We present a one-sex age-structured population dynamics deterministic model with a discrete set of offsprings, child care, environmental pressure, and spatial migration. All individuals have pre-reproductive, reproductive, and post-reproductive age intervals. Individuals of reproductive age are divided into fertile single and taking child care groups. All individuals of pre-reproductive age are divided into young (under maternal care) and juvenile (offspring who can live without maternal care) classes. It is assumed that all young offsprings move together with their mother and that after the death of mother all her young offsprings are killed. The model consists of integro-partial differential equations subject to the conditions of the integral type. Number of these equations depends on a biologically possible maximal newborns number of the same generation produced by an individual. The existence and uniqueness theorem is proved, separable solutions are studied, and the long time behavior is examined for the solution with general type of initial distributions in the case of non-dispersing population. Separable and more general (nonseparable) solutions, their large time behavior, and steady-state solutions are studied for the population with spatial dispersal, too.

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BackgroundIt was recently shown that the treatment effect of an antibody can be described by a consolidated parameter which includes the reaction rates of the receptor-toxin-antibody kinetics and the relative concentration of reacting species. As a result, any given value of this parameter determines an associated range of antibody kinetic properties and its relative concentration in order to achieve a desirable therapeutic effect. In the current study we generalize the existing kinetic model by explicitly taking into account the diffusion fluxes of the species.ResultsA refined model of receptor-toxin-antibody (RTA) interaction is studied numerically. The protective properties of an antibody against a given toxin are evaluated for a spherical cell placed into a toxin-antibody solution. The selection of parameters for numerical simulation approximately corresponds to the practically relevant values reported in the literature with the significant ranges in variation to allow demonstration of different regimes of intracellular transport.ConclusionsThe proposed refinement of the RTA model may become important for the consistent evaluation of protective potential of an antibody and for the estimation of the time period during which the application of this antibody becomes the most effective. It can be a useful tool for in vitro selection of potential protective antibodies for progression to in vivo evaluation.

GENETIC MODELDiploid population models which do not take into account pregnancy are well known in mathematical genetics [1,2]. The pregnancy effect for a single-locus population was investigated by the author in [3] (that model did not include the organism restoration period after abortion or delivery). This paper deals with the evolution of autosomal character and takes into consideration pregnancy and possible abortions as well as the organism restoration period after abortion or delivery. We also take into account the interchange of loci, i.e., the crossingover.Assume that the autosomal character of the population is controlled by ! loci of a pair of holonomous chromosomes and a gene of the jth locus can be in any of nj alleles. Letdenote genotype (the homologous pairs of chromosomes) for a male and a female, respectively. We introduce the following notions:xk (t, r:`)

BackgroundThe modelling framework is proposed to study protection properties of antibodies to neutralize the effects of the plant toxin (ricin). The present study extends our previous work by including (i) the model of intracellular transport of toxin to the Endoplasmic Reticulum and (ii) the model of the internalised antibodies (when antibody is delivered directly into the cytosol).MethodSimulation of the receptor-toxin-antibody interaction is implemented by solving the systems of PDEs (advection-diffusion models) or ODEs (rate models) for the underlying transport coupled with mass-action kinetics.ResultsAs the main application of the enhanced framework we present a comparative study of two kinds (external and internalised) of antibodies. This comparison is based on calculation of the non-dimensional protection factor using the same set of parameters (geometry, binding constants, initial concentrations of species, and total initial amount of the antibody).ConclusionThis research will provide a framework for consistent evaluation and comparison of different types of antibodies for toxicological applications.