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# Force of infection

In epidemiology, force of infection (denoted λ) is the rate at which susceptible individuals become infected by an infectious disease. Because it takes account of susceptibility it can be used to compare the rate of transmission between different groups of the population for the same infectious disease, or even between different infectious diseases. That is to say, λ is directly proportional to β; the effective transmission rate.

$\lambda = \frac {\mbox{number of new infections}} {\mbox{number of susceptible persons exposed} \times \mbox{average duration of exposure}}$

Clearly, there are all kinds of practical difficulties with such a calculation, since not all new infections are reported, and it is often difficult to know how many susceptibles were exposed.

However, λ can be calculated for an infectious disease in an endemic state if homogeneous mixing of the population and a rectangular population distribution (such as that generally found in developed countries) is assumed. In this case, λ is given by:

$\lambda = \frac {1} {A}$

where A is the average age of infection.

To understand this relation, think of A as the average time spent in the susceptible group (remembering that for most infectious diseases, before infection an individual is susceptible to the disease but afterwards they have acquired immunity). The rate at which susceptibles are infected, therefore, is simply 1/A (since rate is 1/time). Recall that λ is also the rate at which susceptibles are infected. This gives us the relation above.

The advantage of this method of calculating λ is that data on the average age of infection is very easily obtainable from doctors' reports, even if they are not reporting all cases of the disease.