A framework for reconstructing transmission networks in infectious diseases

Abstract

In this paper, we propose a general framework for the reconstruction of the underlying cross-regional transmission network contributing to the spread of an infectious disease. We employ an autoregressive model that allows to decompose the mean number of infections into three components that describe: intra-locality infections, inter-locality infections, and infections from other sources such as travelers arriving to a country from abroad. This model is commonly used in the identification of spatiotemporal patterns in seasonal infectious diseases and thus in forecasting infection counts. However, our contribution lies in identifying the inter-locality term as a time-evolving network, and rather than using the model for forecasting, we focus on the network properties without any assumption on seasonality or recurrence of the disease. The topology of the network is then studied to get insight into the disease dynamics. Building on this, and particularly on the centrality of the nodes of the identified network, a strategy for intervention and disease control is devised.

Keywords: Autoregressive model; Betweenness centrality; COVID-19; Network reconstruction; Optimal control.

Fig. 1

An example of modular network along with its detected communities (encircled) is shown. The nodes are color-coded based on their memberships to these communities

Fig. 2

Example of a network with $C=1/3$ and $l=8/6$ is shown. It has three triplets, one of which is closed (triangle). The lengths of the paths from $j_2$ to all others is 1, while that from $j_1$ to $j_4$ is 2

Fig. 3

Examples of random, scale-free, and ordered (lattice-like) networks are shown respectively

Fig. 4

Data and fit under the model of Eq. (1) with negative binomial distributed counts. The three colors show the decomposition of the fitted aggregate counts into travel, intra-locality, and inter-locality contributions to infections amounting to $3\%$, $10\%$, and $87\%$ respectively

Fig. 5

Time evolution of the inter-locality parameter ${\varphi }_{\mathit{it}}$ is shown for the regions with the highest centrality (see Sect. 3)

Fig. 6

The network $A^{(15)}(t=300)$ is overlaid on the map of Lebanon to illustrate its complexity. It is evaluated at the 300-th day of the pandemic using all 15 time interval of our collected data, that is the time span $[0,20·15]$

Fig. 7

The modularity of the network $A^{(n)(t)}$ is shown as a function of time $t=20·n$, with $n=1\dots 41$. It is measure of the quality of the division of a graph into subgraphs

Fig. 8

The figure shows the clustering coefficient and average path length for $A^{(41)}(t)$, which are the matrices estimated using the data from the start of the pandemic to the 820-th day ($41·20$) evaluated at $t=20·n$, where n is the index of the time intervals

Fig. 9

Empirical and estimated distributions of the strength for the matrices $A^{(n)}(t=20·n)$ with $n=1,\dots ,20$ are shown

Fig. 10

Empirical and estimated distributions of the strength for the matrices $A^{(n)}(t=20·n)$ with $n=21,\dots ,35$ are shown

Fig. 11

Estimated values $\widehat{\alpha }$ of the exponent of the power-law distributions of the strengths of the nodes of the 41 matrices $A^{(n)}(t=20·n)$, $n=1,\dots ,41$, which are represented in Fig. 9. Vertical bars indicate $\pm 2{\widehat{\sigma }}_{\widehat{\alpha }}$

Fig. 12

The figure illustrates the iterative scheme. First, the node with the highest centrality $j_2$ is disconnected by removing all its links, as it is the node with the highest number of shortest paths. The resulting network has $j_1$ and $j_3$ with equal centralities and either one can be disconnected. This leads to a total loss of connectivity in the network at the end of the process

Fig. 13

The loss of connectivity versus the fraction of removed nodes for the cascading and non-cascading strategies

Fig. 14

The map shows the localities with the highest centrality whose removals lead to $80\%$ loss of connectivity

Fig. 15

The figure shows the counts, along with the fitted model, for twelve localities ordered by decreasing betweenness centrality