A Novel compartmental VSLIT model used to analyze the dynamics of tuberculosis in Algeria and Ukraine and the assessment of vaccination and treatment effects

Bouchra Chennaf; Mohammed Salah Abdelouahab; René Lozi

doi:10.3842/umzh.v75i12.7658

Bouchra Chennaf Laboratory of Mathematics and Their Interactions, Abdelhafid Boussouf University Center, Mila, Algeria
Mohammed Salah Abdelouahab Laboratory of Mathematics and Their Interactions, Abdelhafid Boussouf University Center, Mila, Algeria
René Lozi Laboratoire J. A. Dieudonné, CNRS, Université Côte d'Azur, Nice, France

DOI: https://doi.org/10.3842/umzh.v75i12.7658

Keywords: Tuberculosis, epidemic, vaccination, parameter estimation, reproduction number

Abstract

UDC 517.9

Despite having low rates of tuberculosis (TB) mortality in many countries, like China, Europe, and the United States, other countries, like India continue to struggle to contain the epidemic. This study intends to examine the effects of vaccinations and treatments on the dynamics of TB in two countries, Ukraine and Algeria, with contrasted demographic profiles. A mathematical model called the VSLIT model is considered for this purpose. The stability of both disease-free and endemic equilibrium is discussed qualitatively. For numerical simulations, the parameters are evaluated by the least squares approach according to the TB-reported data for Algeria and Ukraine from 1990 to 2020.

References

D. Bernoulli, Essai D'une Nouvelle Analyse De La Mortalité Causée Par La Petite Vérole Et Des Avantages De~L'inoculation Pour La Prévenir., Mem. Math. Phys. Acad. Roy. Sci. Paris, 1–45 (1760).

R. Ross, The prevention of malaria, 2nd ed., Murray, London (1911).

W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London A, 115, 700–721 (1927).

H. Waaler, A. Geser, S. Andersen, The use of mathematical models in the study of the epidemiology of tuberculosis, Amer. J. Public Health Nation's Health, 52, 1002–1013 (1962).

Charles S. Revelle, Walter R. Lynn, Floyd Feldmann, Mathematical models for the economic allocation of tuberculosis control activities in developing nations, Amer. Rev. Respir. Dis., 893–909 (1967).

M. S. Abdelouahab, A. Arama, R. Lozi, Bifurcation analysis of a model of tuberculosis epidemic with treatment of wider population suggesting a possible role in the seasonality of this disease, Chaos, 31, № 12, 123–125 (2021).

P. Andersen, T. M. Doherty, The success and failure of BCG-implications for novel tuberculosis vaccine, Nature, 3, № 8, 656–662 (2005).

Y. Ucakan, S. Gulen, K. Koklu, Analysing of tuberculosis in Turkey through SIR, SEIR and BSEIR mathematical models, Math. and Comput. Model. Dyn. Syst., 27, № 1, 179–202 (2021).

Y. Yang, S. Tang, R. E. N. Xlaohong, H. Zhao, C. Guo, Global stability and optimal control for a tuberculosis model with vaccination and treatment, Discrete and Contin. Dyn. Syst. Ser. B, 21, № 3 (2016).

A. O. Egonmwan, D. Okuonghae, Mathematical analysis of a tuberculosis model with imperfect vaccine, Int. J. Biomath., 12, № 7, Aritcle~1950073 (2019).

WHO. Global tuberculosis report, World Health Organization; https://extranet.who.int/tme/generateCSV.asp?ds=notifications.

O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, 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).

P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29–48 (2002).

Y. Wang, J. Cao, Global stability of general cholera models with nonlinear incidence and removal rates, J. Franklin Inst., 352, № 6, 2464–2485 (2015).

Population growth in Algeria; https://www.donneesmondiales.com/afrique/algerie/croissance-population.php.

Trading Economics. Immunization, BCG (% of one-year-old children) Algeria; https://www.tradingeconomics.com/ algeria/immunization-bcg-percent-of-one-year-old-children-wb-data.html/.

Trading Economics. Immunization, BCG (% of one-year-old children) Ukraine; https://www.tradingeconomics.com/ ukraine/immunization-bcg-percent-of-one-year-old-children-wb-data.html/.

A. L. Katelaris, C. Jackson, J. Southern, R. K. Gupta, F. Drobniewski, A. Lalvani, M. Lipman, P. Mangtani, I. Abubakar, Effectiveness of BCG vaccination against mycobacterium tuberculosis infection in adults: a cross-sectional analysis of a UK-based cohort, J. Infectious Diseases, 221, 146–155 (2020).

The World Bank Group. Tuberculosis treatment success rate (% of new cases) --- Algeria; https://data.worldbank.org/ indicator/SH.TBS.CURE.ZS?locations=DZ.

The World Bank Group. Tuberculosis treatment success rate (% of new cases) --- Ukraine; https://data.worldbank.org/ indicator/SH.TBS.CURE.ZS?locations=UA.

Population growth in Ukraine; https://www.donneesmondiales.com/europe/ukraine/croissance-population.php.