The Curve

Let a be the ratio of infected people, then 1-a is the ratio of uninfected people in a given population.

Is it reasonable to assume that infection is bilinear in infected and uninfected people? I'd say yes. It is linear in infected people, twice as many infected people in a given space can infect twice as much, but it is also linear in uninfected people, twice as many uninfected people in a given space can be infected twice as much.

Then we may wish to consider different degrees of infectiousness, designated by the letter r for rate.

Anyway, if a is a solution of the quadratic differential equation

a' = a(1-a)

then a(rt), t for time, is a solution of

a' = ra(1-a).

The former equation is called the logistic equation and its solution is called the logistic function, i.e.

a(t)=exp(t)/(1+exp(t)).

Let us verify that. We use the product rule and the chain rule. a(1-a)=-a²+a. The linear term of that results from the derivation of the first factor of the proposed solution, since the exponential function is its own derivative. And the quadratic term results from the derivation of the second factor, since x^-1'=-x^-2 and (1+exp(t))'=exp(t).

These days we see a lot of curves not unlike the Gaussian bell curve. For that reason let us write down the formula of a(1-a) as well

a'(t) = exp(t)/(1+exp(t)) - exp(2t)/(exp(2t)+2exp(t)+1)

and reintroducing r

a'(t) = r exp(rt)/(1+exp(rt)) - r exp(2rt)/(exp(2rt)+2exp(rt)+1).

So, for r=1 we get the following graphs, for greater values of r we only need to stretch and compress them, that is compress the former horizontally and stretch the latter vertically and compress it horizontally, and for smaller values of r accordingly.

What have we done here?

We have assumed that there is no immunity. We have assumed that we're looking at a situation like the crawling spread of an open tuberculosis infecting everyone.

Let's see

• Robert Louis Stevenson (died age 44)
• Bernhard Riemann (died age 39)
• Friedrich Schiller (died age 45)
• Niels Henrik Abel (died age 26)

and that are just the ones I remember. Wikipedia provides further examples.

This is no trifling matter:

If we observe the above curve then there is no immunity to the corona virus.

And if so, what does it mean to flatten the curve?

1. to gain time.
2. to facilitate the suppression of the public realisation of the true nature of the disease.

There are a great many things that allow for different assessments, but this one will become obvious at some point. I personally have suffered minor symptoms, i.e. blood in sputum and mucus, inflammation like pain in bowels and airways including the lungs, pleural effusion, coughing, wheezing, stabbing pain in the lungs, since the second half of January, that is for almost 3 months now. I haven't developed a fever so far and my sense of smell has actually become better. I'm not disabled in any way, more annoyed than anything. It would be interesting to know whether I'm still infectious after 3 months, but there is no way for me to find out other than doing my own medical research, which I won't. I guess, I have to console myself with the hope that I turn out to be more stable than the virus. And since it seems impossible to go back to how it was before, I try to look at possible positive changes.