Stochastic Modeling of the Persistence of HIV: Early Population Dynamics.

摘要

Mathematical modeling of biological systems is crucial to effectively and efficiently developing treatments for medical conditions that plague humanity. Systems of differential equations are the traditional tools used to theoretically describe the spread of disease within the body. In this project we consider the dynamics of the Human Immunodeficiency Virus (HIV) in vivo during the initial stages of infection. Both mathematical and biological results support the idea that contact with the HIV retrovirus does not automatically imply permanent infection. Given factors such as the CD4+ T-cell growth rate, infection rate, and viral clearance rate, it is possible to correctly predict the end viral state in a deterministic model. While this is useful, such a model lacks the randomness inherent in physical processes and parameter estimation. To account for this, our project examines both discrete and continuous stochastic models for the early stages of HIV infection. These models use the knowledge of biological interactions and fundamental mathematical principles. We also examine the well-known three-component deterministic model in greater detail, proving existence and uniqueness of the solutions. Furthermore, we prove that the solutions remain biologically meaningful, and perform a thorough stability analysis for the equilibrium states of the system. Finally, we develop two new stochastic models and obtain extensive numerical results to measure the probability of infection given the transmission of the virus to a new individual. To simulate the dynamics of the virus, we employ Runge-Kutta methods and the Euler-Maruyama scheme.