In order to explore the difference of epidemic spreading between ourwith otherincluding the classic SISon a single layerand a static multiplexin Fig., we compare theseunder different initial conditions. The multiplexof which thestructure of epidemic spreading layer is the same as that of the single layeris the one defined in Fig.. As shown in Fig., with the help of awareness spreading, not only the epidemic threshold, but also the final density of the infected nodesis smaller than that of the single layer. However, compared with the static multiplexit is obvious that the time variability of the awareness layer weakens the suppression effects on the spreading of epidemics, since the epidemic threshold is smaller and the final epidemic size is larger than that of the static multiplex

From the results above, it is clear that the MMCA method has a high accuracy to predict the epidemic threshold, no matter what values other parameters are set to be. Besides, there are also some interesting phenomena revealed by ourWhen the value ofis large, while the value ofis small, the value of aware probabilityhas an obvious effect on the epidemic threshold. The reason is that at this condition, it is easy for the aware nodes to stay in the aware state, while at the same time, the recovery ability of the nodes is strong. These two effects lead the epidemics to be difficult to outbreak. Hence, if we increase, which means that the unaware nodes can become aware more easily, the epidemic threshold becomes larger. However, ifis small andis large, it is easy for the epidemics to outbreak. And then, even if we increaseand nodes become aware with higher probability, the large value ofmakes nodes forget the epidemics in a short time, which will decrease the spread speed of awareness. That is why in this case,has a little effect on the epidemic threshold. Furthermore, note that there also exists a region bounded bywhere the metacritical point is localized, as illustrated in Ref.

Here, to show the validity of the approach discussed above, we have performed extensiveon multiplexwith 5000 nodes on each layer. The activity drivenwhich is used to generate the information spreading layer, is configured as follows: the power law distribution of the activity ratesatisfies, and the edge numberthat one active node can have is 8. Besides, the rescaling factoris set to be 10 and. The other layer is a scale freeof which the degree distribution. After having these settings, it is easy for us to obtain the boundary of the metacritical point, which equals (0.2677, 0.1390) in our case. In Fig., we show the comparison between analytic results calculated by Eq. (7) andunder various conditions. All the simulations start from a fractionof randomly chosen infected nodes andis fixed to be 0.2. At each time step, all the neighbours of an infected node become infected with the same probabilityand the infected node recovers at a rate. The same process fits for the spreading of information—what we need to do is just replace the infected probabilityand recovery ratewith the aware probabilityand forgetting rate, respectively. Iterate the rules of the coupled dynamical processes with parallel updating until the density of infected nodesis steady. In order to reduce the fluctuation of the density, we make time average that satisfiesand take T = 100 ().

B. Threshold model for awareness information diffusion process on time-varying network

topology structure of network, the way information spreads is also very important. Since in reality, individuals exhibit herd-like behavior because they are making decisions based on the actions of other individuals, which is called information cascade, 39 1, 3 (2013). 39. J. Borge-Holthoefer, R. A. Baños, S. González-Bailón, and Y. Moreno, J. Complex Networks, 3 (2013). https://doi.org/10.1093/comnet/cnt006 model, we use a threshold model to simulate the spreading of information. In the threshold model, for an unaware individual, awareness can come from two sources: the ratio between the number of aware neighbors and all its connections which is also known as node degree surpasses the critical value (local threshold α) or it is already infected. In Fig. 5 model defined on the time varying layer. As for the information spreading, apart from thestructure ofthe way information spreads is also very important. Since in reality, individuals exhibit herd-like behavior because they are making decisions based on the actions of other individuals, which is called information cascade,instead of the classic SISwe use a thresholdto simulate the spreading of information. In the thresholdfor an unaware individual, awareness can come from two sources: the ratio between the number of aware neighbors and all its connections which is also known as node degree surpasses the critical value (local threshold) or it is already infected. In Fig., we illustrate the thresholddefined on the time varying layer.

network is made up of two layers with different models on each layer, as described in Ref. 28 91, 012822 (2015). 28. Q. Guo, X. Jiang, Y. Lei, M. Li, Y. Ma, and Z. Zheng, Phys. Rev. E, 012822 (2015). https://doi.org/10.1103/PhysRevE.91.012822 r i as follows: r i ( t ) = H ( α − ∑ j a j i ( t ) p j A ( t ) k i ( t ) ) , (9) where a j i ( t ) = 1 if there is a connection between node i and node j at time t, otherwise a j i ( t ) = 0 . Besides, k i ( t ) is the degree, the number of its connections, of node i at time t. H(x) is a Heaviside step function, i.e., if x > 0, H(x) = 1, else H(x) = 0. Note that since function q i A ( t ) and q i U ( t ) (Eq. β c = μ Λ max , (10) where Λ max is the same as before, i.e., the largest eigenvalue of the matrix H defined following Eq. 6 MC simulations on top of the same multiplex network defined above. The results show that the agreement between MMCA method and MC simulations is quite well. Besides, as illustrated in Ref. 28 91, 012822 (2015). 28. Q. Guo, X. Jiang, Y. Lei, M. Li, Y. Ma, and Z. Zheng, Phys. Rev. E, 012822 (2015). https://doi.org/10.1103/PhysRevE.91.012822 α ≈ 0.5 , also exists in our model irrespective of the value of δ and μ. Since in our model, the information spreading layer is a time varying network, which is very different from the scale-free network or Erdős-Rényi network, the existence of the two stage effects verifies again that the phenomenon does not rely on the structure of the network, but is a result of the coupled dynamical processes itself. In addition, there is another interesting phenomenon that in Figs. 3 6 MC simulations seems to degrade as μ increases and δ decreases. Actually, a large value of μ and a small value of δ can lower the final epidemic size and heighten the epidemic threshold. If we focus on the absolute discrepancy between the MMCA method and MC simulations, it seems that the discrepancy is large on the occasion. However, from a macro point of view, for example, the error rate which equals β M C − β M M C A β M C , we find the MMCA method always has a good performance whatever μ and δ is since the error rate locates in the range of 0%–5%. Therefore, the multiplexis made up of two layers with differenton each layer, as described in Ref.. For the purpose of calculating the epidemic threshold, we need to change the expression of functionas follows:whereif there is a connection between node i and node j at time t, otherwise. Besides,is the degree, the number of its connections, of node i at time t.(x) is a Heaviside step function, i.e., if> 0,(x) = 1, else(x) = 0. Note that since functionand(Eq. (1) ) represent the transformation between state S and state I, there is no need to change these two functions. Then after the same derivation, we can also get the numerical result of the epidemic thresholdwhere Λis the same as before, i.e., the largestof the matrixdefined following Eq. (6) . In Fig., we crosscheck the numerical results with extensiveon top of the same multiplexdefined above. The results show that the agreement between MMCA method andis quite well. Besides, as illustrated in Ref., the two stage effects, which means that there exists a sharp like transition for the epidemic threshold at, also exists in ourirrespective of the value ofand. Since in ourthe information spreading layer is a time varyingwhich is very different from the scale-freeor Erdős-Rényithe existence of the two stage effects verifies again that the phenomenon does not rely on the structure of thebut is a result of the coupled dynamical processes itself. In addition, there is another interesting phenomenon that in Figs.and, the agreement between the MMCA approximation and theseems to degrade asincreases anddecreases. Actually, a large value ofand a small value ofcan lower the final epidemic size and heighten the epidemic threshold. If we focus on the absolute discrepancy between the MMCA method andit seems that the discrepancy is large on the occasion. However, from a macro point of view, for example, the error rate which equals, we find the MMCA method always has a good performance whateverandis since the error rate locates in the range of 0%–5%.

model on the multiplex network with other models, including the SIS model on the single layer network and the threshold model on the static multiplex network, as can be seen in Fig. 7 model is larger than that of the single layer case. The final epidemic size is also the smaller one, which means that information spreading can always suppress the spreading of epidemics. The differences of the spreading process between the static multiplex network and the time varying case also crosscheck the findings about time variability of the awareness layer topology, as discussed in Fig. 4 Moreover, we also compare ouron the multiplexwith otherincluding the SISon the single layerand the thresholdon the static multiplexas can be seen in Fig.. Obviously, as a result of information spreading, the epidemic threshold of ouris larger than that of the single layer case. The final epidemic size is also the smaller one, which means that information spreading can always suppress the spreading of epidemics. The differences of the spreading process between the static multiplexand the time varying case also crosscheck the findings about time variability of the awareness layeras discussed in Fig.

MC simulations. Since according to Eqs. ρI, here in order to have a better understanding of the MMCA method, we plot ρI as a function of β by means of the MMCA method and MC simulations, as illustrated in Fig. 8 MC simulations decreases with the increase in μ and the decrease in δ, which is the same as in Figs. 3 6 From the results above, we can find that the epidemic threshold calculated by the MMCA method has a good agreement withSince according to Eqs. (1)–(3) , it is easy for us to obtain the steady density of the infected nodes, here in order to have a better understanding of the MMCA method, we plotas a function ofby means of the MMCA method andas illustrated in Fig.. The findings reveal that the agreement between the MMCA method anddecreases with the increase inand the decrease in, which is the same as in Figs.and

models to explore the coupled dynamical processes on multiplex network with awareness layer being a time varying network, it is of most interest for us to explore the effects that the varying topology has on different models. As for the activity driven model, m is a critical parameter for the topology structure of the resulting time varying network. Accordingly in the following, we perform many simulations on the two models by setting m to be 4, 7, 10, and 20. Furthermore, in order to study to what extent the coupled dynamical processes is affected by varying topology, a new index named fluctuation ratio is introduced, which is defined as follows: F R ( t ) = f ( t ) − f ( 4 ) f ( 4 ) , (11) where f ( t ) is the final epidemic size of the coupled dynamical processes when m is set to be t. It is clear that for both models, with the increase in m, the epidemic threshold and final epidemic size become smaller. The reason is that m represents the number of edges each active node can connect, the larger the m is, the denser is the network. As a result, the spreading process on the dense network is quicker, which makes the nodes become aware easier. This will then suppress the spreading of epidemics. Moreover, the results also show that the threshold model is more susceptible to random changes in the topology of time varying network, since the fluctuation of the final epidemic size, as well as the epidemic threshold, is much larger in this case (Fig. 9 Since we propose twoto explore the coupled dynamical processes on multiplexwith awareness layer being a time varyingit is of most interest for us to explore the effects that the varyinghas on differentAs for the activity drivenm is a critical parameter for thestructure of the resulting time varyingAccordingly in the following, we perform many simulations on the twoby setting m to be 4, 7, 10, and 20. Furthermore, in order to study to what extent the coupled dynamical processes is affected by varyinga new index named fluctuation ratio is introduced, which is defined as follows:whereis the final epidemic size of the coupled dynamical processes when m is set to be t. It is clear that for bothwith the increase in m, the epidemic threshold and final epidemic size become smaller. The reason is that m represents the number of edges each active node can connect, the larger the m is, the denser is theAs a result, the spreading process on the denseis quicker, which makes the nodes become aware easier. This will then suppress the spreading of epidemics. Moreover, the results also show that the thresholdis more susceptible to random changes in theof time varyingsince the fluctuation of the final epidemic size, as well as the epidemic threshold, is much larger in this case (Fig.).