Mathematical Analysis and Numerical Simulation of the Integrated HIV/TB Co-Infection Model

In this study, we examined an integrated model of HIV and TB co-infection. Specifically, we proposed a system of nonlinear differential equations to describe the co-dynamics of both diseases and conducted a comprehensive stability analysis of its critical points alongside numerical simulations. The stability analysis of the equilibrium points revealed that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number, R_0, is less than one (R_0<1). This result indicates that, under these conditions, both diseases can be eradicated from the population over time. A local sensitivity analysis was performed to identify parameters with the greatest influence on disease dynamics. The analysis demonstrated that the parameter ω_1 is a key driver for tuberculosis (TB) transmission, as evidenced by its high sensitivity index in the TB-specific reproduction number, R_T. From a biological perspective, ω_1 represents a control measure related to reducing aerosol transmission, highlighting the critical importance of respiratory hygiene, such as covering one's mouth when coughing or sneezing. A similar analysis for HIV indicated that the parameter ω_2 exerts the most significant influence on HIV transmission, as shown by its impact on the HIV-specific reproduction number, R_H. Biologically, ω_2 encapsulates behavioral controls, emphasizing the necessity of promoting safe sexual practices and preventing contact with infected bodily fluids to curb HIV spread. Numerical simulations supported our analytical findings. They demonstrated that in a co-infection setting, the disease with the higher basic reproduction number tends to dominate the epidemiological landscape. Furthermore, the simulations illustrated that targeted increases in the control measures ω_1 and ω_2 lead to a substantial reduction in the number of co-infection cases within the community, underscoring the effectiveness of combined, disease-specific intervention strategies.

HIV, basic reproduction number, stability analysis, numerical simulation, tuberculosis

1. A. J. Omale and B. Bolaji, “MATHEMATICAL MODEL FOR TRANSMISSION DYNAMICS OF HIV AND TUBERCULOSIS CO-INFECTION IN KOGI STATE, NIGERIA,” vol. 11, no. 5, pp. 5580–5613, 2021. [Google Scholar] [Crossref]

2. V. K. Singh, B. Kumar, Y. Tiwar, and A. Singh, “THE ERA OF HIV DISEASES: A REVIEW Structure of HIV Virus,” vol. 32, no. 3, pp. 32844–32852. [Google Scholar] [Crossref]

3. L. E. Hanna, “Pathophysiology of CD4 + T-Cell Depletion in Hiv-1 and Hiv-2 infections,” vol. 8, no. May, pp. 1–8, 2017, doi: 10.3389/fimmu.2017.00580. [Google Scholar] [Crossref]

4. H. Albrecht, “Redefining AIDS: Towards a Modification of the Current AIDS Case Definition,” no. July 1996, pp. 64–74, 1997. [Google Scholar] [Crossref]

5. A. L. Bauer, I. B. Hogue, S. Marino, and D. E. Kirschner, “The Effects of HIV–1 Infection on Latent Tuberculosis,” vol. 3, no. 7, pp. 229–266, 2008. [Google Scholar] [Crossref]

6. A. Sorathiya, A. Bracciali, and P. Liò, “BMC Bioinformatics,” vol. 11, pp. 1–11, 2010, doi: 10.1186/1471-2105-11-S1-S67. [Google Scholar] [Crossref]

7. S. Gakkhar and N. Chavda, “A dynamical model for HIV–TB co-infection,” vol. 218, no. 2012, pp. 9261–9270, 2020, doi: 10.1016/j.amc.2012.03.004. [Google Scholar] [Crossref]

8. O. Sharomi, C. N. Podder, and A. B. Gumel, “Montclair State University Digital Commons Mathematical Analysis of the Transmission Dynamics of HIV/TB Coinfection in the Presence of Treatment,” no. 2008, 2022. [Google Scholar] [Crossref]

9. S. M. A. Rahman, “Study of Infectious Diseases by Mathematical Models: Predictions and Controls,” 2016. [Google Scholar] [Crossref]

10. E. O. Omondi, R. W. Mbogo, and L. S. Luboobi, “Letters in Biomathematics Mathematical analysis of sex-structured population model of HIV infection in Kenya,” vol. 7867, 2018, doi: 10.1080/23737867.2018.1506712. [Google Scholar] [Crossref]

11. J. K. Nthiiri and G. O. Lawi, “Mathematical Model of Pneumonia and HIV/AIDS Co-Infection in the Presence of Protection,” vol. 9, no. 42, pp. 2069–2085, 2015. [Google Scholar] [Crossref]

12. N. Kaur and M. Ghosh, “The role of screening and treatment in the transmission dynamics of HIV/AIDS and tuberculosis co-infection: a mathematical study,” pp. 139–166, 2014, doi: 10.1007/s10867-014-9342-3. [Google Scholar] [Crossref]

13. A. Mallela, S. Lenhart, and N. K. Vaidya, “HIV-TB Co-infection Treatment: Modeling and Optimal Control Theory Perspectives,” J. Comput. Appl. Math., 2016, doi: 10.1016/j.cam.2016.02.051. [Google Scholar] [Crossref]

14. H. Huo and R. Chen, “Stability of an HIV/AIDS Treatment Model with Different Stages,” vol. 2015, 2015, doi: 10.1155/2015/630503. [Google Scholar] [Crossref]

15. P. Trodden, “A One-step Approach to Computing a Polytopic Robust Positively Invariant Set,” vol. 9286, no. c, pp. 1–6, 2016, doi: 10.1109/TAC.2016.2541300. [Google Scholar] [Crossref]

16. P. Taylor, A. Teel, R. Kadiyala, P. Kokotovic, S. Sastry, and C. Hall, “Indirect techniques for adaptive input-output linearization of non-linear systems,” no. November 2014, pp. 37–41, 2007, doi: 10.1080/00207179108953617. [Google Scholar] [Crossref]

17. S. Masood, M. A. Al-Bashrawi, M. A. Khan, and Y. K. Dwivedi, “From data to diagnosis: a systematic review on AI-driven approaches to diabetes prediction,” Artif. Intell. Rev., 2026. [Google Scholar] [Crossref]

18. A. Saltelli, P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and S. Tarantola, “Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index,” Comput. Phys. Commun., vol. 181, no. 2, pp. 259–270, 2010, doi: 10.1016/j.cpc.2009.09.018. [Google Scholar] [Crossref]

19. T. T. Yusuf, “On Global Stability of Disease-Free Equilibrium in Epidemiological Models,” no. 3, pp. 37–42. [Google Scholar] [Crossref]

20. C. B. Series, Optimal Control Applied to Biological Models. [Google Scholar] [Crossref]

21. R. Naresh, D. Sharma, and A. Tripathi, “Modelling the effect of tuberculosis on the spread of HIV infection in a population with density-dependent birth and death rate,” Math. Comput. Model., vol. 50, no. 7–8, pp. 1154–1166, 2009, doi: 10.1016/j.mcm.2009.05.033. [Google Scholar] [Crossref]

22. A. Ebrahimzadeh and A. Jajarmi, “Fractional Optimal Control of Anthroponotic Cutaneous Leishmaniasis with Behavioral and Epidemiological Extensions,” 2025. [Google Scholar] [Crossref]

23. O. J. Peter, A. A. Folasade, M. O. Ajisope, F. O. Ajibade, A. I. Abioye, and F. A. Oguntolu, “Global Stability Analysis of Typhoid Fever Model,” pp. 20–31, 2020. [Google Scholar] [Crossref]

24. Z. Ma, J. Liu, and J. Li, “Stability analysis for differential infectivity epidemic models,” vol. 4, pp. 841–856, 2003, doi: 10.1016/S1468-1218(03)00019-1. [Google Scholar] [Crossref]

25. M. A. Safi and S. M. Garba, “Global Stability Analysis of SEIR Model with Holling Type II Incidence Function,” vol. 2012, 2012, doi: 10.1155/2012/826052. [Google Scholar] [Crossref]

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