SIR Model
The SIR Model
The SIR (susceptible - infected - recovered) model is the core model used to capture the spread of diseases. It makes a variety of simplifying assumptions - such as random mixing of individuals - in order to produce basic insights about the shape and speed of disease spread and the power of interventions.
In the model, individuals fall into one of there categories
Susceptible (S): people who do not have the virus but could catch it
Infected (I): people who have the virus and can spread it
Recovered (R): people who have had the virus, have recovered, and can no longer spread the virus.
The SIR model functions as core building block for the more advanced models developed by epidemiologists described on other pages.
The version of the SIR model, we describe assumes that time unfolds in discrete periods. We will refer to these as "days" but they could also be hours or weeks.
The term I(t) denotes the number of people infected on day t, S(t) denotes the number susceptible on day t, and R(t) denotes the number who are recovered by day t. We will assume a total population of 350 million people. Given that every person must be in one of the three categories, the following must hold for any time period.
I(t) + S(t) + R(t) = 350 million
When ten people have the disease and no one has recovered, this equation reads
10 + 349,999,990 + 0 = 350,000,000
When 100,000people have the disease and 10,000 people have recovered, this equation reads
100,000 + 349,890,000 + 10,000= 350,000,000
This equation:
Infected + Susceptible + Recovered = Population
is the key to the entire model. We represent it graphically below:
Model Dynamics
The number infected changes over time through contact. For the disease to spread a susceptible person must make contact with an infected person and the disease must spread. We represent these as two separate variables:
N-Contact: The number of people that each infected person meets in a day
(in most formal in the videos this is written as a probability but we write it as a number here as it is more intuitive.)P-Spread: The probability that when an infected and a susceptible person meet that the virus spreads.
To calculate the number of newly infected people on day t, we compute the product of four quantities
I(t): the number of infected people on day t
N-Contact: the number of people each meets
S(t)/Population: the probability that the person met is susceptible
P-spread: the probability the disease spreads
Newly Infected (t) = I(t) x N-Contact x S(t)/Population x P-Spread
Early in the process, there are very few infected people, so the numbrer of newly infected will be small. As the number infected increases, then the number of newly infected will increase. This causes exponential growth early in the process (click here to see the full mathematical argument). Once many people have the disease and have recovered, the number of susceptible decreases, which results in an S-Shaped curve for the number of people who have had the disease shown in the green line in our first chart.
Recovery
To complete the model, we must include recovery. To incluide this requires one final variable.
P-Recover:the probabiity a susceptible person recovers in a day
For COVID-19, recovery takes up to 14 days, so we can approximate this as 1/14. This construction assumes some people recover in a day and some take much longer. This sort of simplification is okay for a baseline model. Models used for policy contain far more detail.
We can then write the number of people who recover in time t as follows:
Recover on day t = P-Recover x Infected on day t
The Complete SIR Model
We can now write the complete SIR model which writes the number infected at time t+1 in terms of all of the other variables. Here's the full equation:
Infected (t+1) = I(t) + I(t) x N-Contact x S(t)/Population x P-Spread - P-Recover x I(t)
This looks complicated, but it is intuitive if we view it as an unpacking of this simpler equation:
Infected (t+1) = I(t) + Newly Infected (t) - Recovered (t)
The number of newly infected can also be represented graphically
The Basic Reproduction Number (R-Zero)
Using the complete SIR model equation we can determine whether the number infected will increase or decrease. The number will increase if the number newly infected exceeds the number who recover:
Newly Infected on day t > Recover on day t
We can write this formally as follows:
I(t) x N-Contact x S(t)/Population x P-Spread > P-Recover x I(t)
To solve for the Basic Reproduction Number (R-Zero), we note that early in the spread of the virus I(t) and R(t) are small because few people are infected and few people have recovered. This means that S(t) is approximately equal to the Population size. We can therefore simplify the equation as:
I(t) x N-Contact x P-Spread > P-Recover x I(t)
We can then cancel out the I(t) terms to get the following
N-Contact x P-Spread > P-Recover
By rearranging terms, we see that the virus spreads if
N-Contact x P-Spread/ P-Recover > 1
And it does not spread if the expression on the left hand side is less than one. By convention, we define the left hand side of the inequality as the basic reproduction number, R-zero. It equals the ratio of the probability of spreading and the probability of recovering.
R-zero = N-Contact x P-Spread/ P-Recover
The basic reproduction number, R-zero, corresponds to the growth rate of the virus. If R-zero > 1, the virus will spread because for each person who has the virus, the probability of passing on the disease to a new person exceeds the probability of recovery.
R-zero > 1 (spreads)
R-zero < 1 (does not spreads)
The R_Zero value is an infectious disease's basic reproduction number. It describes the intensity of the disease outbreak, denoting the number of cases on average an infected person will cause during their infectious period. The larger R-zero, the steeper the S-shaped curve of the number of people who get the virus:Data from the Michigan Public Health website about
R_Zero estimate values for COVID-19, as well as other infectious diseases are shown below. These data show the mean values for ranges given on the Public Health site. Different models calculate these values differently. Estimates depend on community and context.
COVID-19 averages at a R_Zero value of 2.74, and can have a more intense contagion potential than the flu, ebola, SARS, H1N1, and MERS. However, it on average has a lower R_Zero Value than the Mumps, Polio, Smallpox, and the Measles. Models Estimating R-zero.
There are many estimates of R-zero. Here is a link to one resource that contains models estimating R-zero.
Flattening the Curve
To phrase "flatten the curve" means to slow the spread of COVID-19.
The phrase "flatten the curve" means to slow the spread of COVID-19. Recall from above that R-zero captures how fast a disease spreads. Thus, the more that we reduce the value of R-zero, the flatter the curve. Look again at the expression for R-zero.
N-Contact x P-Spread/ P-Recover
To reduce the size of R-zero, we could (1) increase the rate of recovery (raise P-recover) (2) reduce the number of contacts people make (lower N-contac), or reduce the probability of spreading given a contact (lower P-spread). The first of these requires medical interventions. Most of the policies intended to flatten the curve focus on one of the other two alternatives.
Multiplicative Not Additive Power of Interventions
Models produce formal mathematical expressions that refine our intuition. The SIR model shows that reducing contacts (staying sequestered) and lowering the risk of spread (by washing hands and surfaces) have a multiplicative effect. If we lower each by 30% we reduce R-zero by more than half through the multiplicative effect: (0.7)(0.7) = 0.49.
Videos
Model Thinking Coursera Videos
Introductory videos on diffiusion, the SIR Model and contagion.
Contagion and Diffiusion (here)
Contagion Models and Tipping Points (here)
Troy Tassier Videos
Troy Tassier of Fordham University has made a series of four videos that explain the mathematics.
Understanding Coronavirus (COVID-19) Pandemic: Part 1 - SIR Models
In an effort to increase understanding of the COVID-19 pandemic I am creating a series of 10-15 minute videos that: 1) explain some basic topics of epidemiology (3 videos) and 2) examine the evolving data as the pandemic develops through the lens of these basic epidemiological principles. This is the first video on S-I-R models. Part 2 explains the concept of a reproduction number, or R. Part 3 explains herd immunity. Beginning with part 4, I interpret the evolving data using these three epidemiological concepts.
Understanding Coronavirus (COVID-19) Pandemic: Part 2 - Reproduction Number or R
In an effort to increase understanding of the COVID-19 pandemic I am creating a series of 10-15 minute videos that: 1) explain some basic topics of epidemiology (3 videos) and 2) examine the evolving data as the pandemic develops through the lens of these basic epidemiological principles. This is the second video. It covers the reproduction number, or R. Part 1 explains S-I-R models. Part 3 explains herd immunity. Beginning with part 4, I interpret the evolving data using these three epidemiological concepts.
Understanding Coronavirus (COVID-19) Pandemic: Part 3 - Herd Immunity
In an effort to increase understanding of the COVID-19 pandemic I am creating a series of 10-15 minute videos that: 1) explain some basic topics of epidemiology (3 videos) and 2) examine the evolving data as the pandemic develops through the lens of these basic epidemiological principles. This is the third video. It covers the concept of herd immunity. Part 1 explains S-I-R models. Part 2 explains the concept of a reproduction number, or R. Beginning with part 4, I interpret the evolving data using these three epidemiological concepts.
Understanding Coronavirus (COVID-19) Pandemic: Part 4 – Data and Flattening TWO Curves
In an effort to increase understanding of the COVID-19 pandemic I am creating a series of 10-15 minute videos that: 1) explain some basic topics of epidemiology (3 videos) and 2) examine the evolving data as the pandemic develops through the lens of these basic epidemiological principles. This is the fourth video. It explains the concept of flattening the curve and why we should look at data from the COVID-19 pandemic using a logarithmic scale. It discusses how we can identify the effect of social distancing in COVID-19 data without any advanced statistical knowledge. In previous videos, Part 1 explains S-I-R models. Part 2 explains the concept of a reproduction number, or R. Part 3 explains herd immunity.
Understanding Coronavirus (COVID-19) Pandemic: Part 5 – What if Italy hadn’t implemented social distancing?
In an effort to increase understanding of the COVID-19 pandemic I am creating a series of 10-15 minute videos that: 1) explain some basic topics of epidemiology (3 videos) and 2) examine the evolving data as the pandemic develops through the lens of these basic epidemiological principles. This is the fifth video. It answers the question: what would have happened in Italy without social distancing? In previous videos, Part 1 explains S-I-R models. Part 2 explains the concept of a reproduction number, or R. Part 3 explains herd immunity. Part 4 discusses the concept of flattening the curve, how to look at data using logarithmic scale, and how the effect of social distancing can be identified in epidemiological data.
Contributors: Thejas Suvarna