Abstract: The theory of Poisson-overdispersed count models has been developed in deep, and consequently there are many known "physical mechanisms" leading to overdispersion. For instance, the general families of Mixed Poisson and Compound Poisson distributions are always overdispersed. These physical mechanisms can be interpreted and successfully used for health sciences and biological modelling. There are also some mechanisms leading to underdispersion but they are not very known. In this talk we are going to review some of them and present new methods and applications. The first mechanism considered is a Poisson-type process where the waiting times are not exponentially distributed. Barlow and Proschan in the 1960s showed that Increasing (Decreasing) Failure Rate distributions for the waiting times produce under(over)-dispersed count distributions. Examples of this mechanism are the models of Winkelmann (1995) using Gamma and Weibull waiting times. The second mechanism is the extended Poisson process of Faddy and Bosch (2001) based on the fact that any count distribution can be represented as a pure birth process with non-constant rates. This representation not always has a simple and meaningful interpretation. The third mechanism is provided by the limiting distribution of a M/M/1 queuing model, where the service time depends of the number of individuals in the queue. An example of this is the original development of the COM-Poisson distribution. This mechanism allows to construct new distributions capable to explain the behaviour of the counts of chromosomal aberrations under high doses of radiation (see Pujol et al., 2014). Finally we will introduce some new mechanisms based on the binomial subsampling operation (p-thinning). It is known that the Poisson distribution is closed under p-thinnings, but if p depends of the number of Poisson realizations the resulting distribution can be underdispersed. Several examples of application will be analyzed and discussed.
 Faddy, MJ. and Bosch RJ. (2001). Likelihood-Based Modeling and Analysis of Data Underdispersed Relative to the Poisson Distribution. Biometrics, 57, 620-624.
 Pujol M., Barquinero JF., Puig P., Puig R., Caballin MR., Barrios L. (2014). A New Model of Biodosimetry to Integrate Low and High Doses. PLoSONE, 9(12):e114137.
 Winkelmann, R.(1995). Duration Dependence and Dispersion in Count-Data Models. Journal of Business and Economic Statistics, 13(4), 467-474.