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

Ladoke Akintola University of Technology, (LAUTECH), Ogbomoso, P.M.B. 4000, Ogbomoso, Oyo State, Nigeria. Ladoke Akintola University of Technology, (LAUTECH), Ogbomoso, P.M.B. 4000, Ogbomoso, Oyo State, Nigeria. Ladoke Akintola University of Technology, (LAUTECH), Ogbomoso, P.M.B. 4000, Ogbomoso, Oyo State, Nigeria. Ladoke Akintola University of Technology, (LAUTECH), Ogbomoso, P.M.B. 4000, Ogbomoso, Oyo State, Nigeria.

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.

INTRODUCTION.
Tuberculosis (TB) is an airborne infectious disease caused by Mycobacterium tuberculosis which affects the lung and other parts of the body [3]. TB infection occurs when droplet nuclei containing tubercle bacilli are inhaled into the lungs and deposited in the alveoli. It is spread when individuals with active TB disease cough, sneeze, sing, laugh or through interaction with infectious individual. After a person becomes infected, the tuberculosis bacteria are controlled by the person"s immune system and the infection becomes latent. The infection becomes active, when the bacteria spread out of control most especially when the tubercle bacilli overwhelm the immune system and break out of the tubercles in the alveoli and spread to the lungs and other parts of the body via blood stream [24]. The risk of infection with the tubercle bacilli is directly related to the degree of exposure and less to genetic or other host factors. In those with HIV, the risk of developing active TB increases to nearly 18% per year [3,5]. TB is a major global cause of disability and death, especially in developing countries. Even in many countries where its overall incidence is low, TB remains a problem; In 2006, TB caused an estimated 1.7 million deaths and 8.9 million new cases of infection [23].
The early symptoms of active Tuberculosis include: Weight loss, Fever, loss of appetite, coughing out of blood, night sweats, and back pain among others. TB is curable disease by the implementation of antibiotics, which decrease the mortality of TB to a minimal level, for instance, a 70% reduction in TB related mortality was recorded in the United State of America (USA) between 1945-1955 [2]. Many doctors prefer to hospitalize the patient in order to observe him or her during treatment, the method is called Direct Observed Therapy Short Course (DOTS), huge success has been achieved by this method. [1].
Human Immunodeficiency Virus (HIV) is caused by retrovirus infection; it progresses to Acquired Immunodeficiency Syndrome (AIDS) when proper treatment is not taken. HIV severely weakens immune system by attacks many blood cells, it causes havoc on the T-cell in the blood by destroying and decreasing their number, leading to decline in body's immunity to fight infection [20].. There are multiple modes of HIV transmission including Sexual intercourse, sharing needles with HIV infected persons, or via HIV-contaminated blood transfusions [21]. Infants may acquire HIV at delivery (Birth) or through breast feeding if the mother is HIV positive. This type of transmission is known as vertical transmission [6]. The term AIDS refers to only the last stage of the HIV infection after which death occurs. The HIV epidemic is sweeping the world both developed and developing nations as HIV has infected millions of people all over the world without any restriction of nationality, religion etc [18]. HIV infection in adults (75-80 percent) has been transmitted to one partner through unprotected sexual intercourse when the other partner is infected with HIV. Mother to child transmission (vertical infection accounts for more than 90 percent) of global infection to infants and children. As HIV infection progresses, immunity declines and patients tend to become more susceptible to common or even rare infections. The sexual transmission is believed globally responsible for the majority of new HIV infections [14,17] The related spread of two or more infections has always been a cause of concern for human being. HIV-TB co-infection is the largest cause among them. Tuberculosis (TB) is the most common opportunistic disease affecting HIV positive people and the leading cause of death in patients with AIDS [8,12]. Also, the relative risk of death and development of other opportunistic infections is higher in HIV-TB co-infected patients as compared to those having only one disease out of the two. [15].
Persons co-infected with TB and HIV may spread the disease not only to other HIV-infected persons, but also members of the general population who do not participate in any of the high risk behaviours associated with HIV. [22]. In recent decades, the dramatic spread of the HIV epidemic in sub-Saharan Africa has resulted in notification rates of TB increasing up to 10 -times. The incidence of TB is also increasing in other high HIV prevalence countries where the population with HIV infection and TB overlap. [15].
Patients with latent Mycobacterium tuberculosis infection are at higher risk for progression into active TB if they are coinfected with HIV [15]. HIV significantly increases individual risk of progression to active TB in both primary TB and the reactivation of latent TB. Likewise TB fuels the progression of HIV to full blwn AIDS [10]. It is estimated that among 33.4 million people living with HIV worldwide, about one-third are also infected by MTB [11]. India has about 1.8 million new cases of tuberculosis annually. In 2012, of the estimated 1.2 million TB deaths, about one-quarter were HIV/TB co-infected patients [25]. The interaction of these deadly diseases is sweeping off the population and it is quite unfortunate that much work has not been done to model the co-infection of these deadly diseases at a population level.
Due to the threat poses by the interaction of HIV with TB, we formulated nine (9) new compartmental models to determine the parameters that influence the transmission of these deadly diseases using forward sensitivity method.

Mathematical model formulation.
A non linear mathematical model is formulated and analyzed to study the sensitivity of parameters involved in the basic reproduction number ) ( 0 R on the dynamical spread of HIV-TB coinfection. In modeling the dynamics, the total homogeneously mixing population at time t, denoted by N(t), is divided into (9) nine mutually-exclusive compartments of Susceptible (S(t)) individuals, Latently HIV The Susceptible population is increased by the recruitment of individuals into the population at rate .  the population decrease by natural death rate  and by singly infected transmission individuals. Both singly infected individuals transmit either HIV or TB infection as follows (note that we split the disease transmission process into those generated by singlyinfected infected individuals to make the formulation easier to follow).

TB Modification parameters for classes
is positively-invariant and attracting with respect to the model (1) Proof: Consider the biologically-feasible region D , defined above. The rate of change of the total population, obtained by adding all equations of the model (13), is given by For this model, it can be shown that the region,

Stability of the disease free equilibrium (dfe)
Disease free equilibrium The reproduction number is the dominant eigen values of The DFE of the HIV-only model (15), given by (18) From equation (21), clearly Hence, the corresponding eigen values of equation (15) are; Sinc e all the real roots are negative, real and distinct. Hence, disease free equilibrium of the HIV only model (15) is locally asymptotically stable (LAS).

Global stability of disease free equilibrium (hiv)
Here, the global asymptotic stability (GAS) property of the DEF of the HIV model only will be explored.

Hiv sensitivity analysis
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 o R and parameters that have high impact on o R and these should be targeted by intervention strategies in order to have most effective control of the disease. This analysis tells us how crucial and important each parameter is to disease transmission. We compute the normalized forward sensitivity index of the reproduction number with respect to its parameters. Definition: If a variable "c" depends differentiably on a parameter "w", then, the normalized forward sensitivity index of "c" with respect to "w" is denoted by Xc, which is defined as For this model, it can be shown that the region,

Stability of the disease free equilibrium (dfe)
Disease free equilibrium    , , , , Hence, the corresponding eigen values of equation (26) are; Sin ce all the real roots are negative, real and distinct. Hence, disease free equilibrium of the TB only model (26) is locally asymptotically stable (LAS).

Global stability of disease free equilibrium (TB)
Here, the global asymptotic stability (GAS) property of the DEF of the TB model only will be explored.
For the model (26), the associated reproduction number, denoted by Thus, from equation (35), the characteristic equation is given by     It follows that the linearized differential inequality above is stable . Hence, we have established that the disease free equilibrium is globally asymptotically stable whenever

TB Sensitivity analysis
Also to each parameters involved in T R , the sensitivity indices of T R with respect to each parameter is calculated below, Results obtained were tabulated below as follows:

Analysis of the full model
Consider now the Co-infection model (14) of HIV-TB Disease free equilibrium of the full model is given by

Local stability of disease free equilibrium (dfe) of HIV-TB model
Since all the real roots are negative, real and distinct. Hence, disease free equilibrium of the HIV-TB model (16) is locally asymptotically stable (LAS).

Global stability of disease free equilibrium (HIV-TB)
Theorem 6: The disease free-equilibrium of the system (16)   negative real parts. It follows that the linearized differential inequality above is stable whenever Results obtained were tabulated below as follows:

Discussion of results
We analyzed a HIV-TB mathematical model, by evaluating the sensitivity indices of parameters involved in the basic reproduction number to know the parameters that need more attention. It is used to discover parameters that have a high impact on basic reproduction number and should be targeted by intervention strategies. increases latent infected individuals from 2122 to 2297. However, when progressor of TB increases from 0.2 to 0.9, it increases TB latent infected individuals from 1502 to 1640. Fig. 13, 14, 15, 16, 17 and 18 shows that detection of infected undetected individuals has a pronounced effect on infected detected and undetected individuals. When the detection of HIV infected undetected individuals increases from 0.4 to 0.9, it reduces infected undetected individuals from 608 to 369 but increases infected detected individuals from 594 to 832. Also, when the detection of TB infected undetected individuals increases from 0.4 to 0.9 it reduces infected undetected individuals from 767 to 302 but increases infected detected individuals from 706 to 1166.

Conclusion
In this paper, a deterministic model for the dynamical transmission of HIV-TB co-infection is presented. The objective is to determine the parameters that influence the dynamical spread of the diseases; the model includes Nine (9) compartmental differential equations. N o v e m b e r 2 8 , 2 0 1 5 A threshold parameter 0 R is defined and is shown that the disease will spread only if its value exceeds unity, i.e.
Sensitivity analysis of basic reproduction number 0 R to all the parameters tell us how crucial and important each parameter is to the disease spread in the environment. It showed that parameters representing the transmission rate of HIV and TB have a large impact on the dynamics of the diseases, and which should be targeted by medical practitioners to forestall the spread of the diseases. Also, fast progressor rate of HIV and TB needs to be checked, effort should be put in place medically to improve immunity of susceptible individuals, from the mathematical analysis, we discovered that people with weak immunity are more prone to the two diseases. It also has great significant effect when the two diseases co-exist in a particular patient.
Numerical simulation shows effects of parameters on the dynamical spread of HIV-TB co-infection and this tells us that a change in these parameters can significantly reduce the spread of HIV-TB co-infection disease in the population over time.