Epidemiological Models of Directly Transmitted Diseases: An Approach via Fuzzy Sets Theory

Abstract

In this article, we consider environmental and demographic fuzziness when the varying uncertainties are modelled by Fuzzy Set Theory. In the first case the uncertainties are considered in the parameters of the model, frequently described by differential equations. In the second the state variables and their variation rates are linguistic and related through fuzzy rules. Here we proposed a methodology to study stability of equilibriums for systems whose direction field is partially known and given by a fuzzy rules. From the results it is possible to obtain much relevant information that can help in the study of classic models.However this methodology can be applied to more complex epidemiological mathematical models. With the goal of making comparisons with the results of the classic model, we apply these ideas in the formulation of the SIS model (susceptible-infected-susceptible), which is the model that describes a disease in which individual recovers but does not develop any kind of immunity. The basic reproduction values (R-0) for the fuzzy model and the classic model are compared. For the SIS model described by a system based on fuzzy rules, instead of differential equations, the stability of the equilibrium point is analyzed using the Lyapunov function. Also, parameter estimation from fuzzy modelling is considered.