Abstract
Epidemiological spreading does not only happen from person to neighbouring person but often over wide distances, when infected but asymptomatic persons travel and carry infection to others over wide distances. Superdiffusion has been suggested to model such spreading in spatially restriced contact networks, i.e. there is still a notion of geographical distance, but spreading happens with high probability proportional to large distances. From fractional calculus several ways of describing superdiffusion are know. Here we investigate the representation in Fourier space and which is easily generalizable to higher dimensional space in order to compare with stochastic models of epidemiological spreading.

Abstract
In this paper we analyze the stochastic version of a minimalistic multi-strain model, which captures essential differences between primary and secondary infections in dengue fever epidemiology, and investigate the interplay between stochasticity, seasonality and import. The introduction of stochasticity is needed to explain the fluctuations observed in some of the available data sets, revealing a scenario where noise and complex deterministic skeleton strongly interact. For large enough population size, the stochastic system can be well described by the deterministic skeleton gaining insight on the relevant parameter values purely on topological information of the dynamics, rather than classical parameter estimation of which application is in general restricted to fairly simple dynamical scenarios.

Abstract
For the spatial stochastic epidemic reinfection model SIRI, where susceptibles S can become infected I, then recover and remain only partial immune against reinfection R, we determine the phase transition lines using pair approximation for the moments derived from the master equation. We introduce a scaling argument that allows us to determine analytically an explicit formula for these phase transition lines and prove rigorously the heuristic results obtained previously.

Abstract
Ourmain interest is to study the relevant biological thresholds that appear in epidemic and immunological dynamical models. We compute the thresholds of the SIRI epidemic models that determine the appearance of an epidemic disease. We compute the thresholds of a Tregs immunological model that determine the appearance of an immune response.

Abstract
The basic contact process or the SIS model is a well known epidemic process and have been studied for a wide class of people. In an epidemiological context, many authors worked on the SIS model considering only the dynamic of the first moments of infecteds, i.e., the mean value and the variance of the infected individuals. In this work, we study not only the dynamic of the first moments of infecteds but also on the dynamic of the higher moments. Recursively, we consider the dynamic equations for all the moments of infecteds and, applying the moment closure approximation, we obtain the stationary states of the state variables. We observe that the stationary states of the SIS model, in the moment closure approximation, can be used to obtain good approximations of the quasi-stationary states of the SIS model.

Abstract
Recently, the notion of a reinfection threshold in epidemiological models of only partial immunity has been debated in the literature. We present a rigorous analysis of a model of reinfection which shows a clear threshold behaviour at the parameter point where the reinfection threshold was originally described. Furthermore, we demonstrate that this threshold is the mean field version of a transition in corresponding spatial models of immunization. The reinfection threshold corresponds to the transition between annular growth of an epidemics spreading into a susceptible area leaving recovered behind and compact growth of a susceptible-infected-susceptible region growing into a susceptible area. This transition between annular growth and compact growth was described in the physics literature long before the reinfection threshold debate broke out in the theoretical biology literature.