Two of the worst aspects of the Novel Corona virus and it's spread are:

• People's ignorance of public health, math, and science, and
• The general anti-elitist, anti-science, my-liberty-is-most-important view, manifested most egregiously by Conservatives in general, and especially in Republicans.

I am not naïve, and I know that the very people that are able and willing to understand what follows are the very same ones that are probably acting responsibility now. But here I go anyway...

Exponential Growth

A lot of the public has never heard of this concept. Others have heard of it, but not fully grasped just how profound it is. Simply stated, as a function, with time (t):

describes how over a single 1-unit time-step, the function will "expand" (be multiplied by) by a factor of k. Note, e, known as Euler's Number, roughly 2.71828, has a lot of nice properties in mathematics, but you could replace it by any number of other positive constants and have similar behavior.

So how does this equation behave for various values of k? Interestingly, if you make k=0, then you have a y=e, which is a constant function. But this is a degenerate case, and is what is called critically stable value, meaning that decreasing the value to be slightly negative, or increasing to slightly positive will result in trajectories that are markedly different. This is depicted below:

The blue curve gives a small positive value for k, and shows exponential growth. You could imagine the curve representing the number of Covid cases in a community, starting with a single person at time zero, and growing to about 140 after 10 days. This sounds bad, but still, only 140 people. So who cares, right? How bad would this be if left unchecked for a whole year? Just evaluating the function we get

This number is similar to the estimate for the number of protons and neutrons in the universe, so way beyond the number of people on earth. This just gives one a good idea of how devastating a pandemic can be.

Exponential Decay

So that all sounds dire, but what about the case for when k is negative? This is the orange curve above, and we can see that it rapidly decreases, and then dissipates to zero. Starting with 140 cases, in only 10 to eradicate the 140 cases. This is the good news of exponential growth. But what influences the value of K? Can we change K? I will get to that in a bit. First let me tell you of another system that is governed by exponential growth and can, with care be controlled.

Nuclear Fission

I don't know enough about nuclear engineering to do it justice, but in basic terms it involves chain reactions shown below:

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It is called a chain-reaction for obvious reasons — the three released neutrons then go on to hit another three U-253 atoms and it goes on its exponential growth. But this presumes that the subsequent neutrons will actually hit another uranium atom. What determines if it chain reaction happens or fizzles out? The geometry of the packing of the uranium atoms, and how densely they are packed determines this, and gives rise to the well known term critical mass.

Pack them loosely, like in a rock containing veins of uranium ore, and it sits there safely slowly decaying (though there have been cases where rocks containing uranium come close enough to critical-mass that they become warm!)

In a fission bomb, such as Little Boy, there is are two pieces of uranium-235 that do not have critical mass when they are apart, but when quickly brought into a packed configuration become critical.

Nuclear reactors are really tricky, because in normal operation they want to keep a constant reaction going, which is just like the choice of k=0 from the graph. If k becomes too negative, the reaction fizzles out, which is great for controlled shutdown, but bad for power generation. If k becomes too large, you have a melt-down.

A successful nuclear reactor will use control-rods, which can be inserted or removed to reduce or increase the rate of chain reaction within the core.

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How does this relate to Covid-19?

Well we know the dynamics of the virus spread is just like our exponential growth equation. But the value for k depends on a lot of factors. Epidemiologists and virologists use a mathematical constant R₀ to describe the number of people that an infected person will pass (on average) the infection to. While can be measured as an indication of the speed of spread, it is not an intrinsic quality of Covid — it depends on innate properties of the virus, but also on environmental properties, and social behavior.

To model how the virus spreads in the community consider a set of people interacting with other people, such as in the diagram below.

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The following interactions might occur:

• Red/Red: In this case infected people cause little to no harm to other infected people (though I guess it might increase their viral load, and cause worse health outcomes).
• Red/Blue: This has the potential for spread. This is where it is crucial to avoid contact (safe distance), use of masks (ideally both, but still useful if only one of the people).
• Red/Green: Indications seem to indicate that this would have >90% chance of not transmitting the virus
• Red/Light-Red: Indications seem that immunity for the light-red person is likely to last a couple of years.

Red/Blue interactions are where our behavior can come into play to reduce the value of R₀. By doing what the CDC tells you, you are in fact being that control-rod and quenching the horrible spread.

Herd immunity will be the condition by which we no longer have to worry about the virus any more. We can get this the hard way, though having people go through this horrible illness, and recovering. They may never be the same and have health issues for life. Vaccination is the next best, but requires a lot of logistics. The easiest and most effective is to simple shelter in place and have it all burn itself out. But this can't happen while we have uneducated fools refusing to do their part.

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