The effect of density-dependent infection on ODE models of viral dynamics

Ordinary differential equation (ODE) models of viral infections assume that the components are spatially well-mixed. This is a particularly bad assumption at early times during the infection when high concentrations of virus are located near the few productively infectious cells. Density-dependent infection rates can help account for this spatial heterogeneity while maintaining the ODE framework. We explore the effect of incorporating three different density-dependent infection rates: the saturated incidence, the Beddington-DeAngelis, and the Crowley-Martin models. We characterize the effect of density-dependent infection by measuring a number of viral titer characteristics (peak viral load, time of peak, upslope, downslope, and infection duration), as well as the basic reproduction number and the infecting time. We find that larger density dependence tends to slow down the infection and lowers the basic reproduction number; consistent with the idea that spatial ``clumping'' of the virus early in the infection will limit its access to uninfected cells, slowing down the infection.