Mathematical analysis of a tuberculosis model with imperfect vaccine. (English) Zbl 1426.92040

Egonmwan, A. O.; Okuonghae, D.

Int. J. Biomath. 12, No. 7, Article ID 1950073, 30 p. (2019).

Summary: Since 1921, the Bacille Calmette-Guerin (BCG) vaccine continues to be the most widely used vaccine for the prevention of Tuberculosis (TB). However, the immunity induced by BCG wanes out after some time making the vaccinated individual susceptible to TB infection. In this work, we formulate a mathematical model that incorporates the vaccination of newly born children and older susceptible individuals in the transmission dynamics of TB in a population, with a vaccine that can confer protection on older susceptible individuals. In the absence of disease-induced deaths, the model is shown to undergo the phenomenon of backward bifurcation where a stable disease-free equilibrium (DFE) co-exists with a stable positive (endemic) equilibrium when the associated reproduction number is less than unity. It is shown that this phenomenon does not exist in the absence of imperfect vaccine, exogenous reinfection, and reinfection of previously treated individuals. It is further shown that a special case of the model has a unique endemic equilibrium point (EEP), which is globally asymptotically stable when the associated reproduction number exceeds unity. Uncertainty and sensitivity analysis are carried out to identify key parameters that have the greatest influence on the transmission dynamics of TB in the population using the total population of latently infected individuals, total number of actively infected individuals, disease incidence, and the effective reproduction number as output responses. The analysis shows that the top five parameters of the model that have the greatest influence on the effective reproduction number of the model are the transmission rate, the fraction of fast disease progression, modification parameter which accounts for reduced likelihood to infection by vaccinated individuals due to imperfect vaccine, rate of progression from latent to active TB, and the treatment rate of actively infected individuals, with other key parameters influencing the outcomes of the other output responses. Numerical simulations suggest that with higher vaccination rate of older susceptible individuals, fewer new born children need to be vaccinated, in order to achieve disease eradication.

MSC:

92C60	Medical epidemiology
34D23	Global stability of solutions to ordinary differential equations
35Q92	PDEs in connection with biology, chemistry and other natural sciences
34C23	Bifurcation theory for ordinary differential equations

Keywords:

tuberculosis; BCG; imperfect vaccine; mathematical model; global stability; bifurcation; uncertainty and sensitivity analysis; numerical simulations

References:

[1]	Alexander, M. E., Moghadas, A. M., Rohani, P.et al., Modelling the effect of a booster vaccination on disease epidemiology, J. Math. Biol.52 (2006) 290-306. · Zbl 1086.92010
[2]	Adewale, S. O., Podder, C. N. and Gumel, A. B., Mathematical analysis of a TB transmission model with DOTS, Canadian Appl. Math. Quart.17(1) (2008) 1-36. · Zbl 1195.92047
[3]	Andersen, P. and Doherty, T. M., The success and failure of BCG-implications for novel tuberculosis vaccine, Nature3 (2005) 656-66.
[4]	Andrews, J. R., Noubary, F., Walensky, R. P.et al., Risk of progression to active tuberculosis following reinfection with Mycobacterium tuberculosis, Clinical Inf. Dis.54(6) (2012) 784-91.
[5]	Arino, J., McCluskey, C. C. and van den Driessche, P., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math.64(1) (2003) 260-276. · Zbl 1034.92025
[6]	Barreto, M. L., Pilger, D., Pereira, S. M.et al., Causes of variation in BCG vaccine efficacy: Examining evidence from the BCG REVAC cluster randomized trial to explore the masking and the blocking hypotheses, Vaccine32(30) (2004) 3759-3764.
[7]	Bhargava, S., Choubey, S. and Mishra, S., Vaccine against tuberculosis: A review, Indian J. Tuberculosis63 (2016) 13-18.
[8]	Blower, S. M. and Dowlatabadi, H., Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example, Int. Statist. Rev. / Revue Int. Stat.62(2) (1994) 229-243. · Zbl 0825.62860
[9]	Briassoulis, G., Karabatsou, I., Gogogllou, V.et al., BCG vaccination at three different age groups: Response and effectiveness, J. Immune Based Therap. Vaccines3(1) (2005).
[10]	Carr, J., Applications of Center Manifold Theory (Springer-Verlag, New York, 1981). · Zbl 0464.58001
[11]	Castillo-Chavez, C. and Song, B., Dynamical models of Tuberculosis and their applications, Math. Biosci. Engrg.1(2) (2004) 361-404. · Zbl 1060.92041
[12]	Clark, S., Lanni, F., Marinova, D.et al., Revaccination of guinea pigs with live attenuated Mycobacterium tuberculosis vaccine MTBVAC improves BCG’s protection against tuberculosis, J. Inf. Dis.0000 (2017) 1-8.
[13]	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. Int.4(14) (2007) 523-531.
[14]	Countrymeter, Population of Nigeria, Retrieved on 8th December, 2016 from http://countrymeters.info/en/Nigeria (2016).
[15]	Daily Post, Overall life expectancy in Nigeria drops to 52 years. Retrieved on 5th May, 2017 from https://dailypost.ng/2012/12/01/life-expectancy-nigeria-52-years/ (2015).
[16]	Dushoff, J., Huang, W. and Castillo-Chavez, C., Backward bifurcations and catastrophe in simple models of fatal diseases, J. Math. Anal. Appl.36 (1998) 227-248. · Zbl 0917.92022
[17]	Egonmwan, A. O. and Okuonghae, D., Analysis of a mathematical model for tuberculosis with diagnosis, J. Appl. Math. Comput.59(1-2) (2018) 129-162. · Zbl 1422.34147
[18]	Egonmwan, A. O. and Okuonghae, D., Optimal control measures for Tuberculosis in a population affected with insurgency, In Smith, F., Dutta, H. and Mordeson, J., Mathematics Applied to Engineering, Modelling, and Social Issues. Studies in Systems, Decision and Control, Vol. 200 (Springer, Cham, 2019), pp. 599-627. · Zbl 1430.92097
[19]	Elbasha, E. H., Podder, C. N. and Gumel, A. B., Analyzing the dynamics of an SIRS vaccination model with waning natural and vaccine-induced immunity, Nonlinear Anal.: Real World Appl.12 (2011) 2692-2705. · Zbl 1225.37104
[20]	Feng, Z., Castillo-Chavez, C. and Capurro, A. F., A model for Tuberculosis with exogenous reinfection, Theor. Popul. Biol.57(3) (2000) 235-247. · Zbl 0972.92016
[21]	Gerberry, D. J., Trade-off between BCG vaccination and the ability to detect and treat latent tuberculosis, J. Theoret. Biol.261 (2009) 548-560. · Zbl 1403.92120
[22]	Gerberry, D. J., Practical aspects of backward bifurcation in a mathematical model for tuberculosis, J. Theoret. Biol.388 (2016) 15-36. · Zbl 1343.92473
[23]	Gumel, A. B., Causes of backward bifurcations in some epidemiological models, J. Math. Anal. Appl.395(1) (2012) 355-365. · Zbl 1251.34065
[24]	Kernodle, D. S., The success and failure of BCG-implications for novel tuberculosis vaccine, Clin Infect Dis.51(2) (2010) 177-184.
[25]	Kumar, A. and Srivastava, P. K., Vaccination and treatment as control interventions in an infectious disease model with their cost optimization, Commun. Nonlinear Sci. Numer. Simulat.44 (2017) 334-343. · Zbl 1465.92061
[26]	LaSalle, J. P. and Lefschetz, S., The Stability of Dynamical Systems (SIAM, Philadelphia, 1976). · Zbl 0364.93002
[27]	Lakshmikantham, V., Leela, S. and Martynyuk, A. A., Stability Analysis of Nonlinear Systems SIAM Rev.33(1) (1991) 152-154.
[28]	Larrouy-Maumus, G., Layre, E., Clark, S.et al., Protective efficacy of a lipid antigen vaccine in a guinea pig model of tuberculosis, Vaccinehttp://dx.doi.org/10.1016/j.vaccine.2017.01.079 (2017).
[29]	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) (2008) 178-96. · Zbl 1400.92013
[30]	McCluskey, C. C., Lyapunov functions for tuberculosis models with fast and slow progression, Math Biosci.3(4) (2006) 603-614. · Zbl 1113.92057
[31]	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. Engrg.3(3) (2006) 527-544. · Zbl 1092.92042
[32]	Mishra, B. K. and Srivastava, J., Mathematical model on pulmonary and multidrug-resistant tuberculosis patients with vaccination, J. Egypt. Math. Soc.22(2) (2014) 311-316. · Zbl 1333.92060
[33]	Mlay, G. M., Luboobi, L., Kuznetsov, D.et al., Optimal treatment and vaccination control strategies for the dynamics of pulmonary tuberculosis, Int. J. Adv. Appl. Math. Mech.2(3) (2015) 196-207. · Zbl 1359.92075
[34]	Moualeua, D. P., Weiserb, M., Ehriga, R.et al., Optimal control for a tuberculosis model with undetected cases in Cameroon, Commun. Nonlinear Sci. Numer. Simulat.20 (2015) 986-1003. · Zbl 1333.92061
[35]	Nguipdop-Djomo, P., Heldal, E., Rodrigues, L. C.et al., Duration of BCG protection against tuberculosis and change in effectiveness with time since vaccination in Norway: A retrospective population-based cohort study, Lancet, http://dx.doi.org/10.1016/S1473-3099(15)00400-4 (2015).
[36]	Ozcaglar, C., Shabbeer, A., Vandenberg, S. L.et al., Epidemiological models of Mycobacterium tuberculosis complex infections, Math. Biosci.236(2) (2012) 77-96. · Zbl 1244.92039
[37]	Nguipdop-Djomo, P., Heldal, E., Rodrigues, L. C.et al., Duration of BCG protection against tuberculosis and change in effectiveness with time since vaccination in Norway: A retrospective population-based cohort study, Lancet Infect. Dis.10 (2015) 10-16.
[38]	Okuneye, K. and Gumel, A. B., Analysis of a temperature- and rainfall-dependent model for malaria transmission dynamic, Math. Biosci.http://dx.doi.org/10.1016/ j.mbs.2016.03.013 (2016). · Zbl 1377.92097
[39]	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) (2013) 6786-6808. · Zbl 1426.92082
[40]	Okuonghae, D. and Aihie, V., Case detection and direct observation therapy strategy (DOTS) in Nigeria: Its effect on TB dynamics, J. Biol. Syst.16(1) (2008) 1-31. · Zbl 1149.92019
[41]	Okuonghae, D. and Aihie, V., Optimal control measures for tuberculosis mathematical models including immigration and isolation of infective, J. Biol. Syst.18(1) (2010) 17-54. · Zbl 1342.92270
[42]	Okuonghae, D. and 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., https://doi.org/10.3389/fmicb.2015.01530 (2016).
[43]	Okuonghae, D. and Korobeinikov, A., Dynamics of tuberculosis: The effect of direct observation therapy strategy (DOTS) in Nigeria, Math. Modell. Natural Phenom.2(1) (2007) 101-113. · Zbl 1337.92212
[44]	Okuonghae, D. and Omosigho, S., Determinants of TB case detection in Nigeria: A survey, Global J. Health Sci.2(2) (2010) 123-128.
[45]	Okuonghae, D. and Omosigho, S., Analysis of a mathematical model for tuberculosis: What could be done to increase case detection, J. Theoret. Biol.269 (2011) 31-45. · Zbl 1307.92351
[46]	Rodrigues, L. C., Pereira, S. M., Cunha, S. S.et al., Effect of BCG revaccination on incidence of tuberculosis in school-aged children in Brazil: The BCG-REVAC cluster-randomized trial, Lancet366 (2005) 1290-1295.
[47]	Rodrigues, L. C. and Mangtani, P., How does the level of BCG vaccine protection against tuberculosis fall over time?BMJ343 (2011) d5974.
[48]	Roth, A. E., Benn, C. S., Ravn, H.et al., Effect of revaccination with BCG in early childhood on mortality: Randomised trial in Guinea-Bissau, BMJ340 (2010) c671.
[49]	Roy, R., Eisenhut, M., Harris, R. J.et al., Effect of BCG vaccination against Mycobacterium tuberculosis infection in children: Systematic review and meta-analysis, BMJ349 (2014) g4643.
[50]	Schiroli, C., Carugati, M., Zanini, F.et al., Exogenous reinfection of tuberculosis in a low burden area, Infection, https://doi.org/10.1007/s15010-015-0759-9 (2015).
[51]	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) (2014) 216-225.
[52]	Song, B., Castillo-Chavez, C. and Aparicio, J. P., Tuberculosis models with fast and slow dynamics: The role of close and casual contacts, Math Biosci.180 (2002) 187-205. · Zbl 1015.92025
[53]	Sun, C. and Yang, W., Global results for an SIRS model with vaccination and isolation, Nonlinear Anal.: Real World Appl.11 (2010) 4223-4237. · Zbl 1206.34069
[54]	Trauer, J. M., Denholm, J. T. and McBryde, E. S., Construction of a mathematical model for tuberculosis transmission in highly endemic regions of the Asia-Pacic, J. Theoret. Biol.358 (2014) 74-84. · Zbl 1412.92301
[55]	Uys, P., Brand, H., Warren, R.et al., The risk of tuberculosis reinfection soon after cure of a first disease episode is extremely high in a hyperendemic community, PLoS ONE10(12) (2015) e0144487.
[56]	van den Driessche, P. and Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci.180 (2002) 29-48. · Zbl 1015.92036
[57]	United Nations Programme on HIV/AIDS (UNAIDS), Communications and Global Advocacy Fact Sheet, UNAIDS (2014).
[58]	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) (2005) 1430-1435.
[59]	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 (2016) A7093.
[60]	The World Bank, Population growth (annual
[61]	World Health Organization (WHO), WHO global tuberculosis report 2015, WHO/HTM/TB/2016.22 (2015).
[62]	World Health Organization (WHO), WHO global tuberculosis report 2018, WHO/CDS/TB/2018.25 (2018).
[63]	The World Bank, Population growth (annual
[64]	The New Humanitarian, Retrieved on 1st May, 2017, http://www.thenewhumanitarian.org/report/95529/kenya-babies-miss-out-tb-immunization.
[65]	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 multi-center study, BMC Infect. Dis.14 (2014) 667.