Individual-based approach to epidemic processes on arbitrary dynamic contact networks

The dynamics of contact networks and epidemics of infectious diseases often occur on comparable time scales. Ignoring one of these time scales may provide an incomplete understanding of the population dynamics of the infection process. We develop an individual-based approximation for the susceptible-infected-recovered epidemic model applicable to arbitrary dynamic networks. Our framework provides, at the individual-level, the probability flow over time associated with the infection dynamics. This computationally efficient framework discards the correlation between the states of different nodes, yet provides accurate results in approximating direct numerical simulations. It naturally captures the temporal heterogeneities and correlations of contact sequences, fundamental ingredients regulating the timing and size of an epidemic outbreak, and the number of secondary infections. The high accuracy of our approximation further allows us to detect the index individual of an epidemic outbreak in real-life network data.


Discretisation of the continuous-time IBA
The IBA to the SIR dynamics on static networks in continuous time has been known 1-3 . By translating it to dynamic contact networks, we obtain the following approximate deterministic dynamics in continuous time: where β and µ are the infection and recovery rates, respectively. We distinguish them from the infection probability β and the recovery probability µ in the main text because we assume discrete time in the IBA, whereas Equations (1), (2), and (3) assume continuous time.
We discretise equations (1), (2), and (3) with a time step ∆t to obtain The time discretisation is justified when the probabilities of state-transition events within ∆t are sufficiently small. In the present case, this is equivalent to saying β ∆t, µ ∆t 1. By assuming that ∆t is the duration of the single snapshot of temporal networks, we obtain β = β ∆t and µ = µ ∆t. Because our discrete-time approach is justified only when β, µ 1, the time discretisation of the continuous-time SIR model is consistent with the assumption justifying our discrete-time approach. * Electronic address: luis.rocha@ki.se † Electronic address: naoki.masuda@bristol.ac.uk To make an intuitive understanding and comparison with the IBA easier, we change from the continuous-time to discrete-time notation and replace t − ∆t by t − 1: If we expand equations (3) and (4) in the main text in terms of small parameters β and µ, and only retain the firstorder terms, we obtain the variants of equations (7) and (8), where N i (t − 1) in equations (7) and (8) is replaced by N i (t). This minor difference arose because, in equations (4) and (5), we used the snapshot network at t − ∆t to evolve the dynamics, whereas the snapshot network at t is used to evolve the dynamics in equations (3) and (4) in the main text.

The hospital dynamic contact network
We carried out the analyses presented in the main text to a third real-life dynamic contact network to assess the generalisability of the results. This data set corresponds to face-to-face human interaction between patients and healthcare workers in a hospital ward 4 . The network contains N = 75 nodes, E = 1, 139 unique pairs of contacts and C = 32, 424 temporal contacts distributed in T = 96.57 hours, giving t max = 17, 382. As we show in the following, the results obtained for the conference and museum data sets, used in the main text, qualitatively hold true for this data set as well. Figure S1(a,b) shows the evolution of the fraction of infected individuals, I(t) (i.e. the prevalence) given two different initial conditions (i.e. different seeds). We compare the estimation of the prevalence based on the IBA and the simulations on the original empirical network and its randomised version. In line with the results for the conference and museum data sets shown in the main text, we observe a reasonably good agreement between the IBA and S-DNO, including the reproduction of both peaks. On the other hand, the mismatch between the IBA and the S-SNR is evident, particularly for node 2, in which the peak prevalence is shifted to earlier times. This is a consequence of the uniformly distributed contacts after randomisation. By calculating the root-mean-square between the models Figure S1(c,d), we confirm the qualitative observation that IBA better captures the original contact sequence.  Figure 2 shows the final outbreak size calculated using the IBA approximation. We observe that results are similar for both simulation methods S-DNO (Fig. S2(b)) and S-DNR (Fig. S2(c)). This result indicates that for this particular dataset, randomization of the time-stamps yields little difference in the final outbreak. As we can see in Fig. S1(a,b), though with different shapes, both simulation methods may infect approximatelly the same number of nodes because a higher infection in the first wave balances a relatively smaller infection in the second wave.

Individual reproduction number
The IBA is also accurate in approximating the S-DNO in terms of the individual reproduction number (Fig. S3(a,b)). The agreement is stronger between the IBA and S-DNO than between the IBA and S-DNR, as observed in the other data sets. The correlation coefficients are larger for the hospi-  tal data set than for the conference and museum data sets presented in the main text.

Effective reproduction number
We calculate the effective reproduction number for the IBA (Fig. S4(a)), S-DNO and S-DNR models. By calcu-lation of the root-mean-square, we observe that the results agree between the IBA and S-DNO ( Fig. S4(b)) and between IBA and S-DNR (Fig. S4(c)) in the full range of parameters studied. This good agreement is possibly a result of the range of parameters studied. More significant differences show up for smaller values of µ, as seen on the top right part of Fig. S4(c).

Source detection
We perform the source-detection experiments using the hospital data set. Similarly to results for the conference and museum data sets, the IBA efficiently detects the source of infection given the states of individuals at time t max and the past contact patterns (Fig. S5).