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Dynamical analysis of an age-structured tuberculosis mathematical model with LTBI detectivity. (English) Zbl 1459.92122

Summary: The age-dependent heterogeneity observed in tuberculosis (TB) epidemiology includes susceptibility, infectiousness, contact preferences of an individual. Also, the chance of finding a direct route to infectious pulmonary TB (PTB) of certain vulnerable risk-group and the diagnosis effort to detect latent TB individual (LTBI) are critical factors in TB epidemiology. The current investigation proposes a mathematical model based on a set of coupled partial differential equations (PDE) to encounter these vital characteristics of TB transmission. The analytical study mainly encompasses well-posedness of the PDE system, the asymptotic behavior of the model around the disease-free equilibrium point \(P_0\) and existence criterion of endemic equilibrium point \(P^\ast \). A threshold quantity \(R_0\), called basic reproductive number provides the average size of infected population due to a single infectious individual introduced in the naive community. The current expression of \(R_0\) offers a notable refinement in basic reproduction number compared to previous estimations. Also, theoretically we observe, detectivity of LTBI cases can both increase and decrease the size of \(R_0\) depending upon a parametric condition.

MSC:

92D30 Epidemiology
35B35 Stability in context of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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[1] Trauer, J. M.; Dodd, P. J.; Gomes, M. G.M.; Gomez, G. B.; Houben, R. M.; McBryde, E. S.; Melsew, Y. A.; Menzies, N. A.; Arinaminpathy, N.; Shrestha, S.; Dowdy, D. W., The importance of heterogeneity to the epidemiology of tuberculosis, Clin. Infect. Dis., 69, 1, 159-166 (2018)
[2] Pai, M.; Rodrigues, C., Management of latent tuberculosis infection: an evidence-based approach, Lung India, 32, 3, 205 (2015)
[3] Feng, Z.; Castillo-Chavez, C.; Capurro, A. F., A model for tuberculosis with exogenous reinfection, Theor. Popul. Biol., 57, 3, 235-247 (2000) · Zbl 0972.92016
[4] Castillo-Chavez, C.; Song, B., Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1, 2, 361-404 (2004) · Zbl 1060.92041
[5] Xu, R.; Tian, X.; Zhang, F., Global dynamics of a tuberculosis transmission model with age of infection and incomplete treatment, Adv. Differ. Equ., 2017, 1, Article 1 pp. (2017) · Zbl 1422.92175
[6] Cao, H.; Zhou, Y., The discrete age-structured SEIT model with application to tuberculosis transmission in China, Math. Comput. Model., 55, 3-4, 385-395 (2012) · Zbl 1255.39007
[7] Brooks-Pollock, E.; Cohen, T.; Murray, M., The impact of realistic age structure in simple models of tuberculosis transmission, PLoS ONE, 5, 1, Article e8479 pp. (2010)
[8] Yan, D.; cao, H., The global dynamics for an age-structured tuberculosis transmission model with the exponential progression rate, Appl. Math. Model., 75, 769-786 (2019) · Zbl 1481.92172
[9] Castillo-Chavez, C.; Feng, Z., Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math. Biosci., 151, 2, 135-154 (1998) · Zbl 0981.92029
[10] Castillo-Chavez, C.; Hethcote, H. W.; Andreasen, V.; Levin, S. A.; Liu, W. M., Epidemiological models with age structure, proportionate mixing, and cross-immunity, J. Math. Biol., 27, 3, 233-258 (1989) · Zbl 0715.92028
[11] Diekmann, O.; Heesterbeek, J. A.P.; Metz, J. A.J., On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28, 365-382 (1990) · Zbl 0726.92018
[12] Shim, E.; Feng, Z.; Martcheva, M.; Castillo-chavez, C., An age-structured epidemic model of rotavirus with vaccination, J. Math. Biol., 53, 719-746 (2006) · Zbl 1113.92045
[13] Feng, Z.; Glasser, J. W.; Hill, A. N.; Franko, M. A.; Carlsson, R. M.; Hallander, H.; Tüll, P.; Olin, P., Modeling rates of infection with transient maternal antibodies and waning active immunity: application to Bordetella pertussis in Sweden, J. Theor. Biol., 356, 123-132 (2014) · Zbl 1412.92278
[14] Inaba, H.; Saito, R.; Bacaër, N., An age-structured epidemic model for the demographic transition, J. Math. Biol., 77, 5, 1299-1339 (2018) · Zbl 1400.92495
[15] Liu, L.; Feng, X., A multigroup SEIR epidemic model with age-dependent latency and relapse, Math. Methods Appl. Sci., 41, 16, 6814-6833 (2018) · Zbl 1405.35227
[16] Busenberg, S.; Castillo-Chavez, C., A general solution of the problem of mixing subpopulations, and its application to risk- and age-structure epidemic models for the spread of AIDS, IMA J. Math. Appl. Med. Biol., 8, 1, 1-29 (1991) · Zbl 0764.92017
[17] Bai, Z., A periodic age-structured epidemic model with a wide class of incidence rates, J. Math. Anal. Appl., 393, 2, 367-376 (2012) · Zbl 1259.34079
[18] Glasser, J.; Feng, Z.; Moylan, A.; Del Valle, S.; Castillo-Chavez, C., Mixing in age-structured population models of infectious diseases, Math. Biosci., 235, 1, 1-7 (2012) · Zbl 1320.92075
[19] Horsburgh, C. R.; O’Donnell, M.; Chamblee, S.; Moreland, J. L.; Johnson, J.; Marsh, B. J.; Narita, M.; Johnson, L.; von Reyn, C. F., Revisiting rates of reactivation tuberculosis: a population-based approach, Am. J. Respir. Crit. Care Med., 182, 3, 420-425 (2010)
[20] Mclvor, A.; Koornhof, H.; Kana, D., Relapse, re-infection and mixed infections in tuberculosis disease, Pathog. Dis., 75, 3 (2017)
[21] Yu, T.; Shi, Y.; Yao, W., Dynamic model of tuberculosis considering multi-drug resistance and their applications, Infect. Dis. Model., 3, 362-372 (2018)
[22] Das, D. K.; Khajanchi, S.; Kar, T. K., Transmission dynamics of tuberculosis with multiple re-infections, Chaos Solitons Fractals, 130, Article 109450 pp. (2020) · Zbl 1489.92147
[23] Das, D. K.; Khajanchi, S.; Kar, T. K., The impact of the media awareness and optimal strategy on the prevalence of tuberculosis, Appl. Math. Comput., 366, Article 124732 pp. (2020) · Zbl 1433.92025
[24] Athithan, S.; Ghosh, M., Mathematical modelling of TB with the effects of case detection and treatment, Int. J. Dyn. Control, 1, 3, 223-230 (2013)
[25] Lalli, M.; Hamilton, M.; Pretorius, C.; Pedrazzoli, D.; White, R. G.; Houben, R. M., Investigating the impact of TB case-detection strategies and the consequences of false positive diagnosis through mathematical modelling, BMC Infect. Dis., 18, 1, 340 (2018)
[26] Egonmwan, A. O.; Okuonghae, D., Analysis of a mathematical model for tuberculosis with diagnosis, J. Appl. Math. Comput., 59, 1-2, 129-162 (2019) · Zbl 1422.34147
[27] 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)
[28] Gripenberg, G.; Londen, S.; Staffans, O., Existence of solutions of nonlinear equations, (Volterra Integral and Functional Equations. Volterra Integral and Functional Equations, Encyclopedia of Mathematics and Its Applications (1990), Cambridge University Press), 341-382 · Zbl 0695.45002
[29] Gripenberg, G.; Londen, S.; Staffans, O., Continuous dependence, differentiability, and uniqueness, (Volterra Integral and Functional Equations. Volterra Integral and Functional Equations, Encyclopedia of Mathematics and Its Applications (1990), Cambridge University Press), 383-424 · Zbl 0695.45002
[30] Hoppensteadt, F., An age dependent epidemic model, J. Franklin Inst., 297, 5, 325-333 (1974) · Zbl 0305.92010
[31] Gumel, A. B., Cause of backward bifurcations in some epidemiological models, J. Math. Anal. Appl., 395, 355-365 (2012) · Zbl 1251.34065
[32] Khajanchi, S.; Das, D. K.; Kar, T. K., Dynamics of a tuberculosis transmission with exogenous reinfections and endogenous reactivation, Physica A, 497, 52-71 (2018) · Zbl 1514.92137
[33] (2019), World Health Organization: World Health Organization Geneva, Global tuberculosis report 2019
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