Exponential Growth In the SIR Model

Exponential Growth in SIR Model

The SIR Model produces exponential growth in the number infected and not linear growth, or does so early in the process. Click on the image to the left to see a simulation that shows how exponential growth arises in a simulation.

Mathematical Analysis

To see the mathematical reasoning that drives the exponential growth, we first must 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, when the number infected is small S ~ POP, so we can write the equation for the number infected, recall this is I(1), as:

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

Simplifying to:

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

On day 2, the number of infected can be written as follows:

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

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

And, therefore, the number infected in day t:

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

To reduce the notation, we can define the rate of growth G to equal:

[1+Pcontact Pspread – Precover]

Then, if we assume two people are initially infected, that is, I(0) = 2, the equation above can be written as:

l(t) = 2G^t

The figure below shows the number of infected for 60 days for different growth rates.