This model is a compartmental SIR model. Compartmental means that we place individuals into distinct compartments, according to some characteristics. We then only track the total number of individuals in each of these compartments. In the simplest model, the only characteristic we track is a person’s infection status. We allow for 3 different stages/compartments:
The SIR model is very basic. It could be extendend by introducing further compartments. For instance we could stratify according to gender, which would give us 2 sets of SIR compartments, one for males and one for females. Some of these extensions are implemented in some of the other apps.
In addition to specifying the compartments of a model, we need to specify the dynamics determining the changes for each compartment. Broadly speaking, there are processes that increase the number of individuals in a given compartment/stage, and processes that lead to a reduction. Those processes are sometimes called in-flows and out-flows.
For our system, we specify only 2 processes/flows:
As with the compartments, we can extend the model and allow other processes to occur. For instance we could allow for natural births and deaths, waning immunity, deaths due to disease, etc. We will do that soon. For now, we keep the model as simple as possible.
To allow us to simulate and study this model, we need to implement it on the computer. For that purpose, it is often useful to write the model as mathematical equations (this is not strictly needed, some computer simluation models are never formulated as mathematical models). One way to formulate the simple SIR model is with a set of ordinary differential equations. Each compartment/variable gets an equation. The right side of each equations specifies the processes going on in the system and how they change the numbers in each compartment via inflows and outflows. For the model described above, the equations look like this:
\[\dot S = -\beta SI\] \[\dot I = \beta S I - \gamma I\] \[\dot R = \gamma I\]
Since not everyone likes equations :) a very good way to describe compartmental models is by showing a flow diagram. Such a diagram consists of a box for each compartment, and arrows pointing in and out of boxes to describe flows and interactions. For the simple SIR model, the flow diagram looks as follows:
Flow diagram for simple SIR model.