Epidemic Models 2, Part 5

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David A. Tanzer, August 17, 2020

Analysis of reaction rates – second approach

In the previous article, we analyzed a simple system with a simple reaction:

$$\mathrm{Unpopped} \xrightarrow{\mathrm{Cooking}} \mathrm{Popped}$$

There we saw that the rate of the reaction is proportional both to a constant associated with the reaction, and to the count in compartment Unpopped. This will hold for all reactions that take a single input connection, including recovery in the SIR model:

$$\mathrm{Infected} \xrightarrow{\mathrm{Recovery}} \mathrm{Recovered}$$

  • rate(recovery) = RateConstant(recovery) * count(Infected)

Now what about

$$\mathrm{Susceptible} + \mathrm{Infected} \xrightarrow{\mathrm{Infection}} 2 \ \mathrm{Infected}$$

Here the rate will naturally depend on a rate constant for the reaction, but will also depend on the population counts at both of the inputs, i.e., on both count(Susceptible) and count(Infected).

In fact, under certain assumptions, the rate of the reaction will be obtained by multiplying the counts of all the input populations:

  • rate(infection) = RC(infection) * count(Susceptible) * count(Infected)

Why?

First, for some intuition. If count(Susceptible) = 0, there is nobody who can get infected, and so rate(infection) – which is exactly what the formula tells is.

Similarly if count(Infected) = 0, nobody is infected and so of course rate(infection) – again confirmed by the formula. More generally, if either count(Susceptible) or count(Infected) are small, we expect rate(Infection) to be small, which is again confirmed by the formula.

For the justification of the formula in general, suppose that we hold the Susceptible population count fixed, and we double the count of infected. Then each susceptible person becomes twice as likely to encounter an infected person, and so the rate of infections will double. Similarly, if we fix Infected, and double Susceptible, then each Infected person becomes twice as likely to infect a susceptible (within a given time period) and so do rate(infection) doubles.

This rules for the rate of a reaction is called its kinetics, and this particular formula is called the mass action kinetics.

Next time we take a slight digression into a very similar story in the area of chemical reactions.

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