I got several reactions to the posts on the PDT and epidemics and several questions on the effectiveness of actually applying PDTs in this area. With this post I try to clarify some of the points I made in a more systematic way.

In mathematical terms, an infection is characterised basically by four factors:

- the basic Reproductive Number, R0, indicating the number of people that would be infected by a single person, assuming they are all susceptible of infection (e.g. not vaccinated). The greater this number and the steeper the curve, faster the propagation of the infection (if the value is lower than 1 the epidemics will not spread). For the Covid-19 the consensus is on an R0 between 2 and 2.5, with an accepted figure of 2.2. The formula governing this growth is

(∂N/∂t)=R0N

where N is the number of people in the considered set and t is the time.

The spreading time depends on factors like the time from being infected to the time of becoming contagious. If you are interested in the math behind the R0 click here. - the availability of people to be infected. It may seem obvious but it looks like many are missing this point. There are not, in the real world, an unlimited number of people to infect > there are about 7 billion on the planet and once they are all infected that’s it. This means that the epidemic will necessarily stop once all have been infected. However, the infection curve is not like the one shown in the first diagram where at the top, once all people have got infected it simply plummet to zero. Epidemics are not following an exponential curve, rather a logistic curve (see the second diagram).

For those interested in formulas here is the one governing the logistic curve:

(∂N/∂t)=R0(1- N/Nmax)N

where N is the number of infected people and Nmax is the total number of people. In the first phase of the infection N is small hence the ratio N/Nmax is close to zero and you get the exponential curve with R0 as a factor. As the number of people infected grows (and these people can no longer be infected) the ratio between number of people that can no longer be infected vs total number of people approaches 1, hence the value of R0 decreases, approaching zero. Thus we get the S curve. - the generation time, that is the time it takes from one person being infected to the time it infects another. For example in measles the generation time is 12 days, for the flu is around 2 days. For Covid-19 current studies point to a 4 day generation time. Unfortunately the incubation time (the time from infection to the appearance of symptoms) is longer, around 5 days, meaning that there is a period when a person feels healthy and yet it can infect others (1 day). Notice that these numbers reflect a median value, with ranges much broader, up to 10 days for incubation time, meaning that the window of infectiousness without apparent symptoms can be large.
- influencing factors (like environmental ones) that can change the R0 over time (leading to different
*effective reproductive number*). This leads to a further expansion of the logistic formula. Several viruses, like the flu ones, are susceptible to climate and temperature, becoming more virulent during Winter and relenting in Summer. This leads to a sequence of infection waves that are still regulated by the logistic formula. Since the second wave is acting on a smaller Nmax (those infected by the first wave will no longer be part of the target population) the number of affected people will be lower. Notice, however, that for a given virus, all condition being equal, the total number of people infected in case of a single wave hitting the whole population or a subsequent series of waves hitting decreasing subsets of the total population will result in the same number of infected. In other words, multiple waves just spread the infection over a longer period of time, they do not change the number of people affected.

The *effect* of an epidemic on people’s health depends on the severity of the disease (often varying in different demographics, in Covid-19 the elderly are more compromised) and on the possibility of cure. This in turns depends on the existence (obvious) of curing protocols and their availability. As an example, in the current epidemics most people presenting severe symptoms require the availability of ICU and respiratory support. Since these are in a limited number healthcare institutions and Governments are deploying measures to spread over time the number of people needing this kind of support, basically they want to make sure that for any new person requiring an ICU bed there is a person recovering that can be dismissed from the ICU. That is the peak of epidemics that can be sustained by healthcare resources. Going over this peak means some people can not be assisted.

Controlling the spread is therefore crucial to decrease the strain on resources, as well as finding better ways to relieve symptoms, eventually hoping that a vaccine will be found and this will decrease the Nmax, the number of people that can be affected (leading as a side effect, logistic curve, to a herd immunity).

Having set the stage we can consider how PDTs can affect the various parameters:

- PDTs can increase the single individual awareness, increasing safe behaviour. This will not hold true for all people, you will still have people that consider themselves as superman, above the virus, and will keep an unsafe behaviour. However, a significant portion of the people will take preventative actions to avoid being infected. Statistically this is equivalent to a reduction in R0, and any reduction in R0 leads to a slower infection rate (the steepness of the curve in the first graphic decreases) giving more time to the healthcare system to take countermeasures (increase the number of resources to support the infected with severe symptoms) and making resources available to more people over a longer period of time.
- PDTs can provide timely information on the status of their physical twin, including the probability of being contagious and data analytics on communities of individuals can lead to identify individuals that might have been infected. This generates both individual awareness (R0 – see previous point) and community awareness with imposition of lock down measures by the authority (following direction from the healthcare institutions) leading to a decrease of the Nmax and thus to an acceleration of the shape change in the S curve (logistic curve). In other words, data analytics made possible by PDTs can help in containment measures. This probability risk of being infected can correlate to the probability of being infectious and as noted before this would be crucial in containment. This is particularly so in the case of Covid-19 where the incubation time is longer than the latency time (a person becomes infectious before experiencing symptoms). The PDT can act as a watch dog in this period. Locking down a community with possibly infectious members provides a big help in overall containment. Besides, data analytics can also point out the probability of infection in different communities providing the authority the means to locking down those sets of individuals that are not just most likely to infect others but also activate the lock down in a way the minimises cross contagion in a community. This kind of information is also crucial at the time of relenting lock down measures. This can be done in a selective way to decrease risk of an uptake of the contagion, or at least to minimise its effect.
- PDTs can release data on location of the physical twins that has been identified as potentially contagious (or definitely contagious having been found positive to a test) that is not allowed (out of his quarantine zone). This data can be used by the authority to enforce quarantine, getting police to pick up the person and persecuting him. Notice that the certainty of being caught and persecuted is a most strong deterrent to avoid this kind of behaviour in the first place. This contribute to decrease the R0.

These three ways of using PDTs have been presented in a growing intensity of privacy reduction. Whilst

- the first is basically preserving all the individual privacy (it is only the physical twin than gets to use hiis data and the analytics derived from the whole data space but without becoming aware of other people’s data, thus not infringing on other people privacy),
- the second act on a community level and does not disclose personal data although the fact of becoming part of a community implies the sharing of commonalities and therefore of some of personal data,
- the third provides full exposure of certain personal data to the authority, like health data and location/activity.

Clearly, the above assumes the existence of a security scheme that protects data from non authorised parties.

There is a fourth contribution from PDTs: intelligence emergence. The data harvested at micro scale by each PDT can be used by machine learning and data analytics correlating them to actions taken (like creation of locked down community, use of pharmaceutical protocols -including masks-, social distancing, broadcasted warnings, …) to assess their effectiveness. These data do not need to be visible in the micro, only the result of the analytics is of importance, hence privacy can be preserved.

The machine learning part is very important both in the understanding of the ongoing epidemics and the effectiveness of countermeasures AND in proactively prepare for new epidemics interception and management.

This is the reason why I have been asking over and over to take the opportunity offered by this emergence to create a safety net to remain in place so that future possible epidemics impacts could be prevented. The idea of implementing some patches under the strain of the current situation and then removing them once the emergency is over is not a good approach. As mentioned, a PDT approach can offer a way to balance private desire of privacy with societal need-to-know.