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.

On Derivation of the Probability of Occurrence of an Epidemic with Application to HIV/AIDS Spread Given Tuberculosis Co-infections in the Presence of Treatment

“…(R ht > 1) = Prob. (ℵ [6.28, 3.7412 ] > 1] = 0.079 =7.9% (10) Probability of 7.9% corresponds to the occurrence of a rare event, and this concurs with what is expected of epidemics in general.…”

“…An HIV positive person infected with Mycobacterium tuberculosis has a 50% chance of developing active TB against a 10% chance for the HIV negative [3]. Worldwide, TB is the most common opportunistic infection affecting HIV positive individuals [4] and it remains to be the most common cause of death in patients with AIDS [5]. It enhances HIV replication leading to increased viral load by accelerating the natural evolution of HIV infection [6] hence increasing the risk of progression from HIV to AIDS ( [7], [8]).…”

“…In 2013, about 9.0 million people had developed TB while 1.5 million died from the disease with 360 000 being HIV-positive [9]. Although deterministic models have been developed and analyzed to estimate the basic reproduction number for HIV and TB coinfection ( [10], [11], [12]), not much has been done to estimate the basic reproduction number as a random variable using probabilistic approach. Probabilistic modeling is a statistical analysis tool that estimates on the basis of past (historical) data the probability of an event occurring again.…”

A Systematic Literature Review of Mathematical Models for Coinfections: Tuberculosis, Malaria, and HIV/AIDS

“… 33 One of this research featured highly intricate model that posed challenges for mathematical analysis. 26–30 , 37 , 39 , 52 Furthermore, articles performed a sensitivity analysis on model, employing an indexing method to assess the impact on basic reproduction number. Basic Reproduction Number is a threshold value that can reference whether a disease will spread or be eliminated from a population.…”

“…Several articles employ these methods, incorporating different types of prevention and intervention controls. 28 , 31 , 34 , 38 , 39 , 45 , 49 , 58 Certain articles determine the best strategy to reduce infection by altering parameter values. 26 , 33 , 40 However, some articles did not incorporate any control measures in their analysis.…”