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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

Georgi Kapitanov 1,

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.
Keywords: partial differential equations, tuberculosis, basic reproduction number, co-infection, invasion reproduction number, age-structure., HIV, population dynamics, mathematical epidemiology.
Mathematics Subject Classification: Primary: 92B05, 92D30; Secondary: 92D2.
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.  Google Scholar

[2]

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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.  Google Scholar

[5]

D. Kirschner, Dynamics of co-infection with m. tuberculosis and hiv-1, Theor Popul Biol., 55 (1999), 94-109. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

, W. H. Organization,, Global tuberculosis report, (2013).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[5]

D. Kirschner, Dynamics of co-infection with m. tuberculosis and hiv-1, Theor Popul Biol., 55 (1999), 94-109. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

, W. H. Organization,, Global tuberculosis report, (2013).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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