# Deterministic and Stochastic models for early viral infection within a host

2010-12

## Abstract

Stochastic models are formulated and applied to intra-host viral and cellular dynamics. Specifically, two Itˆo stochastic differential equation models for early viral infection of host cells are formulated. The stochastic models are based on an underlying deterministic model that was originally formulated for Human Immunodeficiency Virus, type 1 (HIV-1), the most common strain of the virus. However, the deterministic and stochastic models apply to more general viral infections, during the early stages of infection, prior to activation of the immune response. The underlying deterministic model is a system of ordinary differential equations (ODEs) that includes variables for the healthy CD4+ T cells, the target cells of HIV-1, latently infected T cells, actively infected T cells and free virions. The first stochastic model assumes that after viral entry into the host cell and subsequent reproduction, the virus bursts from the cell, killing the host cell (burst model). The second model assumes the virus continually buds off from the host cell until the infected cell dies (budding model). The basic reproduction number R0 is calculated for the underlying deterministic model and it is shown that if R0 < 1, then the disease-free equilibrium (DFE) is both locally and globally asymptotically stable. For the stochastic models, application of Itˆo’s formula allows calculation of the moments corresponding to the distributions in the stochastic models. Because the moment differential equations form an infinite system of differential equations, each moment depending on higher-order moments, they cannot be solved unless some distributional assumption is made. Under the assumption of normality, the mean and variance for the target cell population are calculated. Numerical examples compare the dynamics of the deterministic model to the mean of the two stochastic models when R0 > 1. In addition, the standard deviation is computed and compared in the stochastic models