# Assessing the impact of non-vaccinators: quantifying the average length of infection chains in outbreaks of vaccine-preventable disease

## TL;DR

This paper introduces the concept of the average number of descendant infections (ANDI) to quantify infection chain lengths in outbreaks, providing new insights into disease spread and vaccination impact.

## Contribution

It develops analytical estimates of ANDI using deterministic and stochastic models, highlighting its significance in understanding outbreak severity and vaccination strategies.

## Key findings

- ANDI can be large when R0 > 1, even in small communities.
- High probabilities of adverse outcomes occur down average infection chains.
- Effective vaccination reduces the likelihood of long infection chains.

## Abstract

Analytical expressions for the basic reproduction number, R0, have been obtained in the past for a wide variety of mathematical models for infectious disease spread, along with expressions for the expected final size of an outbreak. However, what has so far not been studied is the average number of infections that descend down the chains of infection begun by each of the individuals infected in an outbreak (we refer to this quantity as the "average number of descendant infections" per infectious individual, or ANDI). ANDI includes not only the number of people that an individual directly contacts and infects, but also the number of people that those go on to infect, and so on until that particular chain of infection dies out. Quantification of ANDI has relevance to the vaccination debate, since with ANDI one can calculate the probability that one or more people are hospitalised (or die) from a disease down an average chain of infection descending from an infected un-vaccinated individual. Here we obtain estimates of ANDI using both deterministic and stochastic modelling formalisms. With both formalisms we find that even for relatively small community sizes and under most scenarios for R0 and initial fraction vaccinated, ANDI can be surprisingly large when the effective reproduction number is >1, leading to high probabilities of adverse outcomes for one or more people down an average chain of infection in outbreaks of diseases like measles.