Model Calculations Background
The SIR Model Can Be Represented with Mathematical Equations
When modeling a large population, it’s difficult to keep track of each individual like in the “Outbreak Simulator” (“SIR Model Basics” tab). So, instead of tracking changes at the individual level, we can use mathematical equations to simulate changes at the population level.
The “Epidemic Simulator” uses a set of mathematical equations to represent population-level processes in the SIR model. These equations describe the two main ways that individuals move between groups in the population:
- Transmission: When susceptible individuals become infected, they move from the susceptible group (S) to the infectious group (I).
- Recovery: When infectious individuals recover, they move from the infectious group (I) to the removed group (R).
These same movements were introduced in the “SIR Background” section (“SIR Model Basics” tab). The figure shows how these movements are represented by mathematical expressions in our “Epidemic Simulator” model.
Expression 1: (t times S over N times I) This math expression models how individuals move from susceptible to infectious (S to I).
Expression 2: (r times I) This math expression models how individuals move from infectious to recovered (I to R).
How many individuals move from susceptible to infectious (S to I) on a given day depends on:
- t: the transmission rate, which represents the average likelihood, per day, that a susceptible individual becomes infected (constant in our model). A larger t means transmission is more likely.
- N: the total number of individuals in the population (constant)
- S: the number of susceptible individuals (varies by day)
- I: the number of infectious individuals (varies by day)
How many individuals move from infectious to removed (I to R) on a given day depends on:
- r: the recovery rate, which represents the average likelihood, per day, that an infectious individual recovers (constant in our model). A larger r means recovery is more likely.
- I: the number of infectious individuals (varies by day)
These Equations Model Changes in Each Group
Our model uses these equations to calculate how the numbers of individuals in each group (S, I, and R) change from one day to the next.
- Positive changes reflect individuals moving into a group, which increases the number of individuals in that group.
- Negative changes reflect individuals moving out of a group, which decreases the number of individuals in that group.
Changes in the number of susceptible individuals ( S )
S changes only when some individuals in S move to I (some susceptible individuals become infected). This decreases S by .
Change in S = − (How many in S move to I)
individuals per day (varies depending on S and I)
Changes in the number of infectious individuals ( I )
I changes based on two processes:
- Some individuals in S move to I (some susceptible individuals become infected). This increases I by .
- Some individuals in I move to R (some infectious individuals recover). This decreases I by .
Change in I = (How many in S move to I) − (How many in I move to R)
individuals per day (varies depending on S and I)
Changes in the number of removed individuals ( R )
R changes only when some individuals in I move to R (some infectious individuals recover). This increases R by .
Change in R = (How many in I move to R)
individuals per day (varies depending on I)
The “Epidemic Simulator” uses these equations to simulate how the number of individuals in each group will change each day. The numbers for each day are displayed in the “SIR Graph.”