The SIR model 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 and 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.
The SIR model is dynamic in three senses 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 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 susceptible individuals 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 susceptible compartment.
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.
Transition rates For the full specification of the model, the arrows should be labeled with the transition rates between compartments.Between S and I, the transition rate is β I, where β is the contact rate, which -roughly speaking - takes into the account the probability of getting the disease in a contact between a susceptible and an infectious subject.Between I and R, the transition rate is ν (simply the rate of recovery). If the duration of the infection is denoted D, then ν = 1/D, since an individual experiences one recovery in D units of time.It is important to stress that here we assume that the permanence of each single subject in the epidemic states is a random variable with exponential distribution. More complex and realistic distributions (such as Erlang distribution) can be equally used with few modifications.
Bio-mathematical deterministic treatment of the SIR model
The SIR model without vital dynamics A single epidemic outbreak is usually far more rapid than the vital dynamics of a population, thus, if the aim is to study the immediate consequences of a single epidemic, one may neglect the birth-death processes. In this case the SIR system described above can be expressed by the set of ordinary differential equations