Classical and fractional analysis of the effects of silicosis in a mining community

Abstract

A mathematical model for the transmission dynamics of silicosis in a mining environment is designed and its qualitative analysis is given. The model takes into account the severity of silica dust exposure in a mining environment. The whole analysis is done in both fractional differentiation and classical integer calculus. In the former case, the Haar wavelet numerical scheme is used to solve the model and perform graphical representations. In the integer calculus case, it is shown that the disease free and endemic equilibria are globally asymptotically stable in the absence as well as in the presence of silica dust particles in the air, respectively. The epidemiological implications of these results are discussed. Numerical simulations are presented to support the theoretical analysis. In fractional differentiation, we show graphically via the Haar wavelet scheme the convergence to the disease free-equilibrium and the global stability of the endemic equilibrium, results successfully confirmed analytically via the classical integer analysis