From data to model
Epidemiological data takes many forms. Here, we focus on two common types of data that can be used to estimate rates and probabilities for disease models.
A crude rate is an estimate of the exponential rate \(\lambda\), assuming the process is memoryless.
Rates from data should be calculated as the number of events during a time period divided by the total person-time at risk during that period. While events like new infections or births are relatively easy to count, the total person-time at risk is often not directly observed and must be estimated.
There are different ways to go about estimating person-time, but a common approach is to use the at-risk population size at the midpoint of the observation period multiplied by the length of the observation period. Again, crude rates estimate rates and should be used as rates in disease models.
Alternatively, data may report the fraction of individuals that transitioned from one state to another during a specified time period. This data represents a transition probability over the specified time period, not a rate. However, we can easily convert this probability to a rate using the formulas derived above:
where \(q = n/N \in [0, 1)\) is the proportion of individuals that transitioned, \(n\) out of \(N\) initially at risk, during the time period \(T\). Note that if all individuals transitioned, \(q=1\), then the rate \(\hat{\lambda}\) is undefined over \(T\).