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

The Dynamics of SIR

Recall that for dynamics, we want to get at the way that the compartments S, I, R evolve over tune,

So far, we’ve defined the rates at which the reactions fire.

So now let’s summarize the rates for the SIR model:

• Infection: Susceptible + Infected —> 2 Infected (with rate constant 1)
• Recovery: Infected —> Recovered (with rate constant 2)

Then:

• Rate(infection) = RC1 * Count(Susceptible) * Count(Infected)
• Rate(recovery) = RC1 * Count(Infected)

It’s but a small step from this to the goal.

Recall the metaphor of the pumps. The equations above say fast the pumps are running, given the population counts.

Take: recovery: Infected —> Recovered.

Every time the pump fires, Infected goes down by one and recovered goes up by one. So if the pump is firing 10 times per minute, the infected population is going down by 10 per minute, and recovered is going up by 10 per minute.

Now take

$$S + I \xrightarrow{\mathrm{Infection}} I + I$$

Every time this reaction fires, S goes down by one and I goes up by one.

The interesting case here is I, because the infection pump is making it grow, while the recovery pump is making it shrink.

So the rate at which Infected increases will be the rate of infection minus the rate of recovery.

• Flow(Susceptible) = – rate(infection)
• Flow(Infected) = rate(infection) – rate(recovery)
• Flow(Recovered) = rate(recovery)

Let’s say the same thing, using standard mathematical terminology. Let S(t), I(t), R(t) be the counts of susceptible, infected and recovered individuals at time $t$.

Then Flow(Susceptible) is the derivative S'(t), and similarly for the other containers.

So:

• S'(t) = -rate(infection)
• I'(t) = rate(infection) – rate(recovery)
• R'(t) = rate(recovery)

Inserting formulas from before, we get:

• S'(t) = -RC1 S(t) I(t)
• I'(t) = RC1 S(t) I(t) – RC2 I(t)
• R'(t) = RC2 I(t)

This is a system of three differential equations, called the rate equation for the SIR model.

The same line of reasoning can be used to determine the rate equations for any reaction network.

Next time we will touch upon the comparison between continuous and discrete models for reaction networks.

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