Time distribution for persistent viral infection

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 a non-negligible portion of individuals with an infection time, significantly longer than the mean, which may have implications on, e.g., when HIV tests should be performed.