# Time Distribution for Persistent Viral Infection

## TL;DR

This paper models the early stages of viral infection to analyze the distribution of times until persistence, revealing a highly skewed distribution with implications for testing strategies.

## Contribution

It introduces a 2D virus-infected cell model and reduces it to an effective 1D model, providing analytical and numerical solutions for infection time distribution.

## Key findings

- Infection time distribution has a fat exponential tail.
- The 2D and 1D models are shown to be equivalent.
- The distribution's skewness affects testing timing.

## Abstract

We study the early stages of viral infection, and the distribution of times to obtain a persistent infection. The virus population proliferates by entering and reproducing inside a target cell until a sufficient number of new virus particles are released via a burst, with a given burst size distribution, which results in the death of the infected cell. Starting with a 2D model describing the joint dynamics of the virus and infected cell populations, we analyze the corresponding master equation using the probability generating function formalism. Exploiting time-scale separation between the virus and infected cell dynamics, the 2D model can be cast into an effective 1D model. To this end, we solve the 1D model analytically for a particular choice of burst size distribution. In the general case, we solve the model numerically by performing extensive Monte-Carlo simulations, and demonstrate the equivalence between the 2D and 1D models by measuring the Kullback-Leibler divergence between the corresponding distributions. Importantly, we find that the distribution of infection times is highly skewed with a "fat" exponential right tail. This indicates that there is non-negligible portion of individuals with an infection time, significantly longer than the mean, which may have implications on when HIV tests should be performed.