Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 129–162

# Analysis of a mathematical model for tuberculosis with diagnosis

• A. O. Egonmwan
• D. Okuonghae
Original Research

## Abstract

This work presents a new mathematical model that investigates the impact of diagnosis and treatment of both latent tuberculosis infections and active cases on the transmission dynamics of the disease in a population. Mathematical analysis reveal that the model undergoes the phenomenon of backward bifurcation where a stable disease-free equilibrium co-exist with a stable endemic (positive) equilibrium when the associated reproduction number is less than unity. It is shown that this phenomenon does not exist in the absence of exogenous re-infection. In the absence of exogenous re-infection, the disease-free solution of the model is shown to be globally asymptotically stable when the associated reproduction number is less than unity. It is further shown that a special case of the model has a unique endemic equilibrium point, which is globally asymptotically stable when the associated reproduction number exceeds unity. Uncertainty and sensitivity analysis is carried out to identify key parameters that have the greatest influence on the transmission dynamics of TB in the population using the reproduction number of the model, incidence of the disease and the total number of infected individuals in the various infective classes as output responses. The analysis shows that the top three parameters of the model that have the most influence on the reproduction number of the model are the transmission rate, the fraction of fast disease progression and the rate of detection of active TB cases, with other key parameters influencing the outcomes of the other output responses. Numerical simulations of the model show that the treatment rates for latent and active TB cases significantly determines the impact of the fraction of new latent TB cases diagnosed (and the fraction of active TB cases that promptly receives treatment) on the burden of the disease in a population. The simulations suggest that, with availability of treatment for both latent and active TB cases, increasing the fraction of latent TB cases that are diagnosed and treated (even with a small fraction of active TB cases promptly receiving treatment) will result in a reduction in the TB burden in the population.

## Keywords

Tuberculosis Latent Active Delay treatment Mathematical model Global stability Bifurcation Uncertainty and sensitivity analysis Numerical simulations

## References

1. 1.
Adewale, S.O., Podder, C.N., Gumel, A.B.: Mathematical analysis of a TB transmission model with DOTS. Can. Appl. Math. Quat. 17(1), 1–36 (2009)
2. 2.
Al-Darraji, H.A.A., Altice, F.L., Kamarulzaman, A.: Undiagnosed pulmonary tuberculosis among prisoners in Malaysia: an overlooked risk for tuberculosis in the community. Tropical Medicine and International Health (2016).
3. 3.
Andrews, J.R., Noubary, F., Walensky, R.P., et al.: Risk of progression to active tuberculosis following reinfection with Mycobacterium tuberculosis. Clin. Infect. Dis. 54(6), 784–791 (2012)Google Scholar
4. 4.
Aparicio, J.P., Castillo-Chavez, C.: Mathematical modelling of tuberculosis epidemics. Math. Biosci. Eng. 6(2), 209–37 (2009)
5. 5.
Asefa, A., Teshome, W.: Total delay in treatment among smear positive pulmonary tuberculosis patients in five primary health centers, southern Ethiopia: a cross sectional study. PLoS ONE 9(7), e102884 (2014). Google Scholar
6. 6.
Bam, T.S., Enarson, D.A., Hinderaker, S.G., et al.: Longer delay in accessing treatment among current smokers with new sputum smear-positive tuberculosis in Nepal. Union: Int. J. Tuberc. Lung Dis. 16(6), 822–827 (2012)Google Scholar
7. 7.
Belay, M., Bjune, G., Ameni, G., et al.: Diagnostic and treatment delay among Tuberculosis patients in Afar Region, Ethiopia: a cross-sectional study. BMC Public Health 12, 369 (2012)Google Scholar
8. 8.
Blower, S.M., Dowlatabadi, H.: Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. Int. Stat. Rev./Rev. Int. Stat. 62(2), 229–243 (1994)
9. 9.
Borgdorff, M.W.: New measurable indicator for tuberculosis case detection. Emerg. Infect. Dis. 10(9), 1523–1528 (2004)Google Scholar
10. 10.
Cain, K.P., Marano, N., Kamene, M., et al.: The movement of multidrug-resistant tuberculosis across borders in East Africa needs a regional and global solution. PLoS Med. (2015). Google Scholar
11. 11.
Carr, J.: Applications of Center Manifold Theory. Springer, New York (1981)
12. 12.
Castillo-Chavez, C., Song, B.: Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1(2), 361–404 (2004)
13. 13.
Cattamanchi, A., Miller, C.R., Tapley, A., et al.: Health worker perspectives on barriers to delivery of routine tuberculosis diagnostic evaluation services in Uganda: a qualitative study to guide clinic-based interventions. BMC Health Serv. Res. 15, 10 (2015)Google Scholar
14. 14.
Centers for Disease Control and Prevention (CDC): Testing for TB infection (2016). http://www.cdc.gov/tb/topic/testing/. Accessed on 23 Sept 2016
15. 15.
Cohen, T., Colijn, C., Finklea, B., et al.: Exogenous re-infection and the dynamics of tuberculosis epidemics: local effects in a network model of transmission. J. R. Soc. Interface 4(14), 523–531 (2007)Google Scholar
16. 16.
Countrymeter Population of Nigeria: Retrieved on 8th December, 2016 from (2015). http://countrymeters.info/en/Nigeria
17. 17.
Delogu, G., Sali, M., Fadda, G.: The biology of mycobacterium tuberculosis infection. Mediterr. J. Hematol. Infect. Dis. (2013). Google Scholar
18. 18.
Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J.: On the definition and the computation of the basic reproduction ratio $$R_o$$ in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990)
19. 19.
Dushoff, J., Huang, W., Castillo-Chavez, C.: Backward bifurcations and catastrophe in simple models of fata diseases. J. Math. Anal. Appl. 36, 227–248 (1998)
20. 20.
Dye, C., Garnett, G.P., Sleeman, K., et al.: Prospects for worldwide tuberculosis control under the WHO DOTS strategy. Directly observed short-course therapy. Lancet 352(9144), 1886–91 (1998)Google Scholar
21. 21.
Esmail, H., Barry, C.E., Young, D.B., et al.: The ongoing challenge of latent tuberculosis. Philos. Trans. R. Soc. B: Biol. Sci. 369(1645), 20130437 (2014). Google Scholar
22. 22.
Fatima, N., Shameem, M., Khan, F., et al.: Tuberculosis: laboratory diagnosis and dots strategy outcome in an urban setting: a retrospective study. J. Tubercul. Res. 2, 106–110 (2014)Google Scholar
23. 23.
Feng, Z., Castillo-Chavez, C., Capurro, A.F.: A model for tuberculosis with exogenous reinfection. Theor. Popul. Biol. 57(3), 235–47 (2000)
24. 24.
Gumel, A.B.: Causes of backward bifurcation in some epidemiological models. J. Math. Anal. Appl. 395(1), 355–365 (2012)
25. 25.
International Union Against Tuberculosis and Lung Disease (The Union): 45th World Union Conference on Lung Health, Barcelona (2014). http://barcelona.worldlunghealth.org/
26. 26.
Issarowa, C.M., Muldera, N., Wood, R.: Modelling the risk of airborne infectious disease using exhaled air. J. Theor. Biol. 372, 100–106 (2015)
27. 27.
Kuznetsov, V.N., Grjibovski, A.M., Mariandyshev, A.O., et al.: Two vicious circles contributing to a diagnostic delay for tuberculosis patients in Arkhangelsk. Emerg. Health Threats J. (2014). Google Scholar
28. 28.
Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Stability analysis of nonlinear systems. SIAM Rev. 33(1), 152–154 (1991)Google Scholar
29. 29.
LaSalle, J.P., Lefschetz, S.: The Stability of Dynamical Systems. SIAM, Philadelphia (1976)Google Scholar
30. 30.
Lee, S.H.: Diagnosis and treatment of latent tuberculosis infection. Tubercul. Respir. Dis. 78, 56–63 (2015)Google Scholar
31. 31.
Lin, Y., Enarson, D.A., Chiang, C.Y., et al.: Patient delay in the diagnosis and treatment of tuberculosis in China: findings of case detection projects. Union: Public Health Action 5(1), 65–69 (2015)Google Scholar
32. 32.
Lin, S.Y., Hwang, S.C., Yang, Y.C., et al.: Early detection of Mycobacterium tuberculosis complex in BACTEC MGIT cultures using nucleic acid amplification. Eur. J. Clin. Microbiol. Infect. Dis. 35(6), 977–984 (2016)Google Scholar
33. 33.
Makwakwa, L., Sheu, M.I., Chiang, C.Y., et al.: Patient and health system delays in the diagnosis and treatment of new and retreatment pulmonary tuberculosis cases in Malawi. BMC Infect. Dis. 14, 132 (2014)Google Scholar
34. 34.
Marino, S., Hogue, I.B., Ray, C.J., et al.: A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol. 254(1), 178–96 (2008)
35. 35.
McLeod, R.G., Brewster, J.F., Gumel, A.B., et al.: Sensitivity and uncertainty analyses for a sars model with time-varying inputs and outputs. Math. Biosci. Eng. 3, 527–44 (2006)
36. 36.
Mesfin, M.M., Newell, J.N., Madeley, R.J., et al.: Cost implications of delays to tuberculosis diagnosis among pulmonary tuberculosis patients in Ethiopia. BMC Public Health 10, 173 (2010)Google Scholar
37. 37.
Mishra, B.K., Srivastava, J.: Mathematical model on pulmonary and multidrug-resistant tuberculosis patients with vaccination. J. Egypt. Math. Soc. 22(2), 311–316 (2014)
38. 38.
Moualeua, D.P., Weiserb, M., Ehriga, R., et al.: Optimal control for a tuberculosis model with undetected cases in Cameroon. Commun. Nonlinear Sci. Numer. Simul. 20, 986–1003 (2015)
39. 39.
Okuneye, K., Gumel, A.B.: Analysis of a temperature- and rainfall-dependent model for malaria transmission dynamic. Math. Biosci. (2016).
40. 40.
Okuonghae, D.: A mathematical model of tuberculosis transmission with heterogeneity in disease susceptibility and progression under a treatment regime for infectious cases. Appl. Math. Model. 37(10–11), 6786–6808 (2013)
41. 41.
Okuonghae, D., Aihie, V.: Case detection and direct observation therapy strategy (DOTS) in Nigeria: its effect on TB dynamics. J. Biol. Syst. 16(1), 1–31 (2008)
42. 42.
Okuonghae, D., Aihie, V.: Optimal control measures for tuberculosis mathematical models including immigration and isolation of infective. J. Biol. Syst. 18(1), 17–54 (2010)
43. 43.
Okuonghae, D., Ikhimwin, B.O.: Dynamics of a mathematical model for tuberculosis with variability in susceptibility and disease progressions due to difference in awareness level. Front. Microbiol. 6, 1530 (2016). Google Scholar
44. 44.
Okuonghae, D., Omosigho, S.: Determinants of TB case detection in Nigeria: a survey. Glob. J. Health Sci. 2(2), 123–128 (2010)Google Scholar
45. 45.
Okuonghae, D., Omosigho, S.: Analysis of a mathematical model for tuberculosis: what could be done to increase case detection. J. Theor. Biol. 269, 31–45 (2011)
46. 46.
Omar, T., Variava, E., Moroe, E., et al.: Undiagnosed TB in adults dying at home fromnatural causes in a high TB burden setting: a post-mortem study. Union: Int. J. Tuberc. Lung Dis. 19(11), 1320–1325 (2015)Google Scholar
47. 47.
Paul, S., Akter, R., Aftab, A., et al.: Knowledge and attitude of key community members towards tuberculosis: mixed method study from BRAC TB control areas in Bangladesh. BMC Public Health 15, 52 (2015)Google Scholar
48. 48.
Pullar, N.D., Steinum, H., Bruun, J.N., et al.: HIV patients with latent tuberculosis living in a low-endemic country do not develop active disease during a 2 year follow-up: a Norwegian prospective multicenter study. BMC Infect. Dis. 14, 667 (2014)Google Scholar
49. 49.
Saifodine, A., Gudo, P.S., Sidat, M., et al.: Patient and health system delay among patients with pulmonary tuberculosis in Beira city, Mozambique. BMC Public Health 13, 559 (2013)Google Scholar
50. 50.
Shea, K.M., Kammerer, J.S., Winston, C.A., et al.: Estimated rate of reactivation of latent tuberculosis infection in the United States, overall and by population subgroup. Am. J. Epidemiol. 179(2), 216–225 (2014)Google Scholar
51. 51.
Shero, K.C., Legesse, M., Medhin, G., et al.: Re-assessing tuberculin skin test (TST) for the diagnosis of tuberculosis (TB) among African migrants in western Europe and USA. J. Tubercul. Res. 4, 4–15 (2014)Google Scholar
52. 52.
Song, B., Castillo-Chavez, C., Aparicio, J.P.: Tuberculosis models with fast and slow dynamics: the role of close and casual contacts. Math. Biosci. 180, 187–205 (2002)
53. 53.
Storla, D.G., Yimer, S., Bjune, G.A.: A systematic review of delay in the diagnosis and treatment of tuberculosis. BMC Public Health 8, 15 (2008)Google Scholar
54. 54.
Trauera, J.M., Denholm, J.T., McBryde, E.S.: Construction of a mathematical model for tuberculosis transmission in highly endemic regions of the Asia-pacific. J. Theor. Biol. 358, 74–84 (2014)
55. 55.
Ukwaja, K.N., Alobu, I., Nweke, C.O., et al.: Healthcare-seeking behavior, treatment delays and its determinants among pulmonary tuberculosis patients in rural Nigeria: a cross-sectional study. BMC Health Serv. Res. 13, 25 (2013)Google Scholar
56. 56.
United Nations Programme on HIV/AIDS (UNAIDS): Communications and global advocacy fact sheet. UNAIDS (2014)Google Scholar
57. 57.
van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)
58. 58.
Verhagen, L.M., Kapinga, R., van Rosmalen-Nooijens, K.A.W.L.: Factors underlying diagnostic delay in tuberculosis patients in a rural area in Tanzania: a qualitative approach. Clin. Epidemiol. Study: Infect. 38, 433–446 (2010)Google Scholar
59. 59.
Verver, S., Warren, R.M., Beyers, N., et al.: Rate of reinfection tuberculosis after successful treatment is higher than rate of new tuberculosis. Am. J. Respir. Crit. Care Med. 171(12), 1430–5 (2005)Google Scholar
60. 60.
Wang, M., FitzGerald, J.M., Richardson, K., et al.: Is the delay in diagnosis of pulmonary tuberculosis related to exposure to fluoroquinolones or any antibiotic? Union: Int. J. Tuberc. Lung Dis. 15(8), 1062–1068 (2011)Google Scholar
61. 61.
Wong, J., Lowenthal, P., Flood, J., et al.: Progression to active tuberculosis among immigrants and refugees with abnormal overseas chest radiographs—California, 1999–2012. Am. J. Respir. Crit. Care Med. 193, A7093 (2016)Google Scholar
62. 62.
World Health Organization (WHO): Global tuberculosis report. WHO report (2013)Google Scholar
63. 63.
World Health Organization (WHO): Guidelines on the management of latent tuberculosis infection (2014)Google Scholar
64. 64.
World Health Organization (WHO): Global tuberculosis report. WHO report (2016)Google Scholar
65. 65.
Yang, W.T., Gounder, C.R., Akande, T., et al.: Barriers and delays in tuberculosis diagnosis and treatment services: does gender matter? Tubercul. Res. Treat. (2014). Google Scholar
66. 66.
Yuen, C.M., Amanullah, F., Dharmadhikari, A., et al.: Turning the tap: stopping tuberculosis transmission through active case-finding and prompt elective treatment. Lancet 386(10010), 2334–2343 (2015). Google Scholar
67. 67.
Zhang, Z., Feng, G.: Global stability for a tuberculosis model with isolation and incomplete treatment. Comput. Appl. Math. (2014). Google Scholar