A mathematical model for simulating the spread of a disease through a country divided into geographical regions with different population densities.

The SIR (susceptible-infectious-recovered) model is a well known method for predicting the number of people (or animals) in a population who become infected by and then recover from a disease. Modifications can include categories such people who have been exposed to the disease but are not yet infectious or those who die from the disease. However, the model has nearly always been applied to the entire population of a country or state but there is considerable observational evidence that diseases can spread at different rates in densely populated urban regions and sparsely populated rural areas. This work presents a new approach that applies a SIR type model to a country or state that has been divided into a number of geographical regions, and uses different infection rates in each region which depend on the population density in that region. Further, the model contains a simple matrix based method for simulating the movement of people between different regions. The model is applied to the spread of disease in the United Kingdom and the state of Rio Grande do Sul in Brazil.

General discrete modelling of lung sound production in normal subjects.

Lung sound analysis is of a major importance in diagnostic malfunctions of the respiratory system. In normal subjects, it is known that these sounds are caused by the interaction of the respiratory flows with the bronchial tree structure. However, the detailed knowledge of the reasons for the spectral characteristics of such sounds remains to be elucidated. In this paper we propose a model for normal lung sound production based on a discretization of air flow in particle-like elements. Their transport with the involved interactions is implemented using a pseudo-molecular dynamics Monte Carlo procedure. General physical principles were considered for the interaction of these elements with the bronchial tree as well as a two-body interaction potential. The particle-tree interactions and the particle-particle interactions represent the flow-tree and the internal flow interactions, respectively. According to the model, sound is produced in each bronchus with the pitch frequency inversely proportional to its dimensions and with amplitude proportional to the intensity of the interaction, also a function of the bronchus dimensions. The lung sound is then the composition of the sounds produced in each bronchus. The model was successful in approximating the spectral characteristics reported by Gavriely et al (1981, 1995) as a direct consequence of the fractal properties of the bronchial tree and the considered internal fluid interactions. Thus, the reported high-frequency spectrum with its affine property as well as the low-frequency irregularity could be reproduced.

General fractal-discrete scheme for high-frequency lung sound production.

A general scheme is proposed to explain the observed spectral properties of high-frequency human respiratory sounds in terms of the interaction between the respiratory flux and a bronchial tree of fractal properties. The air flux is treated as composed of discrete decoupled elements while the tree is assumed to have a Cantor-based geometry. According to this model, the affine behavior often observed in the high-frequency (log-log) spectral range is a direct consequence of the fractal geometry of the bronchial tree in both qualitative and quantitative aspects. This strongly indicates that the dynamics underlying the high-frequency sound generation must have at most nondominant couplings between the relevant fluid components.