Modeling the Spread of COVID-19: Part 3 - Mitigating Strategies

In the previous columns on this topic, I have looked at the problematic criterion for lifting the lockdown in Pennsylvania, and the statistics behind whether you get infected if exposed.  In this column, I will discuss the efficacy of mitigating strategies like mask wearing and social distancing.

The most visible of the public health strategies thus far has been the lockdown.  We have all been hunkered down in our houses, afraid of our groceries, and suddenly, we're all home baking bread because gluten intolerance is no longer a thing (/s).  Additionally, the skies are clearing and due to a lack of traffic, there is verifiably less pollution all around the world.  All of this is in the name of social distancing.  If you aren't exposed to the virus, you can't get it.  Lockdowns work.  Period.  While there is some debate about how long they should last to be effective, there is evidence that a five week lockdown could end the pandemic once and for all (if you can't get it and it runs its course, you can't spread it!).  Others feel that because such lockdowns can never be total (where will you get food? who keeps the lights on? etc.) it will be more like 18 months or about when a viable vaccine is developed.

Social distancing measures can also be used in a less all-or-nothing manner.  The Centers for Disease Control (CDC) recommends, among other measures, when in public spaces keeping a 2-meter distance from others who aren't in your household in order to prevent the spread of COVID-19.  However, if someone who is infected is not wearing a mask, their region of influence, that is, how far they can spread the virus, is not limited to two meters.  Jayaweera et al just published a paper titled: "Transmission of COVID-19 virus by droplets and aerosols: A critical review on the unresolved dichotomy" which calls into question the 2-meter doctrine, especially in closed spaces.

Other supporting work includes that of Qureshi et al (2020) in a paper titled: "What is the evidence to support the 2-metre social distancing rule to reduce COVID-19 transmission?"  Their results can be summarized via their graphic here:

The paper is a fantastic review of prior work by other authors and was conducted by members of the Center for Evidence Based Medicine at Oxford University.    Another work included in the summary is published in the Lancet by Chu et al (2020).  This work looks at the odds ratios (risks) of contracting COVID-19 and considers a number of factors.  We'll be using some of the numbers from this review later in this column.  In the graph below, we can see how social distancing can decrease the risk of infection.

As you might imagine from the Jayaweera et al graphic, mask wearing is one of the other major mitigation strategies.  Masks work mainly by keeping the infection in (you cough into your mask, and the virus is trapped) rather than by keeping the virus out.  How well do they work?  Well that depends on whether you are social distancing, what type of mask you have, and how well it fits.  This is, again, covered in the Chu et al paper in the Lancet, but the bottom line is that N95 masks work much better than surgical masks, but any mask is better than no mask.  Thicker masks that fit well protect you better than thin masks with gaps.  Most masks are relatively effective in stopping YOU from spreading your germs.

OK, so now that we have some background, let's try to model what happens with or without social distancing, and with or without mask wearing.  Let's do some probability calculations to get a sense of how our prior probabilities are modified by either wearing a mask -OR- social distancing (or both!).  In the last column, the probability of contracting COVID-19 for an 18-44 year old was modeled based on the number of contacts one had.  Recall that for this group, the probability of infection at close quarters was 0.022 or 2.2%  On the x-axis is the number of infected people one is exposed to, while on the y-axis is the probability (0.50 = 50%). 

This result was a little bit scary in that it implies that without any precautions, one can very quickly get infected.  It does not model the infectiousness of a given carrier, the time you have spent in their proximity or anything of that nature.  It assumes that all carriers are equal and you've been exposed at close quarters.

The Lancet paper offers some estimates of how masks and social distancing affect infection using what is called an "odds-ratio".  The odds-ratio is the probability that an event happens divided by the probability that an event doesn't happen.  In the case of rolling a 6-sided die and trying for a one, the odds ratio would be: (1/6)/(5/6) = 1/5 = 0.20.  So while your probability of rolling a one is 1/6 = 0.166666..., your odds of rolling a one are 1 in 5.  In the Lancet paper, the odds-ratio for how mask wearing reduces transmission (compared to not wearing a mask at all) is 0.15 with error bounds of 0.07 to 0.34.  The odds-ratio of social distancing at a distance of one meter or more is 0.18 with error bounds of 0.09 to 0.38.   To be clear, there really are a continuum of values here dependent on a number of factors, but starting with the mean values is a way to get started.

Let's consider the social distancing scenario.  If we use the mean odds-ratio of 0.18, we could calculate the probability, but that would be extra work in this case.  We already know that compared to NOT social distancing the ratio probabilities of getting COVID-19 with social distancing to getting COVID-19 without social distancing is 0.18/1.0.    This means that we can just modify any probability we calculate by multiplying it by the odds-ratio. 

• Probability of getting COVID-19 if exposed to an infected individual = 1.0*0.022 = 0.022 or 2.2%.
• Probability of getting COVID-19 if exposed to an infected individual, BUT with at least 1.0 meter of social distancing: 0.18*0.022 = 0.00396 = 0.396%

Likewise, we can determine how wearing a mask affects things. 

• Probability of getting COVID-19 if exposed to an infected individual, BUT with a decent mask on: 0.15*0.022 = 0.0033 = 0.33%

And, we can compute how the two effects work together, presuming they are truly independent:

• Probability of getting COVID-19 if exposed to an infected individual, BUT with a decent mask on AND with social distancing: 0.15*0.18*0.022 = 0.000059 = 0.0059%

OK, this is looking pretty good.  So instead of having a 2.2% chance of getting COVID-19 with every contact with a sick person (this is a bit more than a 1 in 50 chance), we have reduced our risk to something like 59 chances out of a million or a bit less than 3 in 50,000. 

Now, let's put all this in context with the number of contacts we have who have COVID-19.  Here are the initial parameters.

p = 0.022    // probability of getting COVID-19 from close contact
pNot = 1 - p    // probability of NOT get it
pSD = 0.18;    // Odds ratio of social distancing converted to probability
pM = 0.15;    // Odds ratio of mask wearing converted to probability

Then when we calculate the probabilities for the number of contacts (i+1) the code will be as follows:

  pInf[i] = (1 - pNot**(i+1));
  pMask[i] = pM*pInf[i];
  pDist[i] = pSD*pInf[i];
  pMD[i] = pM*pSD*pInf[i];

Recall that the a**b notation in JavaScript means a to the b power.  Other languages like using the "^" character. 

When we put all this together and rerun our simulation with average values, our results look like this:

Look at that!  By using these mitigating measures, especially in combination, we can really substantially reduce our risk.  And this is when we are pretending that everyone we meet is infected.  This sounds great!   So let's go with the social distancing bit and all wear masks, and we're probably pretty good!  Again not so fast!  While these measures make a difference, they don't make you bulletproof.

In the next column, we'll look at the incidence of COVID-19 in the population, and how this affects the model.  Hint - currently the incidence is quite low, so all these numbers will go down.  But at that point, we'll be ready to start considering the ensemble risk due to a group of people, and how risk increases from day to day.

In the meantime, the take home message for today is wear a mask and social distance!  Especially since the number of new cases in Pennsylvania for June 22 is 456.  The epidemic is far from being under control.

References:

M. Jayaweera, H. Perera, B. Gunawardana, J. Manatunge, Transmission of COVID-19 virus by droplets and aerosols: A critical review on the unresolved dichotomy, Environ Res. 188 (2020) 109819. https://doi.org/10.1016/j.envres.2020.109819.

Zeshan Qureshi, Nicholas Jones, Robert Temple, Jessica PJ Larwood, Trisha Greenhalgh, Lydia Bourouiba. What is the evidence to support the 2-metre social distancing rule to reduce COVID-19 transmission? - CEBM, (June 22, 2020) https://www.cebm.net/covid-19/what-is-the-evidence-to-support-the-2-metre-social-distancing-rule-to-reduce-covid-19-transmission/ .

D.K. Chu, E.A. Akl, S. Duda, K. Solo, S. Yaacoub, H.J. Schünemann, D.K. Chu, E.A. Akl, A. El-harakeh, A. Bognanni, T. Lotfi, M. Loeb, A. Hajizadeh, A. Bak, A. Izcovich, C.A. Cuello-Garcia, C. Chen, D.J. Harris, E. Borowiack, F. Chamseddine, F. Schünemann, G.P. Morgano, G.E.U.M. Schünemann, G. Chen, H. Zhao, I. Neumann, J. Chan, J. Khabsa, L. Hneiny, L. Harrison, M. Smith, N. Rizk, P.G. Rossi, P. AbiHanna, R. El-khoury, R. Stalteri, T. Baldeh, T. Piggott, Y. Zhang, Z. Saad, A. Khamis, M. Reinap, S. Duda, K. Solo, S. Yaacoub, H.J. Schünemann, Physical distancing, face masks, and eye protection to prevent person-to-person transmission of SARS-CoV-2 and COVID-19: a systematic review and meta-analysis, The Lancet. 0 (2020). https://doi.org/10.1016/S0140-6736(20)31142-9.