Fourier analysis can provide policymakers useful information for analysing the pandemic behaviours. This paper proposes a Fourier analysis approach for examining the cycle length and the power spectrum of the pandemic by converting the number of deaths due to coronavirus disease 2019 in the US to the frequency domain. Policymakers can control the pandemic by using observed cycle length whether they should strengthen their policy or not. The proposed Fourier method is useful for analysing waves in other medical applications.

The mathematical model we consider here is the classical Bienaymé–Galton–Watson branching process modified with immigration in the state zero.

We study properties of the waiting time to explosion of the supercritical modified process, i.e. that time until all beginning cycles which die out have disappeared. We then derive the expected total progeny of a cycle and show how higher moments can be computed. With a view to applications the main goal is to show that any statistical inference from observed cycle lengths or estimates of total progeny on the fertility rate of the process must be treated with care. As an example we discuss population experiments with trout.

The random splitting model of Lloyd and Williams (1988) is generalised, to allow first beta splitting distributions and secondly, discrete splitting distributions. Analogues of some earlier results are developed. In particular, the equality of the average value of the largest split and the probability that this is achieved at the first split continues to hold.