Connecting the DOTS - How Masks Get Us to Herd Immunity Immediately

Herd immunity for Covid!  This term is frequently bandied about in the news.

It basically means that enough of the population has gotten immune to the virus (either through recovering from it or getting a vaccine) that any new infections will quickly die out instead of spread.

Epidemiologists have estimated herd immunity for SARS-CoV-2 (the virus that causes Covid-19) to be right around 60% or so.  Once 60% of us are immune to Covid, the rest of us are protected because the 40% remaining is not enough fuel for the virus to continue to burn through the population.

Herd immunity is a function of how fast the virus can spread.  The term that specifies the spreadability of the virus is called Basic Reproduction Number or R(0) (you can learn more about this in the appendix if you are interested).

Masks and other interventions that slow the virus down will reduce R(0).  

The question is, by how much.

I.E., how many people will have to wear masks (and how effective do the masks have to be) so R(0) can be reduced enough to result in herd immunity?

To figure this out, we need a mathematical model for R(0).  We can build such a model with DOTS.

DOTS

How fast the virus spreads (or R(0)) is dependent on four factors which can be summarized with the acronym "DOTS".

DOTS stands for:

D - Duration
O - Opportunity 
T - Transmissibility
S - Susceptibility

Here is what each of these terms means:
  • Duration is the length of time that a person is contagious
  • Opportunity is how big a chance that the virus has to jump to another person
  • Transmissibility is how transferrable that the virus is from one person to another
  • Susceptibility is how easily a non-immune person can catch the virus
(A more detailed definition of each term is in the appendix.)

Modeling the virus with DOTS

Multiply all four DOTS terms together and you get how fast the virus will spread.

I've modeled DOTS for two places in the country - Maryland and New York City.

Here are the results (you can see the details of the model in the appendix):


By reading the chart, you can see that if 80% of us wears masks (covering both noise and mouth) while out in public, the herd immunity threshold goes to zero.  That's right.  Zero.  It means that if anyone comes into our area with the virus, the resulting infection will die out.  It will not spread further than a few people.

And if you live in NYC, only 70% of the population needs to be masked to assure herd immunity.  This is because they already have a very high prevalence of antibodies (20%) and that helps considerably.

There are some other cool things that can be explored with this model, for example, understanding the variation in the infection levels in various states at the present time.  I will perhaps explore this in a future blog post.

In the meantime, wear your mask!

Appendix: Modeling with DOTS

The variation in the prevalence of Covid-19 is amazingly high.  For example, New York State went from test positivity rates of as much as 50% to their current value which is very stable at 1%.  Florida has gone from 2% in early June to almost 20% currently.  Furthermore, today you can find states with test positivity of just about any value in between these extremes.

In order to understand the sources of this variation, I created a simplified model for the virus transmission.  Before we get into the specifics of the model, we'll need to cover some basic concepts.

Basic Reproduction Number
Here is a definition of R(0), or the basic reproduction number of a virus, according to the University of Oxford:
  • The basic reproduction number is defined as the number of cases that are expected to occur on average in a homogeneous population as a result of infection by a single individual, when the population is susceptible at the start of an epidemic, before widespread immunity starts to develop and before any attempt has been made at immunization.So, for example, if a person contracts a virus and passes it on to two other people, the R(0) is 2.  

The best estimate for the initial R(0) value for SARS-CoV-2 (the virus that causes Covid-19) when it first hit Maryland is 3.0.  We will use that value as a starting point.

Effective Reproduction Number
Measures taken by governments and citizens alike will have a dramatic impact on R(0).  Any action that slows the transmission of the virus will cause the R factor to decrease.  We will use R(e) to denote the Effective Reproduction Number which is R(0) after it has been decreased due to preventative actions.  

The relationship between R(0) and R(e) can be modeled by taking into consideration the impacts of these preventative actions, which we will do in the next section.  But before we do, let's look at the relationship between R(e) and herd immunity.

Herd Immunity
Here is a definition of Herd Immunity from the Mayo Clinic
  • Herd immunity occurs when a large portion of a community (the herd) becomes immune to a disease, making the spread of disease from person to person unlikely. As a result, the whole community becomes protected — not just those who are immune.
  • Often, a percentage of the population must be capable of getting a disease in order for it to spread. This is called a threshold proportion. If the proportion of the population that is immune to the disease is greater than this threshold, the spread of the disease will decline. This is known as the herd immunity threshold
The relationship between the herd immunity threshold T(h) and R(e) is an important one, and will be used in our model. 
  • T(h) = 1 - 1/R(e)

Modeling the Effective Reproduction Number and Herd Immunity
I constructed a simple model to understand the variation of the reproductive number R(e) and herd immunity threshold T(h) as a function of the preventative actions that can be taken against the virus.

There are four components that make up R(e).  They can be summarized as DOTS.

DOTS stands for:

D - Duration
O - Opportunity 
T - Transmissibility
S - Susceptibility

These four components will attenuate R(0).  I will express them as percentages - where 100% is fully effective and 0% is no impact.
  • "D" is the duration that a person is infectious with the virus.  I'm assuming 11 days will be 100% of the duration of infectiveness.   
    • We will leave the parameter D at 100%, as there is no evidence that the duration of the infectiousness has changed and there is no known way to reduce it as of this point in time.
  • "O" is the opportunity that the virus has for infecting a new person.  There are several ways to characterize this, but I am simplifying it to represent the degree of lockdown in our state.  
    • I am assuming as of today, July 23, we are about 10% locked down (e.g. 90% back to our normal ways of going out and about, shopping, for example).
  • "T" is the transmissibility of the virus.  I am assuming that masks are the primary way we are preventing the transmission of the virus in normal circumstances.  T is composed of two components W (% people wearing masks) and M (mask effectiveness in %) such that T = 1- M*W.
    • M = 80% Mask Effectiveness.  This is the combined percent of virus that would hypothetically make it through both the wearer's mask and the person's mask who is the target of infection.  Note that this simplified model only accounts for virus transfer between two masked people or for transfer between two unmasked people.  It doesn't account for transfer between one masked person and one unmasked person.
    • W = 50% initially for the % people wearing masks.  But we will later let W float as the dependent variable so we can plot it.
  • "S" is the susceptibility of a person in the population to the virus.  The primary change in susceptibility is whether the person has acquired immunity thru recovery from Covid-19 or, in the future, has gotten the vaccine.  The CDC recently reported on serological surveys done over the previous few months, and they have found roughly 3.6% of the population has antibodies to the virus (in the Philadelphia area, which is the closest survey point to Maryland), and that a surprising 20% of people in NYC have antibodies. Note that in our simplified model, susceptibility S = 1 - A, where A is the level of antibodies of a population.
To calculate R(e):
  • R(e) = D*O*T*S*R(0)
To calculate the herd immunity threshold T(h):
  • T(h) = 1-1/R(e) = 1 - 1/(D*O*T*S*R(0))
The results of a run of the model with the above assumptions are given in the table below.




You can see in the table that with an R(0) of 3.0 and 50% of the population wearing masks that are 80% effective, the herd immunity threshold drops from 67% to 36%.  This number (36%) already assumes that 3.6% of the population already has antibodies, so the final T(h) would be the sum of those two numbers or 39.6%. 

To show the effectiveness of masks in reducing the herd immunity threshold, I've plotted the herd immunity threshold as a function of percent mask compliance below.  This assumes an average of 80% mask effectiveness (which assumes that people generally wear their masks correctly).  I've shown two curves, one for Maryland (3.6% have antibodies) and New York City (20% have antibodies).



You can see that even the modest use of masks dramatically reduces the herd immunity threshold.

If you only believe masks are 50% effective, here are the resulting curves: 











 

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