HIV/AIDS and HBV co-infection with optimal control strategies and cost-effectiveness analyses using integer order model

Abstract

Hepatitis B virus (HBV) and HIV/AIDS co-infection is a common infectious disease that has been spreading through different nations in the world. The main objective of the study is to minimize the number of HBV and HIV co-infected individuals in the community and the cost incurred to the effort applied towards protection and treatment control strategies. In this study, a novel HBV and HIV/AIDS co-infection model has been formulated and analyzed to investigate the effects of protection and treatment control strategies on the spreading dynamics of the HBV and HIV/AIDS co-infection in the community. In the qualitative analyses we have computed all the models disease-free equilibrium points, all the models effective reproduction numbers and unique endemic equilibrium points, we have proved the two sub-models disease-free equilibrium points are locally as well as globally asymptotically stable whenever their associated effective reproduction numbers are less than one. In our understanding no one is formulated and analyzed the HBV and HIV/AIDS co-infection model with optimal control and cost-effective analyses so that we have formulated the associated optimal control problem and carried out the optimal control analysis on the HBV and HIV/AIDS co-infection model by implementing the Pontryagin’s minimum principle. Numerical simulations with various combinations of the control efforts implemented are then carried out to investigate the impacts of protection and treatments to tackle the HBV and HIV/AIDS co-infection diseases spreading in the community. Finally, we also carried out cost-effectiveness analysis for the implemented control strategies. From the findings of the numerical simulations we observed that implementing all the proposed controlling strategies simultaneously has fundamental impact to minimize and control the HBV and HIV/AIDS co-infection spreading in the community but cost-effectiveness analysis proved that implemented strategy 4 (implemented the HIV protection and HBV treatment controls simultaneously) is the most cost-effective strategy as compared with all other implemented strategies and we recommend for the health stake holders and policy makers to implement this strategy to tackle the HBV and HIV/AIDS co-infection spreading problem in the community.

Introduction

Hepatitis B virus (HBV) is a contagious viral disease that highly influencing the normal work of individuals’ liver¹ and about two billions individuals throughout the world infected with HBV epidemic among which chronic HBV affects more than 350 million individuals throughout nations in the world^2–6. Since 1990 the mortality rate of viral hepatitis has been increased by 63% and now it is ranked as the seventh leading cause of death and millions of people have been died with chronic HBV stages (liver cirrhosis and cancer) and it spreads through direct contact and indirect transmission like through blood contact or during birth^1,2. The two most common stages of HBV disease are acute and chronic hepatitis stages⁷. Through the first 180 days after individuals exposed to the hepatitis B virus their immune system may be able to remove the HBV virus and resulting in a complete recovery. However, sometimes the HBV infection may progress to the chronic HBV infection stage⁷. It is a blood-borne infectious disease and its modes of spreading includes vertical transmission (from mother to child during birth), sexual contact, drug injections, needles sharing and sharp instruments sharing. There is no well-known treatment intervention for acute hepatitis B, but, the chronic hepatitis B infection can be treated by nucleotide analogues (NAS) and pegylated interferon (PEG-IFN)¹.

Human immunodeficiency virus (HIV) one of the major lives threatening retrovirus spreading throughout the world and is the main the cause of a highly infectious called acquired deficiency syndrome (AIDS)^8–11. HIV/AIDS has been remained the most common cause for worldwide health and economic problem and since 1981 it has been declared as a global pandemic and one of the most destructive epidemic diseases through history^10,11. Approximately 70 million people throughout nations in the world have been affected by this infectious disease⁸. According to UNAIDS report in 2016, 36.7 million individuals were living with HIV/AIDS and more than two thirds of those individuals in the world is living in sub-Saharan African countries^9,11. The most common infection stages of HIV are acute, dormancy and AIDS stages¹². HIV/AIDS can be transmitted from an infected individual to a healthy individual through direct or indirect transmission and its possible control measures are preventive measures and treatment regimens¹¹. Until today there is no vaccine, and there are many obstacles in the AIDS treatment (antiretroviral therapy or ART), but recently, the most prevalent treatment strategy for HIV infected individuals is highly active antiretroviral therapies (HAART), which can prolong the life spans and improve their life quality of individuals¹⁰.

A co-infection is the co-existence of two or more pathogens (infections) on a single individual at the population level⁹. HIV/AIDS and HBV are the most common viral infectious diseases and since they have the same modes of transmission the HBV and HIV/AIDS co-infection disease is common in various nations of the world^2,13,14. More than 10% of individuals infected with HIV/AIDS have been reported as chronically infected with HBV and the HBV and HIV/AIDS co-infection highly increases the risk for liver related morbidity and mortality as compared with the HIV mono-infection^1,15,16. HIV/AIDS infection continues to be one of the most common public health problems with additional risk of HCV and/or HBV co-infection and also the chronic HBV infection prevalence with HIV infection is 6–14%^3,13,17. On the other hand, the interactions between two infectious diseases specially the HBV and HIV/AIDS have a negative impact on the community and become a worldwide concern now a day².

Mathematical modeling is the process of representing real world situations in mathematical terms and expressions and it has been used extensively in studying the behavior of infectious diseases, including their co-infections². From the various branches of mathematical modeling one can motivate to study about eco-epidemiological, ecological, and epidemiological modeling. Epidemiological modeling is the study of the infectious diseases transmission dynamics at the population level and it plays a crucial role in the study of transmission dynamics such as HBV and HIV infectious diseases¹⁸. Nowadays, mathematical model formulations of real world situations with optimal control strategies play a fundamental role in showcasing the impacts and effectiveness of different optimal control intervention strategies¹⁹. Various researchers have constructed and analyzed autonomous compartmental mathematical models with either integer order derivatives see the works of^20–28 or fractional order derivatives see the works of^29–37 to investigate the spreading dynamics of different single infection or co-infection infectious diseases throughout different nations in the world. Jan, Rashid et al.²⁹ formulated and analyzed the dynamical behavior and chaotic phenomena of HIV infection through fractional order derivative mainly with Atangana–Baleanu derivative in the Caputo sense to investigate the dynamics of CD4 + T-cells in HIV infection. Huo et al.¹⁰ investigated the stability of an HIV/AIDS treatment model with different HIV infection stages. The final results of the model analysis reveals that early HIV infection stage treatment for people in asymptomatic HIV infection stage or the pre-AIDS stage is very crucial to tackle the HIV/AIDS spreading in the community. O. Omondi, et al.¹¹ formulated and analyzed a sex-structured population HIV infection model using real data from Kenya. The result showed that prevention of HIV infection still remain the most fundamental to tackle its spread and ART treatment is also crucial to minimize the transmission dynamics in the community. Malede Atnaw Belay et al.¹ formulated and analyzed a compartmental model on the spreading dynamics of Hepatitis B disease with optimal control and cost-effectiveness. The finding verified that the use of prevention control efforts is the most cost benefit strategy to minimize the HBV spreading in the community. Bowong et al.³ formulated and presented the HBV and HIV co-infection deterministic model and they carried out numerical simulations for the full co-infected model to verify the analytical results. The model formulation did not consider optimal control strategies and cost effectiveness analysis. Awoke, T. D., and M. K. Semu³⁸ formulated a TB and HIV co-infection model with optimal control theory in the presence of behavior modification. They investigated the optimal impacts of their proposed control strategies and from their cost effectiveness analysis results they found that the treatment control measure is more effective than the preventive control strategies. Endashaw, E. E., & Mekonnen, T. T³⁹ investigated the impacts of HBV vaccination and HBV and HIV treatments on the spreading dynamics of HBV and HIV/AIDS co-infection. Their findings revealed that implementing HBV vaccination, HBV, HIV/AIDS and HBV and HIV/AIDS co-infection treatments at the optimal possible rate is recommended to control the transmission of HBV and HIV/AIDS co-infection in the community. Endashaw et al.⁴⁰ modified the HBV and HIV/AIDS co-infection model³⁹ by incorporated the vertical transmission i.e., transmission from mother to child and medical interventions. From the findings of the numerical simulations increasing the HBV and HIV mother to child vertical transmission rates exacerbated the HBV and HIV/AIDS co-infection. Omame et al.²⁶ investigated the impacts of optimal strategies on the control of the co-circulation of COVID-19, Dengue and HIV using mathematical model. Din et al.²⁵ formulated and analyzed HBV and COVID-19 co-infection model using stochastic approach COVIID-19 co-infection model in resource limitation settings.

However, to the best of our knowledge, no one has formulated and analyzed the HBV and HIV/AIDS co-infection compartmental model by considering stages of HIV infection with optimal control and cost-effectiveness analyses. Therefore, the results of the study will contribute to the existing body of knowledge on the spreading dynamics of HVB and HIV/AIDS co-infection in the community. It will also assist public health care authorities and the government to determine the most effective allocation of limited resources for protections, and treatments control strategies to minimize the co-infection cases. Thus, in this study, we formulated and analyzed a compartmental model on the HBV and HIV/AIDS co-infection spreading dynamics with optimal control strategies and cost-effectiveness analysis where its main objective is to investigate the impacts of the proposed optimal control strategies and the cost benefit of the implemented control strategies on the HBV and HIV/AIDS co-infection spreading dynamics in the community. Motivated by the reviewed papers biological and mathematical aspects we need to formulate analyze the HBV and HIV/AIDS co-infection model with optimal control strategies and cost-effectiveness.

The remaining sections of this study have been structured as: the second section formulates a nine compartmental model of the HBV and HIV/AIDS co-infection spreading dynamics with proofs of positivity and boundedness of its solutions, the third, fourth and the fifth sections qualitatively analyzed all the models, the optimal control problem, sensitivity analysis and numerical simulations respectively, section six investigated cost-effectiveness analysis and Sect. 7 gives the discussion, conclusion and future directions of the study respectively.

The HBV and HIV/AIDS co-infection model construction

Descriptions

In this section, to construct the HBV and HIV-infection stages co-infection integer order model we partitioned the total number of human population at a given time t denoted by into nine mutually exclusive compartments according to the infection status of individuals: susceptible individuals to either HBV or HIV infection denoted by (, protected individuals against both HBV and HIV infections denoted by , HBV infected individuals denoted by , HIV infected individuals denoted by AIDS patients are denoted by , HVB and HIV co-infected individuals denoted by , HBV and AIDS co-infected individuals denoted by , HBV infected treated individuals denoted by and HIV treated individuals, HBV and HIV/AIDS co-infected treated individuals, and HBV and AIDS co-infected treated individuals denoted by ( such that

1

According to their epidemiological definitions both HBV and HIV are chronic infectious diseases and hence individuals who are at risk (or susceptible) to either HBV are HIV infection acquire HBV and HIV infection at the force of infection rate represented respectively as:

2

3

where the constants given by and are the parameters that modify the degree of infectiousness with the HBV and HIV co-infected individuals, and the HBV and AIDS co-infected individuals respectively, the constants given by , and are the parameters that modify the degree of infectiousness with AIDS patients, the HBV and HIV co-infected individuals, and the HBV and AIDS co-infected individuals with HBV and HIV infections respectively and the parameters represented by and, are the spreading rates of HBV and HIV infected individuals respectively.

In addition to the above descriptions we have the following model assumptions, the model state variables definitions, and the parameter interpretations respectably.

Model assumptions

• Individuals in each compartment are homogeneously mixed.

• All the HIV, the HBV and the co-infected individuals are aware of their infection status and do not involve in the transmission process (they do not transmit their infection to others).

• Individuals in each compartment are subjected to natural mortality rate.

• The total human population is not constant.

• There is no vertical transmission for both infections.

• There is no simultaneous dual-infection transmission.

• The HBV and HIV protections are not 100% effective.

• There is permanent HBV infection.

• There are interactions between the two infections.

• To protect individuals against HBV and HIV infections, we incorporate sexual abstinence, one mate relationship with an uninfected partner, and using condom.

• We assume treatment of the chronic hepatitis B infection by nucleotide analogues (NAS) and pegylated interferon (PEG-IFN).

• We incorporate the HIV treatment (antiretroviral therapy or ART).

Flow chart of the co-infection model (4)

Based on the model descriptions described in the Section "descriptions", the model assumptions described in the Section "Model assumptions", the state variable definitions illustrated in the Section "Model state variables definitions" (Table 1), and the interpretations of parameters stated in the Section "Model parameters interpretations" (Table 2) the HBV and HIV co-infection transmission dynamics flow chart is represented by Fig. 1 below.

Table 1.

The definitions of the co-infection model state variables.

Variables	Definitions
	The total number of susceptible individuals
	The total number of protected individuals from both HBV and HIV infections
	The total number of HIV infected individuals
	The total number of AIDS patients
	The total number of HBV infected individuals
	The total number of HBV and HIV co-infected individuals
	The total number of HBV and AIDS co-infected individuals
	The total number of HBV infected treated individuals
	The total number of HIV infected, AIDS patients, and co-infected treated individuals

Table 2.

Interpretations of the proposed model parameters.

Interpretations	Biological interpretation	Values	References
	Natural death rate of individual	0.01	12
	Recruitment rate of individuals	250	41
	The portion of HBV or/and HIV protection	0.006	42
	Protection losing rate	0.59	43
	Modification parameter	1.2	Assume
	Modification parameter	1.1	Assume
	Modification parameter	1.2	Assume
	Modification parameter	1.1	Assume
	Death rate by HBV infection	0.1	40
	Death rate by HIV infection	0.333	8
	Death rate by AIDS	0.333	9
	Death rate by HBV and HIV co-infection	0.01	39
	Death rate by HBV and AIDS co-infection	0.02	Assume
	Re-infection rate of HBV treated individuals	0.2	Assume
	HBV infection treatment rate	0.3	40
	HBV infection spreading rate	0.3425	44
	HIV infection spreading rate	0.04	40
	HIV infection treatment rate	0.3	39
	AIDS patients treatment rate	0.13	9
	HBV and HIV co-infected treatment rate	0.015	39
	HBV and AIDS co-infected treatment rate	0.012	Assume
	Progression rate from HIV infection to AIDS patients	0.08	9
	Progression rate of co-infections	0.09	Assume

Fig. 1.

The flow chart of the HBV and HIV/AIDS co-infection dynamics where and are stated in Eqs. (2) and (3) respectively.

The HBV and HIV/AIDS Co-infection Model Using the model flow chart represented by Fig. 1 the HBV and HIV co-infection dynamical system (model) is represented by the systems of differential equations illustrated by:

4

where the initial population is quantified as , , , ,, and .

Non-negativity and boundedness of the co-infection model solutions

Each state variables of the co-infection dynamical system (4) represents human population; therefore it is necessary to prove that all are non-negative and bounded.

Theorem 1

The proposed HBV and HIV co-infection dynamical system (4) solutions are non-negative, unique, and bounded in the region represented by:

5

Proof

Every function in the right-hand side of HBV and HIV co-infection dynamical system (4) are on Hence, by the Picard–Lindel f theorem the model has a unique solution. Let the system of differential equations in be written in the form where and is the right hand sides of the model Based on the Picard–Lindel f theorem the functions has the property illustrated by.

6

where and . Because there exists a unique solution for the dynamical system (4), it follows that for all , whenever . The rate of change of number of total population.

is governed by the equation given by:

7

Using the total differential equation described by (7) and applying the Comparison theorem used in⁴⁵ we have computed the result for determined by . Thus, for the initial population illustrated in Eq. (4) with the property , we have determined the final result represented by . Therefore, the solution of the HBV and HIV co-infection model described in (4) exists, unique, and bounded in a feasible region .

Analysis of the models without optimal control strategies

In this section, the dynamical behaviors of the HBV and HIV co-infection model (4) without control strategies are investigated. The HBV-only sub-model, the HIV-only sub-model as well as the HBV and HIV/AIDS co-infection dynamical system shall be investigated respectively.

The HBV-only sub-model analysis

Since the proposed HBV and HIV/AIDS co-infection model is highly complex and non-linear, some rigorous analyses, that may not be feasible in mathematically aspects for the co-infection model (4), and here we assume there is no HIV/AIDS diseases ( hence no interactions between the two diseases). Thus, the HBV sub-model is determined whenever the HIV disease is not an issue in the community, that is, when , and is represented by:

8

where the initial population is quantified as , B , and , and HBV-only force of infection is described as.

9

where

The HBV-only sub-model will be analyzed in the feasible region represented by:

. Using the same criteria we have applied to prove Theorem 1, can be proved it is positively invariant.

Disease-free equilibrium point

Disease-free equilibrium point refers to a scenario in which the HBV infection is not present in a population. The disease-free equilibrium point of the HBV-only sub-model (8) is obtained by setting each of the dynamical system (8) to zero and hence the HBV-only sub-model (8) disease-free equilibrium point is calculated and is represented by

The HBV-only sub-model effective reproduction number

The HBV-only sub-model (8) effective reproduction number (the threshold quantity represented by ) is the mean number of secondary infected individuals produced by a typical infected individual who is living in a community where some of them are susceptible and others are non-susceptible due to some protection strategies in his entire life⁹. Let us apply the next generation matrix operator criteria stated in⁴⁶, and suppose represent the rate at which individuals entering into the compartment i, represent the rate at which individuals leave the compartment i, and represent the rate at which new infectious arise in compartment i. Thus,, and , where and are matrix with is the number of infected compartments. Then the spectral radius of the next generation matrix is the required HBV-only sub-model effective reproduction number (). Applying the same criteria used by³⁷ and after some calculations we have computed the final results described by:

And

10

Thus, using the result described in (10), the effective reproduction number of the HBV-only sub-model (8) is represented by

Next, we investigate both the local and global asymptotic stability of the HBV-only disease-free equilibrium point for the HBV-only sub-model (8), to examine whether there is small perturbations away from the disease-free equilibrium point will going up or going down through time.

Theorem 2

The HBV-only sub-model (8) disease-free equilibrium point denoted by is locally asymptotically stable if and unstable if .

Proof

The HBV-only sub-model (8) Jacobian matrix at the corresponding disease-free equilibrium point is calculated and represented by:

The characteristics equation for the corresponding Jacobian matrix is computed as:

11

where

And

The first two factors of the characteristics equation described in (11) are linear factors represent the associated eigenvalues computed as , and . Applying the Routh-Hurwitz stability conditions every root of the quadratic equation has negative real part. Thus, each of the eigenvalue for the corresponding Jacobian matrix has negative real part and hence the HBV-only sub-model disease-free equilibrium point is locally asymptotically stable if and unstable if

Note: The HBV-only sub-model (8) disease-free equilibrium point is represented by is locally asymptotically stable whenever its effective reproduction is less than unity means that the solutions of the dynamical system (8) with the given initial population values close to this disease-free equilibrium point remain close to the equilibrium and approach the equilibrium as time increases. It also means that the disease-free local stability concerns only for the close neighborhood of the disease-free equilibrium point. In other words the HBV disease-free equilibrium point is locally stable implies that a small number of HBV infected individuals is introduced in to the community then after some time the system will return to the disease-free equilibrium point. Biologically, the local asymptotic stability implies that the HBV infection can be eliminated from the community whenever which is only true if the initial sizes of the sub-population of the HBV sub-model are in the basin of attraction of .

Based on the study⁴⁷, the two most commonly used sufficient conditions that guarantee the HBV-only sub-model disease-free equilibrium point global stability are written as in the form:

12

where represents the number of individuals who are HBV uninfected and represents the number of individuals who are HBV infected. Applying the same method used by⁹, the HBV-only sub-model (8) disease-free equilibrium point given by has global asymptotic stability if the two conditions described below holds:

For the system , has a global asymptotic stability,

, for where is an -matrix and is the region where the dynamical system (8) makes biological sense.

Theorem 3

The HBV-only sub-dynamical system (8) disease-free equilibrium point denoted and given by is globally asymptotically stable if and the two criteria and stated above holds.

Proof

Let be the vector with its components are the uninfected group of individuals and be the vector with its components are the infected group of individuals. Here we have . For the dynamical system (8) the corresponding stated in (14) is given by . Hence, . Obviously, is globally asymptotically stable for system . Then for the other criteria we can determine the expression . Also and from this expression we have determined the result here since and hence . Therefore, implies that the HBV disease-free equilibrium point is globally asymptotically stable whenever .

Note: The HBV-only sub-model (8) disease-free equilibrium point is represented by is globally asymptotically stable whenever its effective reproduction is less than unity means that all the model solutions must approach to an equilibrium point under all initial conditions (there is no nearby conditions). In other words the HBV disease-free equilibrium point is globally stable implies that no matter the size of the perturbation, the disease will not be able to persist in the population. Biologically, the global asymptotic stability implies that the HBV infection can be dies out from the community whenever .

Existence of HBV-only endemic equilibrium point (s)

Endemic equilibrium point is a disease-persistent state that refers to a scenario in which the HBV infection persists in the population. In this sub-section, we determined the HBV sub-model (8) endemic equilibrium point, that is, a point at which the sub-model individual state variables are present at a non-zero level. To compute the sub-model endemic equilibrium point we need to solve the system of equations given by:

13

where

14

and

Considering (13) and (14) solve for and obtained the results given by:

15

Using the expression given in (14) we have such that.

. After simplifying the result, we obtained the final result represented by:

16

if and only if

Therefore, the HBV-only sub-model (8) has a unique endemic equilibrium point if and only if

The HIV/AIDS-only sub-model analysis

Here let us consider there is no hepatitis B virus (HBV) disease in the community (no interactions between the two diseases) and hence from the HBV and HIV (human immunodeficiency virus) co-infection model (4), we determine the HIV (human immunodeficiency virus)/AIDS sub-model by making , which is illustrated by:

17

where the initial human population is quantified by , , ,and , and HIV force of infection is given by.

18

Where the total population considered is

The HIV-only sub-model will be analyzed in the feasible region denoted by:

. Applying the same approach used to prove Theorem 1 the region can be proved it is positively invariant.

Disease-free equilibrium point

Disease-free equilibrium point refers to a scenario in which the HIV/AIDS infection is not present in a population. The HIV-only sub-model (17) disease-free equilibrium point is calculated and is given by

The effective reproduction number

The HIV effective reproduction number (the threshold quantity represented by ) is the mean number of secondary infected individuals produced by one infected individual who is living in a community where some of them are susceptible and others are non-susceptible due to some protection measures in his entire life⁹. Using the same approach used in the Section "The HBV-only sub-model effective reproduction number" we have computed the results given by:

Then, after calculating and simplifying we have obtained the next generation matrix represented by

19

Therefore, the effective reproduction number of the HIV-only sub-model (17) is denoted and represented by

Next, we need to investigate both the local and global asymptotic stability of the HIV-only sub-model disease-free equilibrium point for the sub-model (17), and also to examine whether there is small perturbations away from the disease-free equilibrium point will increase or decrease through time.

Theorem 4

The HIV-only sub-model (17) disease-free equilibrium point denoted by is locally asymptotically stable if and unstable if .

Proof

The HIV-only sub-model (17) Jacobian matrix at the corresponding disease-free equilibrium point is derived and is represented by:

The characteristics equation for the Jacobian matrix at is represented by:

20

where

Using the equation represented (20) we computed the results described by , and , ,

and for the expression illustrated by + in Eq. (20) we applied the Routh-Hurwiz stability criteria we have proved both the eigenvalues have negative real part whenever

Thus, each of the eigenvalue for the Jacobian matrix has negative real part if an only if and hence the HIV-only sub-model disease-free equilibrium point is locally asymptotically stable if and unstable if

Note: The HIV/AIDS-only sub-model (17) disease-free equilibrium point is represented by is locally asymptotically stable whenever its effective reproduction is less than unity means that the solutions of the dynamical system (17) with the given initial population values close to this disease-free equilibrium point remain close to the equilibrium and approach the equilibrium as time increases. It also means that the disease-free local stability concerns only for the close neighborhood of the disease-free equilibrium point. In other words the HIV/AIDS disease-free equilibrium point is locally stable implies that a small number of HIV/AIDS infected individuals is introduced in to the community then after some time the system will return to the disease-free equilibrium point. Biologically, the local asymptotic stability implies that the HIV/AIDS infection can be eliminated from the community whenever which is only true if the initial sizes of the sub-population of the HBV sub-model are in the basin of attraction of .

Theorem 5

The HIV-only sub-model (14) disease-free equilibrium point denoted and represented by is globally asymptotically stable whenever and the two criteria and described in Theorem 4 above holds.

Proof

Let be the vector with its components are the HIV uninfected group of individuals and be the vector with its components are the HIV infected group of individuals. Here we have . For the HIV-only dynamical system (14) the corresponding stated in (14) is given by . Hence, . Obviously, is globally asymptotically stable for system . Then for the other criteria we can determine the expression . Also and from this expression we have determined the result here since and hence . Therefore, implies that the HIV-only sub-model disease-free equilibrium point is globally asymptotically stable if .

Note: The HBV-only sub-model (17) disease-free equilibrium point is represented by is globally asymptotically stable whenever its effective reproduction is less than unity means that all the model solutions must approach to an equilibrium point under all initial conditions (there is no nearby conditions). In other words the HBV disease-free equilibrium point is globally stable implies that no matter the size of the perturbation, the disease will not be able to persist in the population. Biologically, the global asymptotic stability implies that the HBV infection can be dies out from the community whenever .

Existence of the HIV/AIDS sub-model (14) endemic equilibrium point

Endemic equilibrium point is a disease-persistent state that refers to a scenario in which the HIV/AIDS infection persists in the population. In this sub-section, we determined the HIV-only sub-model (17) endemic equilibrium point, that is, a point at which the sub-model individual state variables are present at a non-zero level. To compute the sub-model endemic equilibrium point we need to solve the system of equations given by:

21

where

and

Applying (21) and (22) we solve for , and obtained the results given by:

23

Since and , we substitute (23) in this expression to compute the non-zero expression given by::

= , if and only if ,

where

Therefore, the HIV-only sub-model (17) has a unique endemic equilibrium point if and only if .

Disease-free equilibrium point

The complete HBV and HIV co-infection dynamical system (4) disease-free equilibrium point denoted by is determined by making all the right hand side equations equal to zero, and assuming that there is no disease spread in community (i.e.). Thus, the disease-free equilibrium point of the co-infection dynamical system described in equation is represented by:

24

The co-infection model effective reproduction number

Using the same procedures applied in the sub-subsection and we have computed the results represented by:

And

25

Then, we have calculated the next generation matrix.

26

And the associated eigenvalues of are given by:

Thus, the effective reproduction number of the complete HBV and HIV co-infection model is given by

27

where

is the HBV-only sub-model effective reproduction number and.

is the HIV-only sub-model effective reproduction number illustrated in (10) and (19) respectively.

Theorem 6

The complete HBV and HIV co-infection model (4) disease-free equilibrium point is locally asymptotically stable if the effective reproduction number and is unstable if .

Proof

The Jacobian matrix of the complete HBV and HIV co-infection model (4) concerning at the co-infection disease-free equilibrium point is given by:

28

where , , , , , , ).

Then characteristic equation at HBV and HIV/AIDS co-infection model (4) disease-free equilibrium point is given by

Then the solutions of corresponding characteristic equation are.

if ,

Or

29

Here, all the eigenvalues except eigenvalues in the expression of equation are negative and for remaining two eigenvalues in the expression (29) the Routh-Hurwitz stability criteria is applied and proved that the first column of the Routh-Hurwitz array has no sign change whenever . Hence, the co-infection model disease-free equilibrium point (DFE) is locally asymptotically stable whenever . The disease-free equilibrium of the model is locally asymptotically stable whenever its corresponding effective reproduction number value is less than unity⁴².

The co-infection model (4) endemic equilibrium point

The HBV and HIV/AIDS co-infection model (4) endemic equilibrium point is computed by making the right side of the dynamical system (4) equal to zero provided that ,, and . Let the co-infection model (4) endemic equilibrium point be and the forces of infection for HBV and HIV/AIDS respectively are:

and, and After solving and simplifying the result we determined the results given by

30

, , and .

Analysis of the optimal control problem

In this section, we proposed the control strategies and modified the HBV and HIV co-infection dynamical system (4) with the following proposed time-dependent optimal control strategies represented by:

• Since both HBV and HIV are sexually transmitted viral infectious diseases they have the same protection mechanisms and the control functions and represent the efforts to protect HBV and HIV single infections spreading by implementing sexual abstinence, one mate relationship with an uninfected partner, and condom use by sexually active susceptible individuals) aimed at preventing the HBV and HIV infections respectively.

• the control function is related to treatment of HIV/AIDS infected individuals to increase their recovery rate and recovery period

• the control function is related to treatment of HBV infected individuals to increase their recovery rate and recovery period ,

• the control function is related to treatment of the HBV and HIV/AIDS co-infected individuals to increase their recovery rate and recovery period.

After incorporated the five proposed time-dependent control strategies into the complete HBV and HIV co-infection model described in (4) to investigate the optimal control strategy for minimizing the co-infections of the two diseases the corresponding optimal control problem is represented by:

31

with initial conditions given by , , , , , , and the HBV and HIV forces of infections represented by:

The proposed control functions represented by and such that are bounded and Lebesgue integrable. Here the optimal control problem (31) involves a situation where the number of HBV-infected, HIV/AIDS-infected, the HBV and HIV/AIDS co-infections cases and the cost of implementing protective and treatment control functions respectively given by and are minimized subject to the dynamical system (31).

According to the discussions of the optimal control problem above the objective functional represented below is considered.

32

where is the final time , , , and are weight constants of the HIV/AIDS infected the HBV infected and the HIV and HBV co-infected individuals respectively while for are weight constants for each individual time-dependent control strategy. We choose a nonlinear cost on the control strategies based on the assumption that the costs take nonlinear form as applied in references^41,43,44.

The optimal control strategies are to be found such that.

33

where such that.

34

is the control set.

Theorem 7

Given the objective functional defined on the control set and subject to the dynamical system (31) with the given non-negative initial conditions at time then there exist an optimal control function and corresponding optimal solutions to the initial value problem (31)-(33) by , that minimizes over .

Proof

To verify the following four basic conditions required for the set of admissible controls C we can use the Fleming and Rishel’s theorem stated in⁴⁸.

The set of the model state variables to the system (31)–(34) that correspond to the control functions in is non-empty.

The control set c is closed and convex.

Each right hand side of the state system is continuous, is bounded above by a sum of the bounded control and the state, and can be written as a linear function of with coefficients depending on time and the state.

The integrand of the objective functional given in Eq. (32) is convex.

The first required condition () can be verified by using Picard-Lindelöf’s theorem. If the solutions to the co-infection dynamical system equations solutions are bounded, continuous and satisfies Lipschitz conditions in the model state variables, then there is a unique model solution corresponding to each admissible control function (strategy) in the control set . We have proved that the total number of human population at time is bounded as also each of the model state variables is bounded. Hence the model state variables are continuous and bounded. Similarly we can prove the boundedness of the partial derivatives with respect to the state variables in the model, which establishes that the model is Lipschitz with respect to the co-infection model state variables. This completes the verification that condition holds.

By applying definition stated in references^49–52, the control set is convex and closed this proved the required condition Condition is verified by observing the linear dependence of the model equations on the control variables .

Eventually, to justify the required condition use definition stated in^48,51 that says any constant, linear and quadratic functions are convex. Hence, since the integrand of the objective functional given by is a quadratic function that is convex on . To show the bound on use definition of the control function and then we have since and hence

and . This completes the proof of Theorem 8 stated above.

The necessary conditions that an optimal solution must satisfy come from Pontryagin’s minimum principle (PMP). This principle converts (31)–(34) in to a problem of minimizing a Hamiltonian, with respect to together with the state equation and the adjoint condition.

The Hamiltonian function is illustrated by

35

, where for are the adjoint variables.

Theorem 8

For an optimal control set that minimizes over , there are adjoint variables represented by , …, satisfying the condition , with transversality conditions , where .

Further,

36

Proof:

Suppose is an optimal control and are the corresponding solutions of the dynamical system. Then applying the Pontryagin’s Maximum Principle there exist adjoint variables satisfying:

37

The behaviour of the control can be determined by differentiating the Hamiltonian, with respect to the controls at t. On the interior of the control set, where for all and hence the first conditions that we will consider from the Pontryagin’s Maximum/Minimum principle applied in⁵² are the minimization of the Hamiltonian with respect to the control functions . Since the cost function is convex, if the optimal control occurs in the interior region we must have the following basic necessary and sufficient optimality conditions for the optimal control problem (31) as:

38

Then we have the following expressions

39

Therefore, solving (39) and determined the following results

40

Finally, we solve and simplify the results in (39) we have determined the final optimal control strategies results given by:

41

Theorem 9

For any the bounded solutions to the optimality system are unique. We can refer⁵⁰, for the proof of this theorem.

Numerical simulations and sensitivity analysis

To approximate the solutions of ordinary differential equations (ODEs) of the formulated co-infection model without and with optimal control strategies using numerical simulations (curve fitting), the classical fourth order Runge–Kutta (RK4) numerical method (the forward backward sweep) is utilized with MATLAB. Since it is straightforward and accurate, this method of solving ODEs is frequently utilized. The RK4 technique evaluates the derivative function at several intermediate points within the step interval in order to determine the values of the dependent variables at each step. To get an estimate of the derivative at the present step, it then take the weighted averages of these intermediate evaluations. The dependent variable values are updated using this estimate, and the procedure is repeated iteratively until the intended endpoint is reached. By considering multiple intermediate evaluations, the RK4 method provides a more accurate approximation compared to other simple numerical methods, making it a popular choice for numerical ODE integration.

Sensitivity analysis

In this sub-section of the study, we investigated the significance of some of the co-infection model parameters on the HBV and HIV co-infection transmission in the community. Implementing the same procedures applied in references^8,53, we carried out the sensitivity analysis of the co-infection model parameters to assess their impact on the effective reproduction number of the model. The calculated results revealed that the most sensitive parameter is the parameter that exhibited a larger magnitude in its sensitivity index. Basically, we focused on sensitivity analyses of the model parameters included in the HIV sub-model effective reproduction number and the HBV sub-model effective reproduction number . We carried out the sensitivity analysis for the HBV and HIV co-infection model effective reproduction number based on each relevant parameter, and we have revealed that how crucial each parameter is to the spreading of the HBV and HIV co-infection and the HBV and HIV single infection diseases in the community. For the effective reproduction numbers and , respectively the sensitivity analyses listed below have been carried out.

• 

• 

• 

• 

• 

• .

• .

• 

• .

• .

• 

• 

• 

• .

The simulation curve illustrated by Fig. 2 is performed by considering meaning that, when the HBV and HIV co-infection disease spreads throughout the community. In this study, we have used the parameter values illustrated in Table 1 and computed the sensitivity indices described in Fig. 2. The diseases transmission rates treatment rates and portion of protection have the most significant impact on the co-infection model effective reproduction number.

Fig. 2.

Numerical simulation curves to show sensitivity indices of the model parameters.

Numerical simulation for the optimal control problem

Numerical simulation of the optimal control problem (31) is a critical aspect of this manuscript, offering a computational approach to solve complex systems where analytical solutions are often impractical. Optimal control involves determining the best control inputs over time to achieve a desired objective while adhering to constraints. Moreover, numerical simulation of the optimal control problem enhances the manuscript by providing insights into the system dynamics, performance optimization, and the robustness of control strategies across various applications. We now simulate the optimal control problem (31) numerically using the parameter estimates in Table 2, so that the HBV and HIV/AIDS co-infection model effective reproduction number, (unless otherwise described), to investigate the potential impact of different control strategies on the spreading dynamics of HBV and HIV/AIDS in the population. The inclusion of an optimal control framework in the research is of utmost significance, as it introduces five controls designed to manage the dynamics of HBV and HIV co-infection. These controls include strategies to prevent HBV and HIV infections, improve recovery in cases of each infection, and provide treatment for co-infected individuals. This section highlights the critical importance of these control strategies, both collectively and individually, underscoring their role in shaping effective approaches to address the complexities of HBV and HIV co-infection dynamics. To verify the effect of the proposed control strategies and to verify the analytical results of the optimal control problem (31) we carried out numerical simulation by considering the following equal weight factors (since it is difficult to get values for weight constants of the HIV/AIDS infected the HBV infected and the HIV and HBV co-infected individuals respectively and also weight constants for each individual time-dependent control strategy from related published papers), let us assume the weight constants as , and initial population along with the parameter values illustrated in Table 2, and the initial population is taken as , , , , , . Let us consider the following seven possible control strategies for numerical simulation and cost-effectiveness analysis:

• Strategy 1: Implement the controls (HBV protection and HIV protection simultaneously.

• Strategy 2: Implement the controls (HBV protection and HBV treatment simultaneously.

• Strategy 3: Implement the controls (HBV protection and HIV treatment simultaneously.

• Strategy 4: Implement the controls (HIV protection and HBV treatment simultaneously.

• Strategy 5: Implement the controls (HIV protection and HIV treatment simultaneously Strategy 6: Implement all the treatment controls (,, simultaneously.

• Strategy 7: Applying all the five proposed control measures simultaneously.

Strategy 1: HBV protection ( and HIV protection ( Controls

The numerical simulation to investigate the total number of HBV and HIV co-infected population in the optimal control system (31) when HBV protection () and HIV protection () controls are implemented, is illustrated by Fig. 3. It is observed that when this control strategy is implemented, there is a significant reduction in the total number of HBV and HIV co-infected individuals. Eventually, after five years the total number of HBV and HIV/AIDS co-infected population declining to zero.

Fig. 3.

Impact of HBV and HIV protections on the total co-infected population.

Strategy 2: HBV protection ( and HBV treatment ( controls

In this sub-section numerical simulation is carried out to investigate the impacts of the HBV protection () and HBV treatment () control strategy on the total number of HBV and HIV/AIDS co-infected population in the optimal control system (31) which is described by Fig. 4. It is observed that when this control strategy is implemented, there is a significant reduction in the total number of HBV and HIV/AIDS co-infected individuals as compared to the simulation result without control strategy. Eventually, after five years the total number of HBV and HIV/AIDS co-infected population declining to zero.

Fig. 4.

Impact of HBV protection and HBV treatment on the total co-infected population.

Strategy 3: HBV protection ( and HIV treatment ( controls

In this sub-section numerical simulation is carried out to investigate the impacts of the HBV protection () and HIV treatment () control strategy on the total number of HBV and HIV/AIDS co-infected population in the optimal control system (31) which is described by Fig. 5. It is observed that when this control strategy is implemented, there is a significant reduction in the total number of HBV and HIV/AIDS co-infected individuals as compared to the simulation result without control strategy. Eventually, after five years the total number of HBV and HIV/AIDS co-infected population decreases.

Fig. 5.

Impact of HBV protection and HIV treatment on the total co-infected population.

Strategy 4: HIV protection ( and HBV treatment ( controls

In this sub-section numerical simulation is carried out to investigate the impacts of the HIV protection () and HBV treatment () control strategy on the total number of HBV and HIV/AIDS co-infected population in the optimal control system (31) which is described by Fig. 6. It is observed that when this control strategy is implemented, there is a significant reduction in the total number of HBV and HIV/AIDS co-infected individuals as compared to the simulation result without control strategy. Eventually, after five years the total number of HBV and HIV/AIDS co-infected population decreases to zero.

Fig. 6.

Impact of HIV protection and HBV treatment on the total co-infected population.

Strategy 5: HIV protection ( and HBV treatment ( controls

In this sub-section numerical simulation is carried out to investigate the impacts of the HIV protection () and HIV treatment () control strategy on the total number of HBV and HIV/AIDS co-infected population in the optimal control system (31) which is described by Fig. 7. It is observed that when this control strategy is implemented, there is a significant reduction in the total number of HBV and HIV/AIDS co-infected individuals as compared to the simulation result without control strategy. Eventually, after five years the total number of HBV and HIV/AIDS co-infected population decreases to zero.

Fig. 7.

Impact of HIV protection and HIV treatment on the total co-infected population.

Strategy 6: treatment ( controls

In this sub-section numerical simulation is carried out to investigate the impacts of the HIV, HBV, and the co-infection treatments () control strategy (strategy 6) on the total number of HBV and HIV/AIDS co-infected population in the optimal control system (31) which is described by Fig. 8. It is observed that when this control strategy is implemented, there is a significant reduction in the total number of HBV and HIV/AIDS co-infected individuals as compared to the simulation result without control strategy. Eventually, after five years the total number of HBV and HIV/AIDS co-infected population decreases.

Fig. 8.

Impact of HBV, HIV and co-infection treatments on the total co-infected population.

Strategy 7: Implement all (, controls

The numerical simulation curve illustrated in Fig. 9 reveals the impact of the control strategy 7 (implementing all the proposed control mechanisms simultaneously) emphasizing a significant reduction of the HBV and HIV/AIDS co-infected population as compared to a scenario implemented in Figs. 3, 4, 5, 6, 7 and 8 and where there is no control mechanism implemented.

Fig. 9.

Impact of protections and treatments strategy on the total co-infected population.

Cost-effectiveness analysis

In this section, we carried out a cost-effectiveness analysis in order to verify the costs corresponding to health intervention(s) or strategy (strategies) such as protections, or treatment intervention strategies and the associated cost benefits are usually evaluated using cost-effectiveness analysis approach used in⁴⁵. In this section we will consider the incremental cost-effectiveness ratio (ICER) approaches to analyze cost-effectiveness and evaluate the real cost benefits corresponding to the proposed health intervention strategies. The ICER is defined as:

Where ICER numerator includes the differences in disease averted costs, costs of prevented cases, intervention costs, among others. While the denominator of ICER accounts for the differences in health outcome, including the total number of infections averted or the total number of susceptibility cases prevented. The criteria used to evaluate the cost-effectiveness of different interventions is analyzed the cost-effectiveness by ranking the control strategies in increasing order of effective- ness in terms of the number of infected averted and remove the strategy with dominant ICER value.

In this section, we calculated the total number of HBV and HIV/AIDS co-infection cases averted and the total cost of the proposed strategies applied in Table 3. The total number of HBV and HIV/AIDS co-infection cases prevented is obtained by calculating the total number of individuals when controls are applied and the total number when there is no control implemented. Similarly, we apply the cost functions , over time, to compute the total cost for the various control strategies implemented. Thus, we applied similar approach used in several previous studies like^47,54, the incremental cost-effectiveness ratio (ICER) is calculated to determine the most cost-effective strategy of all the different control intervention strategies considered in this study.

Table 3.

Seven Strategies’ ICER values with total number of infections averted and cost incurred.

Strategy	Total number of infections averted	Total cost incurred ($)	ICER
Strategy 7		3.1810
Strategy 4	5.1006	0.091 ×
Strategy 3	5.9614	0.099 ×
Strategy 5	6.1352	0.54
Strategy 2	2.8256	2.34
Strategy 1	3.2317	0.87
Strategy 6	4.1512	0.96

In this part, using the seven proposed control strategies described in the numerical simulation part for the optimal control problem (31), namely, Strategy 1 (implemented and i.e., to protect individuals against HBV and HIV infections respectively we considered sexual abstinence, one mate relationship with an uninfected partner, and condom), Strategy 2 (implement and i.e., to protect individuals against HBV infection we implemented sexual abstinence, one mate relationship with an uninfected partner, and condom and treat HBV infected individuals simultaneously) Strategy 3 (implement and i.e., to protect individuals against HBV infection we considered sexual abstinence, one mate relationship with an uninfected partner, and condom and treat HIV/AIDS infected individuals simultaneously), Strategy 4 (implement and i.e., to protect individuals against HIV/AIDS infection we considered sexual abstinence, one mate relationship with an uninfected partner, and condom and treat HBV infected individuals simultaneously), Strategy 5 (implement and i.e., to protect individuals against HIV/AIDS infection we considered sexual abstinence, one mate relationship with an uninfected partner, and condom and treat HIV/AIDS infected individuals simultaneously), Strategy 6 (implement and i.e., to minimized the number of individuals infected with HVB, HIV/AIDS and HBV and HIV/AIDS co-infection we considered possible treatment mechanisms for each infection simultaneously), and Strategy 7 implemented all the control strategies (,,) simultaneously are ranked in ascending order with respect to the total number of infections averted as illustrated in Table 3 below.

According to the ICER values computed and described in Table 3 above ICER (Strategy 7) and ICER Strategy 4), it is observed that the ICER for strategy 7 is greater than the ICER for strategy 4. This implies that strategy 7 strongly dominates strategy 4, indicating that strategy 4 is less costly and more effective in comparison with strategy 7. As a result, strategy 7 is eliminated from subsequent ICER computations because the strategy was too expensive and less effective, we excluded from other alternative strategies that were competing for limited resources, resulting in the re-computed ICER for Strategy 4 and Strategy 3.

We employed a similar methodology and, based on the data presented in Table 4, we determined that Strategy 3 was removed since its ICER value was higher than Strategy 4’s. Table 5 below shows the results of the computation we performed to compare Strategies 4 and 5.

Table 4.

Six Strategies’ ICER values with total number of infections averted and cost incurred.

Strategy	Total number of infections averted	Total cost incurred ($)	ICER
Strategy 4	5.1006	0.091 ×
Strategy 3	5.9614	0.099 ×
Strategy 5	6.1352	0.54
Strategy 2	2.8256	2.34
Strategy 1	3.2317	0.87
Strategy 6	4.1512	0.96

Table 5.

Five Strategies’ ICER values with total number of infections averted and cost incurred.

Strategy	Total number of infections averted	Total cost incurred ($)	ICER
Strategy 4	5.1006	0.091 ×
Strategy 5	6.1352	0.54
Strategy 2	2.8256	2.34
Strategy 1	3.2317	0.87
Strategy 6	4.1512	0.96

Here we have to eliminate Strategy 5 and proceeded with the procedures to compare Strategy 4 and Strategy 2, as shown in Table 6 below, as the results shown in Table 5 above indicate that Strategy 5 is more cost-effective than Strategy 4.

Table 6.

Four Strategies’ ICER values with total number of infections averted and cost incurred.

Strategy	Total number of infections averted	Total cost incurred ($)	ICER
Strategy 4	5.1006	0.091 ×
Strategy 2	2.8256	2.34
Strategy 1	3.2317	0.87
Strategy 6	4.1512	0.96

Since Strategy 2 exceeds Strategy 4 in terms of cost, as indicated by the result shown in Table 6 above, we eliminated Strategy 2 and carried out the steps to compare Strategy 4 and Strategy 1, which is explained in Table 7 below.

Table 7.

Three strategies’ ICER values with total number of infections averted and cost incurred.

Strategy	Total number of infections averted	Total cost incurred ($)	ICER
Strategy 4	5.1006	0.091 ×
Strategy 1	3.2317	0.87
Strategy 6	4.1512	0.96

Since Strategy 1 is more cost-effective than Strategy 4 according to the results shown in Table 7 above, we eliminated Strategy 1 and carried out the steps to compare Strategy 4 and Strategy 6 as shown in Table 8 below.

Table 8.

Three strategies’ ICER values with total number of infections averted and cost incurred.

Strategy	Total number of infections averted	Total cost incurred ($)	ICER
Strategy 4	5.1006	0.091 ×
Strategy 6	4.1512	0.96

Finally, we found that Strategy 6 is highly dominated in terms of cost-effectiveness compared to Strategy 4 since the ICER (Strategy 6) is bigger than the ICER (Strategy 4) utilizing the results of the cost-effectiveness study shown in Table 8 above. The analysis reveals that, out of the seven proposed controlling strategies, Strategy 4, which involves protecting HIV/AIDS infected individuals and treating HBV-infected individuals simultaneously is the most economical (cost benefit) strategy we recommend the public health stakeholders to implement this strategy to tackle the HBV and HIV/AIDS co-infection spreading in the community.

Therefore, Strategy 4 (implementing and i.e., to protect individuals against HIV/AIDS infection we considered sexual abstinence, one mate relationship with an uninfected partner, and condom use and treat HBV infected individuals simultaneously) is the most cost-effective strategy.

Discussions, conclusions and future directions of the study

HBV and HIV/AIDS co-infection is a common infectious disease and has been affected millions of individuals throughout the world. In this study, we constructed and analyzed HBV and HIV/AIDS co-infection model with optimal control and cost-effectiveness analyses by considering the two HIV infection stages. In the models qualitative analysis we have proved the model solutions existence, uniqueness, non-negativity and boundedness, we applied the next generation matrix approach we have calculated all the models effective reproduction numbers, using Routh-Hurwitz stability criteria we have proved the local stability of the models disease-free equilibrium points, using the approach in Castillo-Chavez criteria the disease-free equilibrium of the sub-model was proven to be globally asymptotically stable whenever the associated effective reproduction number is less than unity. The co-infection model (4) can be combined with optimal control theory to identify the most effective intervention strategies for the HBV and HIV/AIDS co-infection spreading in the community. The goal is to find the optimal trajectory of these controls that minimizes the HBV and HIV/AIDS co-infection and costs incurred. This involves solving the system of differential equations for the model dynamics and applying Pontryagin’s Minimum Principle to derive necessary conditions for an optimal solution. By analyzing the Hamiltonian and adjoint equations, we computed the optimal combination of intervention strategies at each time point during the co-epidemic. This allows public health authorities to determine the most effective allocation of limited resources for protection, education campaigns, and healthcare to minimize the co-infection cases. Optimal control provides a framework for finding the best co-epidemic mitigation strategy that balances infection reduction with practical constraints on implementation. Moreover, we performed sensitivity analysis of the co-infection model parameters and both the HBV and HIV spreading rates are the most influential parameters on the co-infection spreading. Furthermore, we carried out numerical simulations and simulating of the co-infection model, we used a well-known and more efficient numerical scheme the classical Runge Kutta fourth order (RK4) forward numerical methods with MATLAB, the numerical results are given in the numerical results section. The findings of the study reveals that implementing all the proposed control strategies simultaneously is crucial strategy used to minimize and control the HBV and HIV/AIDS co-infection spreading in the community but cost-effectiveness analysis proved that the HIV protection and HBV treatment strategy is the most cost-effective strategy as compared with all other implemented strategies and we recommend for the health stake holders to implement this strategy to tackle the HBV and HIV co-infection transmission problem in the community. Therefore, this study is useful for the understanding of the HBV and HIV/AIDS co-infection spreading behavior, to implement the right control measures to tackle and minimize the disease transmission in the community. Hence, we recommend that efforts should be made by government and public health stakeholders to protect the HIV/AIDS infection and to treat the HBV infection with low cost in order to bring the burden of the HBV and HIV/AIDS co-infection very low at the community level.

This study did not considered the stochastic approach, the fractional order approach, the age structure of individuals, the HBV infection stages, the environmental factors, and validation of the model with real data collected from the study area etc. These are some of main limitations of this study where interested researchers can consider and modify this study.

Author contributions

Shewafera Wondimagegnhu Teklu formulated and analyzed both the model without optimal controls and with optimal controls qualitatively as well as numerically. And Abushet Hayalu Worike edited and approved the optimal control problem of the model and carried out its qualitative analysis.

Funding

There is no funding for the study.

Data availability

Data used to support the findings of this study are included in the article.

Declarations

Competing interests

The authors declare no competing interests.