V, (iii) V ! S, and (iv) I ! S. Letting p i ðtÞ (respectively, q i ðtÞ), i ¼ 1; …; N, be the probability that a node i is infectious (respectively, vaccinated) at time t, our model under consideration reads as follows: p i ðt þ 1Þ ¼ ð1 À cÞp i ðtÞ þ ½1 À p i ðtÞ À q i ðtÞProb i ðS ! IÞ; q i ðt þ 1Þ ¼ ð1 À dÞq i ðtÞ þ ½1 À p i ðtÞ À q i ðtÞProb i ðS ! VÞ; Here k i denotes the degree of a node i (i.e., the number of neighbors of a node i); s i , ranging from 0 to k i , 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.
(I) The parameter c 2 ½0; 1 in (1a) denotes the recovering probability for infectious individuals for the whole time period. (II) The parameter d 2 ½0; 1 in (1a) denotes the probability of vaccine failure for vaccinated individuals. (III) The term Prob i ðS ! IÞ, defined in (1b), gives the probability that a node i changes its state from S to I. The quantity k 2 ½0; 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 (i) The term wðk i Þ 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 wðkÞ is decreasing in the degree k. In Ref. 4, wðkÞ is chosen to be k Àb for some nonnegative constant b. The larger the quantity b is, the smaller the wðkÞ 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 wðkÞ is independent of k (i.e., b ¼ 0), then there is no contact infection awareness in the epidemic spreading. (ii) The term ð1 À bpðtÞÞ in (1b) is a decreasing function of p(t) and b. 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, b 2 ½0; 1 describes the strength of the average risk assessment from global awareness and hence b is to be termed as the intensity of global infection awareness. For b ¼ 0, this means that global infection awareness is not considered in the epidemic spreading. (iii) The term ð1 À aðs i =k i ÞÞ in (1b) is a decreasing function of ðs i =k i Þ and a. Thus, this term indicates that as the local infectious density ðs i =k i Þ 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 a 2 ½0; 1 indicates the strength of the average risk assessment from local awareness, and hence a is to be termed as the intensity of local infection awareness. Note that a ¼ 0 means that there is no local infection awareness in the epidemic spreading. (iv) The term represents the probability that a susceptible node i will not get infected when it makes a contact with exactly one infectious individual. Thus is the probability that a node i will not change its state from susceptible to infectious when it makes contacts with s i infectious individuals. Thus, Prob i ðS ! IÞ, defined in (1b), gives the probability that a node i changes its state from susceptible to infectious. (IV) The term Prob i ðS ! VÞ, defined in (1c), gives the probability that a node i changes its state from S to V when the epidemic disease is spreading. 5 (i) Similarly, the termwðk i Þ 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 func-tionwðkÞ is decreasing in k. We also set wðkÞ ¼ k Àb for some nonnegative constantb, where the parameterb is to be termed as the intensity of contact immunization awareness. That is, for higherb, a susceptible individual tends to have a higher probability of getting vaccinated.
The terms ð1 ÀbpðtÞÞ andb 2 ½0; 1 in (1c) describe the global immunization awareness and the intensity of global immunization awareness, respectively. Note that Prob i ðS ! VÞ is increasing inb. Consequently, for higherb, a susceptible individual is more likely to get vaccinated.
(iii) The term ð1 Àãðs i =k i ÞÞ in (1c) represents the probability that a node i changes its state from susceptible to vaccinated due to its local awareness. Note that Prob i ðS ! VÞ is increasing inã. Likewise, ð1 Àãðs i =k i ÞÞ and the constantã 2 ½0; 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.
To investigate the effect of the heterogeneity of networks on the epidemic dynamics, we next make a coarsegraining approximation on (1) to derive a continuous-time degree-based mean-field model [1][2][3]35,36 by assuming that (i) individuals with the same degree have the same property of dynamical behaviors, (ii) the variable s i 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 p k ðtÞ and q k ðtÞ, respectively. The model then reads as follows:  (1). Note that a node is less (respectively, more) likely to get infected (respectively, immunized) with the increase of intensity a, b, or b (respectively,ã;b, orb).

Description Parameters
Spreading rate k Recovering probability c Probability of vaccine failure d

Awareness Infection Immunization
Intensity of local awareness aã Intensity of global awareness bb Contact awareness wðkÞwðkÞ Intensity of contact awareness bb 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 p k ð0Þ and q k ð0Þ satisfy 0 < p k ð0Þ; q k ð0Þ, and p k ð0Þ þ q k ð0Þ < 1, for all k, then p k ðtÞ and q k ðtÞ also satisfy 0 < p k ðtÞ; q k ðtÞ, and p k ðtÞ þ q k ðtÞ < 1, for all k and t > 0. The proof of Proposition 1 is given in Appendix B. Proposition 1. Define Then D 2n is positively an invariant for (2).

III. STABILITY ANALYSIS AND THE EPIDEMIC THRESHOLD
In this section, we study the epidemic spreading model (2) and compute the threshold for effective spreading ratê k ð:¼ k=cÞ. 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 d ¼ 0 and eitherã 2 þb 2 > 0 orwðkÞ 6 ¼ 1, the epidemic dies out. In particular, every solution ðp k ðtÞ; q k ðtÞÞ 1 k n of (2) converges to a disease free equilibrium (DFE) ð0; q Ã k Þ 1 k n for some q Ã k in ½0; 1. (ii) In other cases, there exists an epidemic thresholdk c , as given in the following: where such that the epidemic dies out when the effective spreading ratek ð:¼ k c Þ is smaller thank c ; otherwise, the disease breaks out.
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 d > 0 which depends on the contact type of awareness (w andw), local infection awareness (a) and is independent of local immunization awareness (ã) and global type of awareness (b andb). It should also be noted, though, thatã, b, andb play the role on decreasing the final epidemic size (lim t!1 pðtÞ) when the disease breaks out (see, e.g., Table II and Fig. 1 in Section V).

IV. EFFECT OF THE DEGREE DISTRIBUTION ON THE EPIDEMIC THRESHOLD
In this section, we consider the impact of awareness on the epidemic thresholdk c as given in (5). Specifically, we assume the contact type of awareness to be given as follows: 4 It means that individuals with quite a large degree have negligible probability of being infected. Such assumption is also  (2) for the ten cases given in Table II. 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 k 0 are much more inclined to take immunization.

A. Heterogeneous networks
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 PðkÞ $ k Àr and, typically, the exponent r 2 ð2; 3 and k 2 ½m; 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 hki is finite but the variance hk 2 i À hki 2 is infinite as M approaches to infinity. Here hk 2 i :¼ P k k 2 PðkÞ. Remark that, throughout the paper, we denote by hf ðkÞi ¼ P k f ðkÞPðkÞ for any f(k).
For a SF network, if neither type of awareness is considered, that is, a ¼ã ¼ b ¼b ¼ 0 and wðkÞ ¼wðkÞ 1, then we have, via (5), thatk cðSFÞ ¼ hki hk 2 i and hence lim M!1kcðSFÞ ¼ 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 d of vaccine failure is positive. Then the epidemic thresholdk cðSFÞ for a SF network with the exponent r 2 ð2; 3 and k 2 ½m; M tends to be 0 as M approaches to 1 if and only if b 3 À r.
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 thresholdk c 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 thresholdk c then in the limit becomes a positive finite value.

B. Homogeneous networks
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 hki. 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, k ' hki. Networks of this property are called the homogeneous networks. Paradigmatic homogenous networks are the Erd€ os-R enyi 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 iŝ Note that, when neither type of awareness is considered, that is,b ¼ b ¼ a ¼ 0, we have, via (7), thatk cðHomoÞ ¼ hki=hki 2 . Consequentlŷ k cðSFÞ ð¼ hki=hk 2 iÞ <k cðHomoÞ ð¼ hki=hki 2 Þ; (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 thresholdk c 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 logconvex functions, are recorded in Appendix B. hk 1Àb 2 iÀhki 1Àb 2 ¼ a.
The theorem is essentially amount to saying that, for b ¼ 0, there exists a nonempty and finite interval J ¼ ð1; b 2 Þ 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 6 2 ½1; b 2 , then the epidemic disease is easier to break out in the SF network.
In the next theorem, we show that, if the intensityb of contact immunization awareness is nonzero, then the corresponding interval J could be empty, nonempty and finite, or infinite.
¼ a: Note that Jb ;a;d;m could be either of the form (a, b), 0 < a < b < 1, or of the form ða; 1Þ, or the empty set.
For m, the minimum degree among nodes in the SF network, sufficiently large and d 6 ¼ 0, we are able to show that Jb ;a;d;m must be an open, connected, nonempty, and finite interval. In particular, we prove that, given any set of parameters ðb;b; aÞ 2 R þ Â R þ Â ½0; 1, its corresponding sign of k cðSFÞ Àk cðHomoÞ can be determined as long as m is sufficiently large and d 6 ¼ 0. The above mentioned result is recorded in the following theorem.
Theorem 5. Considerb; d > 0 and let A be the union of the sets Here R þ ¼ ½0; 1Þ. Then the following holds.

V. SIMULATIONS
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), PðkÞ ¼ k Àr =c; k ¼ m; …; M, where r ¼ 2.85, m ¼ 2, M ¼ 1000, and c ¼ P M k¼m k Àr , 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 thresholdsk c computed by (5) and the finial epidemic sizes p 1 ð:¼ lim t!1 pðtÞÞ 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, thenk ð¼1:93Þ >k c ð%1:90Þ (respectively, k ð¼1:87Þ <k c ð%1:90Þ). Consequently, we have that the epidemic disease breaks out (respectively, dies out) and its corresponding p 1 % 0:71%. Meanwhile, if we increase the intensityb, b, or a of awareness or decrease the probability d 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 b,b, orã 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 ofk cðSFÞ andk cðHomoÞ claimed in Theorems 4 and 5. To this end, we fix a ¼ 0:6 and b ¼ d ¼ 0:25. In addition, to computek cðSFÞ , we set (5), and, to computek cðHomoÞ , we set hki ¼ P M k¼m k 1Àr =c in (7). Here, the parameters m and b are treated as the testing ones, and their choices and their corresponding epidemic thresholdsk c are given in Table III. In Table III, we see that the sign ofk cðSFÞ Àk cðHomoÞ changes twice from case 1 to 3 as b varies from b ¼ 0.1 to b ¼ 0.8 to b ¼ 2.8. This indicates the existence of the interval Jb ;a;d;m as claimed in Theorem 4. On the other hand, the computed epidemic thresholdsk c for cases 2, 4, and 5 seem to suggest that k cðSFÞ <k cðHomoÞ whenever m ! 2 ð¼ m 0 Þ as claimed in Theorem 5(i).
For clarity, we summarize the results of Theorems 3-5 and our observations from the simulations in the following Table IV. Let Jb ;a;d;m be the set of the parameters b so that k cðSFÞ >k cðHomoÞ . The table gives the range of the parameters ðb; a; d; mÞ for which their corresponding Jb ;a;d;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 Jb ;a;d;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).

VI. RELATED MODELS
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.,ã ¼b ¼ 0;wðkÞ 1 and the vaccine success is perfect, i.e., d ¼ 0, and if, in addition, we let q k ð0Þ ¼ 0, then the model becomes the one considered in Ref. 4. Moreover,k c in (3) is reduced to the following:

VII. CONCLUSION
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 cooperative [40][41][42]  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 discretetime 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 p k ðtÞ and q k ðtÞ 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 k i k; s i s; p i p k and q i q k : It follows that (1b) and (1c) become, respectively: 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 Bðs; kÞ ¼ k s ½HðtÞ s ½1 À HðtÞ kÀs ; where HðtÞ is the probability of a randomly selected link pointing to an infectious individual, and it is assumed to be 1,37 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: since E½s ¼ kH and E½s 2 ¼ k 2 H 2 þ kH À kH 2 . Similarly Next, to derive the continuous version of the epidemic spreading model (2), we first shorten the time interval for the iterations from ½t; t þ 1Þ to ½t; t þ hÞ. In the time interval ½t; t þ hÞ, we assume that the probability of recovering and vaccine failure to be ch and dh, respectively. Similarly, we assume that Prob i;h ðS ! IÞ ¼ hProb i ðS ! IÞ and Prob i;h ðS ! VÞ ¼ hProb i ðS ! VÞ, which means that the probability that an individual changes its state depends linearly on the length of exposure. Then, by (1a) and letting p i p k and q i q k , we have that the discrete-time heterogeneous mean-field model of (1) reads as follows: p k ðt þ hÞ ¼ ð1 À chÞp k ðtÞ þ h½1 À p k ðtÞ À q k ðtÞProb i ðS ! IÞ; q k ðt þ hÞ ¼ ð1 À dhÞq k ðtÞ þ h½1 À p k ðtÞ À q k ðtÞProb i ðS ! VÞ: 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).

APPENDIX B: PROOFS IN THE MAIN TEXT
In this section, we give detailed proofs of Proposition 1, and Theorems 1-5.
To see the first assertion of the theorem, let p ¼ ðp 1 ; p 2 ; …; p n Þ and q ¼ ðq 1 ; q 2 ; …; q n Þ. Consider the Lyapunov candidate function Vðp; qÞ ¼ 1 2 P k ðq k À 1Þ 2 . Then _ V ðp; qÞ ¼ P k ðq k À 1Þ _ q k 0 since, as can been seen in Eq. (2), _ q k ðtÞ ! 0 when d ¼ 0, and q k 1 by Proposition 1. By LaSalle's invariant principle, all solutions of (2) approach the largest invariant S of fðp; qÞ 2 D 2n : _ V ðp; qÞ ¼ 0g. Clearly, when d ¼ 0, _ V ¼ 0 if and only if p k þ q k ¼ 1 or p ¼ 0 by (2) and the definitions of p and H since either a 2 þb 2 > 0 orwðkÞ 6 ¼ 1. However, when p k þ q k ¼ 1; _ p k þ _ q k ¼ Àcp k 0 and the equality holds if and only if p k ¼ 0. It implies that p k ¼ 0 for points in S. Thus, for every solution ðp k ðtÞ; q k ðtÞÞ 1 k n of (2), p k ðtÞ converges to 0 for each k. Moreover, since q k ðtÞ is increasing and bounded above, q k ðtÞ also converges to some fixed point q Ã k in ½0; 1. We next prove the second assertion of the theorem. From (2), it can be easily seen that _ p k ðtÞ ¼ Àcp k ðtÞ þ kwðkÞðk À aÞ½1 À q k ðtÞHðtÞ þ oðpðtÞÞ; _ q k ðtÞ ¼ Àdq k ðtÞ þ ½1 À p k ðtÞ À q k ðtÞ Â ½1 ÀwðkÞð1 ÀãHðtÞ ÀbpðtÞÞ þ oðpðtÞÞ: Hence the Jacobian matrix of (2) at the DFE ðp Ã ; q Ã Þ is where P ¼ ðp kk 0 Þ nÂn with and I, 0, are the identity and zero matrices of size n Â n, respectively. Clearly, k max ðJÞ ¼ maxfk max ðPÞ; k max ðQÞg where k max ðÁÞ takes the maximum real parts of the eigenvalues of a matrix. Define u :¼ ðu 1 ; u 2 ; …; u n Þ T and v :¼ ðv 1 ; v 2 ; …; v n Þ T with u k :¼ wðkÞðk À aÞð1 À q Ã k Þ and v k :¼ kPðkÞ hki . Then, by (B3), we have that P ¼ ÀcI þ kuv T . And so, k max ðPÞ ¼ Àc þ ku T v ¼ Àc þ k P k wðkÞðk À aÞ ð1 À q Ã k Þ kPðkÞ hki . On the other hand, since d;wðkÞ 2 ½0; 1, we have that k max ðQÞ ¼ Àmin k ½d þ ð1 ÀwðkÞÞ < 0. Hence the DFE ðp Ã ; q Ã Þ of model (2) is stable if and only if k max ðPÞ < 0, or equivalently, k c <k c , wherek c is defined as in (3). Moreover, since the epidemic disease breaks out when the stability of DFE of model (2) is lost,k c is indeed the threshold for the outbreak of epidemic disease.