Causes of Backward Bifurcation in a Tuberculosis-Schistosomiasis Co-infection Dynamics

  • Ignatius Ako Department of Mathematics, University of Benin, Benin City, Nigeria
  • Owin Olowu Department of Mathematics, University of Benin, Benin City, Nigeria
Keywords: tuberculosis, schistosomiasis, co-infection, backward bifurcation


To obtain a thorough understanding of the influence of schistosomiasis infections on the transmission dynamics of tuberculosis, a deterministic mathematical model for the transmission dynamics of tuberculosis (TB) co-infection with schistosomiasis is created and examined. The aim of the research is to examine the reasons behind the backward bifurcation in the co-infection dynamics of tuberculosis and schistosomiasis. The backward bifurcation phenomena can be caused by the following parameters, according to the model's analysis (when the associated reproduction number is less than one), other than the well established route of exogeneous re-infection of latently infected TB individuals, the relative rates at which humans with latent schistosomiasis ($\eta_1$) and active schistosomiasis ($\eta_2$) are infected with TB, respectively, the lowered rate of reinfection with schistosomiasis ($\psi$), the fraction of individuals who experience fast progression to active TB ($p$), the adjustment parameter which accounts for the increased probability of infectiousness of humans with active TB and latent schistosomiasis ($\Pi_1$), the treatment rate of people infected with active TB exposed to schistosomiasis ($\zeta_{T1}$) and the rate of progression to active TB and exposed to schistosomiasis to active TB and active schistosomiasis ($\sigma$).


Ako, I. I., Akhaze, R. U., & Olowu, O. O. (2021). The impact of reduced re-infection on schistosomiasis transmission dynamics at population level: a theoretical study. Journal of the Nigerian Association of Mathematical Physics, 59, 61-74.

Andrews, J. A., Noubary, F., Walensky, R. P., Cerda, R., Losina, E., & Horsburgh, R. (2012). Risk of progression to active tuberculosis following reinfection with mycobacterium tuberculosis. Clinical Infectious Diseases, 54(6), 784-791.

Athithan, S., & Ghosh, M. (2013). Mathematical modelling of TB with the effects of case detection and treatment. International Journal of Dynamics and Control, 1, 223-230.

Barbour, A.D. (1982). Schistosomiasis. In R.M. Anderson (Ed.), Population dynamics of infectious diseases (pp. 180-208). Chapman and Hall.

Bhunu, C. P. (2011). Mathematical analysis of a three-strain tuberculosis transmission model. Applied Mathematical Modelling, 35(35), 4647-4660.

Bhunu, C. P., Garira, W., & Magombedze, G. (2009). Mathematical analysis of a two-strain HIV/AIDS model with antiretroviral treatment. Acta Biotheoretica, 57(3), 361-381.

Blower, S. M., McLean, A. R., Porco, T. C., Small, P. M., Hopwell, P. C., Sanchez, M. A., & Moss, A. R. (1995). The intrinsic transmission dynamics of tuberculosis epidemics. Nature Medicine, 1, 815-821.

Castillo-Chavez, C., Feng, Z., & Xu, D. (2008). A schistosomiasis model with mating structure and time delay. Mathematical Biosciences, 211, 333-341.

Castillo-Chavez, C., & Song, B. (2004). Dynamical models of tuberculosis and their applications. Mathematical Biosciences and Engineering, 1(2), 361-404.

Chatterjee, S., & Nutman, T. B. (2015). Helminth-induced immune regulation: implications for immune responses to tuberculosis. PLoS Pathogens, 11(1), e1004582.

Chen, Z., Zou, L., Shen, D., Zhang, W., & Ruan, S. (2010). Mathematical modelling and control of schistosomiasis in Hubei Province, China. Acta Tropica, 115, 119-125.

Chitsulo, L., Engels, D., Montresor, A., & Savioli, L. (2000). The global status of schistosomiasis and its control. Acta Tropica, 77, 41-51.

Chiyaka, E., & Garira, W. (2009). Mathematical analysis of the transmission dynamics of schistosomiasis in the human-snail hosts. Journal of Biological Systems, 17, 397-423.

Cohen, J.E. (1977). Mathematical models of schistosomiasis. Annual Review of Ecology and Systematics, 8, 209-233.

Countrymeters. (2017). Nigeria Population. Retrieved from (accessed on July 19, 2018).

Desaleng, D., & Koya, P. R. (2016). Modeling and analysis of multi-drug-resistant tuberculosis in densely populated areas. American Journal of Applied Mathematics, 4(1), 1-10.

Diaby, M. A., & Iggidr, A. (2016). A mathematical analysis of a model with mating structure. Proceedings of CARI, 246, 402-411.

Diaby, M. A., Iggidr, A., Sy, M., & Sene, A. (2014). Global analysis of a schistosomiasis infection model with biological control. Applied Mathematics and Computation, 246, 731-742.

Feng, Z., Castillo-Chavez, C., & Capurro, A. F. (2000). A model for tuberculosis with exogenous reinfection. Theoretical Population Biology, 57, 235-247.

Feng, Z., Curtis, J., & Minchella, D. J. (2001). The influence of drug treatment on the maintenance of schistosome genetic diversity. Journal of Mathematical Biology, 43, 52-68.

Feng, Z., Li, C.-C., & Milner, F. A. (2002). Schistosomiasis models with density dependence and age of infection in snail dynamics. Mathematical Biosciences, 177-178, 271-286.

Feng, Z., Eppert, A., Milner, F. A., & Minchella, D . J. (2004). Estimation of parameters governing the transmission dynamics of schistosomes. Applied Mathematics Letters, 17, 1105-1112.

Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Review, 42(4), 599-653.

Inobaya, M. T., Olveda, R. M., Chau, T. N. P., Olveda D. U., & Ross A. G. P. (2014). Prevention and control of schistosomiasis: a current perspective. Research and Reports in Tropical Medicine, 5, 65-75.

Lakshmikantham, V., Leela, S., & Martynyuk, A. A. (1991). Stability analysis of nonlinear systems. SIAM Review, 33(1), 152-154.

Li, X. X., & Zhou, X. N. (2013). Coinfection of tuberculosis and parasitic diseases in humans: a systematic review. Parasites & Vectors, 6, 79.

Macdonald, G. (1965). The dynamics of Helminth infections with special reference to schistosomes. Transactions of the Royal Society of Tropical Medicine and Hygiene, 59(5), 489-506.

Milner, F. A., & Zhao, R. (2008). A deterministic model of schistosomiasis with spatial structure. Mathematical Biosciences and Engineering, 5(3), 505-522. []

Monin, L., Griffiths, K. L., Lam, W. Y., Gopal, R., Kang, D. D., Ahmed, M., Rajamanickam, A., Cruz-Lagunas, A., Zuniga, J., Babu, S., Kolls, J. K., Mitreva, M., Rosa, B. A., Ramos-Payan, R., Morrison, T. E., Murray, P. J., Rangel-Moreno, J., Pearce, E. J., & Khader, S. A. (2015). Helminth-induced arginase-1 exacerbates lung inflammation and disease severity in tuberculosis. The Journal of Clinical Investigation, 125(12), 4699-4713.

Moualeu, D. P., Weiser, M., Ehrig, R., et al. (2015). Optimal control for tuberculosis model with undetected cases in Cameroon. Communications in Nonlinear Science and Numerical Simulation, 20, 986-1003.

Mushayabasa, S., & Bhunu, C. P. (2011). Modeling schistosomiasis and HIV/AIDS dynamics. Computational and Mathematical Methods in Medicine, 2011, Article ID 846174.

Ngarakana-Gwasira, E. T., Bhunu, C. P., Masocha, M., & Mashonjowa, E. (2016). Transmission dynamics of schistosomiasis in Zimbabwe: a mathematical and GIS approach. Communications in Nonlinear Science and Numerical Simulation, 35, 137-147.

Okosun, K. O., & Smith?, R. (2017). Optimal control analysis of malaria-schistosomiasis co-infection dynamics. Mathematical Biosciences & Engineering, 14(2), 377-405.

Okuonghae, D. (2013). A mathematical model of tuberculosis transmission with heterogeneity in disease susceptibility and progression under a treatment regime for infectious cases. Applied Mathematical Modelling, 37, 6786-6808.

Okuonghae, D. (2014). Lyapunov functions and global properties of some tuberculosis models. Journal of Applied Mathematics and Computing.

Okuonghae, D., & Aihie, V. (2008). Case detection and direct observation therapy strategy (DOTS) in Nigeria: its effect on TB dynamics. Journal of Biological Systems, 16(1), 1-31.

Okuonghae, D., & Aihie, V. U. (2010). Optimal control measures for tuberculosis mathematical models including immigration and isolation of infective. Journal of Biological Systems, 18(01), 17-54.

Okuonghae, D., & Ikhimwin, B. O. (2016). Dynamics of a mathematical model for tuberculosis with variability in susceptibility and disease progressions due to difference in awareness level. Frontiers in Microbiology, 6, 1530.

Okuonghae, D., & Korobeinikov, A. (2007). Dynamics of tuberculosis: the effect of direct observation therapy strategy (DOTS) in Nigeria. Mathematical Modelling of Natural Phenomena, 2(1), 101-113.

Okuonghae, D., & Omosigho, S.E. (2011). Analysis of a mathematical model for tuberculosis: What could be done to increase case detection. Journal of Theoretical Biology, 269, 31-45.

Olowu, O., & Ako, I. (2023). Computational investigation of the impact of availability and efficacy of control on the transmission dynamics schistosomiasis. International Journal of Mathematical Trends and Technology, 69(8), 1-9.

Olowu, O., Ako, I. I., & Akhaze, R. I. (2021). Theoretical study of a two patch metapopulation schistosomiasis model. Transactions of the Nigerian Association Mathematical Physics, 14, 53-68.

Olowu, O., Ako, I. I., & Akhaze, R. I. (2021b). On the analysis of a two patch schistosomiasis model. Transactions of the Nigerian Association Mathematical Physics, 14, 69-78.

Omame, A., Umana, R.A., Okuonghae, D., & Inyama, S.C. (2018). Mathematical analysis of a two-sex human papillomavirus (HPV) model. Int. J. Biol., 11(7).

Osada, Y., & Kanazawa, T. (2011). Schistosome: Its benefit and harm in patients suffering from concomitant diseases. Journal of Biomedicine and Biotechnology, 2011.

Pangaribuan, R.M., et al. (2016). Threshold dynamic for quasi-endemic equilibrium from co-epidemic HIV-TB model with re-infection TB in heterosexual population. International Conference on Mathematics, Engineering and Industrial Applications, 030041.

Porco, T.C., & Blower, S.M. (1998). Quantifying the intrinsic transmission dynamics of tuberculosis. Theor Pop Biol, 54, 117-132.

Potian, J.A., Rafi, W., Bhatt, K., McBride, A., Gause, W.C., & Salgame, P. (2011). Preexisting helminth infection induces inhibition of innate pulmonary anti-tuberculosis defense by engaging the IL-4 receptor pathway. J. Exp. Med., 208(9), 1863-1874.

Qi, L., & Cui, J. (2013). A schistosomiasis model with mating structure. Abstract and Applied Analysis, 2013.

Qi, L., et al. (2014). Mathematical model of schistosomiasis under flood in Anhui Province. Abstract and Applied Analysis, 2014.

Qi, L., et al. (2018). Schistosomiasis model and its control in Anhui Province. Bulletin of Mathematical Biology, 80, 2435-2451.

Sharomi, O., & Malik, T. (2017). A model to assess the effect of vaccine compliance on human papillomavirus infection and cervical cancer. Appl. Math. Model., 47, 528-550.

Sharomi, O., Podder, C.N., & Gumel, A.B. (2008). Mathematical analysis of the transmission dynamics of HIV/TB co-infection in the presence of treatment. Math. Biosci. Eng., 5(1), 145-174.

Simon, G.G. (2016). Impacts of neglected tropical disease on incidence and progression of HIV/AIDS, tuberculosis, and malaria: scientific links. International Journal of Infectious Diseases, 42, 54-57.

UNAIDS-WHO (2004). Epidemiological fact sheet. Retrieved from

van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180, 29-48.

Waaler, H., Geser, A., & Andersen, S. (1962). The use of mathematical models in the study of the epidemiology of tuberculosis. Am. J. Public Health Nations Health., 52(6), 1002-1013.

WHO (World Health Organization). (2008). Global Tuberculosis Control-Surveillance, Planning, Financing. Geneva, Switzerland: WHO Press.

WHO (World Health Organization). (2015). Global tuberculosis report. Geneva, Switzerland: WHO Press.

WHO (World Health Organization). (2016). Global tuberculosis report 2016. Geneva, Switzerland: WHO Press.

WHO (World Health Organization). (2017). Schistosomiasis Factsheet 2017. Geneva, Switzerland: WHO Press.

WHO (World Health Organization). (2018). Tuberculosis Factsheet 2018. Geneva, Switzerland: WHO Press.

WHO (World Health Organization). (2019). Schistosomiasis Factsheet 2019. Geneva, Switzerland: WHO Press.

Woolhouse, M.E.J. (1991). On the application of mathematical models of schistosome transmission dynamics I: natural transmission. Acta Trop., 49, 241.

Yang, H.M. (2003). Comparison between schistosomiasis transmission modeling considering acquired immunity and age-structured contact pattern with infested water. Mathematical Biosciences, 184, 1-26.

Zhao, R., & Milner, F.A. (2008). A mathematical model of schistosoma mansoni in Biomphalaria glabrata with control strategies. Bulletin of Mathematical Biology, 70(7), 1886-1905.

Zou, L., & Ruan, S. (2015). Schistosomiasis transmission and control in China. Acta Tropica, 143, 51-57.

How to Cite
Ako, I., & Olowu, O. (2024). Causes of Backward Bifurcation in a Tuberculosis-Schistosomiasis Co-infection Dynamics. Earthline Journal of Mathematical Sciences, 14(4), 655-695.