Percorrer por autor "Stollenwerk, Nico"
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- Applications of fractional calculus to epidemiological modelsPublication . Skwara, Urszula; Martins, José; Ghaffari, Peyman; Aguiar, Maíra; Boto, João; Stollenwerk, NicoEpidemiological 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.
- Dynamics of Epidemiological ModelsPublication . Pinto, Alberto; Aguiar, Maíra; Martins, José; Stollenwerk, NicoWe study the SIS and SIRI epidemic models discussing different approaches to compute the thresholds that determine the appearance of an epidemic disease. The stochastic SIS model is a well known mathematical model, studied in several contexts. Here, we present recursively derivations of the dynamic equations for all the moments and we derive the stationary states of the state variables using the moment closure method. We observe that the steady states give a good approximation of the quasi-stationary states of the SIS model. We present the relation between the SIS stochastic model and the contact process introducing creation and annihilation operators. 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 present the phase transition lines using the mean field and the pair approximation for the moments. We use a scaling argument that allow us to determine analytically an explicit formula for the phase transition lines in pair approximation.
- The Higher Moments Dynamic on SIS ModelPublication . Pinto, Alberto; Gouveia Martins, José Maria; Stollenwerk, NicoThe 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.
- The maximum curvature reinfection thresholdPublication . Martins, José; Pinto, Alberto; Stollenwerk, Nico
- On the series expansion of the spatial SIS evolution operatorPublication . Martins, José; Aguiar, Maíra; Pinto, Alberto; Stollenwerk, NicoFor the spatial stochastic susceptible–infected–susceptible model, we consider the perturbative series expansion of the gap between the dominant and subdominant eigenvalues of the evolution operator. We compute explicitly the first terms of the series expansion of the gap with difference equations for the calculation of states.
- A scaling analysis in the SIRI epidemiological modelPublication . Pinto, Alberto; Stollenwerk, Nico; Gouveia Martins, José Maria; Martins, JoséFor 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.
- A spatially stochastic epidemic model with partial immunization shows in mean field approximation the reinfection thresholdPublication . Stollenwerk, Nico; van Noort, Sander; Martins, José; Aguiar, Maíra; Hilker, Frank; Pinto, Alberto; Gomes, GabrielaRecently, 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.