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
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.

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