Here we used the recursive point process and SEIR model to describe the transmission dynamics of pertussis. The recursive point process is a refined version of the Hawkes point process which defines productivity as a function of conditional intensity. The SEIR model is a variant of the basic compartmental model, SIR, which additionally takes into account a period when an individual is infected but not infectious yet. We attempted to fit the models to the reported cases of pertussis in Nevada since 1940. The evaluation of the model fit and prediction errors suggests that the recursive point process performs better in modeling the spread of pertussis and predicting the weekly incidence. However, we still recommend using it along with the SEIR model considering that the compartmental model has been more commonly used for an epidemic study.

Mumps has been long gone from public attention due to developing vaccine programs. In recent year, however, there are random outbreaks of mumps in US, all of which are strongly associated with college campus settings. It is necessary to find out and develop a statistical model with strong forecasting ability to help avoid massive contagion and help surveillance of this epidemic disease in the future. Pennsylvania mumps data is collected from Project Tycho to fit both the point process and SEIR models. Different methods of model evaluation are applied to help determine which one is the best performed. All three methods show quite similarity, but the Recursive model outweighs all others slightly. It is chosen to perform an 75% training vs 25% testing forecasting to see if it is able to catch the dispersal trend of mumps in Pennsylvania. Although the Recursive model predicts well in general, the whole comparison process shed some insights about what could be further done to evaluate and utilize various models.

We investigated the extent to which a SEIR compartmental model, two Hawkes point process, each with a different trigger density function, and a recursive point process could characterize the transmission dynamics of invasive Streptococcus pneumoniae. All models were parameterized using surveillance data from Florida between 2010 to 2014. The maximum likelihood estimates of the parameters were calculated, and weekly counts were predicted using a thinning technique for the point processes and adaptive tau-leaping method for the SEIR model. Results suggest that the point processes performed better than the SEIR model. When comparing goodness of fit and prediction errors between the point processes, the recursive point process appeared to perform reasonably well on both. The recursive point process had an RMSE almost as small as the Hawkes with power-law decaying trigger density, which had the lowest RMSE, and the highest log-likelihood of all the models that were evaluated.

For a single geyser one eruption may inhibit another eruption. The objective is to estimate the inhibition function of geyser eruptions using a non parametric algorithm by extending the non parametric estimation method of Marsan and Lenglin?(2008) for clustered Hawkes processes to the case where there may be inhibition. The proposed method is tested using simulated geyser eruption data from known densities: Exponential, Pareto, Normal, and Uniform. The method is then applied ot the actual data from the Lone Pine Geyser in Yellowstone National Park. The data consists of 163 eruptions from 2011. The results indicate that geyser eruptions do inhibit other eruptions to some degree.