Why SIR Produces Exponential Growth

Why the SIR Model Produces Exponential Growth Early in the Process

Overview

Earlier, we saw how exponential growth led to a much faster increase in the number of infected than linear growth. We now show why the SIR Model produces exponential growth in the number infected and not linear growth, or does so early in the process. Recall the equation for the number of infected in the SIR model.

I(t+1) = I(t) + (Pcontact P infect S(t)/POP – Precover)I(t)

Early in the process, say day one, the number infected is small S is approximately equal to POP. We can then write the equation as

I(1) =I(0) + [Pcontact Pspread – Precover]I(0)

We can write this as follows:

I(1) = I(0)[1+Pcontact Pspread – Precover]

In time 2, the number of infected can be written as:

I(2) = I(1)[1+Pcontact Pspread – Precover]

Note that

I(2) = I(0)[1+Pcontact Pspread – Precover]²

And, therefore, the number infected in time t can be written in the following form:

I(t) = I(0)[1+Pcontact Pspread – Precover]^t

This is an exponential function where the rate of growth G equal [1+Pcontact Pspread – Precover]. For example, if we assume two people are infected initially, I(t) = 2G^t , an exponential function with a growth rate of G. The graph below shows the number of infected for the first 80 days. assuming a growth rate of 1.25. After 15 days, only 45 people are infected. After 30 days nearly 1300 are infected. By day 60, over a million are infected and by day 80, more than 90 million are infected.

This exponential growth is often modeled in so-called S-curves (named for the shape of the curves). And this model appears to apply to the confirmed case, death, and recovery rates associated with coronavirus. (Note that the date information is in a day / month format in the below graphs)

Contributors: Michael Lin, Jonathan Hochberg