Laécio Carvalho de Barros

Fuzzy modelling in population dynamics

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.

Fuzzy differential equations and the extension principle

We study the Cauchy problem for differential equations, considering its parameters and/or initial conditions given by fuzzy sets. These fuzzy differential equations are approached in two different ways: (a) by using a family of differential inclusions; and (b) the Zadeh extension principle for the solution of the model. We conclude that the solutions of the Cauchy problem obtained by both are the same. We also provide some illustrative examples.

A note on Zadeh's extensions☆

Let f:Rn→Rn and f:F(Rn)→F(R)n be Zadehs extension of f to the space of fuzzy compact sets F(Rn). The aim of this paper is to show that if f is continuous, then f:(F(Rn),D)→(F(Rn),D) is also continuous, D being the supremum over Hausdorff distances between their corresponding level sets.

Fuzzy gradual rules in epidemiology

This paper presents an application of the fuzzy gradual rules in an epidemic study of canine rabies in Sao Paulo city, Brazil. A linguistic epidemiological model was elaborated through fuzzy rules built by the Extension Principle. We used both the inference method of Mamdani and of Dubois et al. The results were compared with real data from Sao Paulo and with another MISO Model, which is entirely based on expert knowledge presented in a previous work. Questions about application of fuzzy techniques in epidemiology, different inference methods and the Dubois et al. methodology are discussed.

Attractors and asymptotic stability for fuzzy dynamical systems

Abstract In this work we study the asymptotic properties of maps on fuzzy spaces which are extensions of maps on . The main results are in Section 4 (see Theorem 21) and we give an illustrative example in the last section.

Fuzzy differential equations: An approach via fuzzification of the derivative operator

Abstract In this paper we study fuzzy differential equations (FDEs) in terms of derivative for fuzzy functions, in a different way from the traditional Hukuhara derivative defined for set valued functions. The derivative we use is obtained by means of fuzzification of the classical derivative operator for standard functions. We discuss the relation of this approach to fuzzy differential inclusions (FDIs) and Hukuhara and strongly generalized derivatives. A theorem of existence of a solution is studied, with hypothesis similar to those assumed for FDIs. Some examples are explored in order to illustrate the theory.

STABILITY OF FUZZY DYNAMIC SYSTEMS

In this work we are study the Fuzzy Initial Value Problem (FIVP) with parameters and/or initial conditions given by fuzzy sets. Starting from the flow equation of the deterministic Initial Value Problem (IVP) associates to FIVP, we obtain the FIVP flow, through the principle of Zadeh. Follow, we introduce the concept of fuzzy equilibrium stability of FIVP and some examples are given.

On fuzzy solutions for partial differential equations

In this study we investigate heat, wave and Poisson equations as classical models of partial differential equations (PDEs) with uncertain parameters, considering the parameters as fuzzy numbers. The fuzzy solution is built from fuzzification of the deterministic solution. The continuity of the Zadeh extension is used to obtain qualitative properties on regular @a-cuts of the fuzzy solution. We prove the stability with respect to the initial boundary data, and show that as time goes to zero, the diameter of the fuzzy solution converges to zero and, as a consequence, to the cylindrical surface determined by the curve of the degree of membership. Numerical simulations are used to obtain a graphical representation of the fuzzy solution and a defuzzification of this solution is obtained using the center of gravity method. We theoretically show that the surface obtained by defuzzification with the plane determined by fixing time is indeed the solution of the same initial boundary problem for this time-point for the heat and Poisson equations and, in a particular case, for the wave equation. The deterministic solution and the defuzzified surface intercept are numerically compared using the Euclidean distance.

Fuzzy Differential Equations in Various Approaches

1. Introduction. -2. Basic Concepts. -3. Fuzzy Calculus. -4. Fuzzy Differential Equations. -Mathematical Background. -Index.

Methodology to determine the evolution of asymptomatic HIV population using fuzzy set theory

The aim of this paper is to o study the evolution of postive 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 is 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.