Optimal Control Strategy to Reduce the Infection of the Pandemic HIV Associated with Tuberculosis in Bangladesh

Tuberculosis (TB) associates with HIV/AIDS at any stage of the total infection period. In recent years, HIV and TB have become the leading causes of death due to infectious and long lasting disease. On the basis of the characteristics of disease transmission of HIV and TB, we formulate a mathematical model. The model consists of ten compartments with nonlinear ordinary differential equations. The model is locally asymptotically stable in case of disease free equilibrium point when basic reproduction number is less than unity and locally asymptotically stable in case of endemic equilibrium point when the basic reproduction number is greater than unity. We apply optimal control theory to the co-infection model in the form of Pontryagin’s Maximum Principle by introducing the treatment control and the vaccination control to decrease the transmission of the disease burden. Finally, we perform the numerical simulations of the optimal control model to observe the dynamics of the system in support of analytical findings.


Introduction
Infectious diseases have been threatening throughout the world day by day and these diseases have a great influence on the human population. The history referrers that plague, nipah, swine flu, hepatitis B, HIV/AIDS, TB, COVID-19 etc are most dangerous infectious diseases that cause a huge loss of human life. From its very beginning, scientists have been trying to control these kinds of fatal diseases. After the manifestation of any infectious virus it starts to increase very fast sensibly or insensibly and people need the effective method to resist the outbreak of the disease or at least to control the number of infections (AIDS Report 2016).
Transmission diseases have ever been a great concern of human being from the very beginning of our history. Millions of people die annually from tuberculosis and HIV/AIDS and billions other are being infected. AIDS indeed is a terribly dangerous disease found almost everywhere in human society. But it has not affected all the societies equally throughout the world (Biswas 2012).
People having AIDS are always under the threat of death and this disease is caused by the Human Immunodeficiency Virus usually known as HIV. HIV is a kind of virus which makes the human body almost incapable of fighting any kind of infections. HIV virus causes gradual destruction of immune system of human body by reflecting a decrease of CD4+T cell counts (Nunn et al. 2005). Significant immune destruction helps opportunistic processes develop and thereby infections start taking place in the body. Our human body heavily depends on the immune system to fight infections, the same way it depends on the digestive system to digest food. The immune system also prevents some types of infections from taking place in the body. If the immune system is totally destroyed then a person will have to die even from a simple infection like cold or flu (Sahani et al. 2017).
Tuberculosis more commonly known as TB is a bacterial infectious disease which can co-infect with HIV. TB can be latent (inactive) or active (Bowong and Tewa 2010). Close to one third of the world's population has dormant or latent TB. People living with TB gradually loss the immune system and as a result it causes a great health hazard. A complex interaction exists between TB and human immunodeficiency virus (Samad and Biswas 2018). The HIV and TB co-infection can spread out at an individual and community level. Human immunodeficiency virus and Tuberculosis jointly become the serious global mass health challenge. The incarnation of AIDS make the relation between HIV and TB. The co-infection of HIV and TB exist when individual are HIV positive and are either exposed or active to TB. Getting infected with TB bacteria is not automatic for HIV infected individuals. Similarly individuals infected with TB do not automatically get HIV stage unless in contact with HIV positive individuals (Brosch 2000). Till today the world's most serious epidemic as well as leading infectious killer disease is AIDS/HIV and TB, the ultimate destiny of which is the early death of the infected patient (Nura 2013).
TB continues to be a major source of illness and death in Bangladesh, which ranks sixth in the globe in terms of TB patient burden (Mahmood 2010). In Bangladesh, it is estimated that 70,000 people die from tuberculosis each year, with 300,000 new cases appearing each year. Prior to 2014, the case notification rates for new smear-positive cases and all forms of TB cases were 68 and 122 per 100,000 population, respectively, but by the end of 2014, the number of all types of TB cases had increased, with a significant increase in the number of extra-pulmonary cases due to better case detection. However, overall progress in case detection was slow and steady in 2016, with a case notification rate of 138 per 100,000 people (Kuddus et al. 2021). In 1989, the first case of HIV was discovered in Bangladesh. Every year since then, 10, 20, 100, or 200 additional instances have been discovered (Risingbd.com 2022). According to a government source, a total of 1500 cases of HIV/AIDS were confirmed and recorded in 2009, up from 1207 cases in 2007 and 1495 cases in 2008. The number of instances that go undetected is substantially larger. According to a recent poll, 50% of males tested positive for HIV at a city hospital are students, and 80% of those infected caught the infection through interacting with prostitutes (Mahmood 2010). According to the most recent figures, a total of 6,104 HIV-positive persons have been treated. The remaining 1,125 AIDS patients are still unable to receive therapy. The Bangladesh National AIDS/STD Control Program, on the other hand, claims that the number of AIDS patients receiving treatment is growing every year. In 2019, 52 percent of people were aware of the problem, which increased to 63 percent in 2021. Two years ago, 65 percent of AIDS-infected persons were receiving treatment, which has since risen to 77 percent (Risingbd.com 2022). (Bhunu et al. 2009) proposed a mathematical model of HIV/AIDS and tuberculosis co-infection and showed that the dynamics of different stages of the model. They also applied anti-retroviral therapy for the AIDS case and treatment for active and latent form of TB. (Biswas et al. 2019) proposed a nonlinear model of HIV immunology to describe the interaction between virus and human along with optimal therapy strategy using Pontryagin maximum principle. (Fatmawati and Tasman  Being motivated with the severity of HIV/AIDS and TB worldwide as well as in Bangladesh, we are intended to do the work about it. We extended the model (Bhunu et al. 2009) in context of Bangladesh. Bhunu et al. (2009) applied anti-retroviral therapy for the AIDS case and treatment for active and latent form of TB, but we applied vaccination for latent form of TB and treatment for co-infection taking the weight in Bangladesh perspective. The parameter specification has been carried out by estimation from secondary data. However, describing the epidemiology of these infectious diseases and adopting any control strategy is not an easy task. By using mathematical modeling, we analyze the epidemiological insights of the model. We use two control variables for the prevention and minimization of the disease transmission. We perform the numerical calculation of this model by using ODE45 solver in MATHLAB programming language.

Objectives of the Study
The objectives of this study are: 1. to study the biological features and insights of the transmission of HIV and TB diseases and their co-infection on human body.
2. to propose a model in case of HIV/AIDS and TB with control only in human body using compartmental model with ordinary differential equations.
3. to analyze the HIV/AIDS and TB model in presence with one another.
4. to detect the factor that accelerates the different stages of HIV-TB co-infection.
5. to find a control strategy of the pandemic HIV and TB to minimize the transmission.
6. to solve the nonlinear system of ordinary differential equations with optimal control strategy for both analytically and numerically.
7. to create awareness among the general people about HIV/AIDS and TB so that they become careful of getting infection easily.

Materials and methods
The entire population model is divided into ten compartments that are Susceptible individuals (S) Latent TB with no HIV (L T ), Exposed to TB only (L T0 ), Symptom of TB (S R ), Infected with HIV only (I H ), Those who have recover with temporal immunity (R T ), HIV infected individuals (pre-AIDS) exposed to TB (E TH ), HIV infected displaying HIDS symptoms (H S ), AIDS individuals exposed to TB (E TH ) and AIDS individuals dually infected with TB (H DT ), For the treatment strategies of HIV and TB we considered two control variables like u 1 and u 2 . We have u 1 representing the vaccinations that will treat tuberculosis so that the disease transmission can be minimized and u 2 that take HIV and TB treatment simultaneously. So here u 1 is the vaccination control and u 2 is the treatment control. Now we aim to reduce the number of infected individuals of HIV and TB patients along with the cost of the respective control measures. The optimal control model with nonlinear ODEs are: ii) The control set U must be closed and convex iii) The right hand side of the state system is continuous, bounded above by a linear combination of the control and state and can be written as a linear function of u with coefficients defined by the time and state.
iv) The integrand of the objective functional is convex on U and is bounded below by ( ) Theorem 2: There exists optimal control ( ) 1 2 , u u * * that minimizes the objective functional J over U given by ( ) Proof: We search for the characterization of 1 u * where there arise three cases for ( ) In terms of Hamiltonian, we get the optimality conditions,

Results and discussion
In this section we discuss the numerical simulations with the responsible parameters. Our observation is that these two parameters which are the disease contact rate for HIV and TB respectively are mainly liable for dangerous threat. So we intend to control these parameters by applying controls measures, the vaccination control and the treatment control. To illustrate this we have considered 20 years of total time period because the transmission process of HIV and TB is quite long. Fig. 1 describes the dynamics of susceptible individuals with the optimal vaccination control where a close look reveals that vaccination has slight impact on the susceptible population that decrease it. Fig. 2 states the dynamics of latent TB individuals applying the vaccination control which has a great impact to decrease the patients with latent TB. That means if we apply vaccination to the patients who have latent (13) and (14)   3-4, we see that after applying the control strategy, the disease has no mode to decrease it means vaccination control has no impact on HIV infected population and HIV infected population with AIDS symptom in this model. Figs. 5 and 6 describe the dynamics of HIV infected individuals exposed to TB and AIDS infected individuals exposed to TB along with optimal control. Both Figs 5-6 show that HIV infected exposed to TB patients and AIDS infected exposed to TB have been decreased from the community when vaccination is induced. Figs. 7 and 8 show the dynamic behavior of AIDS individuals dually infected with TB and exposed to TB individuals with optimal control and without optimal con- trol. Vaccination is equally effective to these two compartments which elicit that AIDS dually infected with TB and exposed to TB patients have decreased after applying the strategy of vaccination control. Fig. 9 shows that the dynamic pattern of symptom of TB is decreasing and vaccination has direct impact on this compartment. If the patients who have symptom of TB get vaccination, the numbers of patients decrease gradually from the community. From the Fig. 10, we get the idea of dynamics of recovered population along with the impact of vaccination control. When the vaccination is applied the recovered population increased with a great extends. This means considerable recovery from the co-infection of HIV and TB is possible by applying vaccination among the general mass population.

Conclusion
In the modern day's public health scenarios, HIV and TB are top at the list of infectious diseases worldwide. So to eradicate or to minimize these kinds of contagious diseases is the gigantic challenge through the world. So proper program for HIV-TB, to create opportunity for reaches on HIV-TB and create awareness among the mass population can be helpful to control the infection rates. Counseling for HIV-TB coinfected persons to a great extent, availability of treatment and antidote, early detection of HIV-TB and plan to reduce the virus transmission should be emphasized. The government and the infectious diseases institutes should take all round steps against HIV and TB to eliminate this curse from the human planet.