Numerical methods integrated with fuzzy logic and stochastic method for solving PDEs: An application to dengue

Abstract

Currently dengue epidemics are of great relevance in Brazil and other countries of tropical and subtropical climates, because it is a disease that infects a large number of people and in its most severe form can lead to death. In this work we proposed an integrated mathematical model (SIR-type: susceptible, infected, recovered) to study the evolution both in space and time of dengue disease. The model is given by partial differential equations (PDEs) whose numerical solutions are obtained by hybrid schemes, fuzzy logic and stochastic methods. We use the hybrid explicit numerical schemes WENO-5 (weighted essentially non-oscillatory schemes, fifth order) for regions not smooth of the map and centered finite difference schemes of high order for the regions smooth in space discretization. Also a lifting scheme was made to define smoothness or not in the regions. For the time evolution, we have chosen the third order Runge–Kutta TVD (Total Variation Diminishing). The uncertain parameters related to the behavior of Aedes aegypti are extremely important for development and/or disease control. In this way for incorporating this information into the model, the parameters were estimated using fuzzy rule-based systems and information provided by specialists. Such parameters depend on the people, who provide breeding sites and blood for the maturation of the female's eggs and they depend on rain events, too. This variable, rainfall, presents stochastic dependence on the sampled values and for this reason, we chose the Markov chain method (order 2). Information on the behavior of the disease and the conditions for the proliferation of vectors in the region south of the city of Campinas were researched in the Health Department, Agronomic Institute and with experts of the Medical Sciences Faculty of UNICAMP. Simulations of many situations were performed to obtain scenarios regarding the spread of the disease, considering the characteristics of the region studied.