Crowder: COVID Risk Visualization

Have you ever needed to go somewhere, except you're not sure how safe it is, during these dark times of Covid? We present Crowder, an app which visualizes how risky particular locations are, in terms of catching Covid. The app relies on crowd contribution, so the more users using the app the more accurate it is.

Risk scalar field

"Aren't we all just Gaussian curves, walking around, f**king each other's brains out?" Ongo Gablogian

We compute a positively valued scalar field over the world map (2D plane). The scalar value at a particular point in the field is the covid risk level associated with that particular location, i.e. how risky is it to be at that location.

Each entity in the system (users) is represented as a 2D gaussian curve centered at their current location. This means they are small 'bumps' on the risk field, so it is riskier to be near other users. This represents social distancing.

The scalar field is then the sum of all the gaussian curves in the system. As users move around, this scalar field changes to reflect their new positions.

However, we understand that covid can linger on surfaces which can cause retroactive spreading. To model this, we also add in the older risk fields, scaled so that older ones have a smaller impact than newer ones. Essentially, users leave a trail of 'risk' walking around.

The equations we used are presented below for interested parties.

Some Equations

The Gaussian curve is given by the function below, for the general N-dimensional case.

$f_{\mu,\Sigma}(x) = \frac{1}{\sqrt{|det(\Sigma)|}} \frac{1}{(\sqrt{2\pi})^k} e^{-\frac{1}{2} (x - \mu)^T\Sigma^{-1} (x - \mu) }$

We are interested in the 2-D case $(k = 2)$ where $\mu, z\in \mathbb{R}^{2}$ and $\Sigma \in \mathbb{R}^{2 \times 2}$ .

$\mu$ is the center of the curve, therefore it is the location of the entity on the map.
$x$ is any location on the map we are trying to check the risk level for.
$\Sigma$ is the covariance matrix which represents the shape of the curve (rotation, how wide it's range). This is to be determined based on social distancing guidelines, and will be the same for every person.

Every few seconds, each user will ping the server with it's new location $\mu_i$ . We store each ping along with a timestamp $t_i$ . The risk field is then computed as the scaled sum of gaussian functions associated with each ping:

$R(x) = \sum_i \frac{p}{ct - t_i} f_i(x)$

Where:

$p$ is a constant-valued persistence factor, i.e. how long of a trail we want. Higher $p$ implies a longer trail.
$ct$ is the current time.

user table

keeps track of the list of users in the system | uuid| |:---:| | 0 | | 1 |

locations table

keeps track of location pings, which is used to compute the risk scalar field. Users can move around, so their lat, long can change. The timestamp is used to calculate decay (how fresh & relevant the ping is). See formula below for details.