Index of Seasonal Dispersion (IoSD)

A Measure of Seasonality

We need a metric to describe and compare seasonality across settings. Here we use the variance to mean ratio of the seasonal pattern, also known as the index of dispersion. Since the variance scales with the mean, taking the ratio gives us a scale-free measure that we can use to compare the degree of seasonality.

IoSD - EIR

Here, we define the index of seasonal dispersion (IoSD) for malaria exposure (i.e. for the EIR, \(E(t)\)).

We define the seasonal pattern function, \(S(t)\), as a function of time such that

• \(S(t+365) = S(t)\); and

• the average over a year is 1

\[\frac{1}{365}\int_0^{365} S(t) dt = 1\]

Let \(\bar E\) denote the average annual EIR, and \(E(t)\) denote the daily EIR at time \(t\), and we get:

\[E(t) = \bar E \; S(t).\]

In effect, \(S(t)\) is giving a weight to days of the year in a way that doesn’t change the mean.

We use the index of dispersion – the variance-to-mean ratio – as a simple way of describing the average dispersion of the seasonal pattern. The index of dispersion of the seasonal pattern is defined as \[ \left(\frac{1}{\bar E}\right)^2 \int_0^{365} \frac{(\bar E - E(t))^2}{365} dt = \int_0^{365} \frac{(1 - S(t))^2}{365} dt \]

IoSD - PR

and prevalence (i.e. for the PR, \(x(t)\)).