## Our study

We developed an optimized Digital Contact Tracing (DCT) protocol using network theory to find the minimal number of infected contacts and their secondary contacts to halt the epidemic spreading with minimal social disruptions. The model is tested and calibrated during the ongoing Covid-19 pandemic in the city of Fortaleza, Brazil by using real-time data on individuals geolocalization provided by mobile applications and epidemiological information from government authorities.

We monitor the giant connected component finding that the most optimal strategy is to directly quarantine the maximum kcore of a two-layer contact network seeded by the infected people. We implement this optimized strategy by deploying a contact-tracing App in partnership with the Government of Ceará

Collection and elimination of data

We collect the GPS data for only the past 14 days deleting it after that period ensuring its use only for the app purpose. The data is encrypted and no relation is made between the real person and the data source keeping the process anonymous The novelty of the contact tracing algorithm relies on the estimation of the probability of infection considering a correlated time and space component.

Key Concepts

Contact Area

As the database is labeled in two groups, sources (infected users) and targets (healthy users), a contact is defined as an interaction produced between a source and a target fulfilling specific conditions. Each time stamp the contact area is defined as a circle centred in the position of the source with a radius R. From each timestamp we gather all the targets that are within the circle in a T minutes forward time window. Once all the targets are detected, we compute for them and the infected source the average position and the interval of time they have been within the contact area:

Where d[n] is the euclidean distance between the source and target average positions of the data points within the contact area in the next T minutes from the time stamp and R is the radius of the contact area.

Contact probability of infection

The time component depends on the overlapped amount of time source and target spent within the contact area. The time component probability is proportionally related to the amount of time target and source coexist within the contact area in the T minutes time window.

The probability leans to 1 when the overlapped time is equivalent to the window size and to 0 when there is no overlapped time:

where τ(∆ts,∆tt)[n] is the overlapped time in the time stamp n that depends on the ∆t for the source (∆ts) and the target (∆tt),tsf and tsl are the first an last time data for∆ts respectively,t tf and t tl are the first an last time data for ∆tt respectively and T is the time window size. Note that if the target only has one data point within the circle in the T minutes window the ∆t would be 0 and therefore also the pt. We need at least two points within the circle to have a probability of infection due time. This condition also applies to the source. If the infected user has only one data point there is no time period he remained within the circle so we omit that contact area as the overlapped time with the targets goes to 0.

We obtain the probability of infection for each time stamp:

Finally, we use a recursive expression to obtain a unique probability of infection for eachcontact between source and target:

wherePi[n] is the total probability of infection for a contact between a specific target and source until time stamp n,Pi[n−1] is the total probability of infection in the previous state and pi[n] is the probability of infection for this contact in the current time stamp n.