  Mathematical modeling of biological systems is crucial to effectively and
efficiently developing treatments for medical conditions that plague humanity.
Often, systems of ordinary differential equations are a traditional tool used
to describe the spread of disease within the body. We consider the dynamics of
the Human Immunodeficiency Virus (HIV) in vivo during the initial stage of
infection. In particular, we examine the well-known three-component model and
prove the existence, uniqueness, and boundedness of solutions. Furthermore, we
prove that solutions remain biologically meaningful, i.e., are positivity
preserving, and perform a thorough, local stability analysis for the
equilibrium states of the system. Finally, we incorporate random coefficients
within the model and obtain numerical results to predict the probability of
infection given the transmission of the virus to a new individual.
