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Is there a way to calculate the riskiest places to be infected by COVID-19? My friends and I are having an argument of whether being in a "high traffic-short contact time" situation (public transport) vs "low traffic-high contact time" situation (e.g. place of worship/seminar) is more risky to be infected.

I know it's wise to avoid all of such situations in the practice. But I want to find a way to calculate it even if it requires assigning dummy/guesswork numbers for unknown probability. Where do I begin?

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    $\begingroup$ There are plausible scenarios in which either of you is correct, but the ontological uncertainties (e.g., about the environmental determinants of transmission, about the determinants of individual susceptibility to infection) will make the estimation intervals of infection probability under the various scenarios overlap to a considerable degree. $\endgroup$
    – Alexis
    Mar 12, 2020 at 16:49
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    $\begingroup$ Btw for those who are interested, I found this very interesting simulation exploring the effectiveness of various ways to control epidemics (quarantine, social distance, hygiene etc.) by 3Blue 1Brown on youtube: youtube.com/watch?v=gxAaO2rsdIs $\endgroup$
    – Polo Marco
    Mar 29, 2020 at 10:02

1 Answer 1

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First of all,

The computation is very theoretic and not a good representation or guideline for adapting your behavior (just in case, if that is what you are after). In the comments I had already mentioned several points of critique for this approach:

  • The problem is that these computations will be based on highly subjective estimates about the underlying model/assumptions. Yes, you can compute it.... But do not expect that the answer is rigorous just because it has used mathematics

  • Another problem is that the descriptions of contacts are very complex. How exactly are you gonna be in the description of the 'time of contact'? Are you differentiating just the time of contact or also the type of contact? It is not a deterministic model and you need to deal with with distributions and stochastic behaviour that will make computations more difficult.

    See for example the report about transmission of SARS in airplanes: there were three cases described with an infected person on board. In one case tens of other passengers got infected. In the other two cases it was only one other person (a crew member) that got infected.

  • In addition, are you gonna describe the probability for a single person, or the probability for the public health? In high traffic the probability for a single individual might be low but due to the large number of individuals in those situations there might be a probability that at least one or more persons get infected.

    For public health, the problem is not to compare cases based on the probabilities for individuals to become sick. But instead, the point is to reduce the probability for individuals spreading the virus, make others sick. In general those probabilities (to make others sick) are much higher with high traffic cases. Sick people should not be around many other people.

    There are many of these strange probability effects around. For instance, in Europe there is a lot of focus on people who had contact with the high risk areas; and it seems to be ignored that one may acquire the virus locally as well.
    Indeed, when considering only a contact with a single person, then it is more likely to obtain the virus if this person is from (or had contact with) a high risk area. However, due to the much larger number of contacts with people outside the risk area it may be more likely to acquire the virus from one of those people, albeit the fact that the risk per contact is lower.
    Still, it is not illogical to focus on the high risk areas. But that is more a consideration from the point of view of focusing the limited time, money and materials. Yes, it is more likely to get the coronavirus from somebody that is not from the risk area's. But there are many other viruses from which one may obtain a common cold and we can not deal with all of those cases. When we wish to focus efforts on the most important cases, then the consideration is for which people the common cold is most likely to be due to the coronavirus. In that case it is linked to the high risk areas.


A possible solution (based on a simple model),

Let's consider the (unrealistic) probability of obtaining an infection, conditional on the other person being sick (this is a bit complex, there are different levels of being sick but let's consider this for single cases).

Say, the probability 'to get sick from a single contact of time $t$' is a function of contact time according to some (sort of) homogeneous Poisson process (ie. the waiting time to get hit/sick depends on an exponentially distributed variable and the longer the contact the more likely to get sick)

$$P(\text{sick from contact time $t$}) = 1 - e^{-\lambda t}$$

If you encounter $n$ people, each for a time $t$, sampled from a population of which $p\%$ are sick...

  • then the number of sick people, $S$, that you encounter is binomial distributed $$P(S=s) = {{n}\choose{s}} p^s(1-p)^{n-s}$$

  • the probability of getting sick from those $S$ people is: $$P(\text{sick} \vert t,s) = 1- e^{-\lambda ts}$$

  • the marginal probability of getting sick is $$\begin{array}{} P(\text{sick} \vert t, n) & = & \sum_{s=0}^n \overset{{\substack{\llap{\text{probability}}\rlap{\text{ to encounter}} \\ \llap{\text{$s$ sick }}\rlap{\text{people}} }}}{\overbrace{P(S=s)}^{}} \times \underset{{\substack{\llap{\text{probability to }}\rlap{\text{get sick}} \\ \llap{\text{conditional}} \rlap{\text{ on}} \\ \llap{\text{encountering $s$}} \rlap{\text{ sick people}} }}}{\underbrace{P(\text{sick} \vert t,s)}_{}} \\ \\ &=& 1- \sum_{s= 0}^n {{s}\choose{n}} p^s(1-p)^{n-s}e^{-\lambda ts} \\ &=& 1- \left(1- p + pe^{-\lambda t}\right)^n \end{array}$$ where I solved this last term with wolframalpha.

Note that

$$\lim_{n\to \infty} 1- \left(1- p + pe^{-\lambda t/n}\right)^n = 1 - e^{-\lambda t p} $$

For a given fixed total contact $C = n\times t \times \lambda$ you get an increase as function of $n$. For instance, if $C = 10$ then:

example

Intuitive overview

Below are two graphs that show the value of this term $1- \left(1- p + pe^{-\lambda t}\right)^n $ as function of contact time $t$ and number of contacts $n$. The plots are made for different values of $p$.

Note the following regions:

  • On the right side the contact time is very large and you are almost certainly gonna get infected if the person that you have contact with is sick. More specifically for the lower right corner (if $n=1$ and $t$ is very large) the probability of getting sick will be equal to $p$ (ie. the probability that the other person is sick).
  • More generally for the right side, the region where $\lambda t>1$, a change in the contact time is not gonna change much the probability to get sick from a single person (this curve $1-e^{-\lambda t}$ doesn't change much in value for large $\lambda t$).

    So if $\lambda t>1$ (and you are almost certainly getting sick if the other person is sick) then if you halve the contact time and double the number of contacts, then this is gonna increase the probability to get sick (because the probability to encounter a sick person increases).

  • On the left side for $\lambda t < 1$ you will get that at some point an increase of $n$ with an equal decrease of $t$ will counter each other. On the left side it doesn't matter whether you have high traffic short time vs. low traffic high time.

example p=0.001

example p=0.1


Conclusion

So, say you consider total contact time $n\times t$ being constant, then this should lead to a higher probability for getting sick for higher $n$ (shorter contacts but with more people).

Limitations

However the assumptions will not hold in practice. The time of contact is an abstract concept and also the exponential distribution for the probability of getting sick from a single person is not accurate.

  • Possibly there might be something like contact being more/less intense in the beginning (to compare in the simple model the probability to get sick from a single contact of time $t$ is approximately linear in time $1-e^{\lambda t} \approx \lambda t$)
  • Also when considering infections of groups, instead of individuals, then there might be correlations like when a sick person sneezes then it will hit multiple people at the same time. Think about the cases of superspreaders, e.g. the case of the SARS outbreak in the Amoy Gardens apartment complex where likely a single person infected hundreds of others)

So based on the simple model there is this effect that of for a given total time of contact, $n \times t$, is better to spread it out among less people, $n$. However, there is an opposite effect. At some point, for short $t$, the transmission will be relatively unlikely. For instance, a walk on a busy street means high $n$ but the contacts will not be meaningful to create a high risk. (Potentially you could adapt this first equation $1 - e^{\lambda t}$ but it is very subjective/broad). You could think like something as the '5 seconds rule' (which is actually not correct but gets close to the idea).

Use of the simple model

Although the model used here is very simplistic, it does still help to get a general idea about what sort of measures should be taken and how the principle would work out for a more complex model (it will be more or less analogous to the simple model):

On the right side (of the image), it doesn't help much to change (reduce) the contact time, and it is more important to focus on reducing the number of contacts (e.g. some of the rigorous advise for non-sick family-members that are in quarantine together with sick family members are not much useful since restricting $\lambda t$ for large $\lambda t$ has little effect and it would be better to focus on making less contacts; go cook yourself instead of ordering that pizza)

On the left side, reductions should be weighed against each other. When restrictions that reduce high traffic are gonna lead to low traffic but a longer time then the measures are not gonna help a lot.

A very clear example: I am currently waiting in line to enter the supermarket. They have decided to reduce the number of total people inside the supermarket. But this is entirely useless and possibly detrimental. The total time that we are in contact with other people does not decrease because of this. (And there are secondary effects: partners alone at home with children that have to wait longer. Potential shopping at multiple markets because time is limited at single markets. Etc. It is just silly)

I am letting the older people in the line pass before me since the health effects may be worse for them. And in the meantime I make myself annoyed about this symbolical useless measure (if not even detrimental) and have sufficient time to type this edit in this post, and in the meantime either make other people sick or become sick myself.

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  • $\begingroup$ Maybe the "5 seconds rule" could be modelled as $$1 - e^{\lambda (t - \alpha)}$$ where alpha is the unknown minimum contact time $\endgroup$
    – Polo Marco
    Mar 12, 2020 at 9:43
  • $\begingroup$ There is a closing parenthesis missing in the last equation. I would have fixed it myself, but SE software requires edits to be at least 6 characters. $\endgroup$
    – Igor F.
    Mar 12, 2020 at 10:00
  • $\begingroup$ Superb answer! Thank you $\endgroup$ Mar 12, 2020 at 20:12
  • $\begingroup$ Amazing answer! +1 $\endgroup$
    – Haitao Du
    May 13, 2020 at 9:35

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