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Mathematical modelling in epidemiology


Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic and help inform public health interventions. Models use some basic assumptions and mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of possible interventions, like mass vaccination programmes.

Early pioneers in infectious disease modelling were William Hamer and Ronald Ross, who in the early twentieth century applied the law of mass action to explain epidemic behaviour. Lowell Reed and Wade Hampton Frost developed the Reed–Frost epidemic model to describe the relationship between susceptible, infected and immune individuals in a population.

Models are only as good as the assumptions on which they are based. If a model makes predictions which are out of line with observed results and the mathematics is correct, the initial assumptions must change to make the model useful.

An infectious disease is said to be endemic when it can be sustained in a population without the need for external inputs. This means that, on average, each infected person is infecting exactly one other person (any more and the number of people infected will grow exponentially and there will be an epidemic, any less and the disease will die out). In mathematical terms, that is:

The basic reproduction number (R0) of the disease, assuming everyone is susceptible, multiplied by the proportion of the population that is actually susceptible (S) must be one (since those who are not susceptible do not feature in our calculations as they cannot contract the disease). Notice that this relation means that for a disease to be in the endemic steady state, the higher the basic reproduction number, the lower the proportion of the population susceptible must be, and vice versa.


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