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# Compartmental models in epidemiology

A population comprises a large number of individuals, all of whom are different in various fields. In order to model the progress of an epidemic in such a population this diversity must be reduced to a few key characteristics which are relevant to the infection under consideration. For example, for most common childhood diseases which confer long-lasting immunity it makes sense to divide the population into those who are susceptible to the disease, those who are infected and those who have recovered and are immune. These subdivisions of the population are called compartments.

## 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, 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 two 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 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.

### Transition rates

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 infection rate, 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.

## Mathematical Treatment of the SIR model

The SIR system described above can be expressed by the following set of differential equations:

$\frac{dS}{dt} = - \lambda S I$
$\frac{dI}{dt} = \lambda S I - \delta I$
$\frac{dR}{dt} = \delta I$

This system is non-linear, and so does not admit a generic analytic solution. Nevertheless, significant results can be derived analytically.

Firstly, we note that the basic reproductive rate for the system is given by

$R_0 = \frac{\lambda}{\delta}$

Then dividing the first equation by the third and integrating gives

$S(t) = S(0) e^{-R_0(R(t) - R(0))}$

So in the limit as time goes to infinity, the proportion of recovered individuals obeys the transcendental equation

$R_{\infty} = 1 - S(0)e^{-R_0(R_{\infty} - R(0))}$

Consideration of this equation shows that generically, at the end of an epidemic, not all individuals have recovered, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infected individuals rather than an absolute lack of susceptibles.

## Elaborations on the basic SIR model

### The SEIR model

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).

### The MSIR model

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.

### Carrier state

Some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably Mary Mallon, who infected 22 people with typhoid fever. The carrier compartment is labelled C.

## The SIS model

Some infections, for example the group of those responsible for the common cold, do not confer any long lasting immunity. Such infections do not have a recovered state and individuals become susceptible again after infection.