A singular limit for an age structured mutation problem

TL;DR

This paper demonstrates that, under a specific scaling, the age-structured McKendrick model and the simpler ODE model for trait spread in cell populations become asymptotically equivalent, linking detailed and aggregate descriptions.

Contribution

It establishes a rigorous connection between age-structured PDE models and ODE models for trait dynamics in cell populations through a singular limit analysis.

Findings

Asymptotic equivalence between McKendrick equations and ODE models

Validation of simplified models for trait spread under certain scaling

Mathematical framework for linking detailed and aggregate population models

Abstract

The spread of a particular trait in a cell population often is modelled by an appropriate system of ordinary differential equations describing how the sizes of subpopulations of the cells with the same genome change in time. On the other hand, it is recognized that cells have their own vital dynamics and mutations, leading to changes in their genome, mostly occurring during the cell division at the end of its life cycle. In this context, the process is described by a system of McKendrick type equations which resembles a network transport problem. In this paper we show that, under an appropriate scaling of the latter, these two descriptions are asymptotically equivalent.