Introduction

When a disease suddenly emerges, the dynamical processes of disease1,2,3,4,5,6,7 and information8,9,10 spreading are typically asymmetrically coupled with each other11,12,13,14,15. In particular, the spread of a disease can enhance the crisis awareness and thus facilitates the diffusion of the information about the disease16. Meanwhile, the diffusion of the information promotes more people to take preventive measures and consequently suppresses the epidemic spreading14. To understand the asymmetric interplay between the two kinds of spreading dynamics is of great importance for predicting and controlling epidemics, leading to a new direction of research in complex network science17,18,19. Funk et al. first presented an epidemiological model by incorporating the spread of awareness in a well-mixed population and found that the awareness-based response can markedly reduce the final infection rate. When the awareness is sufficiently strong so as to modify the key parameters associated with the spreading dynamics such as the infection and recovery rates, the epidemic threshold can be enhanced17. Ruan et al. studied a susceptible-infected-recovered (SIR) model with information-driven vaccination and found the epidemic spreading can be significantly suppressed when the information is well spread14.

With the development of technology, the information about disease can quickly diffuse through different channels, such as the word of mouth, news media and online social networks. Usually, the pathways for information spreading are different from the pathways for disease spreading. In view of this, the asymmetric interplay between the information and the epidemic spreading dynamics needs to be considered within multiplex network framework18,19,20,21,22,23. In a multiplex network (multilayer network or overlay network), each network layer for one type of transportation process has an identical set of nodes and a distinct internal structure. And the interplay between multiple layers has diverse characteristics, such as inter-similarity24, multiple support dependence25 and inter degree-degree correlation26, etc. Along this line, Granell et al. established a two susceptible-infected-susceptible (SIS) processes coupled model to investigate the inhibitory effect of awareness spreading on epidemic spreading dynamics in a multiplex network and the results showed that the epidemic threshold was determined by the structures of the two respective networks as well as the effective transmission rate of awareness18. Wang et al. studied the asymmetrically interacting spreading dynamics based on a two susceptible-infected-recovered (SIR) processes coupled model in multiplex networks and found that the outbreak of disease can lead to the propagation of information and rise of epidemic threshold19.

In the asymmetrically interacting spreading dynamics, how an individual being aware of disease responds to the disease can significantly affect the epidemic spreading13,14,27. Sahneh et al. introduced an alter state into the SIS model, where the alerted individuals sensing infection adopt a preventive behavior. When the preventive behavior is implemented timely and effectively, disease cannot survive in the long run and will be completely contained12. Zhang et al. investigated to what extent behavioral responses based on local infection information can affect typical epidemic dynamics and found that such responses can augment significantly the epidemic threshold, regardless of SIS or SIR processes27. All of the previous studies were built on a basic assumption: the behavioral responses to the disease, which is a Markovian process without memory, depend only on current dynamical information such as infected neighbors.

However, behavioral response or behavior adoption is not a simple Markovian process which depends only on current dynamical information. Recent researches on behavior adoption such as innovation28 and healthy activities29 have confirmed that the adoption probability is also affected by previous dynamical information. This is equivalent to social affirmation or reinforcement effect, since multiple confirmation of the credibility and legitimacy of the behavior are always sought30,31,32,33,34. Specifically for an individual, if some of his/her friends have adopted a particular behavior before a given time whereas the other friends newly adopt the behavior, whether he/she adopt the behavior will take all the adopted friends’ adoption into account. Taking the adoption of healthy behavior as an example, Centola has demonstrated that the probability for an individual to adopt a healthy behavior depends on the times of being informed30; in the microblogging retweeting process, the authors have shown that the probability of one individual retweeting a message increases when more friends have retweeted the message35,36. Based on the memory of previous information, this reinforcement effect makes the behavior adoption processes essentially non-Markovian and more complicated.

As we know, taking vaccination against disease may carry some side effects or certain cost37,38, so the decision to take vaccination is worth pondering. Before taking a certain vaccine, people need to confirm the correctness of information which usually relies on the cumulative times of received information and the social reinforcement effect. Thus, the adoption of vaccination can be viewed as a complex adoption behavior. In this paper, the impact of complex vaccination adoption behavior on the two interacting spreading dynamics in a double-layer network is investigated. It is assumed that in physical-contact layer, the probability for an individual to adopt vaccination is determined by the times of the information about disease received in the communication layer and the social reinforcement effect of the cumulative information. It is showed by our findings that the two interacting spreading dynamics is remarkably influenced by this complex adoption behavior. In addition, given that taking vaccination as well as treating infected individuals bear certain costs, we define the social cost as the total cost of vaccination and treatment for infected individuals. Then, the effect of this complex vaccination adoption behavior on social cost is explored and it is found that there are an optimal social reinforcement effect and optimal information transmission rate which entail the minimal social cost.

Results

To present our primary study results, we first described the model of multiplex network, the spreading dynamical process in each layer and the asymmetric interplay between the two spreading processes. Then, we elaborated the theoretical analysis of the asymmetric interacting spreading dynamics in multiplex networks. Finally, we demonstrated the simulation results which are verified by the proposed theory.

Model of multiplex network

A multiplex network with two layers is constructed to represent the contact-communication coupled network. At the beginning, a communication network (labelled A) and a contact network (labelled B) are respectively generated. Supposing that the degree distribution and network size of communication network A are of PA(kA) and N respectively, a random configuration network can be generated according to the given degree distribution, where self-loops or repeated links between a pair of nodes are not allowed5. Meanwhile, layer B is generated in the same way that the network size and degree distribution are given as N and PB(kB), respectively. After that, each node of layer A is matched one-to-one with that of layer B randomly. Moreover, to facilitate the analysis, the constructed double-layer network is an uncorrelated double-layer network and the joint probability distribution of degree kA and degree kB of the same node can be written as PAB(kA, kB) = PA(kA)PB(kB). It means that the degree distribution of one layer is independent of that of the other layer completely. In addition, when the network is very large and sparse, links in the double layers are scarcely overlapped due to random linking in random configuration network model. The theoretical framework of the asymmetric interacting spreading processes in this paper can be easily generalized to the multiplex networks with inter-layer degree correlations19 and overlapping links39.

Two interacting spreading dynamical processes

In such a double-layer network, an infectious disease spreads through physical contact layer (layer B) and the triggered information about the disease diffuses through a communication layer (layer A). In the communication layer (layer A), an improved susceptible-infected-recovered (SIR) model6 is used to describe the dissemination of information about the disease. In this model, each node can be in one of the following three states: (1) susceptible state (S) in which the node has not received any information about the disease; (2) informed state (I), where the node has received the information at least one time and is capable of transmitting the information to other nodes in the same layer. More importantly, let M be the cumulative pieces of information that the node has received from its neighbors, which is used to characterize the memory effect of vaccination adoption behavior31,40; and (3) refractory state (R), in which the node has received the information but is not willing to pass it on to other nodes. During the process of transmission, each informed node (I state) passes the information to all its neighbors in the communication network A at each time step. If a neighbor is in the S state, it will enter I state and update M = 1 with probability βA. If a neighbor is in the I state, it will receive the information again and update M = M + 1 with probability βA. Meanwhile, the informed node enters the R state with probability μA and once the node enters the R state, it will keep in this state forever. Furthermore, a node in layer A will get the information about the disease and update M = 1, once its counterpart node in layer B is infected. As a result, the dissemination of the information over layer A is facilitated by disease transmission in layer B.

The dynamics of epidemic in the contact network B is illustrated by a susceptible-infected-recovery-vaccinated (SIRV) model14, in which a fourth state, the state of vaccination is incorporated into the classical SIR model. The reaction process of the SIR component in layer B is the same as that of the classical SIR model with transmission rate βB and recovery rate μB. Since the behavior of taking vaccination against disease is essentially non-Markovian and complicated, we assume that the probability of a susceptible node turning into vaccinated state in layer B depends on the cumulative times of received information (i.e M) in layer A and the social reinforcement effect. For a susceptible node in layer B, if he receives at least one piece of information at the tth time step and has received M times of the information until time t, the probability that he takes vaccination at time t will be

where ξ1 is the vaccination adoption probability when a node receives the information about disease for the first time. And α means the node’s sensitivity to information, which is used to characterize the strength of social reinforcement effect. When α > 0, the adoption probability ξM increases with the value of M. The memory reinforcement effect disappears once α = 0. For a fixed M, the greater value of α, the stronger the reinforcement effect (i. e., the greater adoption probability ξM). As the adoption of vaccination is determined by the cumulative pieces of received information M and the sensitivity factor of social reinforcement effect α, it is a typical complex adoption behavior. Our main purpose is to investigate the impact of sensitivity factor α on the two interacting epidemic dynamics. The two spreading processes and their dynamical interplay are schematically illustrated in Fig. 1. To simplify our descriptions and differentiate the states of nodes in the two layers, SA (RA) and SB (RB) are defined to be a node in S (R) state in layer A and layer B, respectively. Similarly, IA and IB are set as nodes in informed state and infected state in layer A and B, respectively. And VB is the node in vaccinated state in layer B.

Figure 1
figure 1

Illustration of asymmetrically coupled spreading processes in a double-layered communication-contact network.

(a) Communication and contact networks, denoted respectively as layer A and layer B, each have four nodes. Each node of layer A is matched one-to-one with that of layer B randomly. A node i in layer A is represented as , where the subscript and superscript respectively represent the index of node and the times of received information. (b) At t = 0, node B1 in layer B is randomly chosen as the initial infected node and its counterpart, node A1 in layer A, gains the information and becomes informed state and updates M = 1. While all other pairs of nodes, one from layer A and another from layer B, are in the susceptible state. (c) At t = 1, node B3 in layer B can be infected by infected neighbor B1 with probability βB and if it is indeed infected, its corresponding node A3 in layer A will get the information as well and update M = 1. Within layer A the information is transmitted from A1 to A2, with M = 1 for A2. Since, by this time, A2 is already aware of the infection spreading, whereas its counterpart B2 in layer B takes vaccination with probability ξ1, but fails. At the same time, node A1 in layer A and its counterpart B1 in layer B enter into the refractory state with probability μA and μB, respectively. (d) At t = 2, in layer A, A3 successfully transmits the information to A2. In this case, node A2 updates M = 2. At the same time, its counterpart B2 in layer B takes vaccination with probability ξ2 and successfully becomes a vaccinated node. The spreading dynamics terminate as all infected/informed nodes have entered into the refractory state.

Theoretical analysis

The epidemic threshold and the final infection density are the two key quantities in the dynamics of spreading. Thus, in this paper, a theory is proposed to predict these quantities for both information and epidemic spreading in the double-layer network.

Let PA(kA) [PB(kB)] be the degree distribution of communication layer A (contact layer B) and the average degrees of A and B are and , respectively. Here, our sole focus is the uncorrelated double-layer network, where the joint probability distribution of degree kA and degree kB of a node can be expressed as . Meanwhile, we assume that there is no degree correlations between inner-layer links and inter-layer links. If the specific formula of PAB(kA, kB) is given, the developed theory can be extended to the correlated double-layer networks19,21,26. The variables of , and are used to denote the densities of the susceptible, informed and recovered nodes with degree kA in layer A at time t, respectively. Thereinto, and is the density of IA nodes with degree kA which have received m pieces of information till time t. Similarly, , , and are the densities of the susceptible, infected, recovered and vaccinated nodes with degree kB in layer B at time t, respectively. The effective transmission rates for the two spreading dynamics are respectively expressed as λA = βAA and λB = βBB. Without loss of generality, we set μA = μB = μ, which won’t affect the relative sizes of effective information and disease transmission rates.

The mean-field rate equation of the information spreading in layer A is

where [] denotes the probability of a SA (SB) node with degree kA (kB) in layer A (B) being informed (infected) by its neighbor in the same layer at time t (See Methods for details). The first term in the right hand side (RHS) of Eq. (2) means the loss of SA nodes since they have received information from their neighbors in layer A. And the second term represents the counterpart nodes of SA nodes in layer B are infected by the disease resulting in the decrease of SA nodes. For m = 1, the gain of can only come from SA nodes. But for m > 1, the density of can be increased by the case in which the IA nodes have already received n pieces of information and receive m − n pieces of information again at time t. As a result, the rate equations of when m = 1 and m > 1 should be established, respectively.

When m = 1, the rate equation of is given as

where is the probability of a SA node with degree kA in layer A which has n (n ≤ kA) number of informed neighbors, Bk, mA) denotes the binomial factor and means the probability of an IA node with degree kA being informed again by its neighbors in layer A at time t (See Methods for details). The first and second term in the RHS of Eq. (3) correspond to the case that the SA node receives one piece of information and the case that the SB node is infected by the disease, respectively. The third term means that the informed node (IA) which has only received one piece of information previously receives one or more pieces of information at time t. The fourth term describes the recovery of the IA node.

When m > 1, the rate equation of can be described as

where represents the probability of an IA node with degree kA to have n (n ≤ kA) number of informed neighbors (See Methods for details). The first term in the RHS of Eq. (4) means that a SA node receives m pieces of information at time t. The second term in the RHS of Eq. (4) denotes the case in which the IA node with degree kA has received q () pieces of information previously and then receives m − q pieces of information at time t. The third and the fourth term are the same to those of Eq. (3), which indicate the losses caused by the newly received information and the recovery of IA to RA, respectively. The rate equation for can be written as

The mean-field rate equation of the epidemic spreading in layer B is

where [] refers to the probability that a SA (IA) node with degree kA newly receives information to make its counterpart node in layer B vaccinated (See Methods for details). The first term in the RHS of Eq. (6) means that the SB type nodes are infected by their neighbors in layer B. The second and third terms in the RHS of Eq. (6) represent that the SB nodes’ counterpart nodes are respectively in SA and IA state in layer A, receiving the information about disease and making SB nodes vaccinated.

From Eqs (2, 3, 4, 5, 6, 7, 8, 9), the density associated with each distinct state in layer A or B is given by

where H {A, B}, x {s, ρ, r, v} and kH,min (kH,max) denotes the smallest (largest) degree of layer H. Specially, the density of IA node with degree kA in layer A is . The final densities of the whole system can be obtained by taking the limit t → ∞.

Owing to the complicated interaction between the disease and information spreading process, it is unfeasible to derive the exact threshold values. Thus, a linear approximation method is applied to derive the outbreak threshold of information spreading in layer A (see Supporting Information for details) as

where

and

refer to the outbreak threshold of information spreading in layer A when it is isolated from layer B and the outbreak threshold of epidemic spreading in layer B when the coupling between the two layers is absent, respectively.

For βA < βAu, Eq. (11) shows that the information cannot break out in layer A if layer A and layer B are isolated. When the two spreading dynamics are interacting, near the epidemic threshold, the spread of epidemic in layer B can only lead to a few of counterpart nodes in layer A “infected” with the information and thus these informed nodes in layer A have negligible effect on the epidemic dynamics in layer B since βA < βAu. The above explanation indicates that βBc ≈ βBu when βA < βAu. However, for βA < βAu, the information outbreaks in layer A which makes many counterpart nodes in layer B to be vaccinated and thus hinders the spread of epidemic in layer B. Once a node is in the vaccination state, it will no longer be infected. Usually, we can regard this kind of vaccination as a type of “disease,” and every node in layer B can be in one of the two states: infected or vaccinated. Epidemic spreading and vaccination diffusion (derived by information diffusion) can thus be viewed as a pair of competing “diseases” spreading in layer B41. As pointed out by Karrer and Newman41, when two competing diseases have different growth rates in large size network N, they can be treated as if they were in fact spreading non-concurrently, one after the other.

To clarify the interplay between epidemic and vaccination spreading, we should determine which one is the faster “disease”. At the early stage, the average number of infected and vaccinated nodes in layer B grows exponentially (see Supporting Information). And the ratio of their growth rate can be expressed as

when θ < 1, i.e., , the disease process grows faster than the vaccination process. In this case, we can ignore the effect of vaccination on epidemic spreading. However, when θ > 1, the information process spreads faster than the epidemic process, which is in accord with reality since many on-line social networks and mass media can promote information spreading. Given that vaccination and epidemic can be treated successively and separately, by letting βB = 0 and obtaining the final density of vaccination from Eq. (9), the threshold of epidemic outbreak is given as19

Simulation results

The standard configuration model is used to generate a network with power-law degree distribution42,43 for the communication subnetwork (layer A). The contact subnetwork for layer B is of the Erdös and Rényi (ER) type44. The notation SF-ER is adopted to denote the double-layer network. The network sizes of both layers are set to be and their average degrees are 〈kA〉 = 〈kB〉 = 8. The degree distribution of communication layer A is expressed as , when the coefficient is and the maximum degree is . The degree distribution of contact layer B is . Without loss of generality, we set γA = 3.0, ξ1 = 0.05 and μA = μB = μ = 0.5 in the following simulations. To initiate an epidemic spreading process, a node in layer B is randomly infected and its counterpart node in layer A is thus in the informed state, too. The spreading dynamics terminates when all infected/informed nodes in both layers are recovered and the final densities , and are then recorded. We use 2 × 103 independent dynamical realizations in a fixed double-layer network and average on 30 different double-layer networks to obtain these final densities of each state.

In ref. 45, the variability measure has been verified to be very effective in identifying the SIR epidemic thresholds on various networks. However, for the interacting spreading dynamics, the interplay between them introduces a large external fluctuation into the respective spreading dynamics46, thus invalidate the variability measure. Therefore, we only qualitatively analyze the impact of the value of α (depicting the social reinforcement effect) on the outbreaks of information and disease. In the following simulations, we respectively define the reference information threshold (λAe) and the reference epidemic threshold (λBe) to valuate the outbreak possibility. At the reference threshold, the outbreak rate just reaches a reference value (e.g., 0.01 or 0.05) by using a tolerance47. The larger the value of reference information (epidemic) threshold, the harder the outbreak of the information (epidemic).

From Fig. 2(a,b), it can be seen that the impacts of the value of α on the reference information threshold λAe in layer A can almost be ignored. Nevertheless, it is shown by Fig. 2(c,d) that α has a remarkable influence on the reference epidemic threshold λBe in layer B when the information spreads faster than the disease. In particular, the epidemic threshold first increases with the value of α, but then tends to be stable when the value of α increases. The greater value of α leads to the stronger reinforcement effect (i. e., the greater adoption probability ξM) in layer A, which thus can more effectively suppress the outbreak of epidemic in layer B. However, with the increasing of α, the reinforcement effect of multiple information will reach a saturation point due to the restriction of network structure (e.g., mean degree and degree distribution) and information diffusion (e.g., transmission rate and recovery rate). Comparing Fig. 2(d) with Fig. 2(c), it can be seen that a larger value of λA also causes a higher reference epidemic threshold λBe (i.e., the disease transmission probability at which the final infection density reaches a fixed value such as rB(∞) = 0.01, 0.05).

Figure 2
figure 2

The impacts of social reinforcement effect on the outbreak threshold.

For SF-ER double-layer networks, the reference information threshold λAe and the reference epidemic threshold λBe as the function of the value of α are obtained by numerical simulations. Owing to the difficulty of determining the threshold values from numerical predictions, the reference density, for which the final recovery density in layer A (B) are 0.01 (black down triangles) and 0.05 (red squares), are set to be the reference threshold values. The blue solid line is the corresponding theoretical prediction from Eqs (11, 12, 13) and (15). (a) In communication layer A, the reference information threshold λAe performs as a function of α for λB = 0.05; (b) In communication layer A, the reference information threshold λAe performs as a function of α at λB = 0.5; (c) In the physical contact layer B, the reference epidemic threshold λBe performs as a function of α for λA = 0.3; (d) In the physical contact layer B, the reference epidemic threshold λBe performs as a function of α at λA = 0.5.

It is shown by Fig. 3(a–c) that with different values of λB, more nodes in layer B will be vaccinated [see Fig. 3(c)] with the increase of parameter α, leading to the spreading of epidemic in layer B to be reduced or eliminated [see Fig. 3(b)]. Moreover, the reduction of epidemic also decreases the number of informed individuals [see Fig. 3(a)], i.e., rA is reduced too. It can also be seen from Fig. 3(a–c) that α has a big influence on the values of rA, rB and vB when α  (0, 1), but little influence when α  [1, 5]. Figure 3(d–f) demonstrate the effects of λA on rA, rB and vB with different values of α. From Fig. 3(d), it can be found that rA decreases with λA when λA increases from zero, which is somewhat non-intuitive. As we know, when λA increases from zero, the spreading of information quickly inhibits the spreading of epidemic, which also reduces the promotion effect of epidemic on information spreading. Moreover, the competing effects of the two aspects (the enhancement of information spreading due to the increase of λA and the drop of information spreading due to the reduction of epidemic) may lead to the reduction of rA. However, as we further increase the value of λA, the information can spread quickly and more individuals will be informed [see Fig. 3(d)], which induces more people to be correspondingly vaccinated [see Fig. 3(f)], naturally, the number of infected individuals is reduced [see Fig. 3(e)]. It is noted that there are some discrepancies between the theoretical predictions and simulation results in Fig. 3, because the developed mean field theory can’t accurately capture the dynamical correlations between the two layers19.

Figure 3
figure 3

The impacts of social reinforcement effect and information transmission rate on final states.

For SF-ER double-layer network, subfigures (ac) show the values of rA, rB and vB as a function of α with different values of λB (0.3, 0.5 and 0.8) and their analytical predictions are corresponded to the black solid, red dashed and blue doted lines, respectively. Where λA is set as 0.5. Subfigures (df) illustrate the values of rA, rB and vB versus the parameter λA for different values of α (0, 0.2 and 1.0), corresponding to the black solid, red dashed and blue doted lines respectively. When λB is fixed at 0.5.

We then further study the effects of α and λB on the values of rA, rB and vB in Fig. 4. From Fig. 4(a,b), it can be seen that, though the values of rA and rB increase with λB as λB > λBu, their growth rate slows down with larger α. Figure 4 demonstrates that increasing α can stimulate more individuals to take vaccination, thus raising the value of vA. In RR-ER and SF-SF double-layer networks, the impact of social reinforcement effect on asymmetric interacting spreading dynamics is also explored and the obtained conclusion is consistent (see Figs S1–S3 and Figs S5–S7 in Supporting Information).

Figure 4
figure 4

A systematic investigation of the impacts of social reinforcement effect and disease transmission rate on final states.

For SF-ER double-layer network, (a) recovered density rA, (b) recovered density rB, (c) the vaccination density vB versus α and βB for λA = 0.5.

Social cost

Measures to prevent or eliminate diseases48,49,50 often mean certain social cost37,38, such as, the cost of treating infected individuals and vaccinating susceptible individuals, cost of isolation, cost of reducing outgoing and so on. Although the rapid spread of information and the strong social reinforcement effect can effectively promote the vaccination behavior and thus suppress epidemic spreading, the total cost of vaccination will be greatly increased. From an overall perspective, the government wants to control the diseases to the greatest extent with the minimal cost. In doing so, we define the social cost38 as

here, Λ is the set of all nodes in layer B. VB,i = 1 denotes the node i is in V state, otherwise, VB,i = 0. In the same way, RB,i = 1 means node i has recovered from disease, otherwise, RB,i = 0. Since every node in layer B can be in one of the three states: susceptible, recovered or vaccinated, it is impossible for VB, i and RB,i equaling to one at the same time. cV or cR denotes the cost of vaccination or treatment for a node. For the sake of simplicity, we assume the cost of vaccination and the cost of treatment are comparable and set both of them as unit for all individuals51,52, i.e., cV = cR = 1 and in this case, C = rB + vB.

Now we want to know how social reinforcement effect and information diffusion affect the social cost. Figure 5(a,b) present the social cost C as a function of the sensitivity factor α and the effective information transmission rate λA, respectively. As shown in Fig. 5(a), there exists an optimal value of α which can guarantee the minimal social cost when λA is larger than λB (e.g., λA = 0.5). However, with λA < λB, increasing α can reduce the social cost to some extent because the epidemic spreading is suppressed more or less. Also, there exists an optimal value of λA leading to the minimal social cost [see Fig. 5(b)]. When the number of vaccinated nodes is few, each vaccinated node can protect more than one node from infection, i.e., the herd immunity effect can be successfully produced when VB is small. Thus, increasing the value of α or λA stimulates more vaccinated nodes, which can effectively reduce the social cost. With further increasing the number of vaccinated nodes the disease can be controlled to a very low level. Apparently, it is unnecessary to increase the vaccination coverage any more, because the total social cost will be increased again when VB is further increased. Therefore, an optimal vaccination coverage (i.e., optimal values of α and λA) can be gained by employing the two competing effects, thus guaranteeing the minimal social cost. Consistent conclusions are also obtained in analyzing the influence of social reinforcement effect and information diffusion on social cost in RR-ER double-layer and SF-SF double-layer networks (see Fig. S4 and Fig. S8 in Supporting Information). This suggests that reasonably control the social reinforcement effect and the spread of information is very critical to minimizing the total social cost. For the social reinforcement effect, the risk of disease cannot be ignored, neither should it be exaggerated. As to the spread of disease information, the government should not only ensure the rapid spread of it but also avoid the excessive spread of it. In Fig. 5(c,d), with the increase of λA (α), the optimal αoAo) is reduced, which means that with a faster spread of information (a stronger social reinforcement effect), a minimal social cost is required for a weaker social reinforcement effect (a slower spread of information).

Figure 5
figure 5

Impacts of social reinforcement effect and information transmission rate on the social cost and optimal control.

For SF-ER double-layer network, the social cost C is versus the parameters of α and λA in subfigures (a,b), respectively. Here, the value of λB is fixed at 0.3. The optimal αo versus βA and optimal λAo versus α are demonstrated in subfigures (c,d), respectively. In (a) three different values of λA (0.2, 0.3 and 0.5) are selected, corresponding to the black circle solid, red triangle solid and blue square solid lines, respectively. In (b) different values of α (0.25, 0.5 and 1.0) corresponds to the black circle solid, red triangle solid and blue square solid lines, respectively. (c) the αo versus λA and (d) the λAo versus α under different λB (0.2, 0.3 and 0.5) correspond to the black circle solid, red triangle solid and blue square solid lines, respectively.

Usually, different relative costs of vaccination and treatment are required for different diseases38,53,54. Considering the self-interest characteristic of individuals in real society55, the behavior of taking vaccination is unnecessary for individuals if the cost of vaccination surpasses that of treatment. Therefore, the cost of treatment is considered to be greater than that of vaccination52,56. The impacts of different relative costs of vaccination and treatment (e.g., cR/cV = 2 in Fig. S9 and cR/cV = 5 in Fig. S10) on the optimal control are also studied in Supporting Information. It is found that the above conclusion remains unchanged qualitatively, but further study is still required56.

Discussion

In summarize, in this paper, a memory-based complex adoption mechanism was introduced into an asymmetrically interacting, double-layer network model to elucidate the mutual effects among information diffusion, epidemic spreading and the complex vaccination adoption mechanism. In the model, the information propagation and epidemic spreading occur in layer A and layer B, respectively. Moreover, the probability of vaccination for each informed individual depends on the times of information who has received and the social reinforcement effect. A mean-field based analysis was developed to reveal the two intricate spreading dynamics and to verify results of extensive simulations. Our findings show that such a complex vaccination adoption behavior with non-markov characteristics can inhibit the spread of disease and increase the epidemic threshold in the contact layer. Furthermore, when we consider the cost of vaccination and cost of the treatment for infected individuals, we found that there exists an optimal memory reinforcement effect and an optimal transmission rate of information which can minimize the social cost.

The challenges of studying the intricate interplay between social and biological contagions in human populations are generating interesting science57. In this work, we just considered the social reinforcement effect of cumulative information in complex adoption behavior and thus studied its impact on the two interacting spreading dynamics. As a matter of fact, the behavioral response to disease is also affected by socioeconomic factors such as psychological reflection, economic cost and infection status. The adoption behavior thus presents a more complex and diverse response mode, which may remarkably influence the asymmetric interacting spreading dynamics, especially for epidemic spreading. Our efforts along this line would stimulate further studies in the more realistic situation of asymmetric interactions.

Methods

Mean-Field equations for the spreading dynamics in layer A

To derive the mean-field rate equations for the density variables, we considered the probabilities that SA (SB) node is informed (infected) during the small time interval [t, t + dt]. According to the description of information spreading processes in two interacting spreading dynamical processes, it can be known that the loss of (i.e., the density of the susceptible nodes with degree kA) is caused by two aspects: 1) a SA node has received one or more pieces of information from its neighbors in layer A, i.e., the node is informed by its neighbors; 2) a SA node’s counterpart node in layer B is susceptible (i.e., SB) and it is infected by the disease at this time step.

In random configuration networks without degree correlations, for a SA node, the probability that one randomly selected neighbor is in IA state58 is given as

where

is the density of IA nodes with degree at time t and is the density of IA nodes with degree which have received m pieces of information till time t. One should note that, was adopted rather than in Eq. (17). For a SA node, since all of its neighbors cannot be informed by the SA node, one of its infected neighbors with degree concedes a possibility that other links connect to the SA node, excluding the link between this infected neighbor and its parent infected node. If we neglect the dynamical correlations between neighborhood, for a SA node, the probability for the node to have n number of IA neighbors is

where denotes the binomial factor . Based on the above factors, the probability of a SA node with degree kA to receive the information at least once is

Similar to Eqs (17), (18) and (19), for a SA node in layer B, the probability that one randomly selected neighbor is in IB state is

and

is the probability of a SB node with degree kB which has n number of IB nodes in his neighborhood. Moreover, the probability of the SB node with degree kB to be infected is

At time step t, the density of can be altered by two opposite cases: 1) for a node that is in SA state before time step t and simultaneously receives m pieces of information at time t or that has received n (0 < n < m) pieces of information and simultaneously receives m − n pieces of information at time t, leading to the gains of ; 2) for an IA node which has received m pieces of information and then receives one or more pieces of information again at time t, or the IA node recoveries to RA node, resulting in the losses of .

For an IA node, the probability that one selected neighbor is in IA state is given as

Thus, the probability of an IA node with degree kA to have n (n ≤ kA) number of informed neighbors is

As a result, the probability that the IA node has received at least one piece of information is

Mean-field equations for the spreading dynamics in layer B

There are two cases which can lead to the decrease of , as follows: 1) a SB node is infected by its neighbors in layer B with probability ; 2) The SB node goes to VB state because its counterpart node in layer A is informed and is willing to take vaccination. Firstly, we can conclude that a node must be in SB state if its counterpart node in layer A is in SA state. Ignoring the inter-layer degree correlations and dynamical correlations, the probability that the counterpart node of a node with degree kB has degree kA and is in SA state can be written as . Combining Eqs (1) and (19), for a SA node of degree kA which has n number of informed neighbors and has just received q pieces of information at time t, the probability of taking vaccination is determined by the term . Considering the different numbers of n and p, the probability of an individual to adopt vaccination can be obtained as

Secondly, when a node of degree kB is in SB state with probability in layer B, its counterpart node may have already been informed of m pieces of information with probability , if inter-layer degree correlations and dynamical correlations are ignored. Accumulating different cases of , the probability of an individual to take vaccination can be given as

Additional Information

How to cite this article: Liu, Q.-H. et al. Impacts of complex behavioral responses on asymmetric interacting spreading dynamics in multiplex networks. Sci. Rep. 6, 25617; doi: 10.1038/srep25617 (2016).