Epidemic spread and bifurcation effects in two-dimensional network models with viral dynamics

We extend a previous network model of viral dynamics to include host populations distributed in two space dimensions. The basic dynamical equations for the individual viral and immune effector densities within a host are bilinear with a natural threshold condition. In the general model, transmission between individuals is governed by three factors: a saturating function g( small middle dot) describing emission as a function of originating host virion level; a four-dimensional array B that determines transmission from each individual to every other individual; and a nonlinear function F, which describes the absorption of virions by a host for a given net arrival rate. A summary of the properties of the viral-effector dynamical system in a single host is given. In the numerical network studies, individuals are placed at the mesh points of a uniform rectangular grid and are connected with an m(2)xn(2) four-dimensional array with terms that decay exponentially with distance between hosts; g is linear and F has a simple step threshold. In a population of N=mn individuals, N0 are chosen randomly to be initially infected with the virus. We examine the dependence of maximal population viral load on the population dynamical parameters and find threshold effects that can be related to a transcritical bifurcation in the system of equations for individual virus and host effector populations. The effects of varying demographic parameters are also examined. Changes in alpha, which is related to mobility, and contact rate beta also show threshold effects. We also vary the density of (randomly chosen) initially infected individuals. The distribution of final size of the epidemic depends strongly on N0 but is invariably bimodal with mass concentrated mainly near either or both ends of the interval [1,N]. Thus large outbreaks may occur, with small probability, even with only very few initially infected hosts. The effects of immunization of various fractions of the population on the final size of the epidemic are also explored. The distribution of the final percentage infected is estimated by simulation. The mean of this quantity is obtained as a function of immunization rate and shows an almost linear decline for immunization rates up to about 0.2. When the immunization rate is increased past 0.2, the extra benefit accrues more slowly. We include a discussion of some approximations that illuminate threshold effects in demographic parameters and indicate how a mean-field approximation and more detailed studies of various geometries and rates of immunization could be a useful direction for future analysis.