"Modelling of the Pertussis epidemiology - the dicotomy partial immunity vs temporary immunity."

Whooping cough (Pertussis) is a highly infectious bacterial disease caused by Bordetella pertussis. This is a small Gram negative rod capable of colonizing the cilia of the respiratory epithelium and it has been shown that it can invade alveolar macrophages. The transmission occurs via respiration, mainly through contact with respiratory droplets and in lesser extent to airborne droplets. Pertussis is 1 of the 10 most common causes of death from infectious disease worldwide. The introduction of child vaccination in the early-mid 20th century decreased the incidence of pertussis so enormously as to make it an uncommon disease, altough it is still endemic, with epidemic periods of 3-5 years. In fact, immunization is yet to reach the 100% effectiveness limit at the time of vaccination and it becomes less effective over the years. Since the 1980s there as been an increase in the occurunce of pertussis cases, as well as a shift of the average age of infection towards the higher age classes. This may be atributed to the far more effective recognition and diagnosis of the illness among older age groups (asymptomatic), the development of new strains of B. pertussis that are not affected by the existing vaccines (allowing them to remain endemic) or the waning of induced immunity. Over the years mathematical models attempted to reproduce the epidemiology of the disease and to some extent find out which of the hypothesis (or which combination) is more plausible, if any of these. There seems to be a focus on the loss of induced immunity, being that all of the models so far have considered that the immunity to pertussis is temporary. Our study, based on recent and more and more frequent genetic studies of the B. pertussis populations, considers the immunity to be a mixture of temporary and partial immunity. Partial because of the existence of antigenically different strains of pertussis and the inability of the vaccine to confer total protection. This approach has proven to be substancially different in terms of the epidemiological patterns it produces. Mathematical tools are being used to estimate the potencial aplication of this approach to real epidemiologic data.

"Rényi continuous entropy of DNA sequences."

Entropy measures of DNA sequences estimate their randomness or, inversely, their repeatability. L-block Shannon discrete entropy accounts for the empirical distribution of all length-L words and has convergence problems for finite sequences. We propose a new entropy measure that extends Shannon' s formalism. Rényi's quadratic entropy calculated with Parzen window density estimation method applied to Chaos Game Representation/Universal Sequence Maps (CGR/USM) of DNA sequences constitute a novel technique to evaluate sequence global randomness without some of the drawbacks of the former method. We have analytically deduced some of the asymptotic behaviour of this new measure and also performed the calculation of entropies for several synthetic and experimental biological sequences. This new technique can be very useful in the study of complexity of DNA sequences and provide additional tools for DNA entropy estimation.