# American Institute of Mathematical Sciences

• Previous Article
Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach
• MBE Home
• This Issue
• Next Article
A singularly perturbed HIV model with treatment and antigenic variation
2015, 12(1): 23-40. doi: 10.3934/mbe.2015.12.23

## A double age-structured model of the co-infection of tuberculosis and HIV

 1 Mathematics Department, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, United States

Received  July 2014 Revised  October 2014 Published  December 2014

After decades on the decline, it is believed that the emergence of HIV is responsible for an increase in the tuberculosis prevalence. The leading infectious disease in the world, tuberculosis is also the leading cause of death among HIV-positive individuals. Each disease progresses through several stages. The current model suggests modeling these stages through a time-since-infection tracking transmission rate function, which, when considering co-infection, introduces a double-age structure in the PDE system. The basic and invasion reproduction numbers for each disease are calculated and the basic ones established as threshold for the disease progression. Numerical results confirm the calculations and a simple treatment scenario suggests the importance of time-since-infection when introducing disease control and treatment in the model.
Citation: 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
##### References:
 [1] A. L. Bauer, I. B. Hogue, S. Marino and D. Kirschner, The effects of hiv-1 infection on latent tuberculosis, Math. Model. Nat. Phenom., 3 (2008), 229-266. doi: 10.1051/mmnp:2008051. [2] C. Currie, B. Williams, R. Cheng and C. Dye, Tuberculosis epidemics driven by hiv: Is prevention better than cure?, AIDS, 17 (2003), 2501-2508. doi: 10.1097/00002030-200311210-00013. [3] J. L. Flynn and J. Chan, Tuberculosis: Latency and reactivation, Infection and Immunity, 69 (2001), 4195-4201, URL http://iai.asm.org/content/69/7/4195.short. [4] T. D. Hollingsworth, R. M. Anderson and C. Fraser, Hiv-1 transmission, by stage of infection, Journal of Infectious Diseases, 198 (2008), 687-693. doi: 10.1086/590501. [5] D. Kirschner, Dynamics of co-infection with m. tuberculosis and hiv-1, Theor Popul Biol., 55 (1999), 94-109. [6] S. D. Lawn, A. D. Kerkhoff, M. Vogt and R. Wood, Hiv-associated tuberculosis: Relationship between disease severity and the sensitivity of new sputum-based and urine-based diagnostic assays, BMC Medicine, 11 (2013), p231. doi: 10.1186/1741-7015-11-231. [7] P. Nunn, A. Reid and K. M. De Cock, Tuberculosis and hiv infection: The global setting, Journal of Infectious Diseases, 196 (2007), S5-S14. doi: 10.1086/518660. [8] , W. H. Organization,, Global tuberculosis report, (2013). [9] A. Pawlowski, M. Jansson, M. Sköld, M. Rottenberg and G. Källenius, Tuberculosis and hiv co-infection, PLoS Pathog, 8 (2012), e1002464. doi: 10.1371/journal.ppat.1002464. [10] L. Roeger, Z. Feng and C. Castillo-Chavez, Modeling tb and hiv co-infections, Math Biosci Eng, 6 (2009), 815-837. doi: 10.3934/mbe.2009.6.815. [11] O. Sharomi, C. Podder, A. Gumel and B. Song, Mathematical analysis of the transmission dynamics of hiv/tb coinfection in the presence of treatment, Math Biosci Eng., 5 (2008), 145-174. doi: 10.3934/mbe.2008.5.145. [12] X. Wang, J. Yang and F. Zhang, Dynamic of a tb-hiv coinfection epidemic model with latent age, Journal of Applied Mathematics, 2013 (2013), Art. ID 429567, 13 pp. [13] G. F. Webb, Theory of Nonlinear Age-Dependant Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics Series 89. Marcel Dekker Inc., 1985.

show all references

##### References:
 [1] A. L. Bauer, I. B. Hogue, S. Marino and D. Kirschner, The effects of hiv-1 infection on latent tuberculosis, Math. Model. Nat. Phenom., 3 (2008), 229-266. doi: 10.1051/mmnp:2008051. [2] C. Currie, B. Williams, R. Cheng and C. Dye, Tuberculosis epidemics driven by hiv: Is prevention better than cure?, AIDS, 17 (2003), 2501-2508. doi: 10.1097/00002030-200311210-00013. [3] J. L. Flynn and J. Chan, Tuberculosis: Latency and reactivation, Infection and Immunity, 69 (2001), 4195-4201, URL http://iai.asm.org/content/69/7/4195.short. [4] T. D. Hollingsworth, R. M. Anderson and C. Fraser, Hiv-1 transmission, by stage of infection, Journal of Infectious Diseases, 198 (2008), 687-693. doi: 10.1086/590501. [5] D. Kirschner, Dynamics of co-infection with m. tuberculosis and hiv-1, Theor Popul Biol., 55 (1999), 94-109. [6] S. D. Lawn, A. D. Kerkhoff, M. Vogt and R. Wood, Hiv-associated tuberculosis: Relationship between disease severity and the sensitivity of new sputum-based and urine-based diagnostic assays, BMC Medicine, 11 (2013), p231. doi: 10.1186/1741-7015-11-231. [7] P. Nunn, A. Reid and K. M. De Cock, Tuberculosis and hiv infection: The global setting, Journal of Infectious Diseases, 196 (2007), S5-S14. doi: 10.1086/518660. [8] , W. H. Organization,, Global tuberculosis report, (2013). [9] A. Pawlowski, M. Jansson, M. Sköld, M. Rottenberg and G. Källenius, Tuberculosis and hiv co-infection, PLoS Pathog, 8 (2012), e1002464. doi: 10.1371/journal.ppat.1002464. [10] L. Roeger, Z. Feng and C. Castillo-Chavez, Modeling tb and hiv co-infections, Math Biosci Eng, 6 (2009), 815-837. doi: 10.3934/mbe.2009.6.815. [11] O. Sharomi, C. Podder, A. Gumel and B. Song, Mathematical analysis of the transmission dynamics of hiv/tb coinfection in the presence of treatment, Math Biosci Eng., 5 (2008), 145-174. doi: 10.3934/mbe.2008.5.145. [12] X. Wang, J. Yang and F. Zhang, Dynamic of a tb-hiv coinfection epidemic model with latent age, Journal of Applied Mathematics, 2013 (2013), Art. ID 429567, 13 pp. [13] G. F. Webb, Theory of Nonlinear Age-Dependant Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics Series 89. Marcel Dekker Inc., 1985.
 [1] 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 [2] 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 [3] 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 [4] Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166 [5] Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170 [6] 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 [7] Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455 [8] 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 [9] Svend Christensen, Preben Klarskov Hansen, Guozheng Qi, Jihuai Wang. The mathematical method of studying the reproduction structure of weeds and its application to Bromus sterilis. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 777-788. doi: 10.3934/dcdsb.2004.4.777 [10] 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 [11] Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure and Applied Analysis, 2021, 20 (2) : 755-762. doi: 10.3934/cpaa.2020288 [12] 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 [13] Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377 [14] Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457 [15] Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565 [16] Gabriela Marinoschi. Identification of transmission rates and reproduction number in a SARS-CoV-2 epidemic model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022128 [17] 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 [18] Chang Gong, Jennifer J. Linderman, Denise Kirschner. A population model capturing dynamics of tuberculosis granulomas predicts host infection outcomes. Mathematical Biosciences & Engineering, 2015, 12 (3) : 625-642. doi: 10.3934/mbe.2015.12.625 [19] Suxia Zhang, Xiaxia Xu. A mathematical model for hepatitis B with infection-age structure. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1329-1346. doi: 10.3934/dcdsb.2016.21.1329 [20] Luca Gerardo-Giorda, Pierre Magal, Shigui Ruan, Ousmane Seydi, Glenn Webb. Preface: Population dynamics in epidemiology and ecology. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : i-ii. doi: 10.3934/dcdsb.2020125

2018 Impact Factor: 1.313

## Metrics

• HTML views (0)
• Cited by (5)

## Other articlesby authors

• on AIMS
• on Google Scholar