David A. Tanzer, August 17, 2020, in unit Epidemic Models 2.

What is meant by the rate of a reaction?

Let’s take a simple model, with just one reaction:

$$\mathrm{Sick} \xrightarrow{\mathit{Recovery}} \mathrm{Healthy}$$

Suppose that everyone starts out ill, and eventually recovers.

Suppose that, on any given day, there is a 10% chance that a sick person will recover. So, viewing the process of recovery as a whole, it will be like a popcorn process, where each individual recovery is like the popping of a kernel.

Question: If the events are random, then at any moment in time, how can we speak of the rate at which the kernels are popping?

Suppose that it is a huge vat, and that these kernels are slow to pop, so that it will take two hours for the whole vat to be popped. After initial heating, the rate popping is fast, but tapers off as there are fewer and fewer kernels remaining that have not popped.

Now, due to the randomness, we can’t say exactly what the speed of popping is at an exact moment in time.

But we can estimate it by breaking time into segments of 10 seconds, and counting the number of pops within each of these windows. Dividing these by 10 will give an estimate for the rate of popping per second.

We could use a one second window, but that would introduce more noise in our estimate of the rate.

Alternatively, we could treat the whole cooking process as one trial of an experiment. We could repeat this 1000 times. To estimate the rate at which kernels are popping 20 seconds into the experiment, we could could count the number of kernels popped between 20 and 21 seconds, and average these over all 1000 trials.

Next time we will begin to analyze the reaction rates.

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