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

The rates matter

In the last series, we introduced the idea of a reaction network, which is a collection of processes that are moving individuals between the compartments. What was missing there, though, was any kind of description of how fast the reactions run. This has a significant impact on the overall behavior of the network.

Let’s make this point concrete with an example. Take the SIS model, which is another addition to the menu of models that we gave in an earlier article. This stands for Susceptible – Infected – Susceptible, a model that applies to the common cold. Here there are just two containers, Susceptible and Infected, and two reactions: infection, which transfers people from Susceptible to Infected, and recovery, which goes the other way.

$$\mathrm{Susceptible} \xrightarrow{\mathit{Infection}} \mathrm{Infected} \xrightarrow{\mathit{Recovery}} \mathrm{Susceptible}$$

We’ve got two opposite processes taking place at the same time. Each individual cycles through the two containers — they get sick, recover, get sick, recover…

So what’s the net effect of this tug-of-war between the two processes?

Generally speaking, the population will settle towards some kind of balance, where a certain percentage of the population is susceptible and a certain percentage is infected. People keep cycling between the two compartments, but the in each compartment stabilizes.

Note: in the real world, conditions change, e.g. the temperature across seasons, which affects the rate at which the infections are transmitted. Equilibrium will only be reached if there are no external factors which alter the (relative) rates of the reactions. When changes do occur, the “equilibrium moves to a new state”, as it were, with different percentages in the containers.

Now, assuming no external changes, what are the equilibrium percentages for Susceptible and Infected?

Answer: it depends on the relative rates of the Illness and Recovery processes.

If illness happens is quickly and recovery take a long time, then at equilibrium most people will be ill. Conversely, if illness is slow and recovery is quick, most people will be healthy.

This motivates us to pin down more precisely what is meant by the rate of a reaction within the network. That is our subject for next time.

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