Piotr Gwiazda

José A. Carrillo, Piotr Gwiazda, Karolina Kropielnicka et al.

The Escalator Boxcar Train (EBT) method is a well known and widely used numerical method for one-dimensional structured population models of McKendrick-von Foerster type. Recently the method, in its full generality, has been applied to aged-structured two-sex population model (Fredrickson-Hoppensteadt model), which consists of three coupled hyperbolic partial differential equations with nonlocal boundary conditions. We derive the simplified EBT method and prove its convergence to the solution of Fredrickson-Hoppensteadt model. The convergence can be proven, however only if we analyse the whole problem in the space of nonnegative Radon measures equipped with bounded Lipschitz distance (flat metric). Numerical simulations are presented to illustrate the results.

Carolin Lindow, Christian Düll, Piotr Gwiazda et al.

In this paper, we propose a numerical scheme for structured population models defined on a separable and complete metric space. In particular, we consider a generalized version of a transport equation with additional growth and non-local interaction terms in the space of nonnegative Radon measures equipped with the flat metric. The solutions, given by families of Radon measures, are approximated by linear combinations of Dirac measures. For this purpose, we introduce a finite-range approximation of the measure-valued model functions, provided that they are linear. By applying an operator splitting technique, we are able to separate the effects of the transport from those of growth and the non-local interaction. We derive the order of convergence of the numerical scheme, which is linear in the spatial discretization parameters and polynomial of order $α$ in the time step size, assuming that the model functions are $α$ Hölder regular in time. In a second step, we show that our proposed algorithm can approximate the posterior measure of Bayesian inverse models, which will allow us to link model parameters to measured data in the future.

Piotr Gwiazda, Błażej Miasojedow, Magdalena Rosińska

Dynamics of the infectious disease transmission is often best understood taking into account the structure of population with respect to specific features, in example age or immunity level. Practical utility of such models depends on the appropriate calibration with the observed data. Here, we discuss the Bayesian approach to data assimilation in case of two-state age-structured model. This kind of models are frequently used to describe the disease dynamics (i.e. force of infection) basing on prevalence data collected at several time points. We demonstrate that, in the case when the explicit solution to the model equation is known, accounting for the data collection process in the Bayesian framework allows to obtain an unbiased posterior distribution for the parameters determining the force of infection. We further show analytically and through numerical tests that the posterior distribution of these parameters is stable with respect to cohort approximation (Escalator Boxcar Train) to the solution. Finally, we apply the technique to calibrate the model based on observed sero-prevalence of varicella in Poland.

Piotr Gwiazda, Piotr Orliński, Agnieszka Ulikowska

In the following paper we reconsider a recently introduced numerical scheme. The method was designed for a wide class of size structured population models as a variation of the Escalator Boxcar Train (EBT) method, which is commonly used in computational biology. The scheme under consideration bases on the kinetic approach and the split-up technique - it approximates a solution by a sum of Dirac measures at each discrete time moment. In the current paper we propose a modification of this algorithm, which prevents (possible) exponential growth of the number of Dirac Deltas approximating the solution. Our approach bases on the finite range approximation of a coefficient which describes birth processes in a population. We provide convergence results, including the convergence speed. Moreover, some results of numerical simulations for several test cases are shown.