Fabio A. Milner

A second order splitting method for the Cahn-Hilliard equation

SummaryA semi-discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary, fourth order in space, Cahn-Hilliard equation. Optimal order error bounds are derived in various norms for an implementation which uses mass lumping. The continuous problem has an energy based Lyapunov functional. It is proved that this property holds for the discrete problem.

Mixed finite element methods for quasilinear second-order elliptic problems

A mixed finite element method is developed to approximate the solution of a quasilinear second-order elliptic partial differential equation. The existence and uniqueness of the approximation are demonstrated and optimal rate error estimates are derived.

A TWO-STRAIN TUBERCULOSIS MODEL WITH AGE OF INFECTION ∗

Long periods of latency and the emergence of antibiotic resistance due to incomplete treatment are very important features of tuberculosis (TB) dynamics. Previous studies of two-strain TB have been performed by ODE models. In this article, we formulate a two-strain TB model with an arbitrarily distributed delay in the latent stage of individuals infected with the drug-sensitive strain and look at the effects of variable periods of latency on the disease dynamics.

Gender-structured population modeling : mathematical methods, numerics, and simulations

Preface 1. Historical perspective of mathematical demography 2. Gender structure and the problem of modeling marriages 3. Well-posedness of the Fredrickson-Hoppensteadt two-sex model 4. Numerical methods 5. Age profiles and exponential growth Appendix Bibliography Index.

Analytical and numerical results for the age-structured S-I-S epidemic model with mixed inter-intracohort transmission

A model which describes the dynamics of an

Rapidly converging numerical algorithms for models of population dynamics

S \to I \to S

Schistosomiasis models with density dependence and age of infection in snail dynamics

epidemic in an age-structured population at the steady state is considered. The model consists of a nonlinear and nonlocal system of equations of hyperbolic type and has already been partly analyzed by other authors. Here, a special form for the force of infection is considered. Explicitly computable threshold conditions are given, and some regularity results for the solutions are proven. An implicit finite difference method of characteristics to approximate the solutions is used. Optimal error estimates are derived and results from numerical simulations are presented. The discrete dynamical system arising from the numerical algorithm, is also analyzed, showing that it shares many properties of the continuous model.

A mixed finite element method for a strongly nonlinear second-order elliptic problem

We propose algorithms for the approximation of the age distributions of populations modeled by the McKendrick-von Foerster and the Gurtin-MacCamy systems both in one- and two-sex versions. For the one-sex model methods of second and fourth order are proposed. For the two-sex model a second order method is described. In each case the convergence is demonstrated. Several numerical examples are given.

Estimation of parameters governing the transmission dynamics of schistosomes

New models for schistosomiasis are developed. These models incorporate several realistic features including drug treatment for human hosts, an infection age in snail hosts, density-dependent birth rate of snails, distribution of schistosomes within human hosts, and disease-induced mortality in both human and snail hosts. The qualitative and quantitative mathematical properties of the models are studied, their biological consequences and some control strategies are discussed, and the results of the new models are compared with those of simpler models. It is shown that the new model may have a bifurcation at which the unique endemic equilibrium changes the stability and stable periodic solutions exist. This is quite different from the simpler models. Explicit thresholds of treatment rate are established above which the infection will be controlled under certain levels. Evaluations of cost-effectiveness are also discussed by analyzing the sensitivity of the mean number of parasites per person to changes of other parameters.

L ∞ -error estimates for mixed methods for elliptic partial differential equations

The approximation of the solution of the first boundary value problem for a strongly nonlinear second-order elliptic problem in divergence form by the mixed finite element method is considered. Existence and uniqueness of the approximation are proved and optimal error estimates in L 2 are established for both the scalar and vector functions approximated by the method. Error estimates are also derived in L q , 2 ≤ q ≤ +∞.