Mathematical epidemiology

Mathematical epidemiology uses mathematical models and statistical inference to study the occurrence, transmission, and control of disease in populations.

Calendar time locates an event on a shared external timeline, in contrast with an individual's age or the time elapsed since an individual event.

Incidence measures the occurrence of new cases in a population during a specified period.

The infection age of an infected individual is the time elapsed since that individual's infection.

An imported infection is acquired outside the population or observation system being modelled and enters it as an external source of incidence.

An infectivity profile describes how an infected individual's transmission rate varies with infection age.

The generation interval is the time between infection of a source individual and infection of a secondary case caused by that source.

The generation-interval distribution gives the probability law of the generation interval. In discrete time, its probabilities $g_{\tau }$ form a nonnegative sequence summing to one.

The infectious disease renewal equation expresses current incidence as a convolution of past incidence with an infectivity profile, multiplied by a time-varying transmission level.

For incidence $I_t$ and discrete generation-interval probabilities $g_{\tau }$, the total infectiousness at time $t$ is ${\Lambda }_t={\sum }_{\tau \ge 1}{g}_{\tau }{I}_{t-\tau }$.

A time-varying reproduction number describes transmission at a specified calendar time while allowing transmission conditions to change during an epidemic.

The instantaneous reproduction number freezes the transmission conditions at calendar time $t$ and counts the expected secondary infections produced over a complete infectious lifetime under those conditions.

The case reproduction number is the expected number of secondary infections actually generated by a person infected at time $t$, allowing transmission conditions to change during that person's infectious lifetime.