Case fatality models for epidemics in growing populations.

The asymptotically homogeneous SIR model of Thieme (1992) for growing populations, with incidence depending in a general way on total population size, is reconsidered with respect to other parameterizations that give clear insight into epidemiological relevant relations and thresholds. One important feature of the present approach is case fatality as opposed to differential mortality. Although case fatality models and differential mortality models are equivalent via a transformation in parameter space, the underlying ideas and the dynamic behaviors are different, e.g. the basic reproduction number depends on differential mortality but not on case fatality. The persistent distributions and exponents of growth of infected solutions are computed and discussed in terms of the parameters. The notion of asymptotically exponentially growing state (as opposed to stationary state or exponential solution) coined by Thieme is interpreted in terms of stability theory. Of some interest are limiting cases of models without recovery where two infected solutions exist.

Michaelis-Menten kinetics, the operator-repressor system, and least squares approaches.

The Michaelis-Menten (MM) function is a fractional linear function depending on two positive parameters. These can be estimated by nonlinear or linear least squares methods. The non-linear methods, based directly on the defect of the MM function, can fail and not produce any minimizer. The linear methods always produce a unique minimizer which, however, may not be positive. Here we give sufficient conditions on the data such that the nonlinear problem has at least one positive minimizer and also conditions for the minimizer of the linear problem to be positive. We discuss in detail the models and equilibrium relations of a classical operator-repressor system, and we extend our approach to the MM problem with leakage and to reversible MM kinetics. The arrangement of the sufficient conditions exhibits the important role of data that have a concavity property (chemically feasible data).

Structured populations with diffusion in state space.

The classical models for populations structured by size have two features which may cause problems in biologically realistic modeling approaches: the structure variable always increases, and individuals in an age cohort that are identical initially stay identical throughout their lives. Here a diffusion term is introduced in the partial differential equation which mathematically amounts to adding viscosity. This approach solves both problems but it requires to identify appropriate boundary (recruitment) conditions. The method is applied to size-structured populations, metapopulations, infectious diseases, and vector-transmitted diseases.

Quantitative analysis of mouse urine volatiles: in search of MHC-dependent differences.

Genes of the major histocompatibility complex (MHC), which play a critical role in immune recognition, influence mating preference and other social behaviors in mice. Training experiments using urine scent from mice differing only in the MHC complex, from MHC class I mutants or from knock-out mice lacking functional MHC class I molecules (beta2m-deficient), suggest that these behavioral effects are mediated by differences in MHC-dependent volatile components. In search for the physical basis of these behavioral studies, we have conducted a comparison of urinary volatiles in three sub-strains of C57BL/6 mice, a beta2m-deficient mutant lacking functional MHC class I expression and two unrelated inbred strains, using the technique of sorptive extraction with polydimethylsiloxan and subsequent analysis by gas chromatography/mass spectrometry. We show (i) that qualitative differences occur between different inbred strains but not in mice with the C57BL/6 background, (ii) that the individual variability in abundance in the same mouse strain is strongly component-dependent, (iii) that C57BL/6 sub-strains obtained from different provenance show a higher fraction of quantitative differences than a sub-strain and its beta2m-mutant obtained from the same source and (iv) that comparison of the spectra of beta2m mice and the corresponding wild type reveals no qualitative differences in close to 200 major and minor components and only minimal differences in a few substances from an ensemble of 69 selected for quantitative analysis. Our data suggest that odor is shaped by ontogenetic, environmental and genetic factors, and the gestalt of this scent may identify a mouse on the individual and population level; but, within the limits of the ensemble of components analysed, the results do not support the notion that functional MHC class I molecules influence the urinary volatile composition.