> You can't have half person infected. At some point, everyone is infected and no new person will be infected.

Yes, of course. We all know that.

Can say the same thing for a lot of mathematical physics and other applied math. E.g., when a ball bounces, with the usual freshman physics view, there is a discontinuity in velocity so can't differentiate to get acceleration. General relativity considers mass-energy as a continuum and does differential geometry and uses differential equations, but a simple view of atoms is that they are points and not part of something continuous or we consider atoms as fuzz balls of uncertainty -- all that gets ignored in general relativity. We just saw a general relativity application: A star orbits a black hole at the center of our galaxy, and each orbit is more than 360 degrees around, like Einstein and not like Newton. That the star is made of lots of discrete atoms we ignore.

Such use of continuous and even differentiable several times functions to approximate discrete situations is standard. I saw that issue as a sophomore physics student. But there are no or not many famous research papers showing how using smooth functions to approximate does or does not work. Instead, people understand and accept that, when the discrete case has numbers large enough, we can use continuous approximations.

Heck, linear programming gets used in economic, production, and logistic situations where the real world items are discrete. Often works fine. Was the source of at least one Nobel prize in economics. Yes, when the numbers are small, especially just 0-1, the continuous math can flop badly, and we can encounter the grand question of P versus NP. But if doing resource allocation at General Motors producing 10 million cars a year, can do continuous math, and no one will care about half a water pump.

We can do predator-prey problems -- deer eating the grass, wolves eating the deer, eagles eating the dead deer and wolves, humans shooting the deer, raccoons eating the eagle eggs, the deer eating all the low forage, humans shooting the deer, etc. So, get a system of ordinary differential equations, right, for discrete deer, wolves, eagles, raccoons, and humans. Such equations need differentiable functions, all of which are continuous and differentiable approximations to discrete quantities that are not differentiable or even continuous. People have done that.

Can also attack such predator-prey problems with a continuous time, discrete state space Markov process (stochastic process).

Once in grad school, I mentioned to a prof that for predator-prey problems, the systems of differential equations and the Markov processes, thus, are approximating each other.

Indeed, once the Navy asked for an evaluation of the survivability of the US SSBN (ballistic missile firing) submarines under a special scenario of global nuclear war limited to sea. They wanted their results in two weeks. So it was lots of weapon types -- SSBNs, attack submarines, airplanes patrolling with magnetic anomaly detectors, planes that can drop sonar buoys and homing torpedoes, long range bombers, destroyer ships with good sonar, depth charges, and torpedoes, aircraft carriers, etc. with some numbers of each for each of Red and Blue. So, lots of Red and Blue things shooting at each other.

So for a solution I used a WWII paper by Koopman and saw a continuous time, multi-dimensional, discrete state space Markov stochastic process, wrote the code to generate sample paths, typically 500 for a particular case, using the Oak Ridge random number generator

X(n+1) = X(n) * 5^15 + 1 Mod 2^47

and the Navy got their results on time. Well, the results, the average decline of the SSBNs, looked continuous! Likely the expectation was continuous. So, we have a discrete stochastic situation that yields continuous expectation results.

Lesson: There are lots of continuous situations approximating discrete ones and expectations of stochastic discrete situation being continuous ones.

We are all supposed to know that lesson.

For my derivation for FedEx revenue, packages, and customers, each of those is discrete -- we know that. TV sets are also discrete yet nicely closely followed a logistic curve.

The virus infects people, who are discrete, but we would be foolish to ignore the role of an exponential which is continuous, differentiable, and infinitely differentiable, none of which the people are.

But as I gave at the end, can also get essentially the same result from a continuous time discrete state space Markov process, and there we can handle people one at a time, discretely, right to the end with the last person who gets infected and either gets well or dies at which time the process has an absorbing state and the exact result you mentioned -- no more continuous, asymptotic, limiting stuff.

So, as for the SSBN work, that Markov process addresses your concerns about the discrete nature of the virus and the last person to have the disease.

But for a curve for the virus, if you want to be careful down to the last person and not work with some curve that might have half a case, then there is no curve. Why? The spread of the virus is a stochastic process (random over time; at each time the value is a random variable) and not deterministic (if you will, special case of the general situation that nothing in the universe is deterministic). So, instead of a curve, all we get is a sample path of the process with discrete numbers. So, for analysis and prediction, we face many different sample paths. So, instead of looking at the blizzard of all of those sample paths, we consider expectations and confidence intervals, likely both of which are continuous, that we approximate with empirical averages, say, from running our model 500 times as I did for the SSBN fleet problem.

Lesson: Even if we handle the discreteness explicitly, due to the stochastic aspect, we are driven back to a continuous curve, an average approximating an expectation, that might end with 1/2 a case of the virus.