In general, for a changing incidence rate (IR(x) at time x):
\[ Risk = 1 - e^{-\int_0^t IR(x) \, \mathrm{d} x} \]
For a constant incidence rate (IR):
\[ Risk = 1 - e^{-(IR)(\Delta t)} \]
Figure 4-3 of Epidemiology: An Introduction
Follow-up of 1000 people with a constant mortality rate of \( \frac {11} {1000} \) per year.
Green line: Assuming no population shrinkage
\[ y = 1000 \times \frac {11} {1000} \times \Delta t \]
Red line: Taking into account the population shrinkage
\[ y = 1000 \times (1 - e^{-\frac {11} {1000} \Delta t}) \]
ggplot(data = data.frame(x = 0:50), aes(x = x)) + stat_function(fun = function(t) {
1000 * 11/1000 * t
}, col = "green", lwd = 2) + stat_function(fun = function(t) {
1000 * (1 - exp(-1 * 11/1000 * t))
}, col = "red", lwd = 2) + xlab("Time in years") + ylab("Cumulative number of deaths (risk if devided by 1000)")
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