Browsing by Subject "Stochastic model"
Now showing 1 - 2 of 2
Results Per Page
Sort Options
Item Deterministic and Stochastic models for early viral infection within a host(2010-12) Vidurupola, Sukhitha W.; Allen, Linda J. S.; Allen, Edward J.; Roeger, Lih-Ing W.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 modelsItem INVESTIGATION OF STOCHASTIC REACTION-DIFFUSION PARTIAL DIFFERENTIAL EQUATIONS AND OF CONSISTENT STOCHASTIC DIFFERENTIAL EQUATION MODELS FOR ONE-LOCUS AND TWO-LOCI POPULATION GENETICS(2011-08) Dogan, Elife; Allen, Edward J.; Allen, Linda J. S.; Ruymgaart, FritsThere are two main parts in this work separated into chapters 2 and 3 and chapters 4 and 5, respectively. In the first part, stochastic partial differential equations are derived for the reaction-diffusion process in one, two and three dimensions. Specifically, stochastic partial differential equations are derived for the random dynamics of particles that are reacting and diffusing in a medium. In the derivation, a discrete stochastic reaction-diffusion equation is first constructed from basic principles, i.e., from the changes that occur in a small time interval. As the time interval goes to zero, the discrete stochastic model leads to a system of Ito stochastic differential equations. As the spatial intervals approach zero, a stochastic partial differential equation is derived for the reaction-diffusion process. The stochastic reaction-diffusion equation can be solved computationally using numerical methods for systems of Ito stochastic differential equations. In the second part, stochastic ordinary and partial differential equations are derived for randomly varying populations of haploid and diploid individuals under genetic changes with one, two and a large number of alleles. Specifically, stochastic differential equations are derived for the genotype population distributions. In the derivation, a discrete stochastic population genetics equation is first constructed from basic principles. A similar procedure is applied to find stochastic ordinary differential equations for population genetics. For a large number of alleles, a stochastic partial differential equation is derived for the population genetics. Stochastic differential equation models for interacting populations undergoing genetic changes provide straightforward unifying models for understanding the dynamics of the population genetics problems. Comparisons between numerical solutions of the stochastic differential equations and independently formulated Monte Carlo calculations for the reaction diffusion and population genetics problems support the accuracy of the derivations.