Social clustering in epidemic spread on coevolving networks

Even though transitivity is a central structural feature of social networks, its influence on epidemic spread on coevolving networks has remained relatively unexplored. Here we introduce and study an adaptive susceptible-infected-susceptible (SIS) epidemic model wherein the infection and network coevolve with nontrivial probability to close triangles during edge rewiring, leading to substantial reinforcement of network transitivity. This model provides an opportunity to study the role of transitivity in altering the SIS dynamics on a coevolving network. Using numerical simulations and approximate master equations (AMEs), we identify and examine a rich set of dynamical features in the model. In many cases, AMEs including transitivity reinforcement provide accurate predictions of stationary-state disease prevalence and network degree distributions. Furthermore, for some parameter settings, the AMEs accurately trace the temporal evolution of the system. We show that higher transitivity reinforcement in the model leads to lower levels of infective individuals in the population, when closing a triangle is the dominant rewiring mechanism. These methods and results may be useful in developing ideas and modeling strategies for controlling SIS-type epidemics.

Figure 1

The level of transitivity varies with time $t$ and the probability $\eta$ of rewiring steps made to susceptible neighbors of neighbors. The (a) average local clustering coefficient $C$ and (c) transitivity $\mathcal{T}$ are shown versus time $t$ for simulations with initially Poisson degree distribution with mean degree $\langle k\rangle =2$. The other parameters of the system are $\beta =0.04, \gamma =0.04, \alpha =0.005$, and $\epsilon =0.1$. Also shown are (b) $C$ and (d) $\mathcal{T}$ at $t=5000$ versus $\eta$ for networks with the same mean degree $\langle k\rangle =2$ but different initial degree distributions $[p_k^{\text{P}}$ (Poisson), triangle; $p_k^{\text{TPL}}$ (truncated power law), circle; and $p_k^{\text{DR}}$ (degree regular), square]. Plotted data correspond to mean values computed over 30 Monte Carlo simulations. In (a) and (c), all cases except $\eta =1$ appear to reach stationarity by time $t=5000$.

Figure 2

Local clustering coefficients $C$ grouped by susceptible nodes $S$ and infected nodes $I$ at time $t=5000$ for (a) $\eta =0.2$, (b) $\eta =0.4$, (c) $\eta =0.8$, and (d) $\eta =1$. The initial degree distribution is Poisson. The other parameters of the system are $\beta =0.04, \gamma =0.04, \alpha =0.005, \epsilon =0.1$, and $\langle k\rangle =2$. At time $t=5000$, all cases except for $\eta =1$ reach their stationary states. Each box plot is obtained from a single Monte Carlo simulation of network size $N=25000$. The notch of the box indicates the median (mostly zero), the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively, whiskers extend to the most extreme data points not considered as outliers, and the outliers are plotted individually (the +'s).

Figure 3

Phase diagrams in parameter space $(\gamma ,\eta )$ for the observed disease prevalence $I$ at time $t=10000$ on networks with initial degree distributions that are (a) Poisson, (b) truncated power law, and (c) degree regular. We set here $\beta =0.04, \alpha =0.005$, and $\epsilon =0.1$. At the selected time $t=10000$ all cases except $\eta =1$ appear to have converged to their stationary states. Results are averaged over 30 simulations at each set of parameters. These diagrams are generated through bilinear interpolation from results on a regular grid, leading to some apparent grid artifacts.

Figure 4

(a) Number of components $N_s$ and (b) largest component fraction $s_1$ versus $\eta$ at $t=5000$. The initial degree distributions of the networks are Poisson (P), truncated power law (TPL), and degree regular (DR) with mean degree $\langle k\rangle =2$. The other parameters in the system are $\gamma =0.04, \beta =0.04, \alpha =0.005$, and $\epsilon =0.1$. Every point in the figure corresponds to the mean value computed over 30 simulations. In these simulations, the mean number of components at $t=0$ for Poisson, truncated power law, and degree regular distributions are 666.17, 3348.27, and 6.03, respectively, and the mean largest component fraction at $t=0$ for Poisson, truncated power law, and degree regular are 0.7971, 0.6828, and 0.7452, respectively. All cases except $\eta =1$ reach stationarity by time $t=5000$.

Figure 5

Bifurcation diagrams of the disease prevalence in the stationary state $I_{\infty }$ versus $\beta$ at different values of $\eta$. Each row of plots corresponds to a specific degree distribution: The first row is Poisson, the second row is truncated power law, and the third row is degree regular. Each column of plots corresponds to a particular value of $\eta$, as indicated at the top of the column. Every point represents a mean of 30 realizations of the system at $t=10000$. The last column belongs to the $\eta =1$ case, noting that the $y$ axis is labeled $I$ (cf. $I_{\infty }$) because the simulations have not reached stationarity. The other parameters of the system are $\gamma =0.02$ and $\alpha =0.005$. (We also study the effect of different $\epsilon$ in initially Poisson networks; see the Appendix pp1.)

Figure 6

Illustration of an $S$ node rewiring to a neighbor's neighbor. (a) Before the rewiring, the center nodes in the diagram are of class $S_{kl}$ and $I_{kl}$, respectively. Consider the two $S$ nodes involved when an $SI$ edge is rewired to a distance-2 neighbor: the active node doing the rewiring and the passive node receiving the newly rewired edge. (b) Diagram representing the case when the center is passively rewired by the action of a distance-2 neighbor, with the center node moving to class $S_{(k+1)l}$. (c) Diagram representing the case when the center actively rewires to a distance-2 neighbor and the center node becomes class $S_{k(l-1)}$. The other processes of the adaptive SIS system can be similarly diagramed.

Figure 7

Disease prevalence $I$ at time $t$. The initial degree distributions are Poisson (diamonds) [at $t=0$; see Eq. (1)], truncated power law (squares) [at $t=0$; see Eq. (2)], and degree regular (circles) [at $t=0$; see Eq. (3)]. Markers correspond to means computed over 1000 simulations and lines are the semianalytical AME results. The parameters used are $\beta =0.04, \gamma =0.04, \alpha =0.005, \epsilon =0.1$, and $\langle k\rangle =2$, with (a) $\eta =0$, (b) $\eta =0.2$, (c) $\eta =0.4$, and (d) $\eta =0.6$.

Figure 8

(a) Disease prevalence $I$ at time $t$. The initial degree distribution is Poisson. Different values of $\eta$ are indicated by color. The other parameters of the system are $\beta =0.04, \gamma =0.04, \alpha =0.005, \epsilon =0.1$, and $\langle k\rangle =2$. Circles correspond to means computed over 1000 simulations. Lines are AME results. (b) Observe that for $\eta =1$ the simulations exhibit slow convergence and the AME fails to capture the correct timescale for this convergence. (c) Plot of the corresponding clustering coefficient $C$ and transitivity $\mathcal{T}$ at $t=5000$ for different $\eta$.

Figure 9

Scaling of disease prevalence $I$ with time when $\eta =1$. As the average degree $\langle k\rangle$ becomes greater than 4, the scaling exponent appears to decrease in magnitude, suggesting that this phenomenon may depend on the level of sparsity of the underlying network. The parameters here are $\beta =0.04, \gamma =0.04, \alpha =0.005, \epsilon =0.1$, and $N=25000$. Markers correspond to means computed over 30 simulations.

Figure 10

Disease prevalence in the stationary state $I_{\infty }$ versus $\eta$. The $I_{\infty }$'s were calculated at time $t=10000$. The initial degree distributions are Poisson (diamonds) [at $t=0$; see Eq. (1)], truncated power law (squares) [at $t=0$; see Eq. (2)], and degree regular (circles) [at $t=0$; see Eq. (3)]. Markers correspond to means computed over 30 simulations and the lines are AME results. The other parameters are $\beta =0.04, \gamma =0.04, \alpha =0.005, \epsilon =0.1$, and $\langle k\rangle =2$.

Figure 11

Joint distributions of node states ($S$ and $I$) and degrees at stationarity. The initial degree distributions are (a) Poisson, (b) truncated power law, and (c) degree regular. Markers correspond to means computed over 200 simulations. Dashed and solid lines are the AME predictions for the $S$ and $I$ states, respectively. The parameters used are $\beta =0.04, \gamma =0.04, \alpha =0.005, \epsilon =0.1, \eta =0.4$, and $\langle k\rangle =2$.

Figure 12

Degree distributions in stationary states. The initial degree distributions are (a) Poisson, (b) truncated power law, and (c) degree regular. Asterisks in each plot are the initial degree distributions. We plot results for $\eta =0$, 0.2, 0.4, 0.6, and 0.8, as indicated by the color bar, though in many cases the results at one value of $\eta$ are obscured by those at other values. The other parameters are $\beta =0.04, \gamma =0.04, \alpha =0.005, \epsilon =0.1$, and $\langle k\rangle =2$. We plot results at $t=5000$, with all systems at these $\eta$ reaching their stationary states by this time. (Note that we did not plot $\eta =1.0$ here.) Markers correspond to means computed over 200 simulations. Lines are the AME predictions for the stationary degree distributions. The thick gray line in the background is the reference Poisson distribution expected for a network with $\langle k\rangle =2$. Observe that for both the initial Poisson and truncated power law the stationary state degree distribution falls on this thick line, implying that the final degree in both these cases is Poisson (at least approximately).

Figure 13

Bifurcation diagrams of stationary disease prevalence level $I_{\infty }$ versus infection rate $\beta$ on adaptive networks with an initial Poisson degree distribution, with $\langle k\rangle =2$ for the top row, $\langle k\rangle =4$ for the middle row, and $\langle k\rangle =8$ for the bottom row. In each row, we plot the case when $\eta =0$, 0.2, 0.4, and 0.6. The other parameters in the system are $\gamma =0.02$ and $\alpha =0.005$. Solid lines are the predictions of our semianalytic method and the markers represent the average of 30 simulations. We ran these Monte Carlo simulations for each value $\epsilon =0.001$, 0.01, 0.05, 0.99, and 0.999. For each run, the initial transient was discarded and the prevalence at stationarity was averaged over at least $10000$ time steps.

Figure 14

Susceptibility $\chi$ versus infected rate $\beta$ in adaptive networks with Poisson initial degree distributions with different network size $N$ at time $t=10000$ for various values of $\eta$: (a) $\eta =0$, (b) $\eta =0.2$, (c) $\eta =0.4$, (d) $\eta =0.6$, (e) $\eta =0.8$, and (f) $\eta =1$. Except for the case $\eta =1$, all networks reached their stationary disease prevalence $I_{\infty }$. The other parameters of the system are $\gamma =0.02, \alpha =0.005, \epsilon =0.1$, and $\langle k\rangle =2$. Lines with different colors correspond to means computed over 30 Monte Carlo simulations of different network sizes.

Figure 15

Stationary disease prevalence level $I_{\infty }$ (as measured at stopping time $t=20000$) versus network size on adaptive networks with an initial Poisson degree distribution with (a) $\langle k\rangle =2$ and (b) $\langle k\rangle =4$. In (a) the infected and rewiring rates chosen are:$\beta =0.04$ and $\gamma =0.04$. In (b) the infected and rewiring rates chosen are $\beta =0.01$ and $\gamma =0.02$. The other parameters in the system are $\alpha =0.005$ and $\epsilon =0.1$. Circles (and error bars) represent the mean (and standard deviation) of outcomes from 30 Monte Carlo simulations. We vary $\eta$ from 0 to 1, recalling that $\eta =1$ does not reach stationarity by $t=20000$. We note that $I_{\infty }$ is consistent across network sizes in the region tested in this paper.

Figure 16

(a) Phase diagram in parameter space $(\beta ,\eta )$ for the observed disease prevalence $I$ at time $t=10000$ on the networks with Poisson initial degree distributions. (b) Derivative of disease prevalence with respect to $\beta$, i.e., $\frac{d{I}_{\infty }}{d\beta }$. The jump in $\frac{d{I}_{\infty }}{d\beta }$ with positive values implies a possible discontinuity of $\frac{d{I}_{\infty }}{d\beta }$. Except for the case $\eta =1$, all networks reached their stationary disease prevalence $I_{\infty }$. We here fix the parameters of the system as $\gamma =0.02, \alpha =0.005, \epsilon =0.1$, and $\langle k\rangle =2$. Each pixel corresponds to the mean computed over 30 Monte Carlo simulations.

Figure 17

Bifurcation diagrams of the disease prevalence $I$ versus $\eta$ on networks with initial degree distributions that are (a) Poisson, (b) truncated power law, and (c) degree regular. The parameters of the system are $\beta =0.04, \gamma =0.04, \alpha =0.005$, and $\langle k\rangle =2$. Lines are the AME predictions and symbols are the means from 30 simulations. We run simulations for each value $\epsilon =0.001$, 0.01, 0.2, 0.4, 0.6, 0.8, 0.99, and, 0.999 and plot them with different colors. We set $t=10000$, with all parameters with $\eta$ not close to 1 reaching their stationary states by this time.

Figure 18

Disease prevalence $I$ at time $t$ with different cutoff degrees $k_c$ in networks with a power law degree distribution at the start of the simulation [see Eq. (2)]. Markers correspond to means with standard deviations computed over 30 simulations. The parameter used are $\gamma =0.04, \beta =0.04, \alpha =0.005, \eta =0.2$, and $\epsilon =0.1$. Different cutoff degrees in the truncated power law distribution have different levels of disease prevalence temporarily; however, as $t$ increases, they lead to the same level of disease prevalence.

Figure 19

Bifurcation diagrams of the disease prevalence in the stationary state $I_{\infty }$ versus $\beta$ at various values of $\eta$ and $\epsilon$. Each column of plots corresponds to a particular value of $\eta$ and each row of plots corresponds to a particular value of $\epsilon$. As in Fig. 5, every point represents a mean of 30 realizations of the system at $t=10000$. The last column belongs to the $\eta =1$ case, noting that the $y$ axis is labeled $I$ (cf. $I_{\infty }$) because the simulations have not reached stationarity. The networks have a Poisson degree distribution initially. The other parameters of the system are $\gamma =0.02$ and $\alpha =0.005$.

Figure 20

Early evolution of the disease prevalence $I$ in adaptive networks with Poisson, truncated power law, and degree regular initial degree distributions. The other parameters of the system are $\beta =0.04, \gamma =0.04, \alpha =0.005, \epsilon =0.1$, and $\langle k\rangle =2$. Different values of $\eta$ are chosen to be (a) $\eta =0$, (b) $\eta =0.2$, (c) $\eta =0.4$, and (d) $\eta =0.6$. Symbols and error bars correspond to means computed over 1000 Monte Carlo simulations and the solid lines are the AME predictions.

Figure 21

Time evolution of the disease prevalence $I$, the fraction of SI links $S_I$, the effective branching factor ${\kappa }_{IS}^S$, and the average number of connections that susceptible nodes share with other susceptible nodes $C_{SS}$, for (a)–(d) $\eta =0$ and (e)–(h) $\eta =0.4$. Diamond markers correspond to means computed over 200 Monte Carlo simulations (error bars would be smaller than the markers here) and the solid lines are the AME predictions. The initial degree distribution is Poisson and other parameters of the system are $\beta =0.04, \gamma =0.04, \alpha =0.005, \epsilon =0.1$, and $\langle k\rangle =2$.