1 Introduction

The two infectious agents HIV and Mycobacterium tuberculosis (Mtb) have been co-existing in humans for decades. We have enough evidence that one of these infections accelerates the progression of the other [1–4]. Thus, we have to pay a great attention about the subject towards exploring HIV-TB co-epidemics.

Tuberculosis (TB) is an infectious disease caused by a bacterium called Mtb which can be spread from person to person. It affects the lungs and is called Pulmonary TB but can also affect other parts of the body like brain, glands, kidney and bones is then called Extra-Pulmonary TB [5, 6]. It is a curable disease, but may cause death unless infectious individuals are treating at the right time. The symptoms are depending on the body parts wherever the TB bacteria grow. The symptoms are 3 weeks cough or longer, chest pain, night sweat, and weight loss [7]. The TB bacteria spreads in to the air when TB infectious people sneeze, cough, speak or sing [8–10].

Human Immunodeficiency Virus (HIV) is a lenti-virus responsible for the Acquired Immunodeficiency Syndrome (AIDS) [11]. HIV is transmitting by infected blood, semen, vaginal secretion, and breastfeeding (for infants) from HIV+ mother without treatment. This virus affects white blood cells of the immune system by diminishing the quantity. Gradually, it destroys their functions and the immune system cannot resist other opportunistic infections. This leads HIV progression to the AIDS stage. AIDS is the supreme disease phase of HIV infection [11, 12]. The symptoms are tiredness, fever, night sweats, muscle aches, rapid weight loss, sore throat, etc… [7, 13]. Nowadays, there is no cure and vaccine for HIV, but the proper treatment with Antiretroviral Therapy (ART) individuals can control the virus to prolong their life [14].

The two pathogens Mtb and HIV may come either by co-infection or super-infection. One can increase the effect of the other and they can accelerate the deterioration of immune system function. People living with HIV are 20 up to 30 times at a higher risk of developing active TB disease than their counterparts [15, 16]. The prevalence rate of TB in HIV+ people is increasing because of exogenous re-infection and endogenous re-activation [17]. Individuals infected by HIV lead to a haggled immune system, consequently their susceptibility to Mtb infection is increasing. It is difficult to diagnose TB in HIV infected people [18]. TB also remains the primary agent of death in HIV infected people, counting for around 1 in 3 deaths associated with AIDS [19].

Globally, there are approximately 16 million people infected by HIV-TB co-infection in 2017 [20]. Recently, WHO 2020 report showed that an estimated number of over 14 million people are co-infected [21]. The report stated that South-East-Asia and Sub-Saharan Africa were taking the highest portion. More than 90% of TB deaths occurred in the two regions [22]. TB is also the highest cause of death from TB–AIDS related deaths, of which 95% happened in developing nations [23, 24].

The burden of TB-HIV co-infection is high in Sub-Saharan Africa (SSA) [25]. As stated by WHO report of 2019, about 84% of the total number of TB-HIV co-infection cases occurred in the region. SSA has 12% of the global population, it has accounted for 30% of the 9 million TB incidence cases and more than 270 000 deaths related to TB [26]. In this region, the HIV prevalence is high which grips to over 50% of the patients were dually infected.

As one of the Sub-Saharan Africa countries, Ethiopia is severely affected by the TB-HIV co-epidemic. As of 2019, an incidence rate of TB was 345 per 100 000 of which 33% are living with HIV [27]. Moreover, the WHO 2020 report stated that Ethiopia is one of the high TB and TB-HIV burden countries in the globe [28]. In 2019, over 23% of people are HIV+ from active TB individuals in Ethiopia [27]. Co-infection of TB with HIV accelerates the possibility of progressing from latent to active stage [29].

HIV and TB have different nature and diverse treatment outcomes. Thus, an integrative treatment program is urgently needs for TB-HIV co-infection [1, 30]. TB can be cured with appropriate treatment for a period of six up to nine months. Mostly, the recommendations for people infected by this co-infection disease is to begin TB treatment immediately. Hereafter, they can start ART treatment on the appropriate starting time suggested by physicians. Initiating ART at or afterwards the start of TB therapy may cause Immune Reconstitution Inflammatory Syndrome (IRIS) [18]. This happens when a high pill burden of antibiotics and ART are existing. If IRIS occurs, it will worse TB infection. This leads TB treatment can be complicated. And also, delaying ART until after TB therapy is completed may increase HIV transmission risk and death caused by HIV. Hence, it is serious to differentiate the actual time where dual treatment is given for HIV-TB infected people.

Mathematical models are essential to explore the co-epidemics disease dynamics and to provide better insights about preventive and controlling regimes. Existing models on HIV–TB co-infection are reviewed in the following way.

Navjot et al. [31] formulated TB-HIV co-infection model to study the role of screening and treatment. They considered active sexual adult people in the model. Their result showed that increasing the rate of screening TB leads to decreasing TB infectious people. The researchers recommend that strong cooperation between the TB and HIV intervention regime is necessary to control the disease. M. M. Ojo et al. [32]. investigated impact of vaccination on the dynamics of tuberculosis using mathematical modelling. They discussed the stabilities of equilbrum points and exhibited the occurrence of backward bifurcation in the model. The authors explored the influence of vaccination rate, vaccine efficacy, and effective contact rate on the active TB infectious individuals. Their findings showed that rising the vaccination rate of susceptible people reduces the TB disease burden. Nevertheless, the vaccine efficacy must remain above 25% to minimize the disease problem effectively. They concluded that, reducing the contact rate with TB infectious and expanding vaccination with high vaccine efficiency will diminish the TB disease prolifically. Similarly, the dynamics of Tuberculosis (TB) Outbreak have been examined using mathematical modelling approach, where effective use of treatment rates and isolation [33]. Fatmawati et al. [14] studied the effect of antibiotics and ART optimally to control the transmission dynamics of HIV-TB co-epidemic. Their numerical result showed that coupling of ART and anti-TB optimal control is the most effective strategy to fight against the disease. However, they suggested that antibiotics are better than ART when only one control is used. Grace et.al [34] explored the impact of HIV on TB infection via considering ART and TB treatment in Kenya. Their result suggested that testing and administering latent TB, both ART and TB treatment, HIV testing for all TB patients and vice versa are very crucial for Kenyan people. Roeger et al. [23] introduced a deterministic model of TB-HIV co-infection. They analysed the model and their numerical result suggested that the presence of HIV leads to increase the cases of co-infectious individuals even if TB reproduction is less than unity. The authors endorsed that; more effort should be given for reducing HIV prevalence to control TB infection in the co-infection populace. Awoke et al. [35] proposed TB-HIV/AIDS co-epidemics model with behavioural modification. They extended the model into an optimal control problem by considering behavioural modification as preventive measures and treatment efforts as controlling strategies. Their numerical result showed that applying both preventive and control measures can reduce the disease and cost burden. The authors declared that the cost of applying preventive effort is very small as compared to treatment, but the cost of administering the infection is huge when the rate of disease transmission is high. They conclude that applying both prevention and treatment efforts at a time is a best effective strategy.

More recently, the modeling of TB-HIV and related diseases transmission dynamics are investigated [36–39]. Hadipour et al. [40] investigated TB–HIV co-epidemics treatment controls. They used a mathematical model along with an optimum sliding mode controller. The researchers applied a multi-objective genetic optimization algorithm to find the optimal values of the control coefficients. Their result showed that when controls are applied new infections, disease deaths, and total burden values are reduced rather than without control. A. Ahmad et al. [41] considered TB-HIV co-infection model for mathematical analysis and numerical simulation. They used non-standard finite difference technique with Mickens approach ϕ(h) = h + O(h²) to analysis the model numerically. The researchers presented the graphical scenarios of each compartment for equilibrium points of the model by varying the step size h. Their result displayed that when the step size is increasing slightly, the susceptibility of individuals with HIV, TB, and their co-infection are increasing, otherwise decreasing. Finally, their result showed that the disease burden is decreased when they projected the time duration. Aggarwal and Raj [42] formulated and analysed a fractional order model in Caputo sense for TB-HIV co-epidemics along with recurrent TB and exogenous reinfection. They justified that, when the reproduction number for TB is less than unity, the existence of the backward bifurcation occurred in a certain domain. The researchers considered the memory effect in the Caputo fractional order model and they analysed the model numerically. They showed that the rate of state trajectories convergence to the equilibrium points is more for higher order derivatives rather than small order. The authors displayed that the number of TB and HIV infectious have lower increment for a smaller fractional order. Finally, they conclude that the fractional order of derivative plays a key role against the disease prevalence along with integration of memory effect. Moreover, the two diseases HIV and TB can also be co-infected with the current COVID-19 global pandemic. M.M. Ojo et al. [43] established a co-infection deterministic mathematical model for TB and COVID-19, to investigate their co-infection nature and each disease impact in the community. They performed various simulations to investigate the effect of transmission rates and threshold quantities on the co-infection disease. Their results addressed that the threshold quantities (both effective and invasion reproduction numbers) are the defining factors for disease incursion in the populace. The authors demonstrated that the disease with the maximum invasion reproduction number would dominate but does not drive the other towards elimination. Their numerical analysis also presented that the rising of co-infection transmission rate would upsurge the TB prevalence. Finally, they endorsed the strategies which are prioritizing to diminution the diseases in the population. K. G. Mekonen et al. [44] studied the optimal control of TB and COVID-19 co-epidemics model. Their analytical result revealed that, the rising of TB infected individuals has a positive impact on the transmission of COVID-19 disease and vice versa. They incorporated prevention efforts against TB and COVID-19, treatment for TB, and medical care for COVID-19 infection in the model. Their numerical results showed that the prevalence of co-epidemic disease can be reduced when applying all controlling strategies at a time. N. Ringa et al. [45] formulated a mathematical model of HIV-COVID-19 co-epidemics disease with optimal control. The researchers incorporated HIV and COIVD-19 prevention mechanisms and treatment for COVID-19 in the model. They analysed the model with and without optimal control analytically. The authors also analysed the model numerically based on the data taken from South Africa. Their result showed that the HIV preventive effort and treatment of COVID-19 can minimize the co-infection disease burden.

As we have seen from literatures made by different scholars, TB and HIV/AIDS co-epidemic is a public health concern particularly in developing countries with limited resource.

In this study, we developed a TB-HIV co-infection mathematical model based on the TB model [46] via HIV/AIDS cohorts taking in to account high risk exposed stage (E) and low risk latent TB by treatment (L). Law-risk and high-risk latent TB infected individuals co-infected with HIV are considered in the model formulation. To the best of our knowledge, no other study has examined this possibility. Likewise, we formulated a co-infection model with optimal control consisting high and low risk latent TB stages which are currently responsible for rising TB infectious cases in Ethiopia [19, 47–49]. Similarly, Guo et al. formulated age-structured HIV-TB co-infection model to investigate how to regulate the TB transmission in china [50]. They considered only detection and treatment of latent TB cases and educational campaign to mitigate the TB disease. Whereas, our model examined four controlling strategies which are preventive efforts of TB, preventive efforts of HIV/AIDS, case finding for TB, and HIV treatment to reduce the TB-HIV co-infectious individuals. These strategies were not examined on a TB-HIV co-epidemic model comprises HIV-TB co-infectious people who cannot be fully recovered from TB, which motivates us to undertake this investigation and fill the gap. The other motivation is also the concept of optimal control theory which is applicable to study controlling mechanisms of a disease. The optimal model analysis can be used to identify an effective strategy to minimize the number of infectious individuals and concurrently to reduce the cost incurred during implementation of strategies. To validate our model, the model solution is fitted to the real data collected from Ethiopia. Hence, we developed and analyzed an optimal HIV-TB co-epidemic model with the four suggested interventions using real data, which is the novelty of this work and makes it different from other approaches in the literature.

Thus, the purpose of this study is to establish optimal HIV/AIDS- TB co-infection mathematical model to explore the best cost-effective measure to mitigate the disease burden. Thus, we used time based controlling efforts and we applied optimal combinations of two or more strategies at a time. As shown in all plots (Figs 4–15), the co-infected individuals are dramatically decreased. The elaboration will be presented in the numerical simulations.

The main contribution of this study is to propose the best cost-effective controlling mechanism in the fight against the TB-HIV/AIDS co-epidemic disease in Ethiopia. We chose the country, Ethiopia, because it is one of the 14^th countries in the world affected by HIV-TB co-infection disease, and also one of the 8^th most affected countries in the African continent [28, 51].

The rest of paper is divided as follows: In the following section, the model construction is presented. Section 3 covers the analytical findings of the model properties. In section 4 the proposed model is extended and analyzed analytically through controlling variables while in section 5 the numerical results of the model are presented. The cost-effectiveness of the proposed strategies and conclusions are displayed in section 6 and 7 respectively.

2 Model formulation

We incorporated the following model assumptions to develop a TB-HIV/AIDS co-infection model.

• Individuals infected with TB cannot fully recover, but to latent TB [46, 52]. The TB bacteria cannot removed 100% from the TB infected individual’s body.
• Individuals co-infected with AIDS and active TB are very ill. They have not transmitted HIV virus due to sexual intercourse [53]. They are protected in treatment and their allure for sex is almost insignificant.
• HIV infected individuals under ART treatment are aware of transmitting the disease [7, 27]. The present model will consider those people who are strictly under care and closely monitored.
• The model does not consider vertical transmission of HIV-AIDS and immigrant individuals [54]. HIV can also transmit vertically from HIV infected mother to their new infants. The disease transmission may occur during pregnancy or breastfeeding. Thus, the model which is formulated in this study has not contemplated inflow of HIV-infected individuals. This endorses that the susceptible class comprises new birth people who are vulnerable to HIV infection.

The theory of such an epidemic disease progress in a large population. Initially, the populations diversity can divide into subgroups considering with the nature and the stages of the disease. The sub-groups of populations are named as compartments.

We developed a TB-HIV co-infection model by mixing HIV/AIDS (susceptible, HIV infection with and without AIDS symptoms, and treated individuals from HIV infection) with TB model [46]. Hence, the model allocated the human populations into the following epidemiological compartments. Namely, susceptible individuals (S), exposed (or a high-risk latent TB) (E) that is infected but not infectious individuals, infectious TB (I), low-risk latent TB (L), HIV-infected individuals with no clinical symptoms of AIDS (H), HIV-infected people under treatment for HIV infection (T), HIV-infected individuals with AIDS clinical symptoms (A), exposed (or a high-risk latent TB) co-infected with HIV (H_E), low risk latent TB individuals co-infected with HIV (H_L), HIV-infected individuals (pre-AIDS) co-infected with active TB disease (H_I), HIV-infected individuals with AIDS symptoms co-infected with active TB (A_I) and low risk latent TB individuals infected by HIV-infection with AIDS symptoms (A_L).

Thus, the total population at time t, denoted by N(t), is given by: (1)

The susceptible population is increased by the recruitment of individuals (new births) at a rate π. These people acquire TB and HIV infection at a variable rate or force of infection: and respectively.

The parameters β₁ and β₂ are rates for TB transmission and HIV transmission respectively. Whereas, the modification parameter η represents the relative infectiousness of people with AIDS symptoms compared to HIV infected people without AIDS symptoms. HIV-infected people (pre-AIDS) are less infectious than people with AIDS symptoms because they have lower viral load and positive relationship among infectiousness and viral load [55]. In general, all parameters involved in the model formulation are described in Table 1.

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Table 1. Descriptions of the parameters.

https://doi.org/10.1371/journal.pone.0312539.t001

The co-infection transmission dynamics flow diagram in Fig 1 can be described by the following deterministic system of non-linear ODE. (2) with inital conditions are: (3)

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Fig 1. Flow diagram of the TB-HIV/AIDS co-infection disease transmission.

https://doi.org/10.1371/journal.pone.0312539.g001

3 Model analysis

3.2 Invariance region

In this portion, we assured that the solutions of the system (2) is bounded with non-negative initial values.

Lemma 1 Let Ω be the biological feasible region such that }. Then Ω is positively invariant set for the system (2) and attracts all positive solutions.

Proof:

We showed this clue by adding all the equations in (2) to get the rate of change of N(t).

(9)

Hence, Eq (9) which is a first order ODE can be written as: (10)

Which yields, (11)

Here, , then we derived , ∀ t ≥ 0. This indicates that the TB-HIV co-infection model (2) is positive invariance and bounded ∀ > 0. Hence, the system is well-posed and biologically realistic.

Hereafter, we analyzed each sub model before exploring the full co-infection model.

3.10 Bifurcation analysis

In order to discuss the nature of bifurcation at the threshold value R₀ = 1, we used the central manifold theory [62]. To apply the technique, the next shifts of variables are made.

Let S = x₁, E = x₂, I = x₃, L = x₄, H = x₅, A = x₆, H_E = x₇, H_I = x₈, H_L = x₉, T = x₁₀, A_L = x₁₁, and A_I = x₁₂.

Thus, the system (2) becomes: (46) where and

The Jacobian matrix J of (46) at DFE point is already articulated.

, with

Hereafter, we can calculate the right eigenvectors of J(ε₀) symbolized by:

u = (u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀, u₁₁, u₁₂)^T corresponding to the zero eigenvalues as follows. (47)

The Eq (47) becomes; (48)

Solving system (48) we get:

.

.

.

, since .

. Again, the left eigenvector of J(ε₀) symbolized by v = (v₁, v₂, v₃, v₄, v₅, v₆, v₇, v₈, v₉, v₁₀, v₁₁, v₁₂)^T is calculated as:

, with

Thus (49)

The Eq (49) becomes;

We calculated a and b using the formula:

, and for r = 1 or 2 [62], where n is the number of compartments, and for i = 1, 2, 3, …, 12 in (46).

Hence, Then, .

We considered the case when R_T > R_H i.e., R₀ = R_T and R₀ = 1.

Choose as the bifurcation parameter.

Now,

Again, , where Δ = v₅u₃ + v₅u₄ + v₇u₅θ + v₇u₂ϵ + v₈u₂ϵ₂−v₂u₂−v₃u₃−v₄u₄−v₅u₅ = u₄(v₅−v₄) + u₅(θv₇−v₅) + u₂(ϵ₂v₈−v₂) + u₃(v₅−v₃) + v₇u₂ϵ > 0, since v₅ > v₄, θv₇ > v₅, ϵ₂v₈ > v₂, v₅ > v₃ as one eigenvector can be expressed by the other. In addition v₁₂ω > v₆ω₆.

Hence, both the values of a and b are positive.

Therefore, the model (2) displays a backward bifurcation at R₀ = 1. As R₀ approaches one, the number of co-infectious individuals jump suddenly from 0 to the large endemic equilibrium. Fig 2 shows the diagram representation of a backward bifurcation phenomena. If this bifurcation happen, the HIV-TB co-infection disease persists in the community, even though R₀ < 1. As shown in the Fig 2, there are three equilibria co-exist when R₀ is in the range 0 < R_c < R₀ < 1, where R_c is a critical value, which in Fig 2 is R_c = 0.4835. In this range, both the DFE and EE points are stable, while the middle equilibrium is unstable. When R₀ < R_c, only the DFE point exists and is stable. This justifies reducing R₀ below unity is only a necessary but not sufficient condition for disease elimination. Thus, always a disease eradication cannot just be achieved by making R₀ < 1.

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Fig 2. Numerical simulation of backward bifurcation at the point R₀ = 1, where the vertical axis represents the HIV-TB co-infectious individuals.

https://doi.org/10.1371/journal.pone.0312539.g002

4 Model with optimal control

We used the following four (two preventive and two controlling) efforts.

1. The preventive effort of TB disease (u₁(t)) represents the effort of protecting susceptible individuals who are becoming infected. Such mechanisms are healthy educational campaigns and early detection as well as isolation of infectious individuals are associated with u₁(t).
2. The preventive effort of HIV/AIDS disease (u₂(t)) implies the effort of protecting susceptible individuals who are contacting to HIV/AIDS infected people. Such mechanisms are the HIV/AIDS educational campaign and early detection of HIV infected individuals are associated with u₂(t).
3. The case finding for TB disease (u₃(t)). The effort u₃(t) illustrates the screening and then treatment of high- risk latent TB. The risk that TB infection will progress to TB disease is greatly reduced by treatment of latent TB. Since finite groups are at a high risk of growing TB disease once infected. This effort is a key mechanism for TB control strategy.
4. The treatment effort for HIV/AIDS disease (u₄(t)). The strategy u₄(t) refers to treating HIV infected people with Antiretroviral therapy (ART). This can decrease the individual’s infectiousness level by reducing their viral load and helping them to recapture their immunity to obtain a better life. This treatment can also curtail HIV-TB co-infection rate.

Thus, incorporating the above strategies in the model (2), we get the following optimal control model of HIV-TB co-epidemic. (50)

The optimal controls are defined in the set U = {u_i(t): 0 ≤ u_i(t) ≤ 1, 0 ≤ t ≤ T}, where i = 1, 2, 3, 4.

Let the objective function can be expressed as: [63, 64]: (51) where b₁, b₂, b₃, b₄, and b₅ are the cost associated with the number of H_E, H_I, H_L, A_L, and A_I compartments respectively. The constants c_i, i = 1, 2, 3, 4 are the costs of executing the strategies from u₁ up to u₄ respectively [65]. We have taken a quadratic form for determining the cost of intervention [66, 67].

Thus, we try to find the optimal controls and satisfying

min {J(u₁, u₂, u₃, u₄)|(u₁, u₂, u₃, u₄) ∈ U}, where U is the set expressed above.

Theorem 6 (Existence of solutions). There exists an optimal control , , , and solutions S, E, I, L, H, A, H_E, H_I, H_L, A_L, T, and A_I to control induced initial value problem (50) that minimizes the objective functional J(u_i(t)), i = 1, 2, 3, 4 of (51) over the set of admissible control U.

Proof:

Look the following three conditions of Fleming and Rishel’s theorem [68].

1. The set of solutions of (50) and (51) that incorporate control variables in U is non-empty.
2. The system of Eq (50) is a linear combination of control functions with coefficients are state variables.
3. The integrand L in Eq (51) becomes; is convex on U and it also fulfills L(x, u, t) ≥ δ₁∣(u₁, u₂, u₃, u₄)∣^β − δ₂, where δ₁ > 0 and β > 1.

Firstly, to proof 1, we mentioned to [69, 70]. If the solutions of (50) and (51) are bounded plus Lipschitz, then they are unique.

Thus, the total population N(t) is also bounded above by and bellow by N₀ ≠ 0. Here, each compartment in N(t) is bounded. In that case the state variables are bounded and continuous. Hence, this displays that there is the boundedness of the partial derivatives with respect to the state variables with in the system [71].

This accomplishes that proof 1 holds.

Secondly, the right hand side equation of the control system (50) can be written as: (52) where (53)

Clearly, the system (52) is linearly dependent on u₁, u₂, u₃, and u₄. Therefore, the second condition holds.

Lastly, to verify condition 3, we need to show for any ϱ ∈ (0, 1) such that: (54) where (55) (56)

Further, (57)

Hence, L(x, u, t) is convex on U.

Hereafter, to prove the boundedness on L, we used the following technique.

, since u₃ ∈ [0, 1].

Thus This implies .

The function L(x, u, t) is expressed as: (58)

Therefore, L(x, u, t) ≥ δ₁∣∣(u₁, u₂, u₃, u₄)∣∣^β − δ₂, where and and β = 2.

Theorem 7 Let x = (S, E, I, L, H, A, H_E, H_I, H_L, A_L, T, A_I) and u = (u₁, u₂, u₃, u₄) is an optimal control pair, then there exists a continuously differentiable vector λ₁(t), …, λ₁₂(t) satisfying the following: (59) with transversality conditions λ_i(t_f) = 0, i = 1, 2, 3, …, 12.

Moreover, we get the control set characterized by where and

Proof:

By using the Pontryagin’s Maximum Principle (PMP) [72], we found a Hamiltonian stated as:

, where λ_i, i = 1, 2, …, 12 are the co-state variables.

H, the Hamiltonian is described by:

Hereafter, the second condition of the PMP sates that, ∃ adjoint variables λ_i, i = 1, 2, …, 12 which fulfill the next equalities.

Thus, the above equalities gives the system (7) of first order ordinary derivative of adjoint variables λ_i, where i = 1, 2, 3, …, 12.

Next, by optimality conditions, we get .

Therefore, , , , and .

These results can be expressed in U as:

5 Numerical simulations

Till now, the TB-HIV co-infection model with or without optimal control is analysed analytically. In this portion, we discussed the numerical results to confirm our analytical findings. The simulation gives a clear image about the involvement of control functions on the disease transmission dynamics. We proposed two or more intervention strategies at a time to minimize both the disease and the cost burden.

We collected the HIV-TB co-infection confirmed cases in Ethiopia from July 2021-2023 as shown in Fig 3. We estimated the initial values of each state variables and the parameter’s value, where the model solutions that are best fit to the real data reported. The populations in classes E = 1.19 × 10⁶, I = 3.73 × 10⁵, and L = 2.18 × 10⁷, collected from National TB and Leprosy strategic plan in Ethiopia [27]. Again, we estimated H = 890 311, A = 305 770, H_E = 213 451, H_I = 250 853, H_L = 290 008, T = 1 253 420, A_L = 207 457, A_I = 120 598 from federal health ministry of Ethiopia, CDC, and WHO annual report [7, 21, 28].

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Fig 3. HIV-TB co-infection confirmed cases versus time t from July 2021-2023.

https://doi.org/10.1371/journal.pone.0312539.g003

The susceptible people is obtained by S = N − (E + I + L + H + A + H_E + H_I + T + H_L + A_L + A_I) where N = 102 468 037, then S = 75 573 169. The recruitment people who are entered to class S is calculated by π = b × N, where the birth rate b = 30.97 /1 000. Hence, π = 3 173 435.1.

To estimate the rest constant parameters in the model (2), we formulated the model as: (60)

Here, z is the state variable and θ is the parameter value to be determined.

Define a least squares objective function: (61) where z(i) is the solution of (60), is the real data, and Ds is the data sample size. We get the optimum parameter values by minimizing the objective function: (62)

The algorithm is presented below:

Algorithm 1:

Step 1. Guess initial parameter values a₀. Set a = a₀.

Step 2. Using MATLAB version 2013a ode45 routine, solve Eq (60) using a

to find the solution z(i).

Step 3. Evaluate error using Eq (61).

Step 4. Use a to minimize Eq (62) using an optimization algorithm nlinfit to

find the parameters with 95% confidence interval. Update .

Step 5. Check for the convergence. If the convergence is not satisfied go to Step 2.

Step 6. On convergence, set .

Using the above algorithm the values of the parameters are estimated and presented in Table 2. However, due to lack of real data, the values of some parameters are taken from other related literatures. The time duration of the study is t_f = 10 years.

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Table 2. Symbols and values of the parameters.

https://doi.org/10.1371/journal.pone.0312539.t002

In addition, we assessed the value of the coefficient parameters (b₁ = 0.65, b₂ = 0.55, b₃ = 0.28, b₄ = 1, and b₅ = 1.72) depend on the way of constants found in [18]. Furthermore, we assumed that the value of weight constants based on the importance level of one intervention over the other. These are c₁ = 10⁴, c₂ = 10⁴, c₃ = 2 × 10⁴, and c₄ = 2 × 10⁴. Some data may also taken as just for numerical purpose. Nevertheless, obtaining sufficient data about the vital elements of the TB-HIV/AIDS co-infection model was one big challenge of the study.

We used MATLAB software to validate the analytical results. Here, we discussed the features of the state trajectories with or without optimal control. The control profiles of each strategy are also plotted. We proposed four strategies based on the suggested intervention approaches. They designed as combination of two or more strategies at a time. However, only one intervention at a time is not an effective method [73, 77]. Thus, we have seen these strategies can highly minimize TB-HIV co-infection disease in the country Ethiopia and the elaborations are continued in the next part.

5.1 Preventive effort of TB disease and treatment of HIV/AIDS disease

We used prevention of TB disease combined with HIV/AIDS treatment as an optimal strategy (i.e.u_i ≠ 0, for i = 1, 4, whereas u₂ = 0, and u₃ = 0). The plot (A − E) of Figs 4–6 illustrates the impact of this optimal approach on HIV/AIDS-TB co-infected individuals. This strategy can be used to decrease the number of low risk latent TB individuals co-infected by HIV with pre-AIDS and AIDS symptoms dramatically rather than without optimal approach.

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Fig 4. Infected individuals in H_L and A_L class when applying combined efforts of prevention of TB and treatment of HIV/AIDS optimally.

https://doi.org/10.1371/journal.pone.0312539.g004

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Fig 5. Infected individuals in H_I and A_I class when applying combined efforts of prevention of TB and treatment of HIV/AIDS optimally.

https://doi.org/10.1371/journal.pone.0312539.g005

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Fig 6. Infected individuals in H_E class when applying combined efforts of prevention of TB and treatment of HIV/AIDS optimally.

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As shown in the Fig 4A and 4B, If optimal control is not applied, the co-infected individual in H_L and A_L is increased at the beginning of the year and reach at a peak value at 3.2999 × 10⁵ and 3.1217 × 10⁵ respectively. Hereafter, the disease burden decreased significantly. However, in case of optimal control the prevention effort has an impact on individuals under H_L class by reducing the high risk latent individuals. In addition, the more co-infected people are moved to the treated class due to HIV treatment.

In Fig 5C, when no optimal control is used the co-infectious people in H_I class is decreased due to the influence of constant treatment rate w₄ for HIV and successful TB treatment rate of 1 − ψ. Conversely, in Fig 5D, the co-infectious people in A_I class are increased at the beginning of the years and reach at a peak value 1.5322 × 10⁴. Hereafter, the disease burden decreased significantly. This is because more TB infected people co-infected with HIV are completed their TB treatment at a constant rate τ and with constant HIV treatment rate w₇. However, the time-based optimal approach seems negligible in the H_I compartment but later on we can observe its impact. This combination strategy has a high effect on the co-infected people by active TB and HIV with AIDS symptoms. For that reason, the prevention effort can minimize the susceptible individuals to become TB infected under high-risk latent stage progress to active stage. Additionally, the more co-infectious people are joined to treated class due to HIV treatment.

In Fig 6E, when optimal control is not applied, the co-infected people in H_E class is decreased due to the influence of constant treatment rate w₃ for HIV and treatment rate of high risk latent TB θ₂. Nevertheless, when optimal control is used the disease burden is decreased rather than without optimal control. The impact of this strategy is visible around five years but seems negligible after a while.

Therefore, this strategy can minimize/eradicate the HIV-TB co-infection disease burden in the country, Ethiopia.

5.2 Preventive effort of HIV/AIDS disease and case finding TB

We used prevention of HIV/AIDS disease combined with case finding effort of TB as an alternative mechanism (i.e.u_i ≠ 0, for i = 2, 3, whereas u₁ = 0, and u₄ = 0). The plot (A − E) of Figs 7–9 illustrates the impact of this optimal approach on HIV/AIDS-TB co-infected individuals.

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Fig 7. Infected individuals in H_L and A_L class when applying combined efforts of prevention of HIV/AIDS and case finding TB optimally.

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Fig 8. Infected individuals in H_I and A_I class when applying combined efforts of prevention of HIV/AIDS and case finding TB optimally.

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Fig 9. Infected individuals in H_E class when applying combined efforts of prevention of HIV/AIDS and case finding TB optimally.

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As shown in the Fig 7A and 7B, if no optimal control is applied, the co-infected individual in H_L and A_L are increased at the beginning of the year. The disease burden decreased dramatically after a while. Nevertheless, in case of optimal control, the co-infected individuals in these two sub-classes are not more decreased as compared from the first strategy. Since, there are more HIV-infected people who have recovered from TB but remain low-risk latent due to case-finding effort. Hence, the co-infected populations in the H_L class are increased and reach a peak value 3.6876 × 10⁵. Conversely, the optimal strategy seems negligible for the first around 1 year in A_L class but it gets a significant influence far ahead.

In Fig 8C, the impact of optimal control strategy seems negligible in the H_I class but it has a visible impact to some extent. While, In Fig 8D, the co-infected people in A_I compartment are raised for the first few months and reach a high value 1.5322 × 10⁵. After this, the number of individuals in the class A_I decreased radically.

In Fig 9E, the co-infected people in H_E class are decreased when optimal control is applied. In this compartment, the influence of this strategy is better than the first approach. Since, the number of HIV people co-infected with TB at a high-risk latent stage is more decreased due to case-finding effort.

Hence, this strategy can minimize/eradicate the HIV-TB co-infection disease burden.

5.3 Case finding for TB disease and HIV treatment

We used case finding for TB disease combined with HIV/AIDS treatment as an alternative measure (i.e.u_i ≠ 0, for i = 3, 4, whereas u₁ = 0, and u₂ = 0). The plot (A − E) of Figs 10–12 illustrates the impact of this optimal approach on HIV/AIDS-TB co-infected individuals. All plots showed that the model with and without optimal control plays a great role to minimize the disease burden.

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Fig 10. Infected individuals in H_L and A_L class when applying combined efforts of case finding TB and HIV treatment optimally.

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Fig 11. Infected individuals in H_I and A_I class when applying combined efforts of case finding TB and HIV treatment optimally.

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Fig 12. Infected individuals in H_E class when applying combined efforts of case finding TB and HIV treatment optimally.

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In the Fig 10A, the numerical results displayed that this strategy is better than the second strategy due to HIV treatment effort. The disease burden seems raised at the first of a few months but later, it decreased intensely. The optimal strategy is also a more effective approach on the co-infected class A_I as shown in the Fig 10B. The graphical result shows the combination optimal approach can reduce the disease burden effectively.

In Fig 11C, one can observe that this combination optimal control has enhanced the impact on the co-infected class H_I rather than the second strategy. The same effect is as shown in the sub-population A_I Fig 11D. Since, there are more co-infected people are moved to the treated class due to HIV treatment effort.

In Fig 12E, the number of co-infected people in H_E class are reduced when optimal control is used. In this class, the effect of this strategy is also better than the first approach.

Thus, the optimal control of case ending effort of TB and HIV/AIDS treatment has great impact to reduce the disease burden.

5.4 Using all the intervention efforts

We used all intervention efforts optimally as an alternative mechanism (i.e.u_i ≠ 0, for i = 1, 2, 3, 4). The plot (A − E) of Figs 13–15 displays the effect of this approach on HIV/AIDS TB co-infected people. All graphical results show that the model with and without optimal control plays a great role to minimize the disease burden. The effect of this mechanism on co-infected individuals seems like to the third strategy, but the visible differences appeared on the amount of cost for implementation. This will be discussed in the cost-effectiveness section, however the trajectories are plotted in Fig 16.

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Fig 13. Infected individuals in H_L and A_L class when applying all strategies optimally.

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Fig 14. Infected individuals in H_I and A_I class when applying all strategies optimally.

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Fig 15. Infected individuals in H_E class when applying all strategies optimally and control profiles.

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Fig 16. Trajectories of cost functions.

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The control profiles that generate this simulation result are as shown in the Fig 15F. The pink color of the trajectory u₁ is concealed by the cyan color of the trajectory u₂. The control plots u₁ and u₂ have displayed the maximum efforts required for the entire duration. The control plot u₃ shows that the high case-finding for TB is needed for the first around 1 year and extremely reduced later. The control plot u₄ shows that high treatment of HIV is desired for around 7.8 years and reduced after a while. Finally, all controls are dropped to zero due to the proposed strategies being expected to be over at the end of the time forecast.

6 Cost-effectiveness analysis

Here, we presented the cost-effectiveness rank of one implemented strategy over the other. We achieved this by (Baba and Makinde [78]); they had declared that

.

We got the total number of infected averted which is the difference between the total infectious people with and without control. We applied this technique by ranked increasing order of effectiveness with respect to the infected averted. Besides to this, the total cost is also mentioned in Table 3.

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Table 3. Total infected averted (increasing order) and total cost.

https://doi.org/10.1371/journal.pone.0312539.t003

We compared the strategy of B and C by computing the ICER: and

The comparison displayed that ICER(C) > ICER(B), which shows that strategy C is strongly dominated and does not consume limited resource.

Hence, we should remove strategy C from the set of choices.

Next, we compare strategy B and D.

Already we calculated ICER(B) = 6.3 .

The comparison showed that strategy D is more costly and less effective than strategy B. Hence, we should remove strategy D from the set of choices.

Finally, we compared strategy B and A.

Now, .

This shows that, we should remove strategy A from the set of options.

Therefore, strategy B is the most cost-effective strategy rather than the rest alternatives.

7 Conclusion

The co-epidemic of the two diseases HIV/AIDS and TB is serious in the country Ethiopia as one disease accelerating the rate of infection of the other and vice versa. The disease burden leads to the country having health and economic challenges. Thus, we expect that a lot of new research findings would be incorporated into the body of knowledge. This study addressed the concept of disease transmission dynamics and further controlling efforts. Hence, we developed a new mathematical model for HIV-TB co-infection transmission dynamics. The model has 12 state variables, which is highly non-linear and it was challenged for qualitative analysis. We discussed the points such as positive invariance of the solution set, equilibria points, basic reproduction number (R₀), stability of equilibria points, and probability of bifurcation. We found that the DFE point is locally asymptotically stable when R₀ < 1 and unstable when R₀ > 1, whereas the EE point is locally asymptotically stable when R₀ > 1. At the threshold value (R₀ = 1), the backward bifurcation is occurred. We also developed an optimal control model, which is an extension of the original one via incorporated prevention effort of both diseases, case finding effort of TB disease and HIV treatment. These interventions through a co-epidemic model which is build on high risk and low risk latent TB cases those are not fully recovered is a palpable gap in previous works. Hence, the existence of optimal control solutions as well as their characterization is presented. We used the distinguished Pontryagin’s Maximum Principle (PMP) which states an essential condition for optimal solutions. Using the MATLAB software, we applied the Runge-Kutta method of order four (RK4-method) to validate the analytical results of the optimal control model. The numerical results for the model without optimal control illustrated that decreasing the contact rate of TB and increasing HIV treatment rate of co-infected people have high contribution to suppress the disease transmission. Whereas, the simulation result of the model with optimal controls confirmed that the combination of two or more strategies at a time can effectively minimize co-infected individuals under H_E, H_I, H_L, A_L, A_I classes. Likewise, these approaches can also decrease the cost incurred during interventions. We have seen that all strategies have certain limitations; however, the couple of preventive effort of HIV/AIDS and case finding for TB disease at a time is the best cost-effective option. Those model results are scrutinized using the real data collected from Ethiopia, which is the novelty of this work and makes it different from other approaches in the literature. As upcoming work, the model can be developed via considering vertical transmission of HIV/AIDS transmission dynamics. The model can also extend into a Tri-epidemics model when the global pandemic COVID-19 will be considered; and a fractional order model to explore the memory effects of the biological systems are suggested for further investigation. Moreover, this study will aid in the fight against tuberculosis, HIV/AIDS, and their co-infection policy makers and other concerned organizations. Finally, our study recommends further attention or emphasis should be given to mathematical modelling for exploring transferable diseases.