Compartmental models
A major utility of compartmental modeling is that these models are relatively simple and easy to implement. There are a number of assumptions involved in their construction, which have implications for the types of results or information you may gain from using them.
First, compartmental models are focused on populations, not individuals. Within each compartment, all people have the same properties or characteristics. In other words, the population is homogeneous, and all people are well-mixed, meaning everyone has an equal chance of exposure or infection.
Compartmental models are deterministic; that is, given the same inputs, they produce the same results every time. They are able to predict the various properties of pathogen spread, can estimate the duration of epidemics, and can be used to understand how different situations or interventions can impact the outcome of pathogen spread.
Compartmental models are governed by a system of ordinary differential equations (ODEs) that track the population as a function of time, stratifying it into different groups based on risk or infection status. The models track the number of people using the following compartments:
- Susceptible
- Individual is able to become infected.
- Exposed
- Individual has been infected with a pathogen, but due to the pathogen’s incubation period, is not yet infectious.
- Infectious
- Individual is infected with a pathogen and is capable of transmitting the pathogen to others.
- Recovered
- Individual is either no longer infectious or “removed” from the population.
Which categories are included into specific models is dependent on the pathogen of interest and its transmission properties. For example, a disease without an incubation period, such as Staphylococcus aureus, is represented by an SIR model and a disease that has lifelong infectiousness, such as Epstein-Barr virus, is represented by an SI model.
SIR model
The most commonly used framework is the SIR model, where individuals move from a susceptible class to an infectious class and then finally to a recovered (or removed) class. If immunity wanes, the model becomes an SIRS model as individuals move back into the susceptible class.
SEIR model
Some pathogens include an extended period of incubation, where individuals have been exposed but are not yet infectious; the framework then becomes a SEIR model, where individuals move through an exposed class prior to the infectious state. Waning immunity in this framework would then create a SEIRS model.
SI model
The final framework of compartmental models is the SI model, where individuals experience lifelong infections and never move into the recovered state. For some pathogens, individuals may become susceptible to infection again, and the framework becomes an SIS model. Compartmental models can be modified in numerous ways, with additional states or compartments added as needed; note that the following pages only address the common frameworks.
Model parameters
The following table describes the parameters used by the ODEs for each of the compartmental models described further in this section.
| Symbol | Description |
|---|---|
| \(S\) | Number of susceptible individuals |
| \(E\) | Number of exposed (infected, not infectious) |
| \(I\) | Number of infectious individuals |
| \(R\) | Number of recovered individuals |
| \(\beta\) | Transmission (infection) rate |
| \(\sigma\) | Incubation (progression) rate |
| \(\gamma\) | Recovery rate |
| \(\xi\) | Rate of loss of immunity (SEIRS only) |
| \(\mu, \nu\) | Birth and death rates (vital dynamics) |