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.

Abstract

Abstract
We study a dichotomous decision model, where individuals can make the decision yes or no and can influence the decisions of others. We characterize all decisions that form Nash equilibria. Taking into account the way individuals influence the decisions of others, we construct the decision tilings where the axes reflect the personal preferences of the individuals for making the decision yes or no. These tilings characterize geometrically all the pure and mixed Nash equilibria. We show, in these tilings, that Nash equilibria form degenerated hysteresis with respect to the dynamics, with the property that the pure Nash equilibria are asymptotically stable and the strict mixed equilibria are unstable. These hysteresis can help to explain the sudden appearance of social, political and economic crises. We observe the existence of limit cycles for the dynamics associated to situations where the individuals keep changing their decisions along time, but exhibiting a periodic repetition in their decisions. We introduce the notion of altruist and individualist leaders and study the way that the leader can affect the individuals to make the decision that the leader pretends. © 2014, American Institute of Mathematical Sciences.

Abstract

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.

Abstract
We use Adler, Tresser and Worfolk decomposition of Anosov automorphisms to give an explicit construction of the stable and unstable C1+ self-renormalizable sequences.