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Publications

Publications by José Martins

2017

Bistability of Evolutionary Stable Vaccination Strategies in the Reinfection SIRI Model

Authors
Martins, J; Pinto, A;

Publication
BULLETIN OF MATHEMATICAL BIOLOGY

Abstract
We use the reinfection SIRI epidemiological model to analyze the impact of education programs and vaccine scares on individuals decisions to vaccinate or not. The presence of the reinfection provokes the novelty of the existence of three Nash equilibria for the same level of the morbidity relative risk instead of a single Nash equilibrium as occurs in the SIR model studied by Bauch and Earn (PNAS 101:13391-13394, 2004). The existence of three Nash equilibria, with two of them being evolutionary stable, introduces two scenarios with relevant and opposite features for the same level of the morbidity relative risk: the low-vaccination scenario corresponding to the evolutionary stable vaccination strategy, where individuals will vaccinate with a low probability; and the high-vaccination scenario corresponding to the evolutionary stable vaccination strategy, where individuals will vaccinate with a high probability. We introduce the evolutionary vaccination dynamics for the SIRI model and we prove that it is bistable. The bistability of the evolutionary dynamics indicates that the damage provoked by false scares on the vaccination perceived morbidity risks can be much higher and much more persistent than in the SIR model. Furthermore, the vaccination education programs to be efficient they need to implement a mechanism to suddenly increase the vaccination coverage level.

2016

A repeated strategy for dumping

Authors
Martins, J; Banik, N; Pinto, AA;

Publication
Springer Proceedings in Mathematics and Statistics

Abstract
In this work, we study the phenomena of dumping in a duopoly market through an infinitely repeated game. We consider two firms of different countries competing in the same country. When both firms are cooperating, if the foreign firm deviates from cooperation this can be interpreted as dumping and a period of punishment can be imposed to the foreign firm. After this, firms can play continuously the deviation-punishment game or compete à la Cournot. Previously, we observe that the repeated strategy of deviation-punishment is not adopted in the case of symmetric demand equations. Here, we observe that this strategy of repeated dumping can appear as the best repeated strategy when the demand equations are non-symmetric. © Springer-Verlag Berlin Heidelberg 2016.

2012

Stationarity in Moment Closure and Quasi-Stationarity of the SIS model

Authors
José Martins; Alberto Pinto; N. Stollenwerk

Publication
Mathematical Biosciences - Mathematical Biosciences, vol.236, no.2, pp.126-131

Abstract

2011

An Economical Model For Dumping by Dumping in a Cournot Model

Authors
Nilanjan Banik; Fernanda.A. Ferreira; José Martins; Alberto Pinto

Publication
vol.2, pp.141-153

Abstract

2010

Dynamics of Epidemiological Models

Authors
Alberto Pinto; Maíra Aguiar; José Martins; Nico Stollenwerk

Publication
ACTA BIOTHEOR - ACTA BIOTHEORETICA, vol.58, no.4, pp.381-389

Abstract

2011

On the series expansion of the spatial SIS evolution operator

Authors
Alberto Pinto; José Martins; M. Aguiar; N. Stollenwerk

Publication
Journal of Difference Equations and Applications - Journal of Difference Equations and Applications, vol.17, no.7, pp.1107-1118

Abstract

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