• Previous Article
    Insights from epidemiological game theory into gender-specific vaccination against rubella
  • MBE Home
  • This Issue
  • Next Article
    HIV/AIDS epidemic in India and predicting the impact of the national response: Mathematical modeling and analysis
2009, 6(4): 815-837. doi: 10.3934/mbe.2009.6.815

Modeling TB and HIV co-infections


Department of Mathematics and Statistics, Box 41042, Texas Tech University, Lubbock, TX 79409-1042, United States


Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395


Department of Mathematics and Statistics, Arizona State University, P.O. Box 871804, Tempe, AZ 85287-1804

Received  October 2007 Revised  June 2009 Published  September 2009

Tuberculosis (TB) is the leading cause of death among individuals infected with the human immunodeficiency virus (HIV). The study of the joint dynamics of HIV and TB present formidable mathematical challenges due to the fact that the models of transmission are quite distinct. Furthermore, although there is overlap in the populations at risk of HIV and TB infections, the magnitude of the proportion of individuals at risk for both diseases is not known. Here, we consider a highly simplified deterministic model that incorporates the joint dynamics of TB and HIV, a model that is quite hard to analyze. We compute independent reproductive numbers for TB ($\R_1$) and HIV ($\R_2$) and the overall reproductive number for the system, $\R =\max \{\R_1, \R_2\}$. The focus is naturally (given the highly simplified nature of the framework) on the qualitative analysis of this model. We find that if $\R <1$ then the disease-free equilibrium is locally asymptotically stable. The TB-only equilibrium $E_T$ is locally asymptotically stable if $\R_1>1$ and $\R_2<1$. However, the symmetric condition, $\R_1<1$ and $\R_2>1$, does not necessarily guarantee the stability of the HIV-only equilibrium $E_H$, and it is possible that TB can coexist with HIV when $\R_2>1$. In other words, in the case when $\R_1<1$ and $\R_2>1$ (or when $\R_1>1$ and $\R_2>1$), we are able to find a stable HIV/TB coexistence equilibrium. Moreover, we show that the prevalence level of TB increases with $\R_2>1$ under certain conditions. Through simulations, we find that i) the increased progression rate from latent to active TB in co-infected individuals may play a significant role in the rising prevalence of TB; and ii) the increased progression rates from HIV to AIDS have not only increased the prevalence level of HIV while decreasing TB prevalence, but also generated damped oscillations in the system.
Citation: Lih-Ing W. Roeger, Z. Feng, Carlos Castillo-Chávez. Modeling TB and HIV co-infections. Mathematical Biosciences & Engineering, 2009, 6 (4) : 815-837. doi: 10.3934/mbe.2009.6.815

Georgi Kapitanov. A double age-structured model of the co-infection of tuberculosis and HIV. Mathematical Biosciences & Engineering, 2015, 12 (1) : 23-40. doi: 10.3934/mbe.2015.12.23


Zhong-Kai Guo, Hai-Feng Huo, Hong Xiang. Analysis of an age-structured model for HIV-TB co-infection. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 199-228. doi: 10.3934/dcdsb.2021037


Zindoga Mukandavire, Abba B. Gumel, Winston Garira, Jean Michel Tchuenche. Mathematical analysis of a model for HIV-malaria co-infection. Mathematical Biosciences & Engineering, 2009, 6 (2) : 333-362. doi: 10.3934/mbe.2009.6.333


Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595


Xiaotian Wu, Daozhou Gao, Zilong Song, Jianhong Wu. Modelling Trypanosoma cruzi-Trypanosoma rangeli co-infection and pathogenic effect on Chagas disease spread. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022110


A. M. Elaiw, N. H. AlShamrani. Global stability of HIV/HTLV co-infection model with CTL-mediated immunity. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1725-1764. doi: 10.3934/dcdsb.2021108


Surabhi Pandey, Ezio Venturino. A TB model: Is disease eradication possible in India?. Mathematical Biosciences & Engineering, 2018, 15 (1) : 233-254. doi: 10.3934/mbe.2018010


Expeditho Mtisi, Herieth Rwezaura, Jean Michel Tchuenche. A mathematical analysis of malaria and tuberculosis co-dynamics. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 827-864. doi: 10.3934/dcdsb.2009.12.827


Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Ebraheem O. Alzahrani. A fractional model for the dynamics of tuberculosis (TB) using Atangana-Baleanu derivative. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 937-956. doi: 10.3934/dcdss.2020055


Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639


Oluwaseun Sharomi, Chandra N. Podder, Abba B. Gumel, Baojun Song. Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment. Mathematical Biosciences & Engineering, 2008, 5 (1) : 145-174. doi: 10.3934/mbe.2008.5.145


Miguel Atencia, Esther García-Garaluz, Gonzalo Joya. The ratio of hidden HIV infection in Cuba. Mathematical Biosciences & Engineering, 2013, 10 (4) : 959-977. doi: 10.3934/mbe.2013.10.959


Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239


Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37


Suman Ganguli, David Gammack, Denise E. Kirschner. A Metapopulation Model Of Granuloma Formation In The Lung During Infection With Mycobacterium Tuberculosis. Mathematical Biosciences & Engineering, 2005, 2 (3) : 535-560. doi: 10.3934/mbe.2005.2.535


Yijun Lou, Li Liu, Daozhou Gao. Modeling co-infection of Ixodes tick-borne pathogens. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1301-1316. doi: 10.3934/mbe.2017067


Kazeem Oare Okosun, Robert Smith?. Optimal control analysis of malaria-schistosomiasis co-infection dynamics. Mathematical Biosciences & Engineering, 2017, 14 (2) : 377-405. doi: 10.3934/mbe.2017024


Salihu Sabiu Musa, Nafiu Hussaini, Shi Zhao, He Daihai. Dynamical analysis of chikungunya and dengue co-infection model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1907-1933. doi: 10.3934/dcdsb.2020009


Danyun He, Qian Wang, Wing-Cheong Lo. Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3387-3413. doi: 10.3934/dcdsb.2018239


Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Zakia Hammouch, Dumitru Baleanu. A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 975-993. doi: 10.3934/dcdss.2020057

2018 Impact Factor: 1.313


  • PDF downloads (192)
  • HTML views (0)
  • Cited by (56)

[Back to Top]