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Publicações

Publicações por José Martins

2009

A scaling analysis in the SIRI epidemiological model.

Autores
Martins, J; Pinto, A; Stollenwerk, N;

Publicação
Journal of biological dynamics

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.

2011

Dynamics and Biological Thresholds

Autores
Burroughs, NJ; Ferreira, M; Martins, J; Oliveira, BMPM; Pinto, AA; Stollenwerk, N;

Publicação
DYNAMICS, GAMES AND SCIENCE I

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.

2009

The Higher Moments Dynamic on SIS Model

Autores
Pinto, A; Martins, J; Stollenwerk, N; Simos, TE; Psihoyios, G; Tsitouras, C;

Publicação
NUMERICAL ANALYSIS AND APPLIED MATHEMATICS, VOLS 1 AND 2

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.

2010

A spatially stochastic epidemic model with partial immunization shows in mean field approximation the reinfection threshold

Autores
Stollenwerk, N; van Noort, S; Martins, J; Aguiar, M; Hilker, F; Pinto, A; Gomes, G;

Publicação
Journal of Biological Dynamics

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.

2011

On the series expansion of the spatial SIS evolution operator

Autores
Martins, J; Aguiar, M; Pinto, A; Stollenwerk, N;

Publicação
JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS

Abstract
For 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.

2007

The phase transition lines in pair approximation for the basic reinfection model SIRI

Autores
Stollenwerk, N; Martins, J; Pinto, A;

Publicação
PHYSICS LETTERS A

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.

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