In contrast to the TIB model, instead of assuming independence of vertices we can approximate the marginal probabilities in terms of combinations of lower order marginals using some form of moment closure (Frasca and Sharkey
2016; Sharkey and Wilkinson
2015). Here, we make an equivalent assumption to that of the message passing approaches (Karrer and Newman
2010; Shrestha et al.
2015). We assume the network contains no time-respecting non-backtracking cycles. In other words, starting at some initial vertex
i that leaves via vertex
j, there is no way to find a time-respecting path returning to this vertex that does not return via
j. This is equivalent to a tree network when the temporal network is viewed in its static embedding of the supra-adjacency representation (Bianconi
2018). This allows us to write all higher order moments in Eq. (
5a) as a combination of pairs
\(\langle S_i^n I_k^n\rangle\). To show why this is possible, consider the three vertices
i,
j,
k connected by two edges through
i. If conditional independence of these vertices is assumed given we have the state of
i, then one can make the following assumption,
$$\begin{aligned} \langle X_i^nX_j^nX_k^n\rangle = \langle X_j^nX_k^n|X_i^n\rangle \langle X_i^n\rangle = \frac{\langle X_i^nX_j^n\rangle \langle X_i^nX_k^n\rangle }{\langle X_i^n\rangle }. \end{aligned}$$
(9)
This has the effect of assuming the network is tree-like in structure as it implies any interaction between vertices
j and
k must occur through vertex
i. Thus, the process is exact on networks that contain no time-respecting non-backtracking cycles and otherwise provides an improved approximation of varying degree, which depends on the true network structure. The result obtained in Eq. (
9) is often referred to as the Kirkwood closure (Kirkwood
1935). Under the assumption that the network is tree like, the following simplification is obtained for Eq. (
5a),
$$\begin{aligned} \langle I_i^{n+1}|S^{n}_i \rangle =1 - \prod _{k\in V}\left( 1 - \beta A^{[n]}_{ki}\frac{ \langle S_i^nI_k^n\rangle }{\langle S_i^n\rangle }\right) . \end{aligned}$$
(10)
However, we run into the problem that we have no description for pairs of vertices. Thus, we derive expressions for their evolution from the RMEs for pairs of vertices which is given by,
$$\begin{aligned} \langle X_i^{n+1}X_j^{n+1} \rangle = \sum _{(X_{i}^n,X_{j}^n)\in \Omega ^2}\langle X_i^{n+1}X_j^{n+1}|X_i^nX_j^n \rangle \langle X_i^nX_j^n \rangle , \end{aligned}$$
(11)
For
\(\langle S_i^{n+1}I_j^{n+1} \rangle\), we obtain
$$\begin{aligned} \langle S_i^{n+1}I_j^{n+1} \rangle&= \langle S_i^{n}I_j^{n}\rangle + \langle S_i^{n+1}I_j^{n+1}|S_i^nS_j^n \rangle \langle S_i^nS_j^n \rangle \\&\quad -\, \langle I_i^{n+1}I_j^{n+1}|S_i^nI_j^n \rangle \langle S_i^nI_j^n \rangle \\&\quad -\, \langle S_i^{n+1}R_j^{n+1}|S_i^nI_j^n \rangle \langle S_i^nI_j^n \rangle \\&\quad -\, \langle I_i^{n+1}R_j^{n+1}|S_i^nI_j^n \rangle \langle S_i^nI_j^n \rangle . \end{aligned}$$
(12)
Note that the RME for
\(\langle S_i^{n}S_j^{n} \rangle\) is also required, which we find to be the following
$$\begin{aligned} \langle S_i^{n+1}S_j^{n+1}\rangle&= \langle S_i^nS_j^n\rangle -\langle S_i^{n+1}I_j^{n+1}|S_i^nS_j^n \rangle \langle S_i^nS_j^n\rangle \\&\quad -\,\langle I_i^{n+1}I_j^{n+1}|S_i^nS_j^n \rangle \langle S_i^nS_j^n\rangle \\&\quad -\,\langle I_i^{n+1}I_j^{n+1}|S_i^nS_j^n \rangle \langle S_i^nS_j^n\rangle . \end{aligned}$$
(13)
Since only the probabilities
\(\langle S_i^nI_j^n \rangle\) and
\(\langle S_i^nS_j^n \rangle\) are needed in order to describe the RMEs in Eq. (
10), we consider those two combinations of states. From Frasca and Sharkey (
2016), we obtain the exact transition rates for pairs of vertices and find that we can factorise the pair-wise transition rates similar to Eq. (
5a). Here, we give the expression for
\(\langle S_i^{n+1}I_j^{n+1}|S_i^nS_j^n \rangle\) only while the rest of the pair-wise transition rates are given in “
Appendix”:
$$\begin{aligned}{}&\langle S_i^{n+1}I_j^{n+1}|S_i^nS_j^n \rangle \nonumber \\&\quad = \frac{1}{\langle S_i^nS_j^n\rangle }\left[ \beta \sum _{k_1\in V}A_{ik_1}^{[n]} \langle S_i^n S_j^nI_{k_1}^n\rangle \right.- \beta ^2\sum _{k_1, k_2\in V} A_{ik_1}^{[n]}A_{ik_2}^{[n]}\langle S_i^nS_j^n I_{k_1}^n I_{k_2}^n\rangle \nonumber \\&\qquad +\dots \nonumber \left. - (-\beta )^{N-2}{\sum _{k_1,\dots ,k_{N-2}\in V}} A_{ik_1}^{[n]} \dots A_{ik_{N -2}}^{[n]}\langle S_i^n S_{j}^n \dots I_{k_{N-2}}^n\rangle \right] \nonumber \\&\qquad \times \left[ 1-\beta \sum _{k_1\in V}A_{ik_1}^{[n]} \langle S_i^n S_j^nI_{k_1}^n\rangle \right. + \beta ^2\sum _{k_1, k_2\in V} A_{ik_1}^{[n]}A_{ik_2}^{[n]}\langle S_i^nS_j^n I_{k_1}^n I_{k_2}^n\rangle \nonumber \\&\qquad -\dots \left. + (-\beta )^{N-2}{\sum _{k_1,\dots ,k_{N-2}\in V}} A_{ik_1}^{[n]}\dots A_{ik_{N -2}}^{[n]}\langle S_i^n S_{j}^n \dots I_{k_{N-2}}^n\rangle \right] . \end{aligned}$$
(14)
In the above equation, the term in the first pair of square brackets corresponds to the probability that vertex
i does not become infected and the term in the second pair of square brackets corresponds to the probability that vertex
j becomes infected. Upon applying our moment closure technique Eq. (
14) may be written as
$$\begin{aligned} \langle S_i^{n+1}&I_j^{n+1}|S_i^nS_j^n \rangle \nonumber \\&=\prod _{\begin{array}{c} k\in V\\ k\ne j \end{array}} \left( 1- \beta A^{[n]}_{ki}\frac{\langle S_i^nI_k^n \rangle }{\langle S_i^n\rangle }\right) \left[ 1 - \prod _{\begin{array}{c} k\in V\\ k\ne i \end{array} }\left( 1- \beta A^{[n]}_{kj}\frac{\langle S_j^nI_k^n \rangle }{\langle S_j^n\rangle }\right) \right] . \end{aligned}$$
(15)
By introducing the following functions, the RMEs for pairs as well as the individual vertices can be written more concisely. The probability that vertex
i does not become infected at time step
\(n+1\), given that
i is not infected at time step
n is denoted by
$$\begin{aligned} \Psi _i^n=\prod _{k\in V}\left( 1 - \beta A^{[n]}_{ki}\frac{ \langle S_i^nI_k^n\rangle }{\langle S_i^n\rangle }\right) . \end{aligned}$$
(16)
Similarly, the probability that vertex
i does not become infected at time step
\(n+1\), given that
i is not infected at time step
n while excluding any interaction with
j, is given by
$$\begin{aligned} \Phi _{ij}^n=\prod _{\begin{array}{c} k\in V\\ k\ne j \end{array}}\left( 1 - \beta A^{[n]}_{ki}\frac{\langle S_i^nI_k^n\rangle }{\langle S_i^n\rangle }\right) . \end{aligned}$$
(17)
Then, the evolution of the state of every vertex in the network is determined by the following closed set of equations,
$$\begin{aligned} \langle S_i^{n+1} \rangle&= \Psi _i^n\langle S_i^n \rangle \end{aligned}$$
(18a)
$$\begin{aligned} \langle I_i^{n+1}\rangle&= (1- \mu )\langle I_i^n\rangle + [1 - \Psi _i^n]\langle S_i^n\rangle \end{aligned}$$
(18b)
$$\begin{aligned} \langle S_i^{n+1}I_j^{n+1}\rangle&= (1-\mu ) (1-\beta A_{ji})\Phi _{ij}^n\langle S_i^nI_j^n\rangle \nonumber \\&\quad +\, \Phi _{ij}^n\left( 1-\Phi _{ji}^n\right) \langle S_i^nS_j^n\rangle \end{aligned}$$
(18c)
$$\begin{aligned} \langle S_i^{n+1}S_j^{n+1}\rangle&=\Phi _{ij}^n\Phi _{ji}^n \langle S_i^nS_j^n\rangle . \end{aligned}$$
(18d)