Tuberculosis transmission with relapse in Indonesia: Susceptible vaccinated infected recovered model

Abstract

Tuberculosis (TB) is a contagious disease prevented by the vaccination of Bacillus Calmette-Geurin (BCG). The sufferer is treated by giving the Directly Observed Treatment Shortcourse (DOTS). The patients sometimes do not take the medicine regularly so that the TB-bacteria do not completely vanish from their body. The bacteria only weaken so the patients can be re-infected (relapse). Mathematical modelling can be used to determine TB transmission based on changes in the number of TB patients. By concerning vaccination and relapse factors, a susceptible vaccinated infected recovered (SVIR) model with relapse in TB transmission need to be developed. Indonesia got the second rank in the new case of TB in the world. In the national program of TB prevention, Indonesia has three indicators i.e. prevalence, incidence, and mortality rate. In this article, we derive SVIR model with relapse in TB transmission, then apply the model in Indonesia and measure its accuracy. Furthermore, we also predict the three indicators of TB in Indonesia. The model is nonlinear first-order differential equations. The model is applied in Indonesia by estimating parameters based on 2004-2014 data. Model accuracy is measured based on relative errors in 2015 and 2016. Absolute values of relative error S, V, I, R, and TB incidence are less than 0.1. Thus, the model is said to be quite accurate and can be used to predict the prevalence, incidence, and mortality rates of TB in Indonesia. The prediction of these three indicators shows a decline year by year with an average decrease of 1.90% for prevalence and mortality rates and 1.68% for incidence rates.