Numerical Method for the Age-structured SIPCV Epidemic Model of Healthy cells, Dysplasia, Cervical Cancer Cells and HPV Dynamics
The numerical method for simulation of an age-structured SIPCV epidemic model with age-structured sub-classes of susceptible, infectious, precancerous and cancer cells and unstructured population of human papilloma virus (HPV) dynamics with incubation period is developed. The model assumes two time-delays: (i) the time between viral entry into a target susceptible cell and the production of new virus particles and (ii) duration of the first stage of delayed immune response to HPV population growing. The model of cell population dynamics is described by the initial-boundary value problem for the semilinear hyperbolic equations with age- and time-dependent coefficients and time delay and the dynamics of HPV virus is described by nonlinear delayed ODE. The model considers the immune functional response of organism by the HPV-density dependent death rate. The numerical method is based on the method of characteristics for the semi-linear hyperbolic equations, trapezoidal rules for integrals and has the second order of approximation.
Convergence of the numerical approximations is studied both theoretically and numerically. We prove the stability and second rate of convergence of the approximate solutions to the exact solution of the SIPCV epidemic nonlinear system. Numerical experiments with vanished mesh spacing illustrate the convergence of numerical solution to the benchmark solution. Simulations illustrate the second order of accuracy of the obtained numerical method and show the various dynamical regimes of population dynamics. Simulations for model parameters of the system reveal two unstable dynamical regimes of SIPCV population which correspond to the cancer tumor growth and formation of cancer metastases.