Effects of General Incidence and Polymer Joining on Nucleated Polymerization in a Model of Prion Proliferation

Source abstract

Two processes are incorporated into a new model for transmissible prion diseases. These are general incidence for the lengthening process of infectious polymers attaching to and converting noninfectious monomers, and the joining of two polymers to form one longer polymer. The model gives rise to a system of three ordinary differential equations, which is shown to exhibit threshold behavior dependent on the value of the parameter combination giving the basic reproduction number ${\mathcal{R}_0}$. For ${\mathcal{R}_0} <1$, infectious polymers die out, whereas for ${\mathcal{R}_0} >1$, the system is locally asymptotic to a positive disease equilibrium. The effect of both general incidence and joining is to decrease the equilibrium value of infectious polymers and to increase the equilibrium value of normal monomers. Since the onset of disease symptoms appears to be related to the number of infectious polymers, both processes may significantly inhibit the course of the disease. With general incidence, the equilibrium distribution of polymer lengths is obtained and shows a sharp decrease in comparison to the distribution resulting from mass action incidence. Qualitative global results on the disease free and disease equilibria are proved analytically. Numerical simulations using parameter values from experiments on mice (reported in the literature) provide quantitative demonstration of the effects of these two processes.