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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

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.

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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.

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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.

[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.

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