1. “Derivative ratio as a measure of effect: Sex over age of occurence of myocardial infarction in Brazil” (com PEREIRA, J. C.R. ; BARROS, L. C. European Journal of Epidemiology, Grã-Bretanha, v. 21, p. 263-266, 2006.

 Abstract: The concept of relative risk (RR) as anincidence rate ratio is extended to a ratio of derivatives of cumulative occurrences of events with respect to any continuous variable whose domain is partitioned in sub-domains of study and referent groups to be contrasted. RR, thus expressed as a function, allows the identification of key values of the independent variable through which effect, as a relation between sub-domains, is modulated. The effect of sex over age of occurrence of myocardial infarction in a sample of Brazilian patients is analysed. Male sex, as contrasted to female, is acknowledged as a risk factor for the disease but it is known that this risk wanes with aging. Derivatives of cumulative occurrences with respect to age are calculated for men and women. Given the normal distribution of these occurrences, derivatives are found to be very similar to the corresponding normal density functions, with difference not greater than 0.005. This provides an easy way of identifying the derivatives whose ratio is likewise easily calculated. Results suggest that, in theBrazilian sample of patients studied, the greatest men to women RR is 1.9 at the age of 23.5 years and that at the age of 61.2 years risks become evens.

Key words: Effect modifier, Epidemiological methods, Mathematical model, Relative risk, Risk assessment.

2. “Selection of patients for myocardial perfusion scintigraphy based on fuzzy

      sets theory applied to clinical-epidemiological data and treadmill test esults”

      (com DUARTE, P. S. ; PEREIRA, J. C.R. ; TONELLI, P. A. ; STROCOLLA,

      L. ; FARSKY, P. ; SAMPAIO, C. ; ORTEGA, N. R. S. . Brazilian Journal of

      Medical and Biological Research, São Paulo, v. 39, p. 9-18, 2006.

Abstract: Coronary artery disease (CAD) is a worldwide leading cause of death. The standard method for evaluating critical partial occlusions is coronary arteriography, a catheterization technique which is invasive, time consuming, and costly. There are noninvasive approaches for the early detection of CAD. The basis for the noninvasive diagnosis of CAD has been laid in a sequential analysis of the risk factors, and the results of the treadmill test and myocardial perfusion scintigraphy (MPS). Many investigators have demonstrated that the diagnostic applications of MPS are appropriate for patients who have an intermediate likelihood of disease. Although this information is useful, it is only partially utilized in clinical practice due to the difficulty to properly classify the patients. Since the seminal work of Lotfi Zadeh, fuzzy logic has been applied in numerous areas. In the present study, we proposed and tested a model to select patients for MPS based on fuzzy sets theory. A group of 1053 patients was used to develop the model and another group of 1045 patients was used to test it. Receiver operating characteristic curves were used to compare the performance of the fuzzy model against expert physician opinions, and showed that the performance of the fuzzy model was equal or superior to that of the physicians. Therefore, we conclude that the fuzzy model could be a useful tool to  assist the general practitioner in the selection of patients for MPS.

Key words Fuzzy model, Coronary disease, Scintigraphy, Myocardial perfusion.

3.  “Sobre Conjuntos Fuzzy”. Revista do Professor de Matemática, São Paulo, v. 56, p. 2-9, 2005.

Resumo: Este é um artigo apenas de divulgação. Nele são apresentados os conceitos de conjuntos fuzzy, operações entre conjuntos fuzzy e alguns exemplos ilustrando o potencial de aplicação da Teoria dos Conjuntos.

5. “Clinical signs of pneumonia in children: association with and prediction of diagnosis by fuzzy sets theory”        (com PEREIRA, J. C.R. ; TONELLI, P. A. ; ORTEGA, N. R. S.) Brazilian Journal of Medical and         Biological Research, Brasil, v. 37, n. 5, p. 701-709, 2004.

       Abstract: The present study compares the performance of stochastic and fuzzy models for the analysis of the relationship between clinical signs and diagnosis. Data obtained for 153 children concerning diagnosis (pneumonia, other non-pneumonia diseases, absence of disease) and seven clinical signs were divided into two samples, one for analysis and other for validation. The former was used to derive relations by multidiscriminant analysis (MDA) and by fuzzy max-min compositions (fuzzy), and the latter was used to assess the predictions drawn from each type of relation. MDA and fuzzy were closely similar in terms of prediction, with correct allocation of 75.7 to 78.3% of patients in the validation sample, and displaying only a single instance of disagreement: a patient with low level of toxemia was mistaken as not diseased by MDA and correctly taken as somehow ill by fuzzy. Concerning relations, each method provided different information, each revealing different aspects of the relations between clinical signs and diagnoses. Both methods agreed on pointing X-ray, dyspnea, and auscultation as better related with pneumonia, but only fuzzy was able to detect relations of heart rate, body temperature, toxemia and respiratory rate with pneumonia. Moreover, only fuzzy was able to detect a relationship between heart rate and absence of disease, which allowed the detection of six malnourished children whose diagnoses as healthy are, indeed, disputable. The conclusion is that even though fuzzy sets theory might not improve prediction, it certainly does enhance clinical knowledge since it detects relationships not visible to stochastic models.

Keywords: Epidemiologic methods,  Stochastic models, Fuzzy models,  Clinical signs,  Diagnosis, Data analysis

6. “Methodology to Determine the Evolution of Asymptomatic HIV Population Using Fuzzy Set Theory “ (com R.M. Jafelice, R. C. Bassanezi e F. Gomide), International Journal of Incertainty, Fuzziness and Knowledge-Based Systems; Vol. 13, No 1. pp. 39-58. Word Scientific Pu.Co. (2005).

Abstract: The aim of this paper is to study the evolution of positive HIV population for manifestation of AIDS, the Acquired Immunodeficiency Syndrome. For this purpose, we suggest a methodology to combine a macroscopic HIV positive population model with an individual microscopic model. The first describes the evolution of the population whereas the second the evolution of HIV in each individual of the population. This methodology is suggested by the way that experts use to conduct public policies, namely, to act at the individual level to observe and verify the manifest population.

The population model we address in a differential equation system whose transference rate from asymptomatic to symptomatic population is found through a fuzzy rule-based system. The transference rate depends on the CD4⁺ level, the main  T lymphocyte attacked by the HIV retrovirus when it reaches the bloodstream. The microscopic model for a characteristic individual in a population is used to obtain the CD4⁺ level at each time instant. From the CD4⁺ level, its fuzzy initial value, and the macroscopic population model, we compute the fuzzy values of the proportion of asymptomatic population at each time instant t using the extension principle. Next, centroid defuzzification is used to obtain a solution that represents the number of infected individuals. This approach provides a method to find a solution of a non-autonomous differential equation from an autonomous equation, a fuzzy initial value, the extension principle, and center of gravity defuzzification. Simulation experiments show that the solution given by the method suggested in this paper fits well to AIDS population data reported in the literature.

Keywords: Epidemiological modeling, HIV population model, dynamic fuzzy modeling, fuzzy set theory.

7. “Fuzzy Differential Inclusion: An Application to Epidemiology” (com Barros, L.C., Oliveira, R.Z.); Advances in Soft Computing – Soft Methodology and Random Information Systems, Springer-cajAstur, Oviedo (Espanha), p.631-637 (2004).

Abstract: In this study we consider the SI epidemiological model with heterogeneity in infected class. In this case, the transmission rate is considered as a fuzzy set. The theory of fuzzy differential inclusion is used to produce a solution of the model and a comparison between this solution and the one proposed by the deterministic model is made too.

Keywords: Epidemiology, fuzzy differential inclusion, mathematical expectancy.

8. “Fuzzy modeling in symptomatic HIV virus infected population” (com R. Zafelice, L. C. Barros e F. Gomide), Bulletin of Mathematical Biology, 66, n.6 pp. 1597-1620. Academic Press. ISSN 0092-8240; (2004).

Abstract: This paper introduces a model of the evolution of positive HIV population and manifestation of AIDS. The focus is on the nature of the transference rate of HIV to AIDS. Expert knowledge indicates that the transference rate is uncertain and depends strongly on the viral load and the CD4⁺ level of the infected individuals. Here, we suggest to  view the transference rate as a fuzzy set of the viral load and CD4⁺ level values. In this case the dynamic model results in a fuzzy model that preserves the biological meaning and nature of the transference rate. The paper also includes a comparison between the fuzzy model and the classic Anderson’s model using data reported in the literature.

Keywords: HIV⁺, transference rate, fuzzy modeling.

9. “Fuzzy Set-Based Model to Compute the Life Expectancy of HIV Infected Populations”

(com Jafelice, R.M., Barros, L.C., Gomide, F.) NAFIPS-North American Fuzzy Information Processing Society; IEEE; Special Session on Biomedical Informatics and Computational Intelligence; Banff (Canadá), p.314-318 (2004).

Abstract: in this paper we study the influence of AIDS in the life expectancy of  a population considering a classical population model completed with expert knowledge information via a fuzzy rule-based system. In the model considered, we assume that AIDS has a direct influence in the mortality rate of a population and derive a fuzzy rule-based system to capture this influence using viral load, CD4⁺, and a parameter that modify the transference rate, as variables. We determine the average number of individuals and the life expectancy for specific population groups with no anti-retroviral therapy. The values found are coherent with actual data reported by UN.

Keywords: Fuzzy sets, AIDS, life expectancy, fuzzy rule-based system.

10. “Um Estudo de Automatos Celulares para o Espalhamento Geográfico de Epidemias com parâmetros fuzzy” (com PEIXOTO, M. S) TEMA -Tend. Mat. Apl. Computacional, Rio de Janeiro, v. 5, n. 1, p. 125-133, 2004.

12. “The SI Epidemiological models with a Fuzzy Transmission Parameter”(com M.B.F. Leite e L.C. Barros), Computers & Mathematics with Applications, Pergamon Press, Elsevier Science Ltd., pp 1619-1628, 2003.

Abstract: Fuzzy theory techniques are applied to susceptible-infectious SI epidemiological models. The transmission coefficient is considered as a fuzzy set and the mean number of infected individuals is compared with the trajectory of the mean virus load I(v,t). Also, the fuzzy basic reproduction value R^f₀ is discussed for this formulation and a control policy of the disease, using R^f₀, is proposed.

Keywords: Epidemiology, fuzzy set, fuzzy expected value, fuzzy basic reproductive value^.

14. “Fuzzy rules in asymptomatic HIV virus infected individuals model”(com Jafelice, R.S.M.; Barros, L.C.; Gomide, F.). Frontiers in Artificial Intelligence and Applications – Advances in Logic, Artificial Intelligence and Robotics, 85, v1,pp. 208-215; IOS Press, LAPTEC. (2002).

Abstract: The purpose of this paper is to study of the evolution of positive HIV population for manifestation of AIDS. Our main interest is to model the transference rate in this stage. For this purpose, we use expert information on transference rate because it strongly depends on the viral load and the level CD4⁺ of the infected individual. More specifically the transference rate is modeled as a fuzzy set that depends of the knowledge on viral load CD4⁺ , respectively. The main difference between this model and the classic one relies on the biological meaning of the transference rate. Medical science uses linguistic notions to characterize disease stages and to specify anti-retroviral therapy. Fuzzy set theory provides the formal framework to model linguistic descriptions such as the transference rate using expert knowledge.

Keywords: Fuzzy sets, positive HIV, rate of transference, viral load, fuzzy rule-based model.

15. “A note on the Zadeh's extensions" (com H. Roman-Flores e L. C. Barros). Fuzzy Sets and Systems, 117 (fev/2001) 327-331.

Abstract: Let be f a function of Rⁿ in Rⁿ  and f^ their Zadeh’s extension on the space of fuzzy compact sets. The aim of this paper is show that if f is continuous, then f^ is also continuous where the metric D is the supremum over Hausdorff distances between their corresponding level sets.

Keywords: Zadeh’s extension, Fuzzy compact sets, continuity.

17. "Attractors and asymptotic stability for fuzzy dynamical systems",(com  L.C.Barros e P.A.Tonelli), Fuzzy Sets and Systems, 113, pp. 473-483. Elsevier Science. Amsterdam (2000).

Abstract: In this work we study the asymptotic properties of maps on fuzzy spaces which are extensions of maps on Rⁿ. We analyze the discrete fuzzy systems and the stability of the fixed points of Zadeh’s extension. The main results are in Section 4. We give an illustrative example in the last section, using the logistic discrete model.

Keywords: Dynamical discrete system, membership functions, Zadeh’s extension.

18. "Fuzzy modelling in population dynamics " (com L.C.Barros e P.A. Tonelli). Ecological Modelling , 128, pp.27-33. Elsevier Science (2000).

Abstract: The aim of this paper is to analyze the behavior of models which describe the population dynamics taking into account the subjectivity in the state variables or in the parameters. The models in this work have demographic and environmental fuzziness. The environmental fuzziness is presented using a life expectancy model where the fuzziness of parameters is considered. The demographic fuzziness  is presented using the continuous Malthus and logistic discrete models. An outstanding result in this case is the emergence of new fixed points and bifurcation values to the discrete logistic model with subjective state variables in form of fuzzy sets. An interpretation is offered for this fact which differs from the deterministic one.

Keywords: Fuzzy modeling, demographic and environmental fuzziness, population dynamics.

19. "A simple model of life expectancy with subjective parameters" (com L.C. Barros), Kybernetes: Intern. Journal Systems & Cybernetics, 24, 7, pp.91-99 (1995).

Abstract: The use of deterministic models to describe an observable phenomenon always presents enormous difficulty when evaluating values of the parameters. The usual procedure is to adjust values to results obtained by numerical simulation or by simply guessing their values though subjective considerations. In those cases where randomness of the phenomenon cannot be established, or when there is not sufficient data available, it seems that fuzzy theory is an adequate approach to the problem.

  In this paper the classical stochastic Boltzman model is compared with stochastic fuzzy techiniques. This comparison is carried on a specific case study: the life expectancy of individuals whose “causa mortis” is affected by poverty. It seems natural that the “degree of poverty” together with adverse living conditions must influence the life expectancy of a group of individuals. The model that will be proposed is aimed at measuring this influence.

Keywords: fuzzy sets, life expectancy, degree of poverty, fuzzy expected value.