MATHEMATICAL MODEL FOR TRANSMISSION DYNAMICS OF HIV AND TUBERCULOSIS CO-INFECTION IN KOGI STATE, NIGERIA

Copyright © 2021 the author(s). This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract: In this work, we formulated a deterministic co-infection mathematical model made up of a system of nonlinear differential equations and rigorously analyzed it so as to gain insight into the transmission dynamics of each of the diseases as they were co-circulating in the population. We investigated the existence and stability of equilibria of the co-infection model and we subjected the model to rigorous analysis. The analysis of the sub-models (HIVonly and TB-only models) and that of the co-infection model revealed that their disease-free equilibrium are locally and globally asymptotically stable when their reproduction numbers were each less than unity, showing that the diseases can be put under control under these conditions. We carried out sensitivity analysis of the co-infection model, using data relevant to Kogi state of Nigeria, which revealed that the top ranked parameters that drive the


Human Immunodeficiency Virus (HIV) is a lenti-virus that causes the Acquired
Immunodeficiency Syndrome (AIDs), [1]. The condition in which humans' immune system degrades gradually which allows life threatening opportunistic infections to invade and thrive in the decaying immune system of the infected human is as a result of the disease called AIDS [1,2]. The HIV infection suppresses the body's resistant framework by diminishing the quantity CD4-T Cells [3,4,5,6]. HIV symptoms include chills, diarrhea that lasts for more than a week, tiredness, fever, profuse night sweats, mouth ulcers, muscle aches, rapid weight loss, sore throat [7,8]. The HIV virus can be transmitted through the exchange of a variety of fluids from infected individuals such as blood, breast milk, semen and vaginal secretions. Presently there is no cure and vaccine for HIV infection but effective treatment with antiretroviral drugs can control the virus so that people with HIV can live healthy [9,10]. Tuberculosis (TB) is an infectious disease caused by mycobacterium tuberculosis bacteria [7]. It affects the lungs and virtually other parts of the human body; it can also affect any age group [5,11]. Symptoms of the TB includes chronic cough with blood containing sputum fever, night sweat and weight loss [7]. When people who have active tuberculosis cough, spit,speak,sing or sneeze, they propel and expel tuberculosis germs [11,12,13,14].
HIV and tuberculosis increase the challenge of public health, the synergy between tuberculosis and HIV co-infection in patients is bidirectional on one hand, HIV infection influences the , individuals infected with AIDs (A(t)),this is as a result of the progression of individuals already infected with HIV that refuse to go for treatment and lastly individuals that recovered from tuberculosis infection ( ) In modeling the co-infection of these two diseases, we make the following assumptions: (i) Only individuals infected with TB can be infected with HIV as finding shows that 80% of individuals infected with HIV are practically bound to be infected with TB [20].
(ii) Those infected with TB and HIV are taken for treatment.
(iii) While TB infected individuals are taken for treatment, only those that are treated can recover from the disease, though they are still susceptible to the disease.
(iv) While some proportion of individuals infected with HIV accept to go for treatment, the remaining proportion that refused to go for treatment progress to the AIDs infected class.
(v) Those individuals infected with TB, AIDs on treatment can die due to the diseases. However, the disease-induced death rate due to AIDs is higher as compared to the other. Due to higher efficacy of drugs for TB as compared to antiretroviral drugs available for HIV treatment, disease-induced death rate for TB is assumed to be lower.
(vi) Natural death rate for individuals in all the classes are taken to be the same.
(vii) The transmission rate for the individuals in the susceptible class to be infected with TB is reduced by a factor ) ( 1 u which represents the awareness educational campaign measures available for them that they should always cover their mouth when coughing, sneezing and the need for the infants to be inoculated against the disease. Similarly, the transmission rate for the Susceptible to be infected with HIV is reduced by a factor ), ( 2 u which represents awareness educational campaign measures that susceptible individuals practice safe sexual activities, the need to avoid contacts with infected body fluids and the prevention of mother to child transmission during child birth or breast feeding. Furthermore, infection of individuals with TB and HIV together can be reduced by a factor ) ( 3 u which also represents the awareness educational campaign measure available for the individuals infected with TB to refrain themselves from HIV infection with proper awareness campaign.
Based on the assumptions stated above, the schematic diagram for the co-infection model is as presented below:

MODEL DESCRIPTION
Individuals are recruited into the Susceptible class at the rate (  ), the population reduces at the rate ( ) 1  , the rate at which individuals in the Susceptible class came in contact with Tuberculosis infected individuals in TB infected class ( ) 1 I , this rate is reduced by ( ) 1 u which represents an awareness educational campaign measures for the individuals in the susceptible class to always cover their mouth when coughing, sneezing and the need to always inoculate the infants against TB. The individuals in Susceptible class also decreases at the rate by which individuals carelessness get exposed to HIV infected individuals in class ( ) 2 I and became infected at the contact rate ( ) 2  , this rate can be reduced by a factor ( ) 2 u a control parameter that represents an awareness educational campaign measure available for the individuals in susceptible class to practice safe sexual activities, the need to avoid contact with bodily fluid of infected individuals   and by natural death.
Arising from above, the co-infection model for HIV/AIDs and Tuberculosis is given by the following systems of non-linear differential equations: Below is the model state variables and Parameters description:

POSITIVITY OF SOLUTION
Considering the fact that model (1) is a monitor for human population, there is a compelling need that all its state variables and parameters remains positive for all time, t. Hence, the following non-negativity results for the state variables in model (1) is given rise to: Given that the initial data be of the model (1) with non-negative initial data, will remain non-negative for all time 0  t . Proof: From the first equation of the model: By integrating both sides of this, we have: Taking the exponential of both sides of this, we have: Similarly, it can be shown that other state variables: Consequently, for all non-negative initial conditions, all solutions of the model (1) remain positive.

INVARIANCE PROPERTY
For our model (1) to be meaningful epidemiologically, there is a need to show that all the state variables of modelare positive for all time (t). In other words, the solutions of model (1) with non-negative initial data will remain non-negative for all ≥ 0;theproof of this is as follows: At steady state, when there is no infection in the system, (2) becomes: Thus, the region D is a positively-invariant set under the flow described by model (1)

ANALYSIS OF SUB MODELS
Before analysing full co-infection model (1), it is pertinent to gain some insights into the dynamics of the model with only tuberculosis appearing in the system and the case when only HIV is appearing in the system. This we do in this section.

TUBERCULOSIS-ONLY MODEL
From the general co-infection model (1), neglecting the interaction of HIV infected individuals by setting 0 , we obtain tuberculosis-only infected model which is as presented below: The flow diagram for tuberculosis alone model is as presented below: 5591 MATHEMATICAL MODEL FOR TRANSMISION DYNAMICS OF HIV

TUBERCULOSIS-ONLY MODEL
The disease-free equilibrium (DFE) of the model (2) is given by: It is not difficult to show that the set 1 D is positively invariant, such that it will attract all positive solution of sub model (2). Consequently, in this region, the given model can be considered to be epidemiologically and mathematically well posed and it is sufficient to consider the dynamics of model (2) in 1 D .
We obtain the DFE of model (2) by setting the right hand side of the equations in the model to zero to obtain: The local asymptotic stability (LAS) of the DFE is shown by using the next generation operator method on (2). By using related notations given byvan den Driessche and Watmough in [21], the matrices F and V for the new infection terms and the remaining transfer terms, are, respectively, given by: From here, it follows from van den Driessche and Watmough [21],that the reproduction number of model (2) is given by: We Claim the result below from theorem (2) given by [21].

EXISTENCE AND LOCAL STABILITY OF ENDEMIC EQUILIBRIUM POINT (EEP) OF MODEL (2)
Let the endemic equilibrium of tuberculosis-only infection model (2) be represented by: and setting the right hand sides of the equations in model (2) to zero, solving for the state variables in terms of the force of infection at steady statewe obtain: By substituting the expression for the endemic equilibrium point into the force of infection at steady state given in (3), we obtain: It follows from (5) and (6) that the polynomial (6) has a unique positive solution whenever 1  T R .

GLOBAL ASYMPTOTIC STABILITY OF THE DFE OF MODEL (2)
In order to show that system (2) does not undergo a backward bifurcation at 1 = T R , we now prove the global asymptotic stability of the DFE of model (2).

Theorem 3.1
The DFE of the tuberculosis-only model (2) Since all the model parameters are non-negative, it follows that Thus, the above theorem shows that the classical epidemiological requirement of 0 1 R  is the necessary and sufficient condition for the elimination of the disease from the community.

HIV-ONLY MODEL
Also from the general co-infection model The flow diagram for HIV-only model is as presented below.

HIV-ONLY MODEL
The disease-free equilibrium (DFE) of the model (7) is given by: It is not difficult to show that the set 2 D is positively invariant, such that it will attract all positive solution of sub model (7). Hence, in this region, the given model can be considered to be epidemiologically and mathematically well posed and it is sufficient to consider the dynamics of model (7) in 2 D .  (7) given in [21].

EXISTENCE AND LOCAL STABILITY OF ENDEMIC EQUILIBRIUM POINT (EEP) OF MODEL (7)
Let the endemic equilibrium of tuberculosis alone infection model (7) It follows from (10) and (11) that the quadratic equation (10) has a unique positive solution

GLOBAL ASYMPTOTIC STABILITY OF THE DFE OF HIV-ONLY MODEL (7)
In order to show that system (7)   (1) The system (7), the HIV-only model can be written in the form: We now denote the disease-free equilibriumof this system (7) by: (2) In order to guarantee the local asymptotic stability of the model (7), the conditions (W1) and (W2) following must be satisfied: U is globally asymptotically stable (GAS).
is an M-matrix (where the off-diagonal elements of A are non-negative) and  is the region where the model makes biological sense. If system (7) satisfies the aforestated necessary and sufficient conditions, then the following theorem holds: The disease-free equilibrium 0 S  of the HIV-only model (7) . Consequently, since the given necessary and sufficient conditions stated above is met, then the disease-free equilibrium 0 S  of the HIV-only model (7) is globally asymptotically stable when 1 This implies that the model (7) does not exhibit backward bifurcation.

ANALYSIS OF FULL HIV-TB CO-INFECTION MODEL
A major requirement of an epidemiological model is that it is stable (local and global asymptotic stability). Therefore, in this section, we do the analysis of model (1) for its stability property.

LOCAL STABILITY OF THE DFE OF HIV-TB CO-INFECTION MODEL (1)
In the absence of disease, otherwise called steady state, the Disease Free Equilibrium refers to a state when there is neither HIV/AIDs nor Tuberculosis infection in the system; as such, When the basic reproduction numbers T R and H R are more than one, the disease invades the population under consideration while when it is kept less than one, the disease will be wiped out of the population with time.

EQUILIBRIUM OF THE CO-INFECTION MODEL
In this section, we investigate the local stability of the co-infection model (1).

Theorem 4.2
The DFE of the co-infection model is locally asymptotically stable if 0 1 t R  and unstable if

Proof:
From co-infection model (1), the Jacobian matrix expression of the DFE ( ) 0  is given by:

MODEL Theorem 4.3
The DFE of the co-infection model (1)

Proof:
Let us define A  T  T  I  I  A  T  T  I  I Then the time derivative of U along the solution of the system (1) is given by:

ENDEMIC EQUILIBRIUM POINT
In this section we investigate the existence of endemic equilibrium for the co-infection model (1).
At the Endemic Equilibrium, there is presence of HIV and tuberculosis in the given system. Thus, we obtain endemic equilibrium when ( ) 0 , , ,

SENSITIVITY ANALYSIS
Sensitivity analysis shows how sensitive a model is to changes in the values of the parameters of the model and the variation in the structure of the model [18].Sensitivity analysis is required to know which parameter should be targeted towards control intervention strategies.
Sensitivity index of a parameter say ( ) 1  depends on the differentiable ( ) T R which is expressed as: The sensitivity index for other parameters can be expressed likewise. The values of sensitivity index corresponding to each of the reproduction numbers for each of the diseases is as presented in the table below.  Sensitivity analysis results shows that parameters with positive sensitivity indexes increase the endemicity of the diseases, while parameters with negative sensitivity indexes decrease the endemicity of the two diseases. From table 2, parameters with negative sensitivity indexes will have significant impact on the reduction of the basic reproduction number of tuberculosis T R to a value less than 1 and as such, these parameters should be targeted in controlling the spread of the tuberculosis in the given population under consideration. Observe that from the sensitivity indexes result, it is evident that educational measure for the susceptible individuals to always cover their mouth when coughing, sneezing and the need for infants to be vaccinated against tuberculosis which is represented by 1 u should be given greater attention towards reduction in the spread of the disease as it has the highest negative sensitivity index -1.00883, this is to be followed by treatment rate of the infected individuals whose index is -0.88272 . From table 3 likewise, parameters with negative indexes will have significant impact on the reduction of the reproduction number of HIV with tuberculosis infection to a value less than 1 and as such it should be targeted towards controlling the spread of the disease. It is also obvious that educational campaign measure for the susceptible individuals to practice safe sexual activities, the need to avoid contact with bodily fluids of infected individuals which is represented by 2 u whose sensitivity index is the highest with -0.71878 will have greater impact on the reduction of the burden of the disease.

NUMERICAL SIMULATION AND DISCUSSION
We carried out numerical simulations of the co-infection model to illustrate some of the theoretical results obtained in this study. This was done using MATLAB code solver. Variables and parameters values used in the simulations are as presented in the table 4 below: 17.  5,0000 [3] The results from the simulations are as presented in the figures below:

TREATMENT STRATEGY
We carried out the simulation of the co-infected model (1)

CONCLUSION
In this study, we formulated a system of non-linear differential equations to gain insight into the transmission dynamics of HIV-TB in a population where they are co-circulating. The sub models (TB-only and HIV-only models) were rigorously analyzed. Each of them has their disease-free equilibrium locally asymptotically stable when their reproduction number were each less than unity; each of them were globally asymptotically stable too when their reproduction number