1. Introduction

2. Background

3. Methodology

3.3. Micro-Level Contact Modeling

This section introduces various micro-level contact models. Initially, we recapitulate naive approaches from our prior work [8], before advancing to innovative contact modeling techniques, which leverage HMMs and Bayesian optimization.

While aggregated mobility data may be available, detailed micro-level movements are typically not available, leaving a gap in accurately modeling the nuanced interaction patterns that influence infection dynamics. By bridging this gap, micro-level models can enhance the precision of infection forecasting. In the context of this study, micro-level encounter modeling aims to replicate individual movement patterns within confined spaces accurately, ensuring the resultant temporal networks reflect infection dynamics and network properties of their real counterparts. All models discussed are designed to input arrival and departure times, producing temporal contact networks that, while maintaining consistent overall node counts, differ in the number and duration of edges due to varying movement patterns.

Real-world datasets used in this study offer actual temporal network as a baseline for validation, essential for the parameterization of HMMs. As already pointed out, we infer arrival and departure times in these cases based on a node i’s initial and final edge appearance, assuming that nodes are not leaving the location between contacts.

Surprisingly, a significant number of brief edge durations can be observed when nodes that have been isolated for longer periods, such as two hours, are completely removed from the network. Apparently, nodes spend a considerable time without any contact recorded, even in school environments. This is likely due to the face-to-face nature of the empirical networks. In [30], a similar experimental-technical infrastructure was employed as with for the empirical networks used in this study, but focusing on temporal networks at scientific conferences. The study found that contacts were rare during presentations, even in crowded rooms, because attendees generally do not face each other. This implies that in other environments, such as offices and high schools, close proximity alone does not guarantee that contacts are recorded by radio devices. To ensure our models accurately represent this aspect, we chose to model nodes continuously, from their first appearance to their last in the empirical network, without removal even if they show no contacts for prolonged periods.

3.3.1. Naive Micro-Level Encounter Models

Baseline approach ($\mathit{BASE}$): Our baseline approach delivers the most simplistic and intuitive way to build contact networks from the arrival and departure times of individuals at certain locations. A similar approach was described by [11]. In essence, this method leverages mobility data and individual-specific time allocations at specific locations to compute intersecting time frames between individuals, subsequently constructing contact networks. Individuals present at the same location are linked by edges in a contact network. Transforming this concept into a temporal dynamic network, we establish edges connecting pairs of individuals who coincide at a given point in time within the same location (see Section 2.3). Under this premise, our approach assumes an equal likelihood of infection for any pair of individuals who share the same duration of stay at a location. In other words $\mathit{BASE}$ constructs a fully connected network between all nodes active at time t. Additionally the contact intensity w influences the weight of an edge. This parameter w is assumed to be constant for all edges and is determined by the type of location, e.g. locations with a lot of social interactions like kindergartens or cafes are assumed to have a higher w value than libraries or supermarkets. A noticeable difference between our approach and the approach suggested by [11] is that in our case nodes are always connected to all other active nodes, while in their case the number of contacts was capped at 20. While in big locations both approaches will generate very different networks, they are identical for small locations where the number of nodes stays below 20 for the majority of the time. This simplified framework forms the foundation of our exploration, serving as a reference point against which we compare our more intricate modeling techniques.

Random graph-based approach ($\mathit{RAND}$): In our random graph-based approach, similar to [9], every possible edge, meaning that node $i,j$ are present at the location at time t, is selected with probability $p_{RAND}$. Additionally, a contact duration is drawn from a Pareto distribution $\mathcal{P}(t_{cd,min}=1,{\alpha }_{cd})$. Contacts, therefore, have a minimum duration of one-time step and follow a power law determined by the shape parameter ${\alpha }_{cd}$. This distribution accounts for the variable nature of interaction durations, resulting in a dynamic and realistic representation of human encounters when the recurrence of contacts is completely random. A possible application would be in locations where interactions form mainly due to uncorrelated movement instead of social relations, like in supermarkets, where the case of two individuals being nearby for the entire shopping trip is rather unlikely, however frequent but short contacts are to be expected.

Clique-based approach ($\mathit{CLI})$: We advance the clique-based strategy of [12] for the purpose of micro-level encounter modeling. To construct cliques, we utilize a combination of spatial dynamics and contact patterns. First, individuals are assigned to sub-spaces within the location, with the parameterizable $Npps$ mirroring the number of individuals per space. Whenever a node is introduced into the location , it stays in its space and forms connections to all nodes present in this space, forming tight cliques. To allow contacts between cliques at every time step, a node changes its space with probability $p_{clique}$ for a duration that is drawn from a normal distribution $\mathcal{N}(\mu ,\sigma )$. Afterward, the node returns to its default space.

Clique-based approach with random substructure ($\mathit{CLI}+\mathit{RAND})$: Cliques from $\mathit{CLI})$ can be imagined as classrooms, offices, or apartments in residential buildings. Large values for $Npps$ will generate a high number of contacts, e.g. a classroom with 30 students already generates 435 edges, at every time frame, due to their fully connected nature. To lower the density of the clique networks and allow for edge changes we add our $\mathit{RAND}$ approach to sit on top of the clique structure. Contacts within cliques are now randomly sampled according to the procedure explained in $\mathit{RAND}$. This leads to the formation of cliques with an adjustable density, where individuals have pronounced edges connecting them within the clique, reflecting intensive interactions. In contrast, connections outside the clique are rare, mirroring more sporadic or distant interactions. The underlying idea of this approach is to encapsulate the nuanced interplay between spatial arrangements and interpersonal encounters.

3.3.2. Human Mobility-based Micro-Level Encounter Models

To build temporal contact networks from HMMs, we used an open source implementation of $\mathit{RWP}$ and $\mathit{TLW}$2 that follows the model description provided in Section 2. The $\mathit{STEPS}$ model was integrated with $\mathit{RWP}$ to form the combined $\mathit{STEPS}+\mathit{RWP}$ model, as detailed in Section 2.2. Since all models need a confined area for nodes to walk in, we build synthetic locations according to our empirical networks. We therefore infer the capacity C of each location. For the primary school, the high school the office networks, we assumed the capacity of the location to be equal to the number of participants in the experiment. The capacity for the supermarket network is defined as the peak number of active nodes across all time steps, which results in a capacity of 44. To determine the area based on a location’s capacity, we utilized values for location-dependent space per person from [12], denoted as $\rho$. We developed a quadratic surface based on these values, calculating the area A as $A=C\times \rho$. For the $\mathit{STEPS}$-based models, we additionally sub-structured the area into $⌈\sqrt{N_V/Npps}⌉$ sub-spaces in horizontal and vertical direction, where $N_V$ denotes the number of nodes. This ensures a uniform distribution of sub-spaces across the entire area, resulting in some sub-spaces potentially containing fewer nodes than the default $Npps$.

In all HMM-based approaches, we continuously track node movements during the simulation. A contact between two nodes i and j is generated if both contact conditions

$$\mathrm{Cond}.1:0\le d({\overrightarrow{r}}_i,{\overrightarrow{r}}_j)\le d_{\max },$$

(2)

$$\mathrm{Cond}.2\mathrm{a}:arccos\left(\frac{{\overrightarrow{v}}_i·{\overrightarrow{r}}_{ij}}{\parallel {\overrightarrow{v}}_i\parallel ·\parallel {\overrightarrow{r}}_{ij}\parallel }\right)\le \frac{{\phi }^{*}}{2},$$

(3)

$$\mathrm{Cond}.2\mathrm{b}:arccos\left(\frac{{\overrightarrow{v}}_j·{\overrightarrow{r}}_{ji}}{\parallel {\overrightarrow{v}}_j\parallel ·\parallel {\overrightarrow{r}}_{ji}\parallel }\right)\le \frac{{\phi }^{*}}{2},$$

(4)

are fulfilled. $Cond.1$ ensures that for nodes positioned at $\overrightarrow{r_i}$ and $\overrightarrow{r_j}$, the distance between them must be smaller than a specified maximum distance $d_{max}$ to result in an edge. To calculate distances between all nodes at every time step, we use the well-known kd-tree algorithm, utilizing standard Euclidean distance. Secondly, we want to determine whether the line of sights of both nodes align or not, accounting for the experimental design outlined in Section 3.2.1. Therefore, we define their line of sight vectors $\overrightarrow{v_i}$, $\overrightarrow{v_j}$ which are always parallel to the latest movement. For all nodes that pass condition $Cond.1$ the angles between their line of sights and the connecting vector $\overrightarrow{r_{ij}}=\overrightarrow{r_j}-\overrightarrow{r_i}$ are calculated. If those angles are smaller or equal to one-half of their field of view ${\phi }^{*}$, then conditions $Cond.2a$ and $Cond.2b$ are fulfilled, and a contact between nodes $i,j$ is generated. Following the conditions from the original studies outlined in Section 3.2, we set the field of view to 120 degrees, a typical value for the human binocular field of view, and the maximum contact distance to 1.5 meters.

Table 1. Parameters used for different locations and network properties.$\rho$ is the density [m${ }^2$/node], $\beta$ represents the SIR transmission rate. $N_V$ denotes the number of nodes and $N_E$ represents the number of edges in the original real-world network.

Table 1. Parameters used for different locations and network properties.$\rho$ is the density [m${ }^2$/node], $\beta$ represents the SIR transmission rate. $N_V$ denotes the number of nodes and $N_E$ represents the number of edges in the original real-world network.

Parameter	High-school	Primary-school	Office	Supermarket
$\rho$ [m${ }^2$/node]	2.0	2.0	10.0	10.0
$\beta$	0.007	0.0013	0.013	0.075
$N_V$	327	242	217	539
$N_E$	47300	60623	12162	6660

3.3.3. Bayesian Optimzation for Hyperparameter Selection

To perform hyperparameter optimization, we used the Optuna framework described in [31]. Optuna employs advanced Bayesian optimization algorithms to identify a set of parameters from a specified search space that minimizes a designated objective function. A comprehensive table detailing all tuned parameters and their respective ranges can be found in Appendix A.1. To evaluate the error generated by a specific model, we focus on metrics that assess the similarity between the infection dynamics of the empirical network $G_{\mathrm{e}}$ and the modeled network $G_{\mathrm{m}}$. The infection dynamics are calculated as outlined in Section 3.1. Here, $I_m$ and $I_e$ denote the number of infected nodes for $G_{\mathrm{m}}$ and $G_{\mathrm{e}}$, respectively. We measure both, the difference in infection peaks, denoted as $\Delta I_{max}$, and the timing difference of these peaks, expressed as $\Delta T_{I_{max}}$, using the :

$$\Delta I_{max}(I_{\mathrm{m}},I_{\mathrm{e}})=\frac{1}{N_V}\left|\underset{t}{max}\left\{I_e\left(t\right)\right\}-\underset{t}{max}\left\{I_m\left(t\right)\right\}\right|,$$

(5)

$$\Delta T_{I_{max}}(I_{\mathrm{m}},I_{\mathrm{e}})=\left|\frac{\underset{t}{argmax}\left\{I_e\left(t\right)\right\}-\underset{t}{argmax}\left\{I_m\left(t\right)\right\}}{\underset{t}{argmax}\left\{I_e\left(t\right)\right\}}\right|,$$

(6)

such that a model achieving a low value for $\Delta I_{max}$ closely replicates the peak number of infections observed in the empirical network. When a network yields a small value for $\Delta T_{I_{max}}$, it suggests that the timing of the peaks in both the model and the empirical network align closely, demonstrating that the model effectively captures the empirical infection dynamics. In general, topologically different networks can generate very similar infection dynamics. To address this, we also take into account the overall number of edges generated as well as the similarity in the contact duration distributions. The relative difference in the total number of edges between two networks is defined as:

$$\Delta N_E(G_{\mathrm{m}},G_{\mathrm{e}})=\left|\frac{N_{E,G_{\mathrm{e}}}-N_{E,G_{\mathrm{m}}}}{N_{E,G_{\mathrm{e}}}}\right|,$$

(7)

where $N_{E,network}$ is calculated using

$$N_{E,network}=\sum _{i,j,t}{a}_{i,j,network}\left(t\right).$$

(8)

To assess the similarity between two contact duration distributions, we utilize the well-known Kolmogorov-Smirnov (KS) test. The KS test quantitatively determines if two underlying one-dimensional probability distributions differ significantly. We compute the difference in the contact duration distribution with

$$\Delta t_{cd}({\mathrm{G}}_{\mathrm{m}},{\mathrm{G}}_{\mathrm{e}})=KS({\mathrm{G}}_{\mathrm{m}},{\mathrm{G}}_{\mathrm{e}}).$$

(9)

After conducting explorative experiments with various weights and observing the typical value spectra for each metric, we defined the objective function $\mathcal{L}\left(\mathrm{mod}\right)$ with the following weights:

$$\mathcal{L}({\mathrm{G}}_{\mathrm{m}},{\mathrm{G}}_{\mathrm{e}})=5·\Delta I_{max}(I_{\mathrm{m}},I_{\mathrm{e}})+3·\Delta T_{I_{max}}(I_{\mathrm{m}},I_{\mathrm{e}})+2·\Delta N_E({\mathrm{G}}_{\mathrm{m}},{\mathrm{G}}_{\mathrm{e}})+\Delta t_{cd}({\mathrm{G}}_{\mathrm{m}},{\mathrm{G}}_{\mathrm{e}}).$$

(10)

All models, except for BASE, in our framework, are stochastic. Consequently, the outcome of $\mathcal{L}({\mathrm{G}}_{\mathrm{m}},{\mathrm{G}}_{\mathrm{e}})$ not only depends on the parameters selected from the search space but also varies around a mean value. Additionally, SIR runs are stochastic and need to be executed multiple times. To balance the stochastic variations of the SIR runs and the network construction, we generate 20 network realizations for a given set of parameters and conduct 250 SIR runs per network, a number of runs that resulted in stable infection peaks during our tests. The mean value of all $\mathcal{L}({\mathrm{G}}_{\mathrm{m}},{\mathrm{G}}_{\mathrm{e}})$ values (1 for each network realization) is considered as the objective function value for that trial. For each model, a total number of 150 trials was computed by scanning the search space for the optimum parameter set.

Due to the stochastic nature of the model, the final parameter set can still yield slight differences in the final objective function value. To account for this, we conduct a final test with 21 networks generated using the optimal parameters. The definitive value for $\mathcal{L}({\mathrm{G}}_{\mathrm{m}},{\mathrm{G}}_{\mathrm{e}})$ is determined as the median among these 21 networks. All model evaluations within our experiments will be applied to these resulting median networks, including final SIR runs with $10\times N_V$ iterations. Appendix A.1 lists all the model parameters used in the Bayesian optimization process.

4. Results

5. Discussion

6. Conclusions

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