Estimation of the lifetime distribution from fluctuations in Bellman-Harris processes

The growth of populations without interactions can often be modeled by branching processes where each individual evolves independently and with the same law. In Bellman-Harris processes, each individual lives a random time and is then replaced by a random number of offspring. We are interested in the estimation of the parameters of this model. Our motivation comes from the estimation of cell division time and we focus on Gamma distribution for lifetime and binary reproduction. The mean of the lifetime is closely related to the growth rate of the population. Going farther and describing lifetime variability from fixed time observations is a challenging task, due to the complexity of the fluctuations of non-Markovian branching processes. Using fine results on these fluctuations, we describe two time-asymptotic regimes and explain how to discriminate between them and estimate the parameters. Then, we consider simulations and biological data to validate and discuss our method. It allows to determine single-cell parameters from time-resolved measurements of populations without the need to track each individual or to know the details of the initial condition. The results can be extended to more general branching processes.