MATHEMATICAL ANALYSIS OF THE ROLE OF DETECTION RATE IN THE DYNAMICAL SPREAD OF HIV-TB CO-INFECTION

Human Immunodeficiency Virus (HIV) co-existing with Tuberculosis (TB) in individuals remains a major global health challenges, with an estimated 1.4 million patients worldwide. These two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. We formulated new fifteen (15) compartmental models to gain more insight into the effect of treatment and detection of infected undetected individuals on the dynamical spread of HIV- TB co-infection. Sub models of HIV and TB only were considered first, followed by the full HIV-TB co-infection model. Existence and uniqueness of HIV and TB only model were analyzed quantitatively, and we shown that HIV model only and TB only model have solutions, moreover, the solutions are unique. 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 ( 0 R ) was analyzed, using next generation matrix method (NGM), and it has been shown that the disease free equilibrium point is locally asymptotically stable whenever 1 0  R and unstable whenever this threshold exceeds unity. i.e. 1 0  R , Numerical simulation was carried out by maple software using differential transformation method, to show the effect of treatment and detection of infected undetected individuals on the dynamical spread of HIV-TB co-infection. Significantly, all the results obtained from this research show the importance of treatment and detection of infected undetected individuals on the dynamical spread of HIV-TB co-infection. Detection rate of infected undetected individuals reduce the spread of HIV-TB co-infections. detection of infected undetected individuals of more due the in the numbers of infected undetected individuals. It was shown fuels the progression of HIV into full blown AIDS in the absence of proper treatment, likewise HIV increases latent TB to active TB in the absence of treatment. The results shown that in the presence of treatment, the rate of Active TB and HIV decreases as the treatment increases and consequently, the recover TB and HIV individuals increase rapidly.


Human Immunodeficiency Virus (HIV) is the etiological agent that causes acquired Immunodeficiency
Syndrome (AIDS) when left untreated. HIV weakens immune system by attacks many blood cells, it wrecks T-cell in the blood by destroying and decreasing their number leading to decline in body's immunity to fight infection [28]. The term AIDS refers to only the last stage of the HIV infection after which death occurs. The HIV epidemic is posing threat to 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 [25]. 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. The sexual transmission is believed globally responsible for the majority of new HIV infections [22,24].
There are multiple modes of HIV transmission including Sexual intercourse, sharing needles and sharp objects with HIV infected persons, or via HIV-contaminated blood transfusions [30]. Infants may acquire HIV at birth or through breast feeding if the mother is HIV positive, this type of transmission is known as vertical transmission [8]. 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 infections. The absolute T-helper cell (CD4 + ) count or percentage is used most often to evaluate the progression of HIV infection and to help clinicians make treatment decisions. [18].

TUBERCULOSIS (TB)
is an airborne infectious disease caused by a bacterium called a MYCOBICTERIUM TUBERCULOSIS (Tubercle bacilli).It typically affects lung and other parts of the body. TB can be acquired when an individual is exposed to tubercle bacilli produced by an infective individual. 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 [3,7]. TB is one of the major global cause of death, especially in developing countries. Even in many countries where its overall incidence is low, TB remains a problem; there has been an outbreak in the New York in the recent past [5], and incidence is currently rising in the United Kingdom ( UK) [16] . In 2004, TB caused an estimated 1.7 million deaths and 8.9 million new cases of infection [36], Worldwide, 9.15 million new infections occurred in the year 2006. In 2009, WHO estimated that there were 9.4 million incident cases of TB and 1.3 million deaths due to TB among HIV negative cases globally [35]. Over 95% of infections are in developing countries, where TB remains a dominant cause of morbidity and mortality [18].
Even today after the development of advanced screening, diagnostic and treatment method for the disease, it is shown that about one third of the world's population has been exposed and is infected with tuberculosis. [27]. anyone can become infected with T.B. bacteria, but people with HIV infectious disease suffers greater risk of getting infected with this disease. T.B. is the second leading cause of death among people with infectious disease after HIV [29].
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. [1] 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 bloodstream. [35]. If not treated active TB can be fatal. 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].

A BRIEF BACKGROUND ON HIV AND TB CO-INFECTIONS
The synergetic effect produced by the interaction of mycobacterium tuberculosis (TB) and the human immune-deficiency syndrome (HIV) are well known [26,19,20,34], where each accelerates the progression of the other. For instance, since its emergence in the 1980s, the HIV/AIDS pandemic continues to play a major role in the resurgence of TB disease which has greatly increased the morbidity and mortality rate of the disease worldwide. While HIV infections increase the risk of many opportunistic infections, the interactions between HIV and several infectious disease agents have caused particular medical and public health concern. [26]. For adults co-infected with HIV and latent TB, the life time risk of developing active TB disease rises from an estimated 10% up to 50%. This resulted in a parallel pandemic of HIV/AIDS and TB disease where HIV prevalence is high. [18]. 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. The largest increase in the number of TB cases has been in men aged 25-44. [33].
HIV fuels progression to active disease in people infected with TB [13]. Rate of recurrence of TB, both due to endogenous reactivation and exogenous re-infection, are increased among people infected with HIV, [18]. For instance, in many countries of eastern and southern Africa, the rates of TB notification have increased by five or more times as a result of HIV infection. Similarly, TB also fuels the progression of HIV infection in the patient to full blown AIDS after which death occurs. [37].
Currently, about 8% of global TB cases are linked to HIV infection, but this number is likely to increase in the future. For instance, the number of HIV positives in India is estimated to be 3.97 million cases, and almost 60% of the reported cases of AIDS had TB. January 28, 2016 The interaction of TB and HIV/AIDS is on the increase and these infections occur mostly in poor economically regions of the world. The immunodeficiency caused by HIV infection reactivates latent TB infection and accelerates the progression of newly acquired TB [15].
The enormous public health burden inflicted by these two deadly diseases necessitates the use of mathematical modeling to gain insights into the effects of treatment on the infected detected individuals and detection of infected undetected individuals on the dynamical spread of the disease.

MATHEMATICAL MODEL FORMULATION
A non linear mathematical model is formulated and analyzed to study the effect of detection of undetected individuals on the dynamical spread of HIV-TB co-infection.
In modeling the dynamics, the total homogeneously mixing population at time t, denoted by N(t), is divided into (15) The population of Treated HIV individuals is increased by those that have received treatment from HIV detected infected individual at the rate ( 1  ) this population reduces by fraction of treated individual that moved back to latently HIV individuals at the rate, (  ) since treatment does not completely clears the bacteria and finally reduced by natural death rate (  ).
The population of active TB induced HIV is increased by the progression from latent stage to active stage at the rate ( TH  ), the population decreased by natural death, induced mortality due to disease at the rate () and () respectively, individual who recovered also moved to recovered TB induced HIV at the rate ( 1    1   accounts for the assumption that latent individual have reduced infection rate, this is to account for the fact that individuals with latent TB infection have partial immunity against exogenous re-infection. [29].The population increased by natural recovery at the rate ( ), number of unsuccessful treated individuals who move to the latent TB individual at the rate ) ( 1   and rate (  r ) at which treated TB individual wanes off the treatment. Then the population is given by The population of undetected infectious individuals is increased by the infection of fast progressor at the rate T   ) 1 ( 1  and the development of symptoms by latent individual at the rate  is the fraction of exposed individuals who develop symptoms and are detected. It is further increased by the exogenous re-infection of expressed individual at the rate  the fraction of is re-infected exposed individuals who are detected, and fraction of unsuccessful treated individuals that move from detected individuals to undetected individuals at the rate (   2 ). This population is decreased by natural recovery at a rate (), detection of undetected individual at a rate ( UH  ), natural death at the rate (µ) and disease induced death at a rate ( Hence, The population of fail treatment compartment is generated by the failure of treated infected detected individuals at the The treatment failure could be due to a number of reasons, such as incomplete compliance to the specified treatment or drug resistance among others. The population decreased by fraction of treated individuals who lose their treatment the population decreased by the rate at which TB individuals who fail treatment move to other classes, the natural death and induced death at the rate  . The population of TB recovered individual is increased by the treatment of detected individual at the rate This population is further decreased by natural death and disease induced death (at a rate RT and   ).
The population of TB induced HIV recovered individual is increased by the treatment of active TB at the ( 1  ) and leather reduced by natural death rate at  and rate ( T  ), at which treated individual lose their treatment induced immunity.

MATHEMATICAL MODEL OF HIV-TB
TB Modification parameters for classes

POSITIVITY OF SOLUTIONS
For HIV-TB model only to be epidemiologically meaningful and well posed, we need to prove that all state variables are non-negative for all The closed set is positively-invariant and attracting with respect to the model (23) 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 (23), is given by

ANALYSIS OF SUB MODEL
Before analyzing the full model (23), it is pertinent to gain insights into the dynamics of the models for HIV only and TB only.

HIV MODEL ONLY
For this model, it can be shown that the region,

DERIVATION OF BASIC REPRODUCTION NUMBER (R O ) FOR HIV ONLY The Next Generation Matrix (F.V -1 ) Method
One of fundamental questions of mathematical epidemiology is to find the threshold conditions that determine whether an infectious disease will spread in a population when the disease is introduced into the population [14].
The basic reproduction number of the model (25) H R is calculated by using the next generation matrix [31]. Using their approach [31], we have, After taking partial derivative F and V, we have

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.

Theorem3.4:
The disease free-equilibrium of the system (25)  at steady state. The proof is based on using the comparison method [21] to prove the global stability.
According to [6,31], all eigen values of the matrix (F -V) have negative real parts.

DISEASE FREE EQUILIBRIUM
For critical points, we set At this free equilibrium, it is assumed that there is no infection, Hence the (DFE) is given as

GLOBAL STABILITY OF DISEASE FREE EQUILIBRIUM (TB)
Theorem3.5: The disease free-equilibrium of the system (35) is globally asymptotically stable whenever

DERIVATION OF BASIC REPRODUCTION NUMBER (R O ) FOR HIV-TB CO-INFECTION
The associated matrices F and V, are given, respectively, by  I  I  I  I  I  I  I (   2  18  19  17  13  20  14  18  13  15  19  15  13  16  20  2  16  20  2 (   2  2  18  19  2  2  16  20  19  18  15  11  20  16  15  11  20  17  14  11  2 1  18  19   17  19  1  16  20  19  18  15  20  16  15  20  17  14  2 (   2  17  19   1  16  20  1  2  18  19  18  10  15  19  15  18  16  20  17  10  20  14  2 Hence it is shown that the associated Reproduction number is given by The new fifteen (15) compartmental non linear differential equations were designed to gain more insight on the effect of epidemiological features on the dynamical spread of HIV-TB co-infection. Numerical simulation of the model was carried out by maple software, using differential transformation method as to be able to determine the dynamical spread pattern of the disease in the community. Graphs and tables feature prominently in the discussion of results.      is about 75% effective. Figure 3 and 4 has shown that detection rate has pronounced effect on HIV undetected infected and detected infected individuals. It shows that the higher the detection rate, the higher the infected detected individuals and the lower the undetected infected individuals. It reduces it to a minimal level when the detection rate UH  is about 0.7. Figure 5 and 6 Shown that treatment of TB has pronounced effect on TB infected detected individuals. It reduces the numbers of infected detected individuals and shows the increase in recovered individuals especially when the treatment rate 2  is about 75% effective.  co-infection; it was observed that the detection of undetected individuals reduced the spread of disease, the disease becomes more endemic due to the increment in the numbers of infected undetected individuals. It was also shown that TB fuels the progression of HIV into full blown AIDS in the absence of proper treatment, likewise HIV increases latent TB to active TB in the absence of treatment. The results shown that in the presence of treatment, the rate of Active TB and HIV decreases as the treatment increases and consequently, the recover TB and HIV individuals increase rapidly.

NUMERICAL SIMULATIONS
In conclusion, epidemiological features such as detection rate of infected undetected individuals and treatment rate of infected detected individuals play vital roles in the control of the spread of HIV-TB co-infection disease in the community. Medical practitioners and policy health makers need to increase the rate at which infected undetected individuals are detected for proper and timely treatment, also, drugs should be made available at a cheaper rate for the consumers.