Compartmental disease models are some of the most widely used models of infectious disease. In these models individuals (or portions of an individual) are considered to be in a compartments based on their disease status. By convention, commonly used compartmental models are designated by the first letters of the compartments and the order people move between them. So, the model for a disease where people start out Susceptible, become Infectious after catching the disease, and then after some time clear the disease and become Susceptible again would be an SIS model. The four most commonly used compartments are Susceptible, Exposed, Infectious, and Removed. Susceptible contains those individuals who can still be infected with disease. Exposed contains individuals who have been infected with the disease, but have not yet become infectious themselves. Infectious contains those individuals who are infected and infectious. Removed (sometimes called Recovered) contains those individuals who have been removed from the system either through death or immunity. The disease state of a population is speccified by knowing the nuber of individuals (or proportion of the population) that is in each compartment.
There are two main types of compartmental models: deterministic and stochastic. In deterministic model, rates of movements between compartments are used to represent the processes by which individuals are infected, become ill and eventually recover from disease. These rates specify a set of differential or difference equations, which can be evaluated to give that state of the model at a future time point. In deterministic models compartments need not contain an integer number of individuals.
Deterministic models only give the expected behavior of the disease in a population under a set of assumptions about the natural history and epidemiology of the disease. Often we would also like to know how the disease behaves in more extreme cases, and what the distribution of this behavior is. Stochastic models provide a method for addressing this issue. In a stochastic model, whole individuals are moved between compartments based on a random draw from some specified distribution. To get informative results from a stochastic compartmental model it is necessary to do hundreds, or even thousands of experiments.
Compartamental models also may have "closed" or "open" populations. In closed populations the set of individuals represented by the model is static, no one comes in and no one comes out. It is of note that in such populations any disease that confers permanent immunity after infection will eventually become extinct. In an open population there is continuous immigration and emigration from the population, usually via birth and death.
STEM comes with deterministic implementations for three commonly used compartmental models: SI(S), SIR(S), and SIER(S). These models are implemented as having open populations, but closed populations can be easily simulated by setting the immigration/emigration rate to zero. Which model is appropriate for a modeling project depends on both the disease of interest and the timescale that we are interested in modeling our results.
SI(S) are useful for modeling diseases with a short latent period (the time between being infected and becoming infectious) that either confer no long term immunity (in the case of SIS models) or result in permanent infection (in the case of SI models). Diseases of this type include the common cold and many macro-parasite infections. Susceptible individuals are infected at the rate of βI, at which time they lose infectiousness at the rate of γ. Mathematical details of this model can be viewed here.
SIR(S) models are useful for modeling diseases with a short latent period that confer immunity after infection, whether permanently (an SIR model), or temporarily (an SIRS model). Such a model might be useful for modeling diseases such as influenza or chicken-pox. Susceptible individuals are infected at the rate of βI. They then loose infectiousness at the rate of γ and then become immune to further infection. Immunity is lost at the rate of σ. Mathematical details of this model can be viewed here.
SEIR(S) models are useful for modeling diseases that have a long latent period in which infected individuals are not themselves infectious. Measles, influenza and smallpox are all diseases that might be appropriately characterized by this model. Susceptible individuals are infected at the rate of βI. These individuals become infectious at the rate of ε, and then loose infectiousness at the rate of γ, becoming immune to further infection. Immunity is lost at the rate of σ. Mathematical details of this model can be viewed here.