A multiobjective impulsive control scheme is proposed to give answers on how optimal vaccination campaigns should be implemented, regarding two conflicting targets: making the total number of infecteds small and the vaccination campaign as handy as possible. In this paper, a stochastic SIR model is used to better depict the characteristics of a disease in practical terms, where little influences may lead to sudden and unpredictable changes in the behavior of transmissions. This model is extended to analyze the effects of impulsive vaccinations in two phases: the transient regime control, taking into account the necessity to reduce the number of infected individuals to an acceptable level in a finite time, and the permanent regime control, that will act with fixed vaccinations to avoid another outbreak. A parallel version of NSGA-II is used as the multiobjective optimization machinery, considering both the probability of eradication and the vaccination campaign costs. The final result using the proposed framework nondominated policies that can guide for public managers to decide which is the best procedure to be taken depending on the present situation.

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This article describes the results obtained with a stochastic SIR (Susceptives-Infectives-Recovered) model with impulsive vaccination campaigns. It tries to give answers to the question of how vaccination campaigns should be implemented, regarding random influences and two main targets which conflict with each other: making the vaccination campaign as cheap and handy as possible and the total number of infected persons as small as possible or the probability of eradication as greater as possible. The target of the analysis is to compare if it is better to consider the probability of eradication before or in the multiobjective optimization procedure. Results show that pre-consider the probability of eradication will be similar to the post-considering regarding their costs and quantity of infected persons, despite being computationally harder.

The crossing point of two different distribution functions may be of interest for different reasons. The comparison of two different production processes with respect to failures may be one field of application, since the point of intersection of the corresponding distribution functions may be used for selecting the production process of superior quality. As a consequence, an estimator for the crossing point is needed. In this paper an estimator sequence is proposed by altering an approach that has been developed by Hawkins and Kochar in 1991. Using an approach suggested by Ferger in 2009, strong consistency and asymptotic normality of the proposed estimator sequence are derived by considering the argmax of a rescaled process which is selected as the scaled estimating error of the estimator sequence. Subsequently, weak convergence of this process to a limit process in the Skorokhod space is shown, where this limit argmax will turn out to satisfy a Gaussian distribution. A similar result has been obtained by Hawkins and Kochar in 1991, but by means of a different approach.