Abstract
The numerical method for simulation dynamics of nonlinear epidemic model of age-structured sub-populations of susceptible, infectious, precancerous and cancer cells and unstructured population of human papilloma virus (HPV) is developed (SIPCV model). Cell population dynamics is described by the initial-boundary value problem for the delayed semi-linear hyperbolic equations with age- and time-dependent coefficients and HPV dynamics is described by the initial problem for nonlinear delayed ODE. The model considers two time-delay parameters: the time between viral entry into a target susceptible cell and the production of new virus particles, and duration of the first stage of delayed immune response to HPV population growing. Using the method of characteristics and method of steps we obtain the exact solution of the SIPCV epidemic model in the form of explicit recurrent formulae. The numerical method designed for this solution and used the trapezoidal rule for integrals in recurrent formulae has a second order of accuracy. Numerical experiments with vanished mesh spacing illustrate the second order of accuracy of numerical solution with respect to the benchmark solution and show the dynamical regimes of cell-HPV population with the different phase portraits.
Highlights
INTRODUCTIONUsing the method of characteristics [3], [4], [5], [6], [8], [9], [11], [24], [53], [55], [58] and method of steps from the theory of delayed differential equations [12], [28], [46], [57], [59], we obtain an exact solution of the SIPCV epidemic model
Human papilloma virus (HPV) is the most common sexually transmitted infection that can cause dysplasia and cervical cancer [27], [36], [45]
Using the method of characteristics [3], [4], [5], [6], [8], [9], [11], [24], [53], [55], [58] and method of steps from the theory of delayed differential equations [12], [28], [46], [57], [59], we obtain an exact solution of the SIPCV epidemic model
Summary
Using the method of characteristics [3], [4], [5], [6], [8], [9], [11], [24], [53], [55], [58] and method of steps from the theory of delayed differential equations [12], [28], [46], [57], [59], we obtain an exact solution of the SIPCV epidemic model This solution is given in form of the recurrent formulae (like in works [3], [4], [8], [55]) in which the densities of all subpopulations are defined through the integrals from solution taken at previous instance of time. Simulations show that the relative numerical error converges pointwise to zero with h → 0 (where h is a mesh spacing) by the quadratic low that illustrates and confirms the second order of accuracy of numerical solution These results are in good agreement with the evaluations of numerical errors obtained earlier in experiments [8] with the numerical method designed for the nonlinear age-structured models of population dynamics by the same approach.
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