# Estimating Mean Viral Load Trajectory From Intermittent Longitudinal Data and Unknown Time Origins

## TL;DR

This paper introduces a statistical method to estimate average viral load over time using intermittent measurements and unknown infection dates.

## Contribution

A novel EM algorithm is proposed to estimate mean viral load trajectories from partial and intermittent data.

## Key findings

- Two viral load measurements per individual can accurately estimate the mean trajectory under parametric assumptions.
- The EM algorithm effectively handles unknown infection times and missing measurements as latent variables.
- The method was successfully applied to SARS-CoV-2 data from Israel to reconstruct daily mean viral load.

## Abstract

Viral load (VL) in the respiratory tract is the leading proxy for assessing infectiousness potential. Understanding the dynamics of disease‐related VL within the host is of great importance, as it helps to determine different policies and health recommendations. However, normally the VL is measured on individuals only once, in order to confirm infection, and furthermore, the infection date is unknown. It is therefore necessary to develop statistical approaches to estimate the typical VL trajectory. We show here that, under plausible parametric assumptions, two measures of VL on infected individuals can be used to accurately estimate the VL mean function. Specifically, we consider a discrete‐time likelihood‐based approach to modeling and estimating partial observed longitudinal samples. We study a multivariate normal model for a function of the VL that accounts for possible correlation between measurements within individuals. We derive an expectation‐maximization (EM) algorithm which treats the unknown time origins and the missing measurements as latent variables. Our main motivation is the reconstruction of the daily mean VL, given measurements on patients whose VLs were measured multiple times on different days. Such data should and can be obtained at the beginning of a pandemic with the specific goal of estimating the VL dynamics. For demonstration purposes, the method is applied to SARS‐Cov‐2 cycle‐threshold‐value data collected in Israel.