# Threshold Dynamics of an HIV-TB Co-infection Model with Multiple Time Delays

## Authors

• M. Pitchaimani Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai- 600 005, Tamil Nadu, India.
• Saranya Devi A Ramanujan Institute for Advanced study in Mathematics, University of Madras https://orcid.org/0000-0002-1551-5088

## Keywords:

Bifurcation, Co-infection, Delay Differential equations, HIV- TB, Stability.

## Abstract

In this article, a mathematical model to study the dynamics of
HIV-TB co-infection with two time delays is proposed and analyzed.
We compute the basic reproduction number for each disease (HIV and
TB) which acts as a threshold parameters. The disease dies out when
the basic reproduction number of both diseases are less than unity
and persists when the basic reproduction number of atleast one of the
disease is greater than unity. A numerical study on the model is also
performed to investigate the influence of certain key parameters on the
spread of the disease. Mathematical analysis of our model shows that
switching co-infection (HIV and TB) to single infection (HIV) can be
achieved by imposing treatment for both the disease simultaneously
as TB eradication is made possible with effective treatment.

## References

Amin Jajarmi and Dumitru Baleanu, A new fractional analysis on the interaction of HIV with CD4+ T-cells, Chaos, Solitons and Fractals, 133 (2018) 221–229.

R. M. Ana, Carvalho and Carla M. A. Pinto, A coinfection model for HIV and HCV, BioSystems, 124, (2014) 46–60.

G. Bolarin and I. U. Omatola, A Mathematial Analysis of HIV/TB Co-infection Model, Journal of Applied Mathematics, 6, (2016) 65–72.

Threshold Dynamics of an HIV-TB Co-infection Model 31 T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations, J. Math. Anal. Appl, 334, (2007) 1130–1145.

Carla M.A. Pinto, Ana R. M. Carvalho, New findings on the dynamics of HIV and TB coinfection models, Applied Mathematics and Computation, 242, (2014) 36–46.

Centers for Disease Control and Prevention, About HIV/AIDS, 2019. Available from : www.cdc.gov.

Centers for Disease Control and Prevention, The TB and Latent TB Infection Reports, 2014. Available from: www.cdc.gov.

Constantinos I. Siettosa and Lucia Russo, Mathematical modeling of infectious disease dynamics, Virulence, 4, (2013) 295–306.

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computa- tion of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol, 28, (1990) 365–382.

Emily C. Griffiths, Amy B. Pedersen, Andy Fenton and Owen L. Petchey, The nature and consequences of coinfection in humans, Journal of Infection, 63, (2011) 200–206.

H. I. Freedman , V. S. H. Rao, The trade-off between mutual interference and time lags in predactor-prey systems, Bull. Math. Biol, 45, (1983) 991–1004.

K. Gopalasamy, B.G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl, 139 (1989) 110–122.

Grace Gakii Muthuri and David M. Malonza, Mathematical Modeling of TB-HIV CoInfection, Case study of Tigania West Sub County, Kenya, Journal of Advances in Mathematics and Computer Science, 27, (2018) 1–18.

J. K. Hale, S. M. V. Lunel, Introduction to Functional Differential Equations, 3rd edition, Springer-Verlag, 1993.

J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproductive ratio, J. R. Soc. Interface, 2, (2005) 281–293.

Jiazhe Lin, Rui Xu and Xiaohong Tian, Threshold dynamics of an HIV-1 virus model with both virus-to-cell and cell-to-cell transmissions, intracellular delay and humoral immunity, Applied mathematics and computation, 315, (2017) 516–530.

H. M. John, W. H. Russell, Complex Analysis for Mathematics and Engineering, 3rd edition, Jones and Bartlett Publishers, Canada, 1997.

P. Krishnapriya and M. Pitchaimani, Analysis of time delay in viral infection model with immune impairment, J. Appl. Math. Comput., 55, (2017) 421–453.

P. Krishnapriya, M. Pitchaimani and Tarynn M. Witten, Mathematical analysis of an in- fluenza A epidemic model with discrete delay, Journal of computational and Applied Math- ematics, 324, (2017) 155–172.

V. Lusiana, P.S. Putra, N. Nuraini, and E. Soewono, Mathematial modeling of transmission co-infection tuberculosis in HIV community, AIP Conference Proceedings, (2017).

C. Monica and M. Pitchaimani, Analysis of stability and Hopf bifurcation for HIV-1 dy- namics with PI and three intacelluar delays, Nonlinear Analysis:Real World Applications, 27, (2016) 55–69.

J. D. Murray, Mathematical Biology, 1stedition, Springer, NewYork, 1989.

R. Naresh and A. Tripathi, Modelling and analysis of HIV-TB Co-infection in a variable size population, Mathematical Modelling and Analysis, 10, (2010) 275–286.

Nita H. Shah and Jyoti Gupta, Modelling of HIV-TB Coinfection Transmission Dynamics, American Journal of Epidemiology and Infectious Disease, 2, (2014) 1–7.

Ram Naresh, Dileep Sharma and Agraj Tripathi, Modelling the effect of tuberculosis on the spread of HIV infection in a population with density-dependent birth and death rate, Mathematical and computer Modelling, 50, (2009) 1154–1166.

Sunitha Gakkhar and Nareshkumar Chavda, A dynamical model for HIV-TB co-infection, Applied Mathematics and Computation, 218, (2012) 9261–9270.

Tsuyoshi Kajiwara, Toru Sasaki and Yasuhiro Takeuchi, Construction of Lyapunov func- tionals for delay differential equations in virology and epidemiology, Nonlinear Analysis: Real World Applications, 13, (2012) 1802–1826.

World Health Organization. The HIV-TB Reports, 2013. Available from : www.cdc.gov.

Yang Kuang, Delay Differential Equations with Applications in Population Dynamics, 1st edition, Academic Press, New York, 1993.

2022-08-01

## How to Cite

Pitchaimani, M., & A, S. D. (2022). Threshold Dynamics of an HIV-TB Co-infection Model with Multiple Time Delays. Tamkang Journal of Mathematics, 53(3), 201–228. https://doi.org/10.5556/j.tkjm.53.2022.3295

Papers