Continuous Monocyclic and Polycyclic Age Structured Models of Population Dynamics

This paper focuses on the study of continuous age-structured models, or more general, physiologically structured models, which used for detailed and accurate study of population dynamics in many ecological, biological applications and medicine. In contrast to simpler unstructured models, these models allow us to relate the individual life-histories described as fertility and mortality rates of an individual at a given age with population dynamics. Depending from the particularity of reproduction mechanism continuous age-structured models are divided into monocyclic (reproduction occurs only at the one fixed age of individuals) and polycyclic (reproduction occurs with age-dependent probability at some age reproductive window) models. The linear monocyclic age-structured models are used often in cell cycles modelling, in population dynamics of plants, etc. In this case continuous age-structured models allow for obtaining the exact analytical solution. Since the linear and non-linear polycyclic age-structured models are more general then monocyclic models, they coverwider  range of applications in life science. But in this case solution of model can be obtained only in the form of recurrent formulae and can be used only in numerical algorithms. Both solutions obtained in this work allow us to study numerically the important dynamical regimes population outbreaks of three types: oscillations with large magnitude, pulse sequence and single pulse. Thus, analysis of continuous age-structured models of population dynamics provides insight into features and particularities of complex dynamical regimes of populations in many applications in biology, ecology and medicine

Age-structured Delayed SIPCV Epidemic Model of HPV and Cervical Cancer Cells Dynamics II. Convergence of Numerical Solution

The numerical method for simulation of age-structured SIPCV epidemic model with age-structured sub-classes of susceptible, infectious, pre-cancerous and cancer cells and unstructured population of human papilloma virus (HPV) dynamics with incubation period is developed. 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. The numerical experiments based on the grid refined method confirm and illustrate the second order of accuracy of the obtained numerical method and show the various dynamical regimes of population dynamics. Simulations for model param- eters of the system reveal two unstable dynamical regimes of SIPCV population which correspond to the cancer tumor growth and formation of metastases in organism

Age-structured Delayed SIPCV Epidemic Model of HPV and Cervical Cancer Cells Dynamics II. Convergence of Numerical Solution

The numerical method for simulation of 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. 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. The numerical experiments based on the grid refined method confirm and 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 metastases in organism