| • Science | • People | • Locations | • Timeline |
Standard convention labels these three compartments S (for susceptible), I (for infectious) and R (for recovered). Therefore, this model is called the SIR model.
This is a good, simple, model for many infectious diseases including measles, mumps and rubella.
The letters also represent the number of people in each compartment at a particular time. To indicate that the numbers might vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t) and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.
As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptibles falls rapidly as more of them are infected and thus enter the infectious and recovered compartments. The disease cannot break out again until the number of susceptibles has built back up as a result of babies being born into the compartment.
The SIR is also dynamic in the sense that individuals are born susceptible, then may acquire the infection (move into the infectious compartment) and finally recover (move into the recovered compartment). Thus each member of the population typically progresses from susceptible to infectious to recovered. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments.
For the full specification of the model, the arrows should be labelled with the transition rates between compartments.
Between S and I, the transition rate is λ, the force of infection, which is simply the rate at which susceptible individuals become infected by an infectious disease.
Between I and R, the transition rate is δ (simply the rate of recovery). If the duration of the infection is denoted D, then δ = 1/D, since an individual experiences one recovery in D units of time.
For many infections there is a period of time during which the individual has been infected but is not yet infectious himself. During this latent period the individual is in compartment E (for exposed).
For many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta or through colostrum). This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.