key: cord-0010887-xokzzmo9
authors: Wu, Bin; Mao, Shanjun; Wang, Jiazeng; Zhou, Da
title: Control of epidemics via social partnership adjustment
date: 2016-12-23
journal: Phys Rev E
DOI: 10.1103/physreve.94.062314
sha: 80d39da31a1cf76edd1da25fe087d6d342077595
doc_id: 10887
cord_uid: xokzzmo9

Epidemic control is of great importance for human society. Adjusting interacting partners is an effective individualized control strategy. Intuitively, it is done either by shortening the interaction time between susceptible and infected individuals or by increasing the opportunities for contact between susceptible individuals. Here, we provide a comparative study on these two control strategies by establishing an epidemic model with nonuniform stochastic interactions. It seems that the two strategies should be similar, since shortening the interaction time between susceptible and infected individuals somehow increases the chances for contact between susceptible individuals. However, analytical results indicate that the effectiveness of the former strategy sensitively depends on the infectious intensity and the combinations of different interaction rates, whereas the latter one is quite robust and efficient. Simulations are shown to verify our analytical predictions. Our work may shed light on the strategic choice of disease control.

Modeling the spreading of infectious diseases has a long history [1] [2] [3] [4] [5] [6] [7] . Mathematical models not only deepen the understanding of epidemic dynamics, but also shed light on the control of diseases. In recent years, much attention has been paid to epidemic control via social relationship adjustment [8] [9] [10] . As a pioneering work, Gross et al. proposed a susceptible-infected-susceptible (SIS) model on an adaptive network. Therein the susceptible individual breaks the link with the infected partner and rewire to another randomly selected susceptible individual [11] . This rewiring rule brings in highly complex dynamics (such as bistability and oscillation) to the classical SIS model. The rewiring dynamics then opens up the avenue on how individualized partnership adjustment alters the epidemic dynamics. On the one hand, besides the SIS model, typical epidemic models have almost been investigated including the susceptible-infectedrecovered-susceptible (SIRS) model [12] , the susceptibleinfected-recovered (SIR) model [13] , and the susceptibleinfected-vaccinated (SIV) model [14] . On the other hand, more realistic and complex link-rewiring rules are proposed. In particular, generalizations of Gross et al.'s rewiring rule are mainly twofold. For one thing, after the disconnection of susceptible-infected (SI) link, the susceptible is assumed to reconnect to a randomly selected member of the population whether it is susceptible or not [15] . For another, the infected is also allowed to switch its partnership from the susceptible to a new randomly selected contact [16] . Besides the rewiring rule which is dependent on the infection process, the rewiring rule that is independent of the infection process was also investigated [17, 18] . In spite of different model assumptions, all these models showed that the infection propagation can be * bin.wu@bupt.edu.cn † Corresponding author: zhouda@xmu.edu.cn greatly influenced by the dynamical networks. In particular, the infection can be effectively suppressed by reducing the interaction opportunities between susceptible and infected individuals.

Besides the above-mentioned link-rewiring models, another type of adaptive network is the link-activation-deactivation model [19] [20] [21] . It assumes that a link can either be broken or recreated on the basis of the infectious states of the two end points of the link. In this model, only local information is required, which could be more realistic [10] . In particular, Guo et al. proposed an ASIS model, in which any SI link can be broken (deactivated). After the disconnection of an SI link, the two disconnected nodes can be reconnected again once both of them become susceptible (activated) [21] . Despite seeming differences, the link-activation-deactivation dynamics is similar to the rewiring dynamics. On the one hand, Guo et al. showed that the ASIS model (initiated on complete graphs) approximates the link-rewiring model [11] . On the other hand, the quasistationary (metastable) fraction of infected individuals can be reduced by increasing the effective breaking rate (proportional to the ratio of deactivating rate to activating rate). This echoes the results based on the link-rewiring models that the disease can be controlled by reducing the contacts between susceptible and infected individuals. Therefore, both types of linking dynamics in epidemic control can be seen as decreasing the interaction rate between susceptible and infected individuals (called SI control). Furthermore, considering that the effective breaking rate in [21] also depends on the activating rate between susceptibles, their work reminds us of the significance of susceptible-susceptible (SS) links in epidemic control.

Intuitively, increasing the interaction time between susceptibles can also be a control strategy (called SS control). Yet it is seldom addressed, compared to the SI control that has been intensively studied in previous literatures. It seems that these two control strategies are two sides of the same coin. Actually, this is true in Gross et al.'s model [11] , since, based on their rewiring rule, the decrease of SI links directly leads to the increase of SS links. However, this is no longer valid in Risau-Gusmán and Zanette's model [16] , since the disconnection of an SI link does not necessarily result in the reconnection of an SS link. Therefore, the SS and SI control strategies are not equivalent in general.

In this work, we provide a comparative study on the SI control and the SS control by proposing a link-rewiring SIS model. Unlike the models only allowing the breaking of SI links, we allow all three types of links [SS, SI, and infectedinfected (II)] to be broken, equipped with three independent parameters to characterize the breaking rates of SS, SI, and II links. Actually, this assumption mimics the intrinsic nature of human mobility [22, 23] , namely, people moving or changing their social relationships due to a variety of reasons, even without the consideration of avoiding infectious diseases. In this way, SI links should not be the only type that is allowed to be broken; both SS and II links can change. For example, in acquired immune deficiency syndrome (AIDS) not only are the susceptible individuals willing to avoid contact from the infected individuals, but the susceptible-susceptible and infected-infected relationships may also be broken up due to unsatisfactory sexual experiences; i.e., the rewiring processes can happen in SS and II links. Besides, we allow all the individuals to be capable of adjusting any of their partners. This mirrors the freedom of social life. It also excludes the central control of epidemics, for example, via organizations. In this way, we could concentrate on how the social partnership adjustment strategies alone alter the fate of epidemics.

We demonstrate analytically that our model captures the epidemic dynamics with nonuniform interaction rates under fast linking dynamics. It is shown that sometimes the SS control is more effective and robust than the SI control. In particular, strengthening the closeness between susceptibles (SS control) effectively eradicates the disease no matter how infectious the disease is. However, the effectiveness of the SI control sensitively depends on the infectious intensity and the intrinsic mobility rate of the population. In other words, there are cases such that the SI control cannot eliminate the disease as efficiently as the SS control. Simulation results are also shown for validating our theoretical predictions. Our findings suggest that, besides the SI control, it is still of concern that the SS control may serve as a better candidate for epidemic control.

In this section, we propose the model of epidemic spreading coupled with a simple stochastic link-rewiring dynamics. Then we theoretically analyze the epidemic model with nonuniform interaction rates based on the time scale separation.

We consider a structured population of N individuals. The population is located on a connected network. We assume that the average degree k is much smaller than the population size N, i.e., k N . Here nodes refer to individuals and links represent social ties between individuals.

We adopt a standard SIS model to study the epidemic spreading. The SIS model assumes that susceptible individuals get infected with a probability proportional to the number of their infected neighbors; infected individuals recover and become susceptible with no immunity to the disease after a period of recovery time. The SIS model has three features: (i) the whole population size N is constant over time, (ii) the transmission of disease only happens via the SI links, and (iii) the recovery of infected individuals is independent of the status of their neighbors.

Let I t be the number of infected individuals at time t; therefore, the mean-field equation of the SIS model on the structured population is given as follows:

Here λ is the transmission rate and μ is the recovery rate.

All through the paper we assume that μ = 1 without loss of generality, and N SI is the number of SI links.

The social relationships between individuals are not eternal but are continuously coevolving. As a typical example, susceptible individuals tend to avoid contacts with infected ones by adjusting their local connections. It has served as the most recognized prototype in the study of epidemic control on dynamical networks. However, individuals may receive miscellaneous information when making rewiring decisions; thus, it is possible for all the individuals to adjust all of their current social relationships. Such a rewiring process captures mobilitylike human behavior.

Here we propose a simple link-rewiring dynamics by extending the dynamical nature from SI links to all types of links in the network. Each individual is either susceptible (S) or infected (I). Thus, there are three types of links: susceptible-susceptible (SS), susceptible-infected (SI), and infected-infected (II) links. To characterize the fragilities of different types of links, we define w XY (XY ∈ {SS,SI,II}) as the probability with which an XY link breaks off in the process of disconnection. In each rewiring step, a link XY is selected randomly from the network. With probability w XY , the link is broken; otherwise, the link remains connected. If it is broken, X or Y is picked as the active individual, who is entitled to reform a new link. Its new neighbor is randomly selected from the individuals who are not in its current neighborhood. Self-connections and double connections are thus not allowed here.

In this way, the link-rewiring dynamics can be modeled as a Markov chain in the state space of {SS,SI,II} [24] [25] [26] . Considering the transition probabilities between states, let us take the transition from SI to SS as an example. This happens only when SI is broken off and S is selected to reform a new link to another susceptible individual. Note that the total population size N is much larger than the average degree k, and the transition probability Q SI→SS is approximately given by w SI × 1 2 × s, where s is the density of susceptibles at the moment. Similarly, we calculate all the other transition probabilities, yielding the transition probability matrix

where i is the density of infected individuals. According to the standard Markov chain theory [27] , there exists a unique limiting distribution satisfying Q = provided Q is irreducible and aperiodic. Namely, when w SS w SI w II is = 0, Q has a unique stationary distribution

where

It is challenging to capture N SI due to the complexity of real social networks [28] [29] [30] . This is already true in static networks, and it becomes even more difficult taking into account the dynamical nature of social networks [8] . Here we overcome this problem by assuming the adiabatic elimination of fast linking dynamics [27] (also called annealed adaptive dynamics [10] ); i.e., the adjustment of social ties is much more frequent than the update of infection states. This assumption implies time scale separation of the two coupled dynamics. In other words, the disease is unlikely to spread until the social configuration tends to the stationary regime. In this way, N SI is approximated as

where Nk/2 is the total number of the links in the network and π SI is the fraction of SI links in the stationary regime. This approximation greatly reduces the complexity of the coupled dynamics. In light of this, the idea of time scale separation has frequently been used in analyzing complex dynamics on adaptive networks (epidemics [18, 19, 31] and evolutionary games [24, 32] ). By substituting Eq. (4) into Eq. (1) we have

In particular, when all the interaction rates are uniform and positive (w SS = w SI = w II > 0), Eq. (6) reduces to

Equation (7) is nothing but the classical SIS model [2] , provided that λ e = kλ is redefined as the effective transmission rate. This implies that the population is as a well-mixed population, if individuals break their partnerships with no social bias. It should be pointed out that, when w SS = w SI = w II = 0, the transition probability matrix Eq. (2) violates the irreducible condition [33] that our analysis relies on. In fact, this case resembles the static network, which has been excluded from our analysis. When the interactions are violated from above social unbias, on the one hand, it results in nonuniform interactions in the population [34] . Therefore, Eq. (6) extends the classical SIS model from uniform interaction rates to nonuniform interaction rates. Noteworthy is that this nonuniform extension is an emergent property from microscopic stochastic linking dynamics, which is not assumed in prior. On the other hand, if we define (i) = kλi A(i)w SI , our model also extends the classical SIS model from density independent transmission rate to density dependent transmission rate [15, 35] . In other words, the dynamical nature of social networks essentially acts as a feedback mechanism on the SIS model. The feedback mechanism, which is taken as the central idea of control, can significantly alter the epidemic dynamics.

Noteworthy is that all the analyses above are based on the time scale separation. Thus it suggests that the link-rewiring event should happen with a sufficiently large probability (close to 1) in each update. Furthermore, we give a more precise lower bound for this probability based on pair approximations: It is found that the time scale separation is at work provided the likelihood of the linking dynamics is greater than (see Appendix A)

For more general cases where the time scale separation is absent, the approximation based on the moment closure could be applied to provide theoretical insights [36] (see Appendix A).

Our main concern in this comparative study is epidemic control via changing the interaction rates in different ways. Based on Eq. (1), it is N SI that determines the spread of infection. The more SI links there are, the more likely the spread of infection. Generally, there are two ways to control N SI . For one thing, it is natural to increase w SI for reducing the interaction rate (1/w SI ) between susceptible and infected individuals (SI control). For another thing, decreasing w SS can also reduce the exposure of susceptibles to infection (SS control). Therefore, we investigate the control of epidemics via these two strategies. More specifically, by taking the uniform interaction rates (w SS = w SI = w II > 0) as the reference case, we would like to provide a comparative study on both the SI control (w SI > w SS = w II ) and the SS control (w SS < w SI = w II ). In the following, we assume that the effective transmission rate is always larger than the recovery rate, i.e., λ e > 1, where the epidemic control is necessary.

To decrease the interaction rate between susceptible and infected individuals, it is equivalent to increase the breaking FIG. 1. SI control of epidemics, i.e., increasing the breaking probability between susceptible and infected individuals. For the uniform interaction case (w SS = w SI = w I I ), the model degenerates to the conventional SIS model. There is only one internal equilibrium and it is stable provided λ e > 1. Here we solely adjust w SI such that the duration time of SI links is shorter than the other two types of links, i.e., w SI > w I I = w SS . The three panels show the phase diagrams in the (w SI ,λ e )-plane. The quality of the SI control is sensitively dependent on the reference uniform breaking probabilities. (a) When they are small (w SS = w I I = 0.05), decreasing the interaction between susceptible and infected individuals makes the phase diagram change from endemic state (red) to bistable state (yellow) and then to final extinct state (blue). (b) When w SS = w I I = 0.5, there is no bistable state (yellow) any more. It becomes harder to eradicate disease when the population is even more mobile. (c) The right panel shows that the SI control is incapable of eradicating the disease provided the population is intrinsically highly mobile (w SS = w I I = 0.95).

probability w SI . Based on the uniform interaction as the reference case, we are interested in how the epidemic dynamics is changed by increasing w SI . Here the uniform interaction can mimic the basic migration rate in the population. To illustrate our main results, we consider three typical cases with different initial values of the uniform interaction rates (see Appendix B for technical details): Fig. 1(a) 

In this case, we set initially the breaking probabilities for all types of links to be 0.05. The disease can be controlled by increasing w SI from 0.05 to 1. In particular, for small infectious rate (i.e., λ e 2), there is a phase transition with the increase of w SI . That is, the final state of epidemics turns from endemic to extinction. For large λ e (i.e., λ e > 2), there is a small region of bistability where the disease persists or dies out due to the initial infected fraction. Compared to the single continuous phase transition in the conventional (uniform) SIS model, the nonuniform SIS model can give rise to multiple phase transitions. The emergent bistability in adaptive SIS model has already been reported in previous studies [11, 15, 37, 38] , but it is quite difficult to approximate the conditions under which bistability is present. For our model, we explicitly provide those analytical conditions under which the bistability emerges based on Eq. Fig. 1(b 

In this case, increasing w SI is not as effective as that in the above small initial case. For small λ e , even though there still exists a phase transition from endemic state to extinct state, the marginal value of w SI that needs to cross the transition line is large. More importantly, when λ e is large enough, increasing w SI is unable to eradicate the disease any more. The disease will persist no matter how large the interaction rates between susceptible and infected individuals are. Moreover, it is shown in Fig. 2 that the endemic level is not sensitive to w SI . In other words, by increasing w SI , the final fraction of infected individuals declines very slowly. That is to say, the increase of w SI can neither qualitatively change the final state of the endemic nor quantitatively inhibit the final fraction of infected individuals.

In this case, the endemic state is always the global stable state provided λ e > 1. That is, the epidemics cannot be eradicated by the SI control. 3. SS control, i.e., decreasing the breaking probability between susceptibles. We start from the uniform case w SS = w SI = w I I . Here the disease is solely controlled by increasing the duration time of the social ties between susceptibles, i.e., w SS < w I I = w SI . These phase diagrams are similar for all the reference uniform interaction rates: i) for small λ e , the SS control makes the disease change from endemic state (red) directly to extinction state (blue); ii) for large λ e , the SS control can still eradicate the disease, but the phase diagram has to pass from endemic (red) to bistable state (yellow) and finally to extinction (blue).

To summarize, the control efficiency via reducing the interaction rate between susceptible and infected individuals strongly depends on the reference breaking probabilities, i.e., the intrinsic population mobility. The more likely the population is mobile, the worse the SI control performs.

Unlike the SI control, increasing the interaction rate between susceptibles is shown as an effective and robust strategy for epidemic control. In fact, no matter what the intrinsic mobility rate of the population is, the SS control successfully eradicates the disease. To this end, we study the three typical reference population mobility cases in the above section (see Appendix B for technical details). Figure 3 shows that the phase diagrams for the three cases are quite similar to each other:

(i) For small λ e (1 < λ e 2), by decreasing w SS , the final state of disease is directly transformed from endemic to extinction.

(ii) For large λ e (λ e > 2), the bistablilty arises in all three cases. That is, no matter how large the initial uniform interaction rates are, with the decrease of w SS , there is an intermediate region where the disease persists or dies out depending on the initial fraction of disease. Furthermore, we analytically obtain that the bistable region is given by

By comparison, the SS control is more effective than the SI control in two ways. On the one hand, the control of w SS is independent of the intrinsic population mobility, i.e., robust control. On the other hand, decreasing w SS can always effectively eradicate the disease regardless of infectious intensity (Fig. 4 illustrates the position of equilibria as a function of w SS in the bistable case).

In this section, we present agent-based simulations and further discuss the efficiency of the time scale separation method based on the comparison between the simulation results and theoretical predictions.

The contact process [39] is adopted to model the epidemic spreading on networks. Let α ∈ (0,1) be the probability of epidemic spreading in each update. The simulation is performed as follows:

(1) Initially, there are N individuals located on a regular graph with degree k, where each individual has exactly k neighbors. Then N 0 infected individuals and N − N 0 susceptible individuals are randomly distributed. Here w SI = w I I = 0.5, and λ e = 7. Increasing the interaction time between susceptibles (i.e. decreasing w SS ) effectively eradicates the disease. In particular, for 0.033 < w SS < 0.08 (bistability), the disease dies out provided the initial infection is few in number. Even when the initial number of infection is large, the final level of infection is still lower than the case with 0.08 < w SS < 0.5. For w SS < 0.033, the disease is eradicated no matter what the initial state is. (2) We generate a random number r ∈ (0,1). If r < α, we perform the contact process. Otherwise (r α), we perform the linking dynamics.

(3) If the contact process occurs, an infected individual (called Bob) is selected randomly. With probability μ k Bob λ+μ Bob becomes susceptible, where k Bob is the degree of Bob. Otherwise a neighbor of Bob is selected at random. This neighbor, namely Jack, is infected with probability λ k Bob λ+μ . It is noteworthy that Jack becomes infected if his status is susceptible. However, this new infection event does not change the state of Jack if Jack has been infected already. Then we return to step 2.

(4) If the linking dynamics occurs, a link is selected randomly. The type of this link is denoted as XY (XY ∈ {SS,SI,II}). With probability w XY , the link is broken; otherwise the link remains connected. If it is broken, X or Y is picked as the active individual, who is entitled to reform a new link. The new neighbor is randomly selected from the individuals who are not in its current neighborhood. Then we return to step 2.

Each data point is averaged over 50 independent samples. In each sample, we run a transient time of 10 6 generations, and we set the mean value over time window of the last 10 3 generations to be the final fraction of infected individuals.

It should be pointed out that the simulation results are robust for all initial connected graphs, provided the number of infected individuals, N 0 , population size N , and the average degree k are fixed. The regular graph here only serves as a prototype for simulations. In fact, our linking dynamics is a Markov chain, which is irreducible and aperiodic. This yields that the limiting behavior is independent of the initial configuration of the network [33] . Furthermore, the assumption of time scale separation allows all the links to converge to the stationary distribution. Therefore, all the links would converge to the stationary distribution no matter what type of graph it is initially.

With the coupled linking dynamics, the final fate of the infection can be of three cases: die out no matter what the initial fraction of the infected individuals is (called extinction), stabilize at a nonzero fraction of infected individuals no matter what the (positive) initial fraction of infected individuals is (called endemicity), or stabilize at a nonzero fraction of infected individuals if the initial fraction of infected individuals exceeds a critical value and die out otherwise (called bistability).

For the extinction cases, simulation results are found to be in good agreement with the analytical predictions. This is true for all the parameter regions predicting extinction for both SI and SS controls (see Fig. 5 ).

For the endemic cases, Fig. 6 shows that the population would end up with a constant fraction of infected individuals, provided there are infected individuals initially. This is exactly in line with the analytical predictions. Furthermore, the inconsistency between the analytical and simulation results is less than 10%, which is acceptable. Considering this 10% disagreement, the analytical predictions systematically overestimate the simulation results. In fact, the agent-based contact process is a Markov process with an absorbing state, where no infected individual is present. In other words, the disease would go extinct eventually if the system evolves over a sufficiently long time. Our analytical results, however, are in the quasistationary time scale [40, 41] . The inconsistency between the analytical and simulation results suggests that the running time of 10 6 is beyond the quasistationary time scale. Thus the system may evolve to the absorbing state with non-negligible chances.

For the bistability cases, the simulation results show qualitative agreement with the analytical predictions. In particular, the critical initial fraction of infected individuals, ensuring a dramatic outbreak of epidemics, is consistent with the unstable fixed point predicted by the analytical result (see the blue dashed lines in Fig. 7) . Disagreements, however, are also present. For example, the theoretical results tend to underestimate the final infection when the infection fraction is rare initially. In fact, this bistable case bears two internal equilibria lying at x * 1 (unstable) and x * 2 (stable) (x * 1 < x * 2 ). For a small initial fraction of infected individuals, the deterministic part of the system drives the infection to extinction based on the analytical investigation. Yet by its intrinsic stochastic nature of the epidemic spreading, the infection would increase in number and be possibly trapped around the stable equilibria from time to time. Even though it is a type of rare event, it takes quite long to escape from this trap. Thus on average it results in a relatively higher level of final fraction of infected individuals given the running time of simulations (here 10 6 generations). In other words, it is the interplay between the stochastic effect and stable equilibrium at zero that results in such inconsistency. It is noteworthy, despite this quantitative inconsistency, that the salient feature of the bistable dynamics is still captured by the analytical predictions.

In Fig. 8 , we investigate how the population size affects the accuracy of the analytical approximation. Theoretically, large population size inhibits the stochasticity arising from the finite population effect, which is closer to the mean-field approximation. Similar discussions can be found in [24] . Figure 8 shows the case with N = 100 still captures the bistable dynamics as the case with N = 500 does.

We have proposed a simple link-rewiring rule to model social partnership adjustment. Therein all the links are about to break, capturing the mobile nature of the population. This simple model paves the way to compare different rewiringbased epidemic control strategies.

We find that, for mild infectious disease, both SI and SS control strategies can eradicate the disease. For strong infectious disease, however, it is more efficient to adopt the SS control than the SI control. This result is counterintuitive. Intuitively, reducing the contacts between susceptible and infected individuals is believed to suppress the disease propagation. Moreover, it seems that decreasing the interaction rate of SI links could naturally result in the increase of SS links [37] . How can these two strategies perform so differently? One of the salient features of our model is the variability of II links, which has seldom been addressed previously. Actually, increasing w SI is equivalent to decreasing both w SS and w II . In other words, the SI control is equivalent to simultaneously strengthening SS links and II links. Similarly, the SS control is equivalent to simultaneously reducing the closeness of Another feature of our reconnection rule is nonselectivity. In other words, individuals are allowed to rewire to a randomly selected member no matter if it is susceptible or not. Compared to the selective rule in [11] , individuals in our model do not necessarily know who gets infected currently, which is more realistic. Actually, the nonselective rule increases the exposure of the susceptible individuals to the infected individuals. This is very likely in the beginning of an epidemic season, where the information on infection status is unaccessible. In particular, even though the SI control increases the breaking possibility of each SI link, a new SI link may be generated again due to the nonselective rule. By contrast, the SS control makes a straightforward intervention during the process of disconnection. That is, by strengthening the closeness between susceptible individuals, the SS strategy reduces the possibility of SI connection effectively. In this way, the nonselective rule has a relatively small impact on the SS control. Therefore, in the framework of the nonselective rewiring rule, the SS strategy is more efficient than the SI strategy.

Concentrating on the relation between the lifespan of each type of links and epidemic spreading, our model does not account for other features that are also considerable in capturing the epidemic dynamics of real world networks. For example, (1) our linking dynamics does not take into account the social interactions with memory, such as friendship and working partners, in which individuals preserve the contacts that they used to make [20, 21] , and (2) the link-rewiring process is a strong simplification of real adaptive networked human behavior. It is not necessarily realistic for individuals who break up a relationship to have a new partner immediately. However, it probably mimics the dynamics of networks in AIDS to some extent: The susceptible individuals break up their (mostly sexual) relationships with their infected partners and switch to other perceived healthy individuals. Moreover, the infected individuals may also rewire their links to other infected individuals.

To sum up, our result captures the causation between the link fragility and the disease control. Furthermore, this model might serve as a starting point to compare different rewiring control strategies for more general models closer to reality. 61603049) and the Fundamental Research Funds for the Central Universities.

Our model couples the linking dynamics and the epidemic dynamics. While the method in the main text is analytically insightful, it requires the time scales of the two dynamics to be separated. In other words, individuals should adjust their partners much faster than the spread of epidemics to make this method applicable. This is, however, not the case in general. We propose another analytical method to overcome this restriction. The method is based on pair approximation and rate equations [36] . Here we concentrate on how the method helps us estimate the condition under which the time scale separation is valid.

Let I and S be the global frequencies of infected and susceptible individuals, i.e., i and s in Eq. (6); and let XY = N XY /N be the frequencies of XY pairs, where XY ∈ {II,SI,SS}. Thus I + S = 1 and II + SI + SS = 1 hold. The system thus is determined by three independent variables: I , SI , and II . The crucial assumption for pair approximation is that a higher order of moments can be captured by moments of pairs. In the following, we write down the rate equations of the three variables under the assumption of pair approximation.

For the evolution of a fraction of the infected, it is only determined by the epidemic dynamics. In this case, the number of infected individuals increases or decreases by one, or stays the same in one time step. By the Kolmogorov forward equation, we have that

In particular, the probability that infected individuals increase by one in number happens when (1) the epidemic spreading is ongoing (with probability α), or (2) a susceptible individual is selected (with probability S ), and it is infected by one of its infected neighbors. The fraction of the infected individuals around a susceptible individual is SI S based on pair approximation. Thus there are on average k SI S infected neighbors around the selected susceptible individual, where k is the average degree of the entire network. Therefore, the infection probability of the susceptible individual within a small time interval t is λ k SI For the evolution of the links, it can be caused both by the linking dynamics and the epidemic spreading. Taking the change of II as an example, when the linking dynamics happens (with probability 1 − α), II links would increase by one if an SI link is selected, then broken, and the infected individual of the SI link is selected, and it switches to another infected individual (with probability I SI w SI 2 ); II links would decrease by one if an II link is selected, then broken, and the selected infected individual switches to a susceptible individual (with probability S II w II ). When the epidemic spreading happens (with probability α), for the recovery event, an infected individual is selected (with probability I ), and it recovers with probability μ t. If the selected I individual has q (0 q k) infected neighbors [with probability k q ( II I ) q (1 − II I ) k−q ], the change of II links is −q. For the infection event, a susceptible individual is selected (with probability S ) if it has h (0 h k) infected neighbors [with probability k h ( SI S ) h (1 − SI S ) k−h ]; the infection happens with probability hλ t, and the change of II links in this case is h. Taking into account the formula of the expectation and the variance of the binomial distribution yieldṡ

With similar arguments we havė

Finally we obtain the equations of moments with closed forms, i.e., Eqs. (A2)-(A4). This method has been used in both evolutionary game theory [42] and epidemic dynamics [10, 36] before. These equations can be employed to investigate the coupled dynamics of links and epidemics for any time scales.

Furthermore, the dynamics of SI and I I can help us figure out the condition under which the time scale separation is valid. The time scale separation requires that the evolution of links is mainly determined by the link-rewiring process. It implies that 2α(k−1)λ 

This necessary condition is a more precise criterion compared with α 1 to ensure the time scale separation. It suggests that the condition for the time scale separation would be more demanding with the increasing of the average degree k. This also supports our assumption in the main text that k should be much smaller than N .

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Stochastic Interacting Systems: Contact, Voter and Exclusion Processes

On the quasi-stationary distribution of the stochastic logistic epidemic

Evolutionary stability and quasistationary strategy in stochastic evolutionary game dynamics

A simple rule for the evolution of cooperation on graphs and social networks

We thank the referees for their helpful comments. Discussions with Professor Ming Tang are greatly acknowledged. D.Z. is grateful for funding by the National Natural Science 

Here we give a rigorous dynamical analysis of Eq. (6) , based on which the main results in Sec. III are obtained. Rewriting Eq. (6) leads towhere the cubic polynomial f (i) is given byThe asymptotic properties of Eq. (6) are totally determined byIf λ e w SS − w SI 0, i = 0 is the only stable fixed point; the infection will finally die out. If λ e w SS − w SI > 0, i = 0 is an unstable fixed point, andbecomes the only stable fixed point, corresponding to endemic infection.It is shown that there exists a phase transition at λ e = w SI /w SS , which is quite similar to the conventional SIS model in which the critical point is located at λ = 1.When 2w SS w II − w SI w II − w SS w SI = 0, it is possible for the model to give rise to bistability. Letting w = w SI /w SS , we have the following: ). (b) If w SI = w II , bistable ⇔ (λ e , 1 w ) ∈ (2,+∞)×( 4 4+λ 2 e , 1 λ e ). To show how we get the above results, we take the case w SS = w II as an example. In this case,, and its discriminant is denoted as = (2w − (2 + λ e )) 2 − 4(1 − w)(λ e − w); then the sufficient and necessary condition for bistability is given byBy solving the above set of inequalities, we obtain that (λ e ,w) ∈ (2,+∞) × (λ e , λ e + √ λ 2 e +(λ e −2) 2 2). Similarly, we get the result for the case w SI = w II .