A Mathematical Model of Acute Herpes Simplex Virus Corneal Infection

Abstract

Abstract: : Purpose: To devise a mathematical model of corneal cell turnover mimicking corneal cell dynamics and to extend this model with further equations to model viral infection, proliferation, and clearance to reproduce observed dynamics of viral plaque forming unit (PFU) production. Methods: The mathematical model is a non–linear system of six differential equations with terms that describe the corneal epithelium’s response to wounding and viral infection. Three equations for cellular dynamics of the corneal epithelium without viral infection describe 1) the rate of cell proliferation p(t), 2) the rate of quiescent cell production q(t), and 3) the combined amount of growth factors c(t). The equations are 1) dp(t)/dt=a₁p(t)c(t)–a₂p(t)c(t)+s(c), where s(c) is the term for stem cell mitosis and it =a₃c(t) if c(t)<kc₀ and s(c)=a₃kc₀ if c(t)≥kc₀; 2) dq(t)/dt=a₂p(t)c(t)–a₅q(t)c(t); and 3) dc(t)/dt=a₇a₉c(t)(q(t)+p(t))–a₁₀c(t). At equilibrium: a₁p₀c₀–a₂p₀c₀+a₃c₀=0; a₂p₀c₀–a₅q₀c₀=0; and a₇–a₉c₀(q₀+p₀)–a₁₀c₀=0. By using the equilibrium equations and estimates from the literature for p₀, q₀ and c₀,we obtain a₁, a₂, a₃, a₅, a₇, a₉, and a₁₀. The system of equations behaves as expected if cells are removed (corneal wounding); proliferation (healing) continues until a stable number of cells is reached (equilibrium reestablished). Three further equations are added to the system to model the dynamics of viral infection: 1) an equation for the rate of cells infected but not yet producing virus di(t)/dt=a₄p(t)v(t)+a₆q(t)v(t)–a₁₁i(t); 2) an equation for cells actively producing virus dj(t)/dt=a₁₁i(t)–a₁₂j(t); and 3) an equation for virus (PFU) in the tears dv(t)/dt=a₁₃j(t)–a₁₄p(t)v(t)–a₁₅q(t)v(t)–a₁₆tv(t)–a₁₇v(t). Results: Given a range of viral inoculation concentrations, the dynamics of PFU observed in experimental data are closely duplicated by the model predictions. Extensive graphic analysis of the model predictions for experimental observations will be presented. Conclusions: Numerical analytic solutions of non–linear equation systems can duplicate the behavior of the corneal epithelium. This model can make accurate predictions under various conditions of PFU load and differences in viral replication efficiency.

Keywords: herpes simplex virus • cornea: epithelium • computational modeling