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

M. Haider Ali Biswas; S. Abdus Samad; Tahera Parvin; M. Tusberul Islam; Asep K. Supriatna

doi:10.5614/cbms.2022.5.1.2

Authors

M. Haider Ali Biswas Mathematics Discipline, Science, Engineering and Technology School, Khulna University, Bangladesh
S. Abdus Samad Mathematics Discipline, Science, Engineering and Technology School, Khulna University, Bangladesh
Tahera Parvin Mathematics Discipline, Science, Engineering and Technology School, Khulna University, Bangladesh
M. Tusberul Islam Mathematics Discipline, Science, Engineering and Technology School, Khulna University, Bangladesh
Asep K. Supriatna Department of Mathematics, Padjadjaran University, Bandung, Indonesia

DOI:

https://doi.org/10.5614/cbms.2022.5.1.2

Keywords:

Mathematical model, HIV and TB co-infection, Optimal control, Pontryagin's Maximum Principle

Abstract

Tuberculosis (TB) and HIV/AIDS has become hazardous among communicable diseases and so as their co-infection in present era. HIV virus gradually weakens immune system in human body, and then TB infects with the assist of HIV/AIDS at any stage of the total infectious period. Today, HIV and tuberculosis (TB) are the main causes of mortality from infectious and chronic diseases. In this Study, we manifest a compartmental co-infection model including HIV and TB on the basis of their characteristics of disease transmission. The model is divided into 10 compartments, each with its own set of nonlinear ordinary differential equations. Using the Pontryagin's Maximum Principle, we investigate the existence of state variables, objective functional and optimum control plans. Identifying the most effective ways for reducing infection among the individuals, the optimal control techniques like vaccination control and treatment control measures are applied. The goal of this study is to lower the rate of HIV-TB co-infection and the cost of treatment. Another objective is to find the better control strategy to prevent HIV/AIDS that invites other pathogen in human body by gradual loosing of immunity. We carried out the investigation both analytically and numerically to divulge the effectiveness of the vaccination and treatment control to lessen the HIV and TB infection among the individuals.

References

Ziet,, B. P., and Dunkelberg, H., The history of the plague and the research on the causative agent Yersinia pestis, ELSEVIER, International Journal of Hygiene and Environmental Health, 207(2), pp. 165-178, 2004.

Biswas, M. H. A., Optimal Control of Nipah Virus Infections: A Bangladesh Scenario, Journal of Pure and Applied Mathematics: Advanced and Applications, 12(1), pp. 77-104, 2014.

Biswas, M. H. A., Model and control strategy of the deadly Nipah virus (NiV) infections in Bangladesh, Research and Review in Bioscience, 6(12), pp. 370-377, 2012.

Rewar, S., Swine Origin Influenza A (H1N1) Virus: an Overview, International Journal of Technical and Research Applications, 3(3), pp. 1-5, 2015.

Khatun, M. S. and Biswas, M. H. A., Optimal Control Strategy for Preventing Hepatitis B Infection and Reducing Chronic Liver Cirrhosis Incidence, Infectious Disease Modeling, 5, pp. 91-110, 2020.

Sahani, S. K. and Biswas, M. H. A., Stability Analysis of Co-infection Model of HIV and Hepatitis B Virus, S. Mukhopadhyay, S. Bhattacharyya, K. Ghosh (Eds.), International Conference on Modeling, Computing and Technological Innovation, Book Chapter, ISBN: 978-93-86256-60-7, pp. 284-291, 2017, Excel India Publishers, New Delhi, India.

Biswas, M. H. A., Haque, M. M., and Mallick, U. K., Optimal Control Strategy for the Immunotherapeutic Treatment of HIV Infection with State Constraint, Optimal Control, Applications and Methods, 40(4), pp. 807-818, 2019.

Biswas, M. H. A., On the Evolution of AIDS/HIV Treatment: An Optimal Control Approach, Current HIV Research, 12(1), pp. 1-12, 2014.

Biswas, M. H. A. On the Immunotherapy of HIV Infections via Optimal Control with Constraint, Proceedings of the 18th International Mathematics Conference 2013 jointly organized by Bangladesh Mathematical Society and Independent University Bangladesh, Bashundhara Campus, Dhaka on 20-22 March 2014, pp. 51-54, 2014.

Biswas, M. H. A., AIDS Epidemic Worldwide and the Millennium Development Strategies: A Light for Lives, HIV & AIDS Review, 11(4), pp. 87-94, 2012.

Samad, S. A. and Biswas, M. H. A., Mathematical Understanding of Dynamical Model for the Transmission of Endemic Tuberculosis in Bangladesh, Advanced Modeling and Optimization, 20(2), pp. 329-344, 2018.

Biswas, M. H. A., Islam, M. A., Akter, S., Mondal, S., Khatun, M. S., Samad, S. A., Paul, A. K. and Khatun, M. R., Modelling the Effect of Self-Immunity and the Impacts of Asymptomatic and Symptomatic Individuals on COVID-19 Outbreak, CMESComputer Modeling in Engineering & Sciences, 125(3), pp. 1033-1060, 2020.

Islam, M. S., Ira, J. I., Gupta, P. K., Biswas, M. H. A., Kabir, K. M. A. and Mobin, M. A., Isolation Effect on Age-Stratified Compartmental Model of the COVID-19, Communications in Nonlinear Analysis, 8(1), pp. 1-12, 2021.

Trinh, Q. M., Nguyen, H. L., Nguyen, V. N., Nhuyen, T. V. A., Sintchenko, V. and Marais, B. J., Tuberculosis and HIV co-infection-focus on the Asia-Pacific region, International Journal of Infectionous Diseases, 32, pp. 170-178, 2015.

Gov. Report, AIDS, A Time Line of HIV/AIDS, 2020.

Silva, C. J. and Delfim, F. M. T., A TB-HIV/AIDS Coinfection Model and Optimal Control Treatment, Discrete and Continuous Dynamical System, 35(9), pp. 1-25, 2015.

Padmapriyadarsini, C., Narendran, G., and Swaminathan, S., Diagnosis & Treatment of Tuberculosis in HIV co-infected Patients, Indian J. Med. Res., 134(6), pp. 850-865, 2011.

Rom, W. N. and Markowitz, S. B., Environmental and Occupational Medicine, Lippincott Williams & Wilkins, 2007.

Global Tuberculosis Institute Report, History of Tuberculosis, 2016.

Mahmud, S. A. I., TB and HIV/AIDS in Bangladesh, Journal of AIDS and HIV Research, 2(4), pp 66-78, 2010.

Nura, R., Modeling and Optimal Control Strategy for HIV/ AIDS Disease in the Presence of Infective Immigrants and Non linear Incidence, M.Sc. thesis, University of Dar-es-Salam, Tanzania, 2013.

Brosch, R., Gordon, V. S., Marmiesse, M., Brodin, P., Buchrieser, C., Eiglmeier, K., Garnier, T., Gutierrez, C., Hewinson, G., Kremer, K., Parsons, L. M., Pym, A. S., Samper, S., Soolingen, V. and Cole, S. T., A new evolutionary scenario for the Mycobacterium tuberculosis complex, Proc Natl Acad Sci USA, 99, pp. 3684-3689, 2000.

Bowong, S. and Tewa, J. J., Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate, Communications in Nonlinear Science and Numerical Simulations, 15(11), pp. 3621-3631, 2010.

Awoke, T. D. and Kassa, S. M., Optimal control Strategy for TB-HIV/AIDS Co-Infection Model in the Presence of Behavior Modification, MDPI, 6(5), p. 48, 2018.

Gupta, V. K., Tiwari, S. K., Sharma, S. and Nagar, L., Mathematical Modeling of Tuberculosis with Drug Resistance to the First and Second Line of Treatment, Journal of New Theory, 21, pp. 94-106, 2018.

Fatmawati and Tasman, H., An optimal Treatment Control of TB-HIV Coinfection, International Journal of Mathematics and Mathematical Sciences, 2016(3), pp. 1-11, 2016.

Biswas, M. H. A., Paiva, L. T. and de Pinho, M. D. R., A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11(4), pp. 761-784, 2014.

Biswas, M. H. A., Necessary conditions for optimal control problems with and without state constraints: a comparative study, Transactions on Systems and Control, 6(6), pp. 217-228, 2011.

Bhunu, C. P., Garira, W. and Mukandavire, Z., Modelling HIV/AIDS and Tuberculosis Coinfection, PMID, 9(2), pp. 1-16, 2002.

Lenhart, S. and Workman, J. T., Optimal Control Applied to Biological Model, Chapman and Hall, New York, 2007.

Roeger, L. W., Feng, Z. and Chavez, C. C., Modeling TB and HIV Co-Infections, Mathematical Bioscience and Engineering, 6(4), pp. 815-837, 2009.

Azim, T., Khan, S. I., Haseen, F., Huq, N. L., Henning, L., Pervez, M. M., Chowdhury, M. E. and Sarafian, I., HIV and AIDS in Bangladesh, Journal of health publications, 26(3), pp. 311-324, 2008.

Constantinos, I., Siettos and Russo, L., Mathematical modeling of infectious disease dynamics, PMC, 4(4), pp. 295-306, 2013.

Islam, M. S., Ira, J. I., Kabir, K. M. A. and Kamrujjaman, M., Effect of lockdown and isolation to suppress the COVID-19 in Bangladesh: an epidemic compartments model, J. Applied Mathematics and Computation, 4(3), pp. 83-93, 2020.

Khatun, M. S. and Biswas, M. H. A., Mathematical analysis and optimal control applied to the treatment of leukemia, Springer, Journal of Applied Mathematics and Computing, 64(1), pp. 331-353, 2020.

Zaman, K., Tuberculosis: A Global Health Problem, Editorial. J. Health Popul, 28(2), pp. 111-113, 2010.

Zaman, K., Hossain, S., Yunus, M., Arifeen, S.E., Mahmud, A., Begum, V., Islam, A., Baqui, A. H. and Luby, S., Tuberculosis in Bangladesh: A 40-Year Review, ICDDR, B, Scientific session, 86, pp. 4-6, 2007.

Zumla, A., Malon, P., Henderson, J. and Grange, J., Impact of HIV infection on tuberculosis, Post Graguate Medical Journal, 76(895), pp. 259-268, 2000.

Marino, A. A., and Frilot, C., Nonlinearity in Biological System: How can Physics Help?, Energy and Information transform in Biological System, 245-263, 2003.