Mathematical stochastics model of tuberculosis using empirical data from Malang city

Abstract

Mathematical models are utilized as a powerful tool to describe the transmission of diseases, especially Tuberculosis. Throughout these models, differential equations systems are commonly implemented. In this study employs a nonlinear differential equation to portray the disease of Tuberculosis, and the model was constructed in deterministic and stochastic systems. According to the deterministic model, we elaborate the positivity and boundedness of solutions, then two fixed points were obtained, which are disease-free equilibrium and endemic equilibrium points, and their stability conditions is established using Routh Hurwitz. Furthermore, the reproduction number is found by demonstrating the maximum of eigen value utilizing the next-generation matrix. Further discussion related to constructing a deterministic model involving two control variables, i.e. an effort to maintain distance between the susceptible population and the infected population, and efforts to treat the infected population. Analysis of optimal control problems by using the Pontryagin principle. Last study establishes about numerical simulations and parameter estimation using genetic algorithms. Both deterministic and stochastic models rely on parameter estimate results for simulation. Applying different disturbance levels to the stochastic model has a significant impact on the dynamical solution. In accordance to the optimal control simulation, the second control variable performs better than the first.