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.
Highlights
Tuberculosis (TB) is an airborne infectious disease caused by Mycobacterium tuberculosis which affects the lung and other parts of the body [3]
It is useful to conduct an investigation to determine how sensitive the threshold quantity basic reproduction number is with respect to its parameters, this will help us to know which of the parameters causes most reduction in Ro and parameters that have high impact on Ro and these should be targeted by intervention strategies in order to have most effective control of the disease
It is observed from the table 2, 3 and 4. that when the parameters with positive sensitivity values increase while the remaining parameters remain constant, the value of basic reproduction number increases, and this increase the endemicity of the disease in the population
Summary
Tuberculosis (TB) is an airborne infectious disease caused by Mycobacterium tuberculosis which affects the lung and other parts of the body [3]. Tuberculosis (TB) is the most common opportunistic disease affecting HIV positive people and the leading cause of death in patients with AIDS [8,12]. The population of Detected infected HIV individual increases by the fraction of latently individuals who develop disease symptoms (at the rate 1 H ) , where 1 is the endogenous reactivation rate and the detection of undetected individual at the rate UH. The population of detected infectious individual increases by the fraction of latent individuals who develop diseases symptoms at a rate 2 T and detection rate for undetected individual at the rate ( UT ) .The population is decreased by those that are treated who later moved to recovered compartment at the rate ( 2), natural death rate ( ) and disease induced death at a rate ( dT ) This gives dTD dt.
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