### BRAINSTORMING LIVESTREAM:

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Today (Monday, April 6) at 3 pm ET

### VERY INCOMPLETE NOTES, BEING UPDATED:

VERY INCOMPLETE NOTES, BEING UPDATED:

# Ideas about Digital Contact Tracing

Ideas about Digital Contact Tracing

## Tracing Delayed Encounters

Tracing Delayed Encounters

Given that someone tests positive for COVID-19, who might they have infected? To be able to answer this question with even reasonable reliability can have a huge effect on the ability to contain the virus.

The most promising approaches seem to rely on privacy-protecting phone-based digital contact tracing, for example based on Bluetooth transactions between phones that are in close proximity.

The first level question is: have two people been in the same place at the same time? If they have, then if one of them tests positive, one can deduce that the other might have been infected in that encounter.

But there is more one would like to know. What if person A leaves virus (aerosolized, on a surface, etc.) where person B could pick it up a bit later? Is there a way to use instantaneous phone proximity data to determine if this can have happened?

I think there is a statistical approach to doing this (that, bizarrely, is similar to things that have come up in work I’ve been doing very recently about the construction of space and time in physics). The basic point is to start from the “encounter network” of instantaneous encounters, then to use the structure of the network to infer “longer-term connections”.

For example, with each person represented by an integer, one might record the encounters:

The most promising approaches seem to rely on privacy-protecting phone-based digital contact tracing, for example based on Bluetooth transactions between phones that are in close proximity.

The first level question is: have two people been in the same place at the same time? If they have, then if one of them tests positive, one can deduce that the other might have been infected in that encounter.

But there is more one would like to know. What if person A leaves virus (aerosolized, on a surface, etc.) where person B could pick it up a bit later? Is there a way to use instantaneous phone proximity data to determine if this can have happened?

I think there is a statistical approach to doing this (that, bizarrely, is similar to things that have come up in work I’ve been doing very recently about the construction of space and time in physics). The basic point is to start from the “encounter network” of instantaneous encounters, then to use the structure of the network to infer “longer-term connections”.

For example, with each person represented by an integer, one might record the encounters:

Out[]=

{{1,2},{1,3},{1,5},{1,6},{2,3},{2,4},{3,4},{3,6},{4,5},{4,6}}

which one can represent by the graph:

Out[]=

But to even begin to know the significance of this graph we need another piece of information: the timestamps for the encounters that correspond to each of the edges of the graph. If 3 is infected, and 3 encounters 4, then 4 encounters 5, this can define a chain of infection from 3 to 5. But if 4 encounters 5 only before 4 encounters 3, this will not define a possible chain of infection. As we shall see, however, to know about the earlier encounter of 4 and 5 can still be useful in reconstructing relevant information about the graph and about other possible chains of infection.

We can imagine two important kinds of “delayed infection”, which we might characterize as object mediated and location mediated.

In the object mediated case, a person 1 might physically hand an object to person 2, who then hands it to person 3, and so on. If person 1 is infected, they can effectively infect the object, which can then infect everyone it is handed to. An extreme example of this might be a relay race, in which person 1 infects the baton, which is then successively passed to a sequence of other people.

Ignoring timestamps, the encounters in the relay race could be recorded as:

We can imagine two important kinds of “delayed infection”, which we might characterize as object mediated and location mediated.

In the object mediated case, a person 1 might physically hand an object to person 2, who then hands it to person 3, and so on. If person 1 is infected, they can effectively infect the object, which can then infect everyone it is handed to. An extreme example of this might be a relay race, in which person 1 infects the baton, which is then successively passed to a sequence of other people.

Ignoring timestamps, the encounters in the relay race could be recorded as:

Out[]=

{{1,2},{2,3},{3,4},{4,5},{5,6},{6,7},{7,8}}

with graph:

Out[]=

If we include timestamps, this would effectively become a directed graph, in which person 1 could infect everyone along the chain.

Consider now the case of a location-mediated infection. In this case, an infected person 1 might put virus onto a surface or into an aerosol, and then some time later, person 2 might touch that surface or inhale the aerosol. (Obviously there are also intermediate cases of “locations that move”, but these distinctions will not in the end be relevant to what we come up with.)

A list of encounters tells us who was at the same place at the same time. But what we now need is to know who was at the same place, but perhaps at a different time.

To understand what is involved here, consider a collection of agents walking randomly on a line (with time plotted down):

A list of encounters tells us who was at the same place at the same time. But what we now need is to know who was at the same place, but perhaps at a different time.

To understand what is involved here, consider a collection of agents walking randomly on a line (with time plotted down):

Out[]=

The points where these lines coincide are the encounters (lines that cross not at grid points are not counted here as “encounters”):

Out[]=

If we ignore timestamps we can use these encounters to deduce a network of connections between agents:

Out[]=

In reality each edge is also tagged with a timestamp, or a list of timestamps. Chains of edges are then sufficient to identify potential object-mediated chains of infection. To deduce potential location-mediated chains of infection we in effect have to reconstruct physical space from the collection of encounters we know. (In a physics analogy, this is like reconstructing space from a causal graph of events.)

(If one had GPS data, it can give additional information, but sometimes this may be difficult to interpret--for example if people are traveling on a train, where absolute GPS coordinates are changing, but people may still effectively encounter the “same locations”. In a physics analogy, we want a comoving frame.)

[[ Construct light cones ]]

(If one had GPS data, it can give additional information, but sometimes this may be difficult to interpret--for example if people are traveling on a train, where absolute GPS coordinates are changing, but people may still effectively encounter the “same locations”. In a physics analogy, we want a comoving frame.)

[[ Construct light cones ]]

[[ Minkowskian graph diffusion ]]

### How People Can Be Connected

How People Can Be Connected

The kind of procedure discussed above will generate “connection paths” between people. These paths will contain chains of the form 123 etc., where each element of each chain has a timestamp. There may be multiple chains that connect two given people.

In practical applications, one needs some kind of norm to apply that determines the “infection distance” between people. This might involve lengths of time that elapse, or numbers of links in each chain, or numbers of distinct chains, or numbers of “ambient people”.

Ultimately one might hope to determine the relevant parameters empirically. But in practice I expect that one can just make reasonable estimates based on simple models.

Ultimately one must make tradeoffs between identifying too many and too few potential infections.

In practical applications, one needs some kind of norm to apply that determines the “infection distance” between people. This might involve lengths of time that elapse, or numbers of links in each chain, or numbers of distinct chains, or numbers of “ambient people”.

Ultimately one might hope to determine the relevant parameters empirically. But in practice I expect that one can just make reasonable estimates based on simple models.

Ultimately one must make tradeoffs between identifying too many and too few potential infections.

[[[ Possible norms ]]]

### Environmental Information

Environmental Information

[[ May be useful to use accelerometer in phone to determine if a person is walking etc. Perhaps a proxy for whether the phone is a large or open space or not ]]

## Can One Use Blockchain?

Can One Use Blockchain?

So far we have just discussed how to use encounter data. But the next issue is how to get encounter data, and how to deliver it to people who need it, while preserving privacy.

[[[ Given an infected person, they must propagate their data not just to their immediate encounterees, but also recursively to a succession of encounterees, up to the point determined by the norm ]]]

[[[ Imagine adding data to a blockchain, so different apps can interoperably read and write it ]]]

[[[ Given an infected person, they must propagate their data not just to their immediate encounterees, but also recursively to a succession of encounterees, up to the point determined by the norm ]]]

[[[ Imagine adding data to a blockchain, so different apps can interoperably read and write it ]]]

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