Population growth in discrete time: a renewal equation oriented survey

TL;DR

This paper surveys population models based on renewal equations focusing on newborns, highlighting their advantages in lower dimensionality and connection to key epidemiological metrics, and relates them to traditional models.

Contribution

It provides a comprehensive analysis of renewal equation models, linking them to classical structured-population models and characterizing their long-term behavior.

Findings

Renewal equations simplify population modeling by focusing on newborns.

The paper characterizes the asymptotic behavior of population models using renewal equations.

It establishes a relationship between renewal models and traditional structured-population models.

Abstract

Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individuals and the model specifies the production of offspring as a function of age. This has two advantages: (i) as a rule, there are far fewer birth states than individual states in general, so the dimension is often low; (ii) it relates seamlessly to the next-generation matrix and the basic reproduction number. Here we start from the renewal equation for the births and use results of Feller and Thieme to characterise the asymptotic large time behaviour. Next we explicitly elaborate the relationship between the two bookkeeping schemes. This allows us to transfer the characterisation of the large time behaviour to traditional structured-population models.