## Journal of Applied Mathematics

### Dynamic of a TB-HIV Coinfection Epidemic Model with Latent Age

#### Abstract

A coepidemic arises when the spread of one infectious disease stimulates the spread of another infectious disease. Recently, this has happened with human immunodeficiency virus (HIV) and tuberculosis (TB). The density of individuals infected with latent tuberculosis is structured by age since latency. The host population is divided into five subclasses of susceptibles, latent TB, active TB (without HIV), HIV infectives (without TB), and coinfection class (infected by both TB and HIV). The model exhibits three boundary equilibria, namely, disease free equilibrium, TB dominated equilibrium, and HIV dominated equilibrium. We discuss the local or global stabilities of boundary equilibria. We prove the persistence of our model. Our simple model of two synergistic infectious disease epidemics illustrates the importance of including the effects of each disease on the transmission and progression of the other disease. We simulate the dynamic behaviors of our model and give medicine explanations.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 429567, 13 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394807906

Digital Object Identifier
doi:10.1155/2013/429567

Mathematical Reviews number (MathSciNet)
MR3039712

Zentralblatt MATH identifier
1266.92058

#### Citation

Wang, Xiaoyan; Yang, Junyuan; Zhang, Fengqin. Dynamic of a TB-HIV Coinfection Epidemic Model with Latent Age. J. Appl. Math. 2013 (2013), Article ID 429567, 13 pages. doi:10.1155/2013/429567. https://projecteuclid.org/euclid.jam/1394807906

#### References

• National Institute of Allergy and Infectious Diseases (NIAID), HIV Infection and AIDS: An Overview, United States National Institutes of Health, Bethesda, Md, USA, 2006.
• G. D. Sanders, A. M. Bayoumi, V. Sundaram et al., “Cost-effectiveness of screening for HIV in the era of highly active antiretroviral therapy,” New England Journal of Medicine, vol. 352, no. 6, pp. 570–585, 2005.
• World Health Organization (WHO), Antiretroviral Therapy (ART), WHO, Geneva, The Switzerland, 2006.
• D. Kirschner, “Dynamics of co-infection with M. tuberculosis and HIV-1,” Theoretical Population Biology, vol. 55, no. 1, pp. 94–109, 1999.
• S. M. Moghadas and A. B. Gumel, “An epidemic model for the transmission dynamics of hiv and another pathogen,” ANZIAM Journal, vol. 45, no. 2, pp. 181–193, 2003.
• B. G. Williams and C. Dye, “Antiretroviral drugs for tuberculosis control in the era of HIV/AIDS,” Science, vol. 301, no. 5639, pp. 1535–1537, 2003.
• T. C. Porco, P. M. Small, and S. M. Blower, “Amplification dynamics: predicting the effect of HIV on tuberculosis outbreaks,” Journal of Acquired Immune Deficiency Syndromes, vol. 28, no. 5, pp. 437–444, 2001.
• R. W. West and J. R. Thompson, “Modeling the impact of HIV on the spread of tuberculosis in the United States,” Mathematical Biosciences, vol. 143, no. 1, pp. 35–60, 1997.
• R. Naresh and A. Tripathi, “Modelling and analysis of HIV-TB co-infection in a variable size P,” Mathematical Modelling and Analysis, vol. 10, no. 3, pp. 275–286, 2005.
• 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,” Mathematical and Computer Modelling, vol. 50, no. 7-8, pp. 1154–1166, 2009.
• E. F. Long, N. K. Vaidya, and M. L. Brandeau, “Controlling Co-epidemics: analysis of HIV and tuberculosis infection dynamics,” Operations Research, vol. 56, no. 6, pp. 1366–1381, 2008.
• M. Martcheva and S. S. Pilyugin, “The role of coinfection in multidisease dynamics,” SIAM Journal on Applied Mathematics, vol. 66, no. 3, pp. 843–872, 2006.
• G. J. Butler and P. Waltman, “Bifurcation from a limit cycle in a two predator-one prey ecosystem modeled on a chemostat,” Journal of Mathematical Biology, vol. 12, no. 3, pp. 295–310, 1981.
• P. Magal, C. C. McCluskey, and G. F. Webb, “Lyapunov functional and global asymptotic stability for an infection-age model,” Applicable Analysis, vol. 89, no. 7, pp. 1109–1140, 2010.
• E. M. C. D'Agata, P. Magal, and S. G. Ruan, “Asymptotic behavior in nosocomial epidemic models with antibiotic resistance,” Differential and Integral Equations, vol. 19, no. 5, pp. 573–600, 2006.
• M. Iannelli, “Mathematical theory of age-structured population dynamics,” in Applied Mathematics Monographs 7, comitato Nazionale per le Scienze Matematiche, Consiglio Nazionale delle Ricerche (C.N.R.), Pisa, Italy, 1995.
• H. R. Thieme, “Persistence under relaxed point-dissipativity (with application to an endemic model),” SIAM Journal on Mathematical Analysis, vol. 24, no. 2, pp. 407–435, 1993.
• J. K. Hale and P. Waltman, “Persistence in finite dimensional systems,” SIAM Journal on Mathematical Analysis, vol. 20, no. 2, pp. 388–395, 1989.
• Centers for Disease Control and Prevention (CDC), Coinfection With HIV and Hepatitis C Virus, CDC, Atlanta, Ga, USA, 2006.