Threshold methods for threshold models in age dependent population dynamics and epidemiology
This paper considers two models: the Gurtin and MacCamy model in nonlinear, age-dependent population dynamics, and an epidemic model of S.I.S type with age structure. First-order finite differences along the characteristics of the differential operator are used to construct a first-order numerical method for approximating the solution of the Gurtin-MacCamy model. Boundedness and monotonicity are assumed for the birth and death rate functions (instead of Lipschitz continuity) to prove convergence of the method. The numerical approach is based on the observation that the solution of a certain differential-delay problem approaches the solution of the original one as the delay tends to zero. Stability of the numerical method is discussed. It is also proved that the solution of the discrete model stabilizes asymptotically towards the solution of a discretization of the differential equation for the equilibrium distribution. The other model describes the spread of a mild epidemic through the population described by the first model. Both intracohort and intercohort transmissions of the infection are considered. A first order method is constructed by extending suitably the one for the Gurtin-MacCamy model. The authors prove again that the discrete solution of the difference scheme has a similar large time behavior as the solution of the continuous model.
nonlinear, age- dependent population dynamics
intracohort and intercohort transmissions
scientific article; zbMATH DE number 165191
scientific article