# Coupled Dynamic Model of Resource Diffusion and Epidemic Spreading in Time-Varying Multiplex Networks

Authors: Ping Huang; Xiao-Long Chen; Ming Tang; Shi-Min Cai
Journal: Complexity
 (2021)

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## Abstract

In the real world, individual resources are crucial for patients when epidemics outbreak. Thus, the coupled dynamics of resource diffusion and epidemic spreading have been widely investigated when the recovery of diseases significantly depends on the resources from neighbors in static social networks. However, the social relationships of individuals are time-varying, which affects such coupled dynamics. For that, we propose a coupled resource-epidemic (RNR-SIS) dynamic model (coupled model for short) on a time-varying multiplex network to synchronously simulate the resource diffusion and epidemic spreading in dynamic social networks. The equilibrium analysis of the coupled model is conducted in a general scenario where the resource generation varies between susceptible and infected states and the recovery rate changes between resourceful and noresource states. By using the microscopic Markov chain approach and Monte Carlo simulations, we determine a probabilistic framework of the intralayer and interlayer dynamic processes of the coupled model and obtain the outbreak threshold of epidemic spreading. Meanwhile, the experimental results show the trivially asymmetric interactions between resource diffusion and epidemic spreading. They also indicate that the stronger activity heterogeneity and the larger contact capacity of individuals in the resource layer can more greatly promote resource diffusion, effectively suppressing epidemic spreading. However, these two individual characters in the epidemic layer can cause more resource depletion, which greatly promotes epidemic spreading. Furthermore, we also find that the contact capacity finitely impacts the coupled dynamics of resource diffusion and epidemic spreading.


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## Body

## 1. Introduction

Most real networks are not isolated and the spreading dynamic processes on such real networks described by classical models such as the susceptible-infected-susceptible model (SIS) [1, 2] and susceptible-infected-recovery model (SIR) [3] may be interrelated and interactive with each other [4–6]. In epidemic spreading, a variety of factors can greatly influence the spread of diseases, such as the particular individuals’ birth rate [7], immunity [8, 9], public policy [10, 11], vaccination [12], awareness [13–19], and resource [20, 21].For example, Brandeau [20] et al. determined the optimal allocation of a limited resource for epidemic control among independent populations, and Chen et al. [21] found that any additional quantity of resources above the critical level would only bring marginal benefits to the containment. These works focus more on how public resource investment affects the spread of disease. However, in a general scenario, many patients tend to depend on the individual resources from their direct or indirect neighbors in social networks because of the limitation of public resource investment. Thus, some researchers naturally investigate the interactions between individual resources and epidemic dynamics. They assume that resource diffusion and epidemic spreading occur on a two-layer interdependent network because resources usually propagate through social links and diseases spread through physical contacts. They find that individual resources play a crucial role in controlling or mitigating epidemic spreading by using an appropriate strategy of resource allocation [22–27].Although many works based on contact network models have been done on the resource allocation for epidemic control, most of them assume that contacts are static through time. However, contact patterns are not static [28–31] and abundant close contacts in heterogeneous social networks promote both epidemic outbreak and spreading [32–35]. In recent years, the activity-driven (AD) temporal network model has attracted a large amount of attention for considering the dynamic connectivity patterns in real networks [19, 36–39]. In such network models, each node is assigned with an activity potential that determines its probability to create pairwise interactions with others randomly selected at each time.Furthermore, some theoretical methods describe the evolution of the dynamic spreading process and analyze the corresponding stylized characters. Two of the widely used approaches are heterogeneous mean-field (HMF) [40–42] and the microscopic Markov chain approach (MMCA) [43]. MMCA can deal with individual-level infection dynamics much better. This approach is first used to derive the expression of the outbreak threshold of epidemic spreading on epidemic-awareness multiplex networks [44, 45]. Very recently, the MMCA is also expanded to understand the interaction between awareness propagation and epidemic spreading on static multiplex networks [46, 47] and time-varying multiplexes [19, 48].In real life, healthy individuals can earn resources (e.g., money) or strive for personal resources. However, patients quickly lose the ability to generate resources. For solving the problem of deficiency of individual resources, patients commonly seek resources from their friends or relatives and even use the crowdfunding platform (e.g., GoFundMe and Waterdrop) to obtain resources provided by healthy individuals. As the resource diffusion and epidemic spreading on dynamic social networks synchronously happen in real society, the present work uses the AD model to construct a generic time-varying multiplex network representing the active contacts (or social relationships) of social networks. Because the dissemination of resources relies on the spreading of news on the crowdfunding platform, we use the Resourceful-Noresource-Resourceful (RNR) model (i.e., a SIS-like model) to describe resource diffusion. The evolutionary process of epidemic spreading is described by the SIS model. These two dynamic models are coupled into the RNR-SIS model by interlayer links of the time-varying multiplex network better to understand the interactions between resource diffusion and disease spreading. Meanwhile, considering the similarity between awareness propagation and resource diffusion, we make efforts to apply the MMCA for constructing a theoretical framework of the coupled resource-epidemic dynamic model (i.e., coupled RNR-SIS model) on a time-varying multiplex network. Theoretically, the equilibrium analysis based on MMCA determines a probabilistic framework of the coupled model and theoretical outbreak thresholds involving different parameters. By performing extensive numerical simulations, we obtain the outbreak thresholds of epidemic spreading via variability measurement, which well agree with the theoretical analysis. Moreover, the experimental results also show that stronger activity heterogeneity and larger contact capacity of individuals greatly influence the coupled dynamics of resource diffusion and epidemic spreading. However, the influence of small contact capacity is finite.Our paper is organized as follows: in Section2, we describe the coupled RNR-SIS model on a time-varying multiplex network. In Section 3, we use MMCA to propose a probabilistic description of the dynamical processes and derive the theoretical expression of the outbreak threshold. In Section 4, we compare numerical simulations with theoretical analysis and investigate the influences of different parameters on resource diffusion and epidemic spreading. In Section 5, we conclude our work.


## 2. Coupled Resource-Epidemic Model on a Time-Varying Multiplex Network

The coupled RNR-SIS model on a time-varying multiplex network is depicted in Figure1. In this model, individuals are mapped into the nodes of an AD temporal two-layer network. The individual resources diffuse on the upper layer (i.e., resource layer), and the spread of disease occurs on the lower layer (i.e., epidemic layer). In the dynamic process of resource diffusion (or epidemic spreading), the nodes are alternatively changing from the noresource (or susceptible) to resourceful (or infected) states according to resource rate diffusion λ (or epidemic transmission rate β) and conversely with resource loss rate δ (or epidemic recovery rate μ). λ is parallel with β. For example, the possibility is that you will give a hand when you browse information about fundraising. Thus, in the coupled dynamic process, the nodes are divided into four categories, noresource and susceptible (NS), resourceful and susceptible (RS), noresource and infected (NI), and resourceful and infected (RI). We assume the trivially asymmetric interactions between resource diffusion and epidemic spreading because the individual resources effectively affect the recovery of disease, but the spread of disease only affects the generation of individual resources. Thus, we further refine the epidemic recovery rate into μN and μR according to the node state in the epidemic layer. Obviously, the infected nodes with the resource can recover faster than those without resources, which indicates μN=ξμR0≤ξ≪1. Table 1 shows the detailed definition of parameters in the coupled model. Most of the parameters are probabilities and rates, and their domain is limited in the interval 0,1.Figure 1
The sketch of coupled RNR-SIS model on an AD temporal two-layer network. The nodes indicate individuals in real society, and the edges depict dynamic contacts and social relationships of individuals in social networks. Individual resources diffuse on the upper layer (i.e., resource layer), leading to the node in a resourceful or noresource state. The spread of disease occurs on the lower layer (i.e., epidemic layer), leading to the node in a susceptible or infected state. The coupled dynamic process evolves from (a) timest1 to (b)t1+Δt, where node 2 is infected from its neighbors, node 4 recovers in the epidemic layer, and node 3 changes from the noresource to resourceful state in the resource layer.
(a)(b)Table 1
Notions of parameters.
SymbolDomainDefinitionN[0,∞]Number of nodes in networkt[0,∞]Time stepδ[0, 1]Resource loss rateλ[0, 1]Resource diffusion rateβ[0, 1]Epidemic transmission rateμ[0, 1]Epidemic recovery rateμN[0, 1]Epidemic recovery rate of the nodes in noresource stateμR[0, 1]Epidemic recovery rate of the nodes in resourceful statem1[0,∞]Contact capacity of active nodes in the resource layerm2[0,∞]Contact capacity of active nodes in the epidemic layerai[0, 1]Activity level of nodei in the resource layerbi[0, 1]Activity level of nodei in the epidemic layerη1[0,∞]Rescaling factor of resource layerη2[0,∞]Rescaling factor of epidemic layerxi[ϵ, 1]Activity potential of nodei in the resource layeryi[ϵ, 1]Activity potential of nodei on epidemic layerɛ(0, 1]The lower cutoff of activity potentialF1x—Distribution of nodes’ activity potential of resource layerF2y—Distribution of nodes’ activity potential of epidemic layerPiX[0, 1]Probability of nodei in state Xrit[0, 1]Probability that nodei has no resources from neighbors at time tqit[0, 1]Probability that nodei will not be infected by any neighbors at time tβc[0, 1]The upper threshold ofβ for the outbreak of epidemicλc[0, 1]The lower threshold ofλ for the outbreak of epidemicMore concretely, we construct the AD networks ofN nodes for each layer regarding different parameters. Each node i is assigned with an activity level (i.e., firing rate) ai=η1xi in the resource layer and bi=η2yi in the epidemic layer, where ηi is a rescaling factor to make the average number of active nodes per unit time be η1xN and η2yN. According to the previous work [49], the activity levels of individuals in social networks usually obey a power-law distribution with a cutoff, and due to the experience from [36], we use the given probability distributions F1x∝x−γ1 and F2y∝y−γ2 to generate the hierarchical activity potential of individuals in the resource and epidemic layer, respectively. Also, we set the lower cutoff ɛ=0.001 of activity potential, that is, ɛ≤xi≤1 (yi is likewise).Figure 2
Transition probability trees of three types of nodes. They describe the transient changes of nodes’ states at each time. The definitions of parameters can be found in Table1.
(a)(b)(c)Based on the AD temporal two-layer network, we describe the evolving process of coupled RNR-SIS model as the following steps:(i)
At each timet, node ii=1,…,N becomes active with probabilities ai and bi in the corresponding layer. If the node is active in the resource layer, it will generate m1(corresponding m2 in the epidemic layer) edges to randomly selected nodes. Otherwise, no connections are created. Both active and inactive nodes can receive connections from active nodes.(ii)
The coupled RNR-SIS dynamic process is run on the constructed two-layer networks. Initially, the nodes only exist in the RS and NI states. The rules are specifically described as follows: (a) only the susceptible nodes can generate resources at the end of any moment. (b) To some extent, the resourceful nodes lose their all resources with probabilityδ or diffuse resources to their direct neighbors in the noresource state with probability λ. (c) The susceptible nodes are infected by their direct neighbors in the infected state with probability β. (d) The infected nodes with (or without) resources recover in the susceptible state with probability μR (or μN). Note that the NS nodes only exist at the transient time but finally disappear at the end of the synchronous-updating coupled dynamic process and the resource diffusion to susceptible nodes is insignificant according to rule (a). Thus, there are only three types of nodes in the coupled dynamic process at each time. Figure 2 shows the transition probability trees of three types of nodes describing the transient changes of their states.(iii)
At next timet+Δt (we set Δt=1), all edges of the previous network in each layer are deleted and we repeat the construction of AD networks. The coupled dynamic process continuously evolves according to the rules and also is terminated when it converges to the stable state.


## 3. Theoretical Analysis Based on Microscopic Markov Chain Approach

According to the model mentioned above, each node can be in one of the four states at timet with the probabilities represented by PiNSt, PiRSt, PiNIt, and PiRIt. According to rule (a), we can easily obtain PiNSt≡0. Note that PiR=PiRS+PiRI and PiI=PiNI+PiRI. Based on MMCA, we give the following theorems.Theorem 1.
rit denotes the probability of node i not getting resources from any direct neighbors, which can be written as(1)rit=∏j1−λm1NaiPjRt+ajPjRt.The partm1/NaiPjRt is the probability that an active node i creates a link with an active node j in the state R and the second part m1/NajPjRt is the probability that an active node i gets a link from an active node j in the state R.Theorem 2.
qit denotes the probability for the active node i not being infected by any neighbors, which can be written as(2)qit=∏j1−βm2NbiPjIt+bjPjIt.The partm2/NbiPjIt is the probability that an active node i creates a link with an active node j in the state I and the second part m2/NbjPjIt is the probability that an active node i gets a link from an active node j in the state I.Theorem 3.
According to Figure2, we can develop the Markov chains equations for each node i as follows:(3)PiRSt+1=PiRStδqit+1−δqit+PiNIt1−ritμR+μNrit+PiRIt1−δμR+δμN,(4)PiNIt+1=PiRStδ1−qit+PiNIt1−μNrit+PiRItδ1−μN,(5)PiRIt+1=PiRS1−δ1−qit+PiNIt1−rit1−μR+PiRIt1−δ1−μR.The theoretical analysis based on MMCA aims to deduce the theoretical outbreak thresholdβc of epidemic spreading through theorems (1)–(3). In the stable state, equations (3)–(5) can satisfy PiRSt+1=PiRS, PiRIt+1=PiRI, and PiNIt+1=PiNI. Near the outbreak threshold of epidemic spreading, the probability of an active node i to be infected is approximately equal to zero, PiI⟶0. Therefore, we can obtain(6)qi≈1−m2βbiρI+θbI,where θbI=∑jbjPjI/N. Next, we give a proof to deduce the theoretical outbreak threshold βc of epidemic spreading.Proof.
By ignoring the higher-order items, we simplify equations (3)–(5) as(7)PiRS=PiRSδqi+1−δqi+PiNI1−riμR+μNri+PiRI1−δμR+δμN,(8)PiNI=PiNI1−μNri+PiRIδ1−μN,(9)PiRI=PiNI1−μR1−ri+PiRI1−δ1−μR.
It is easy to find that the combination of equations (8) and (9) equals equation (7). By substituting equation (8) into equation (7), consequently, we can rewrite equation (7) as follows:(10)PiRS1−qi=μRPiI+μN−μRPiNI1−μN.
Note thatPiRS=1−PiNI−PiRI, and with ignoring higher-order items, the part PiRS1−qi can be simplified as 1−PiRI1−qi. Then, considering ξ⟶0(i.e., μN≪μR<1) and PiNI<PiI≪1, we simplify equation (10) as follows:(11)μRPiI=1−PiRI1−qi.
Then, we note thatθb2RI=∑jbj2PjRI/N. Multiplying bi on both sides of equation (11) and taking average over all nodes, we obtain a similar expression of θbI as(12)μRθbI=1−θbRI1−qi.
Next, we consider equations (11) and (12) as a matrix form(13)μRm2βρIθbI=b−θbRI1−ρRIb2−θb2RIb−θbRIρIθbI,βc is the minimum value of β that satisfies equation (13). And with ρIθbI≠0, we can easily solve the critical point with iteration method through the matrix(14)βc=μRm2b2−θb2RI1−ρRI+b−θbRI−1.
When there is only single layer in which only the epidemic spreads, the outbreak threshold of epidemic spreading could be solved by settingθbRI=0, θb2RI=0, and ρRI=0 ,(15)βc=μm2b2+b.


## 4. Results

To validate our MMCA based theoretical analysis, we run numerical experiments by the Monte Carlo simulations (MC) and compare the experimental results with theoretical ones regarding different parameters. We construct the coupled network with 1,000 nodes by two AD models. Considering the identical social networks and referring to [36], we set the exponents of probability distributions of activity potential with γ1=γ2=2.5, the contact capacities of active nodes with m1=m2=5 , and the rescaling factors with η1=η2=1000. The values of the rescaling factor do not affect the following results, but the activity heterogeneity and contact capacity have a significant influence, both of which are verified by the following experimental results. Note that the values of these parameters do not change unless they are chosen as the variables. We represent the final fraction of infected nodes ρI and the fraction of noresource nodes ρN in the stable states, which characterize the coupled dynamic processes of resource diffusion and epidemic spreading. Initially, we set a small fraction of infected nodes with ρI=0.08. Meanwhile, to highlight the crucial influence of individual resources on the recovery of disease, we herein set ξ=0 (i.e., μN=0 and μ=μR) that means the infected nodes without resource cannot recover. In the following, we investigate the influences of a number of parameters involved with the RNR-SIS dynamic model and AD network model on the coupled dynamic processes by performing hundreds of MC numerical simulations and obtain a number of numerical results.Firstly, we investigate the influence ofβ on the coupled dynamic processes. In Figure 3, it can be found that the changes of ρI and ρN with β indicate that there exists a phase transition of ρI and ρN when β is larger than the critical points, and the critical points of ρI and ρN can be identified by the peak of variability χ calculated as [50](16)χ=ωρ2−ρ2ρ.Figure 3
(a) The final fractionsρI and ρN of the infected nodes and the noresource nodes as a function of epidemic transmission rate β. The blue circle indicates ρN and the red circle indicates ρI. For both ρI and ρN, we present the results of numerical simulation and MMCA approach in full and open circles, respectively, which are in good agreement with each other. (b) The variability χ of ρI and ρN as a function of β. We can determine the critical points of ρI and ρN by the peak of variability and obtain the outbreak threshold of epidemic spreading βc=0.12. The other fixed parameters are δ=0.10, λ=0.64, μ=0.48.
(a)(b)Note that, in Figure3(b), we multiply ω = 1000 to identify the peak more clearly, which does not change the threshold value. For ρI, its critical point βc=0.12 approximately equals the theoretical outbreak threshold of epidemic spreading by solving equation (14). Moreover, as shown in Figure 3(a), the numerical results of ρI and ρN are in good agreement with the theoretical ones obtained by solving equations (1)–(5) (the same in the following figures), which verifies the validation of MMCA based dynamic framework of the coupled model.Then, we investigate the influences ofδ and λ on the coupled dynamic processes, which mainly imply how individual resource diffusion affects the epidemic spreading. Figure 4 shows the changes of ρI and ρN with δ and λ, respectively. Specifically, both ρI and ρN increase with δ in Figure 4(a) but decrease with λ in Figure 4(b). And, the increasing (or decreasing) trend as a function of δ (or λ) is significantly different between resource diffusion and epidemic spreading. Obviously, δ and λ have a greater influence on resource diffusion than epidemic spreading. More interestingly, a smaller λ≃0.2 can dramatically reduce ρI, which, to some extent, suggests the diffusion of individual resources is of importance for suppressing the spread of disease.Figure 4
The final fractionsρI and ρN of the infected nodes and the noresource nodes as a function of the resource loss rate δ (a) and the resource diffusion rate λ (b). The symbols are the same as in Figure 3. The results show that both ρI and ρN increase with δ but decrease with λ. And, a smaller λ∼0.2 can dramatically reduce ρI, indicating the diffusion of individual resources is of importance for suppressing the spread of disease. The other fixed parameters are β=0.70, μ=0.48.
(a)(b)We have investigated the influence of single parameter (i.e.,δ, λ, and β) on the coupled dynamic process. As the dynamic process of epidemic spreading is more important, we need the comprehensively synergistic influences of these parameters on the epidemic spreading and its outbreak threshold. Figure 5 shows the heat map of ρI in respect to a pair of δ (or λ) and β. We can find that, in Figure 5(a), the theoretical outbreak threshold βc weakly decreases with δ, and in Figure 5(b), it quickly increases then barely changes with λ which indicates that the resource diffusion suppresses the outbreak threshold with a lower resource loss rate and larger resource diffusion rate. Meanwhile, when the epidemic outbreaks (i.e., β>βc), it can be seen that smaller δ and larger λ can more effectively suppress the spread of disease, confirming the conclusion found in Figure 4. We also calculate the (simulated) outbreak threshold of λ obtained by the variability measurement in equation (16) based on the Monte Carlo simulations of numerical experiments, which are shown with the transverse curve in Figure 5(b). When λ is less than the simulated outbreak threshold, the epidemic will outbreak even if β is very small. We also note that λc (i.e., the transverse curve) fluctuates only when β is less than 0.2. A smaller β results in a smaller threshold of resource diffusion rate that a lower rate of epidemic spreading can reduce the risk of an epidemic outbreak caused by a low rate of resource diffusion. However, a smaller λ results in a smaller βc making it easy for an epidemic outbreak. These results suggest that the diffusive dynamics of individual resources can affect both the outbreak threshold and the spread of disease.Figure 5
The heat maps ofρI as a function of a pair of δ and β (a) and λ and β (b). The longitudinal curve in Figures 5(a) and 5(b) indicates the theoretical outbreak thresholds of β obtained by solving equation (13), which is the same as those in Figure 7. And, the transverse curve in Figure 5(b) indicates the simulated outbreak thresholds of λ obtained by the variability measurement, which affects the outbreak of epidemic. It trivially fluctuates only when β is less than 0.2. These two subfigures clearly show that δ and λ affect the outbreak thresholds of each other and effectively affect the outbreak of the epidemic, which suggests that resource diffusion can play an important role in the control of epidemic spreading. The other fixed parameters are λ=0.64 for (a), δ=0.10 for (b), μ=0.48.
(a)(b)Next, we make efforts to investigate the influences of time-varying network structure mainly including activity heterogeneity and contact capacity of nodes in two AD networks on the coupled dynamic process. Figures6(a) and 6(b) show that γ1 and m1 in the resource layer, respectively, impact ρI and ρN. They clearly show both ρI and ρN weekly increase with γ1 in Figure 6(a) and decrease with m1 in Figure 6(b). The finding suggests that stronger activity heterogeneity (corresponding to smaller γ1) and larger contact capacity effectively promote resource diffusion (corresponding to smaller ρN) but suppress epidemic spreading (corresponding to smaller ρI). Note that, in comparison with the epidemic spreading, γ1 and m1 have more influence on the dynamic process of resource diffusion.Figure 6
The final fractionsρI and ρN of the infected nodes and the noresource nodes as a function of activity potential γi and the contact capacity mi. The symbols are the same as those in Figure 3. The upper panel presents that ρI and ρN changes with γ1 (a) and m1 (b) in the resource layer, which shows that stronger activity heterogeneity and larger contact capacity of individuals in the resource layer can effectively promote resource diffusion but suppress epidemic spreading. The lower panel presents that ρI and ρN changes with γ2 (a) and m2 (b) in the epidemic layer, which suggests that these two characters of individuals in the epidemic layer cause more resource depletion and thus promote epidemic spreading. The other fixed parameters are δ = 0.10, λ = 0.64, β = 0.70, μ = 0.48, and m1 = 5 for (a), (c), and (d); m2 = 5 for (a), (b), and (c); γ1 = 2.5 for (b), (c), and (d); and γ2 = 2.5 for (a), (b), and (d).
(a)(b)(c)(d)In contrast, we present thatγ2 and m2 in the epidemic layer, respectively, impact ρI and ρN in Figures 6(c) and 6(d). They clearly shows that both ρI and ρN weakly decrease with γ2 in Figure 6(c) and increase with m2 in Figure 6(d). These results are the opposite to the ones found in Figures 6(a) and 6(b), which suggest that stronger activity heterogeneity (corresponding to smaller γ2) and larger contact capacity cause more resource depletion (corresponding to larger ρN) and thus promote epidemic spreading (corresponding to larger ρI). Also, in comparison with the resource diffusion, we note γ2 and m2 having more influences on the dynamic process of epidemic spreading.The following results also focus on the synergistic influences ofγi, mi, and β on the epidemic spreading and its outbreak threshold. It is clearly shown that, in Figure 7(a), the outbreak threshold is almost uncorrelated with γ1 and, in Figure 7(b), it weakly increases and then barely changes with m1. Meanwhile, for β>βc, we show that the spread of disease is obviously suppressed by more heterogeneous activity potential and larger contact capacity of nodes in the resource layer, which confirms the conclusion found in Figures 6(a) and 6(b). These results indicate that the dynamic social relationships of individuals represented by the AD network model in the resource layer cannot inhibit the outbreak of the disease but can effectively suppress the spread of disease.Figure 7
The heat maps ofρI as a function of a pair of γi and β (a–c) and mi and β (b–d). (a, b) γ1 and m1 are approximately uncorrelated with the outbreak threshold and effectively affect the spread of disease, which suggests that the dynamic social relationships of individuals in the resource layer cannot inhibit the outbreak of the disease but can effectively suppress the spreading of disease. (c, d) γ2 and m2 significantly affect the outbreak threshold and the spread of disease, which suggests that the dynamic physical contacts of individuals in the epidemic layer have strong influences on the dynamic process of epidemic spreading. The other fixed parameters are δ = 0.10, λ = 0.64, μ = 0.48, and m1 = 5 for (a), (c), and (d); m2 = 5 for (a), (b), and (c); γ1 = 2.5 for (b), (c), and (d); and γ2 = 2.5 for (a), (b), and (d).
(a)(b)(c)(d)In contrast, in Figures7(c) and 7(d), we can find that the outbreak threshold is strongly affected by γ2 and m2. More specifically, when the activity potentials of individuals in the epidemic layer become more heterogeneous (corresponding to smaller γ2), the outbreak thresholds dramatically decrease. When the contact capacities of individuals in the epidemic layer are much larger, the outbreak thresholds also dramatically decrease. Meanwhile, for β>βc, we find that stronger activity heterogeneity and larger contact capacity of individuals in the epidemic layer significantly promote the spread of disease, which also confirms the conclusion found in Figures 6(c) and 6(d). Besides, combining with the results in Figures 7(b) and 7(d), it can be seen that, in respect to each β>βc, the contact capacity of individuals in both the resource and epidemic layers has a limited influence on the spread of disease.


## 5. Conclusion

In conclusion, the present work has investigated the coupled dynamic processes of resource diffusion and epidemic spreading via the proposed RNR-SIS dynamic model on time-varying multiplex networks. We perform the investigations by assuming a good abstraction from a real scenario, where the diseases spread on a physical contact network, the individual resources diffuse on an online social network, and they dynamically interact with each other. Considering such two networks are temporal due to the time-varying physical contacts or social relationships of individuals, we first construct the time-varying two-layer network by using two AD network models with different parameters. The AD model encodes the connectivity pattern (i.e., the dynamic process of changing physical contact or social relationship) of individuals in the distribution of activity potential following a power-law function empirically measured in real-world networks. This function allows the definition of a simple dynamic process based on the nodal activity level, providing a time-dependent description of the connectivity pattern.Then, to explore the full-phase diagram of the coupled RNR-SIS dynamics on the time-varying two-layer network, we develop a probabilistic framework of the intralayer and interlayer dynamic processes of the coupled model by extending MMCA to time-varying multiplex networks. Based on the theoretical analysis of the probabilistic framework, we obtain the expressions of the fractions of infected nodes and noresource nodes and the outbreak threshold. Monte Carlo simulations are essential to verify the theoretical analysis. Comparing with extensive Monte Carlo simulations of the same system. The approximation using MMCA has an accuracy of up to 1.93% error for the prediction of the epidemic incidence and 21.79% error for the outbreak threshold ofβ. The principal reason for the large error in βc is that we neglect some higher-order terms and other related terms. The approximate agreement between theoretical and numerical results indicates the validation of the determined probabilistic framework. Meanwhile, these results show the trivially asymmetric interactions between the coupled dynamic process of resource diffusion and epidemic spreading. That is, the spread of disease is effectively suppressed by the resource diffusion with a lower resource loss rate and larger resource diffusion rate, but the diffusion of individual resources is trivially affected by the epidemic spreading even with a very larger disease transmission rate. Besides, we note that when epidemic outbreaks, the resource diffusion rate has a limited influence on the outbreak threshold and the fraction of infected nodes and also the disease transmission rate affects the threshold of resource diffusion rate limitedly around the threshold of the disease transmission rate.Next, considering the influence of time-varying two-layer network structure on the coupled dynamics of resource diffusion and epidemic spreading, we mainly focus on how the activity potential and contact capacity of individuals in the resource and epidemic layers affect the final fraction of infected nodes and noresource nodes and the outbreak threshold. The theoretical and numerical results suggest that stronger activity heterogeneity and larger contact capacity of individuals in the resource layer can suppress the spread of disease because they promote the diffusion of individual resources. In contrast, these two characters of individuals in the epidemic layer tend to mitigate the diffusion of individual resources and thus increase epidemic prevalence. Meanwhile, we note that the outbreak threshold is nearly independent of the time-varying network structure in the resource layer, suggesting the resource diffusion of individuals cannot inhibit the outbreak of disease but only can effectively suppress the spreading of disease. Besides, when the epidemic outbreaks, the contact capacity of individuals in both the resource and epidemic layer has a finite influence on the final fraction of infected nodes, suggesting that increasing contact capacity above a very large value makes no sense at all.Through the conclusion mentioned above, it can be seen that the diffusion of individual resources interacts closely with the dynamic process of epidemic spreading. On the one hand, the quicker diffusion of individual resources can effectively suppress epidemic spreading. However, only considering the dynamics of resource diffusion, we easily waste the individual resources due to the limited influence of the resource diffusion rate. On the other hand, we turn to analyze how the time-varying network structure affects the coupled dynamic process. When the epidemic outbreaks, individuals with larger activity potential and contact capacity have a high risk of getting infected but also are able to quickly obtain the resources of the neighbors. An extreme scenario is that the lack of individual resources might happen if the epidemic deteriorates, and then it entered a vicious circle. Thus, the obtained conclusion may be useful to provide methods of controlling large-scale epidemic prevalence, such as strengthening resource diffusion, reducing resource loss, speeding up of individuals’ activities, and enlarging the contact capacity in social networks. Finally, the proposed coupled resource-epidemic model and analytical approach may be instructive to multiplex spreading models in time-varying multilayer networks.



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