Abstract
For a spatial stochastic epidemic model we investigate in the pair approximation scheme the differential equations for the moments. The basic reinfection model of susceptible-infected-recovered-reinfected or SIRI type is analysed, its phase transition lines calculated analytically in this pair approximation.

Abstract
We prove that the stable holonomies of a proper codimension 1 attractor Lambda, for a C-r diffeomorphism f of a surface, are not C1+theta for theta greater than the Hausdorff dimension of the stable leaves of f intersected with Lambda. To prove this result we show that there are no diffeomorphisms of surfaces, with a proper codimension 1 attractor, that are affine on a neighbourhood of the attractor and have affine stable holonomies on the attractor.

Abstract
The Von Mises-Fisher distribution defined on the hypersphere is one of the most used distributions for modelling vectorial data. If we reject the uniform distribution as a model for vectorial data, then the Von Mises-Fisher distribution is an alternative model. So we use this distribution as an alternative hypothesis in the study of the power of some tests of uniformity for some dimensions of the sphere.

Abstract
We derive likelihood ratio tests for the equality of the directional parameters of k bipolar Watson distributions defined on the hypersphere with common concentration parameter. We analyze the power of these tests in the case of two distributions supposing in the alternative hypothesis two directional parameters forming an angle, which varies from 18 degrees to 90 degrees. We also compare the likelihood ratio tests with a high-concentration F-test.

Abstract
As the Watson distribution is frequently used for modeling axial data, it is important to investigate the existence of possible outliers in samples from this distribution. Then, we develop for the bipolar Watson distribution defined on the hypersphere, some tests of discordancy of an outlier or several outliers en bloc based on the likelihood ratio, supposing an alternative model of contamination of slippage type. We evaluate the performance of these tests of discordancy of an outlier and we also compare some tests of discordancy of an outlier available for this distribution.

Abstract
After characterization of preferred conditions for Helicobacter pylori survival in the sessile state, it was observed that the bacterium transforms from spiral to coccoid under mild circumstances, whereas under extreme ones it is unable to undergo shape modification. This strongly supports the view that transformation into the coccoid form is an active, biologically led process, switched on by the bacterium as a protection mechanism.