Mathematical modelling and sensitivity analysis of HIV-TB co-infection.

Abstract

In this paper, we formulated a new nine (9) compartmental mathematical model to have better understanding of parameters that influence the dynamical spread of Human immunodeficiency virus (HIV) interacting with Tuberculosis (TB) in a population. The model is analyzed for all the parameters responsible for the disease spread in order to find the most sensitive parameters out of all. Sub models of HIV and TB only were considered first, followed by the full HIV-TB co-infection model. Stability of HIV model only, TB model only and full model of HIV-TB co-infection were analyzed for the existence of the disease free and endemic equilibrium points.  Basic Reproduction Number (R₀) was obtained using next generation matrix method (NGM), and it has been shown that the disease free equilibrium point is locally asymptotically stable whenever R₀ > 1and unstable whenever this threshold exceeds unity. i.e.. R₀ > 1 The relative sensitivity solutions of the model with respect to each of the parameters is calculated, Parameters are grouped into two categories: sensitive parameters and insensitive parameters. Numerical simulation was carried out by maple software using Runge-kunta method, to show the effect of each parameter on the dynamical spread of HIV-TB co-infection, i.e. detection of infected undetected individuals plays a vital role, it decreases infected undetected individuals. Also, increased in effective contact rate has a pronounced effect on the total population; it decreases susceptible individuals and increases the infected individuals. However, effective contact rate needs to be very low in order to guaranteed disease free environment.