Explainability of the COVID-19 epidemiological model with nonnegative tensor factorization

Abstract

The world is witnessing the devastating effects of the COVID-19 pandemic. Each country responded to contain the spread of the virus in the early stages through diverse response measures. Interpreting these responses and their patterns globally is essential to inform future responses to COVID-19 variants and future pandemics. A stochastic epidemiological model (SEM) is a well-established mathematical tool that helps to analyse the spread of infectious diseases through communities and the effects of various response measures. However, interpreting the outcome of these models is complex and often requires manual effort. In this paper, we propose a novel method to provide the explainability of an epidemiological model. We represent the output of SEM as a tensor model. We then apply nonnegative tensor factorization (NTF) to identify patterns of global response behaviours of countries and cluster the countries based on these patterns. We interpret the patterns and clusters to understand the global response behaviour of countries in the early stages of the pandemic. Our experimental results demonstrate the advantage of clustering using NTF and provide useful insights into the characteristics of country clusters.

Keywords: COVID-19; Clustering; Explainable AI; Nonnegative tensor factorization; Pattern mining; Stochastic epidemiological modelling.

Fig. 1

Overall architecture of the SEM with multiple components [34]. A feedback mechanism also incorporates the community influence in the process

Fig. 2

Example of traditional consolidated matrix representation. Here multiple matrices representing country $\times$ parameter (i.e. $\mathcal{C}\times \mathcal{P}$) over multiple particles are consolidated by element average to derive a single matrix $\mathbf{Y}$

Fig. 3

Example of tensor representation. Instead of consolidation, multiple matrices encoding country $\times$ parameter over multiple particles are represented as a three-dimensional tensor, where particle is the third dimension

Fig. 4

Overall process of NTF for clustering and interpretation. A 3-D tensor is factorized into 3 factor matrices $\mathbf{C},\mathbf{P}$, and $\mathbf{D}$ representing the latent features of countries, parameters, and particles, respectively, after learning dependencies among the modes. Each factor matrix can become input to clustering the countries and interpreting the parameter and particle patterns

Fig. 5

Determining the six clusters containing similar countries based on the factor matrix $\mathbf{C}$

Fig. 6

Interpretation of country clusters using the factor matrix $\mathbf{P}$ which represents the parameter patterns. The darker the colour in each column, the higher the value of the feature, indicating the characteristics of that column

Fig. 7

Countries grouped together in all three datasets. Each colour indicates the set of countries that travel together from T1 to T2 to T3. Some clusters are not included in the figure as they do not travel together with any other countries for all three datasets

Fig. 8

Characteristics of clusters w.r.t. parameters using NTF: a T1(March time-period); b T2 (April time-period); and c T3 (June time-period)

Fig. 9

Characteristics of clusters w.r.t. parameters using NMF: a T1(March time-period); b T2 (April time-period); and c T3 (June time-period)

Fig. 10

Characteristics of clusters w.r.t. particles: a T1 (March period); b T2 (April period); and c T3 (June period)

Fig. 11

Cluster to cluster similarity based on characteristics. Similarity score of 0.5 is set as threshold. For example, the connection between C1 of T1 to C3 of T2 means that C1 of T1 is similar to C3 of T2

Fig. 12

Cluster to cluster similarity based on characteristics. Similarity score of 0.8 is set as threshold