Clustering of Countries Based on the Associated Social Contact Patterns in Epidemiological Modelling

Abstract

Mathematical models have been used to understand the spread patterns of infectious diseases such as coronavirus disease 2019 (COVID-19). The transmission component of the models can be modelled in an age-dependent manner via introducing contact matrix for the population, which describes the contact rates between the age groups. Since social contact patterns vary from country to country, we can compare and group the countries using the corresponding contact matrices. In this paper, we present a framework for clustering countries based on their contact matrices with respect to an underlying epidemic model. Since the pipeline is generic and modular, we demonstrate its application in a COVID-19 model from Röst et al. (2020) which gives a hint about which countries can be compared in a pandemic situation, when only non-pharmaceutical interventions are available.

Appendices

Appendix 1: First Appendix

1.1 List of the European Countries

We considered the following European countries in the demonstration:

Albania, Armenia, Austria, Belarus, Belgium, Bosnia, Bulgaria, Croatia, Cyprus, Czechia, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Montenegro, Netherlands, North Macedonia, Poland, Portugal, Romania, Russia, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Ukraine, United Kingdom.

Appendix 2: Second Appendix

1.1 The Governing Equations of the Epidemic Model

The governing equations of the disease model described in Sect. 2.2 take the form

$$\displaystyle \begin{aligned} {S^i}'(t)={} & -\beta_0 \frac{S^i(t)}{N_i}\cdot\sum_{k=1}^{16} M^{(k,i)}\left[I_{p}^{k}(t) + \mathrm{inf}_a \sum_{j=1}^3 I_{a,j}^{k}(t) + \sum_{j=1}^3 I_{s,j}^{k}(t)\right] \\ {L_1^i}'(t)={} & \beta_0 \frac{S^i(t)}{N_i}\cdot\sum_{k=1}^{16} M^{(k,i)}\left[I_{p}^{k}(t) + \mathrm{inf}_a \sum_{j=1}^3 I_{a,j}^{k}(t) + \sum_{j=1}^3 I_{s,j}^{k}(t)\right] - 2 \alpha_l L^i_1(t) \\ {L_2^i}'(t)={} & 2 \alpha_l L_1^i(t) - 2\alpha_l L_2^i(t),\\ {I_a^i}'(t)={} & 2 \alpha_l L_{2}^{i}(t) - \alpha_{p} I_{p}^{i} (t)\\ {I_{a,1}^i}'(t)={} & p^{i} \alpha_{p} {I}_{p}^{i} (t) - 3 \gamma_{a} I_{a,1}^{i}(t)\\ {I_{a,2}^i}'(t)={} & 3\gamma_{a} I_{a,1}^{i}(t)- 3\gamma_{a} I_{a,2}^{i}(t)\\ {I_{a,3}^i}'(t)={} & 3\gamma_{a} I_{a,2}^{i}(t)- 3\gamma_{a} I_{a,3}^{i}(t)\\ {I_{s,1}^i}'(t)= {}& (1 - p^i) \alpha_{p} I_{p}^{i} - 3 \gamma_{s} I_{s,1}^i(t)\\ {I_{s,2}^i}'(t)= {} & 3 \gamma_{s} I_{s,1}^i(t) - 3 \gamma_{s} I_{s,2}^i(t)\\ {I_{s,3}^i}'(t)= {} & 3 \gamma_{s} I_{s,2}^i(t) - 3 \gamma_{s} I_{s,3}^i(t)\\ {I_h^i}'(t)={} & h^i (1 - \xi^i) 3 \gamma_{s} I_{s,3}^i(t) - \gamma_h I_h^i(t)\\ {I_c^i}'(t)={} & h^i \xi^i 3 \gamma_{s} I_{s,3}^i(t)-\gamma_c I_c^i(t)\\ {I_{\mathrm{cr}}^i}'(t)={} & (1 - \mu^{i}) \gamma_{c} I_{c}^{i} (t) -\gamma_{\mathrm{cr}} I_{\mathrm{cr}}^{i} (t)\\ {R^i}'(t)={} & 3 \gamma_{a} I_{a,3}^{i} (t) + (1- h^{i}) 3 \gamma_{s} I_{s,3}^{i} (t) + \gamma_{h} I_{h}^{i} (t) + \gamma_{\mathrm{cr}} I_{\mathrm{cr}}^{i} (t)\\ {D^i}'(t)={} & \mu^i \gamma_c I_c^i(t), {} \end{aligned} $$

(2)

where the index $i \in {1, . . . ,16}$ represents the corresponding age group. As it can be seen, the model contains age-dependent parameters (probabilities p, h, $\xi$, $\mu$, for which upper index shows the age group) and age-independent ones (fraction $\mathrm {inf}_a$ and transition parameters $\alpha$ and $\gamma _X$, where $X\in \{a,s,h,c,\mathrm {cr}\}$). Notation here is aligned with the parameter file located in the repository of the framework. Here $\mathrm {inf}_a$ denotes the relative infectiousness of $I_a$ compared to $I_s$, for more details about the other parameters and the methodology for parametrization, see [25].

1.2 Next-Generation Matrix

To calculate $\mathcal {R}_{0}$ for the previously mentioned epidemic model, we consider the infectious subsystem for

$$\displaystyle \begin{aligned} {L_{1}^{i}}(t), {L_{2}^{i}}(t), {I_{p}^{i}}(t), {I_{j}^{i}}(t),\end{aligned}$$

with $j\in \boldsymbol \{a, s\} \times \{1,2,3\}, i \in \{1,...,16\}$, thus

$$\displaystyle \begin{aligned} X(t) &= \begin{bmatrix} L_{1}^{i}(t) & L_{2}^{i}(t) & I_{p}^{i}(t) & I_{a, 1}^{i}(t) & I_{a, 2}^{i}(t) & I_{a, 3}^{i}(t) & I_{s, 1}^{i}(t) & I_{s, 2}^{i}(t)& I_{s, 3}^{i}(t) \end{bmatrix}^\top \end{aligned} $$

and linearization gives

$$\displaystyle \begin{aligned} {X^\prime}(t) = (\beta_0 \cdot T + \Sigma) \cdot X(t),\end{aligned}$$

where $T \in \mathbb {R}^{144 \times 144}$ is the transmission part and $\Sigma \in \mathbb {R}^{144 \times 144}$ represents the transition mechanisms in the model. The matrix $\Sigma$ is a block-diagonal matrix, where blocks have size of $9\times 9$ containing transition parameters related to the linear terms of the system. On the other hand, the transmission matrix T is partitioned into blocks of size $9\times 9$, and each of this blocks have nonzero elements only in their first rows, since transmission between individuals affects only the classes $L_1^i, i\in \{1,2,\dots ,16\}$, and these nonzero elements are related to the corresponding elements of the contact matrix and the transmission-related parameter $\mathrm {inf}_a$. The Next-Generation Matrix (shortly NGM) can be calculated as

$$\displaystyle \begin{aligned} \mathrm{NGM} = -\beta_0\cdot T \Sigma^{-1},\end{aligned}$$

and the basic reproduction number is the dominant eigenvalue of the NGM, i.e.,

$$\displaystyle \begin{aligned} \mathcal{R}_0 = \beta_0 \cdot \rho(-T \Sigma^{-1}).\end{aligned}$$

On one hand, the model parameters are the same for all countries in the paper; on the other hand, this calculation has to be executed for each countries separately, since social contact matrices (thus T matrices) are different. For more details about NGM method, see [5].