Stochastic monotonicity and continuity properties of functions defined on Crump-Mode-Jagers branching processes, with application to vaccination in epidemic modelling

TL;DR

This paper studies how vaccination affects epidemic spread modeled by Crump-Mode-Jagers branching processes, establishing properties that inform optimal vaccination strategies to control outbreak duration and size.

Contribution

It introduces stochastic monotonicity and continuity results for functions on these processes, aiding in designing effective vaccination policies.

Findings

Proves monotonicity of outbreak duration with vaccination levels

Establishes continuity of total infections relative to vaccination

Demonstrates application to mumps outbreak control in Bulgaria

Abstract

This paper is concerned with Crump-Mode-Jagers branching processes, describing spread of an epidemic depending on the proportion of the population that is vaccinated. Births in the branching process are aborted independently with a time-dependent probability given by the fraction of the population vaccinated. Stochastic monotonicity and continuity results for a wide class of functions (e.g., extinction time and total number of births over all time) defined on such a branching process are proved using coupling arguments, leading to optimal vaccination schemes to control corresponding functions (e.g., duration and final size) of epidemic outbreaks. The theory is illustrated by applications to the control of the duration of mumps outbreaks in Bulgaria.