Analyzing the Dynamic Characteristics of a Tuberculosis Epidemic Model Using Numerical Methods
Main Article Content
Abstract
Background: This study investigates the dynamic features of a viral model in tuberculosis using various numerical methods. The primary aim is to assess the performance of an analytical model for tuberculosis transmission using different numerical approaches and to compare the effectiveness of these methods in simulating the disease behavior under various conditions.
Methods: In this paper, the dynamics of tuberculosis are analyzed using the VSEIT epidemiological model and the numerical method NSFD. This model describes the community with five primary indicators: Vaccinated (V), Susceptible (S), Exposed (E), Infected (I), and Treatment (T) populations. The results indicate that the Non-Standard Finite Difference (NSFD) method is not only effective in accurately measuring the dynamic characteristics of the proposed model but also confirms the local and global stability of disease equilibrium.
Results: Simulation results demonstrate that the NSFD method is a unique and effective approach for controlling and predicting the spread of tuberculosis. This comparison underscores the effectiveness of this method compared to traditional Euler and fourth-order Runge-Kutta (RK4) methods.
Conclusion: This research contributes to a better understanding of tuberculosis epidemiology and offers potential control strategies for public health risks. The importance and effectiveness of the NSFD method in modeling the dynamic nature of tuberculosis are clearly highlighted. This study serves as a valuable recommendation in the field of controlling and predicting the spread of tuberculosis.
Article Details
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
World Health Organization. Global Tuberculosis Report. Geneva, Switzerland: World Health Organization; 2023. Available from: [https://extranet.who.int/tme/generateCSV.asp?ds=notifications](https://extranet.who.int/tme/generateCSV.asp?ds=notifications) (accessed on 1 January 2023).
Narain R, Nair SS, Naganna K, Chandrasekhar P, Rao GR, Lal P. Problems in defining a "case" of pulmonary tuberculosis in prevalence surveys. Bull World Health Organ. 1968;39(5):701.
Schwartz SH. An overview of the Schwartz theory of basic values. Online readings in Psychology and Culture. 2012;2(1):11.
Hamer WH. The Milroy lectures on epidemic diseases in England: The evidence of variability and of persistency of type; delivered before the royal college of physicians of London, March 1st, 6th, and 8th, 1906. Bedford Press; 1906.
Martini E. Berechnungen und Beobachtungen fur Epidemiology und Bekämpfung der Malaria auf Grand von Balkanerfahrungen. W. Genet. Lota AJ. Elements of physical biology. Williams & Wilkins; 1921, 1925.
Frimodt-Møller J. A community-wide tuberculosis survey in a South Indian rural population, 1950-55. Bull World Health Organ. 1960;22(12):61.
Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc Lond A. 1927;115(772):700-721.
Ullah S, Khan MA, Farooq M, Gul T. Modeling and analysis of tuberculosis (TB) in Khyber Pakhtunkhwa, Pakistan. Math Comput Simul. 2019;165:181-199.
Ucakan Y, Gulen S, Koklu K. Analyzing of tuberculosis in Turkey through SIR, SEIR and BSEIR mathematical models. Math Comput Model Dyn Syst. 2021;27(1):179-202.
Egonmwan AO, Okuonghae D. Mathematical analysis of a tuberculosis model with imperfect vaccine. Int J Biomath. 2019;12(07):1950073.
Revelle CS, Lynn WR, Feldmann F. Mathematical models for the economic allocation of tuberculosis control activities in developing nations. Am Rev Respir Dis. 1967;96(5):893-909.
Hedrich AW. Monthly estimates of the child population "susceptible" to measles, 1900-1931, Baltimore, MD. Am J Epidemiol. 1933;17(3):613-636.
Bernoulli D, Chappelle D. Essay dune nouvelle analyses de la mortality cause par la petite virile, et des advantages de inoculation pours la prévenir. 2023.
Muench H. Catalytic models in epidemiology. Harvard University Press; 1959.
Antczak T, Arana-Jimenez M. Optimality and duality results for new classes of nonconvex quasidifferentiable vector optimization problems. Appl Comput Math. 2022;21(1):21-34.
Hamidoğlu A, Taghiyev M, Weber G. On construction of pursuit-evasion games in discrete control models. Appl Comput Math. 2022;21(1).
He CH, Liu C, He JH, Sedighi HM, Shokri A, Gepreel KA. A fractal model for the internal temperature response of a porous concrete. Appl Comput Math. 2022;21(1):71-77.
Iskandarov S, Komartsova E. On the influence of integral perturbations on the boundedness of solutions of a fourth-order linear differential equation. TWMS J Pure Appl Math. 2022;13:3-9.
Akbay A, Turgay N, Ergüt M. ON SPACE-LIKE GENERALIZED CONSTANT RATIO HYPERSURFACES IN MINKOWSKI SPACES. Twms J Pure Appl Math. 2022;13(1):25-37.
Kalsoom HUMAIRA, Ali MA, Abbas MUJAHID, Budak HÜSEYIN, Murtaza GHULAM. Generalized quantum Montgomery identity and Ostrowski type inequalities for preinvex functions. TWMS J Pure Appl Math. 2022;13(1):72-90.
Raza A, Rafiq M, Awrejcewicz J, Ahmed N, Mohsin M. Dynamical analysis of coronavirus disease with crowding effect, and vaccination: a study of the third strain. Nonlinear Dyn. 2021;107(4):3963-3982.
Alfwzan WF, Raza A, Martin-Vaquero J, Baleanu D, Rafiq M, Ahmed N, Iqbal Z. Modeling and transmission dynamics of Zika virus through efficient numerical method. AIP Advances. 2023;13(9).
Raza A, Awrejcewicz J, Rafiq M, Mohsin M. Breakdown of a nonlinear stochastic nipah virus epidemic models through efficient numerical methods. Entropy. 2021;23(12):1588.
Raza A, Arif MS, Rafiq M. A reliable numerical analysis for stochastic gonorrhea epidemic model with treatment effect. Int J Biomath. 2019;12(06):1950072.
Hamam H, Raza A, Alqarni MM, Awrejcewicz J, Rafiq M, Ahmed N, Mohsin M. Stochastic modelling of Lassa fever epidemic disease. Mathematics. 2022;10(16):2919.
Naveed M, Raza A, Soori AH, Inc M, Rafiq M, Ahmed N, Iqbal MS. A dynamical study of diarrhea delayed epidemic model: application of mathematical biology in infectious diseases. Eur Phys J Plus. 2023;138(7):664.
Avilov KK, Romanyukha AA. Mathematical Models of Tuberculosis Extension and Control of it. Math Biol Bioinform. 2007;2(2):188-318. Mathematical Biology, 2(2).
Hamer WH. The Milroy lectures on epidemic diseases in England: The evidence of variability and of persistency of type; delivered before the royal college of physicians of London, March 1st, 6th, and 8th, 1906. Bedford Press; 1906.
Martini E. Berechnungen und Beobachtungen fur Epidemiology und Bekämpfung der Malaria auf Grand von Balk
anerfahrungen. W. Genet. Lota AJ. Elements of physical biology. Williams & Wilkins; 1921, 1925.
Frimodt-Møller J. A community-wide tuberculosis survey in a South Indian rural population, 1950-55. Bull World Health Organ. 1960;22(1-2):61.
Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc Lond A. 1927;115(772):700-721.
Ullah S, Khan MA, Farooq M, Gul T. Modeling and analysis of tuberculosis (TB) in Khyber Pakhtunkhwa, Pakistan. Math Comput Simul. 2019;165:181-199.
Andersen P, Doherty TM. The success and failure of BCG: implications for a novel tuberculosis vaccine. Nat Rev Microbiol. 2005;3(8):656-662.
Ucakan Y, Gulen S, Koklu K. Analyzing of tuberculosis in Turkey through SIR, SEIR and BSEIR mathematical models. Math Comput Model Dyn Syst. 2021;27(1):179-202.
Egonmwan AO, Okuonghae D. Mathematical analysis of a tuberculosis model with imperfect vaccine. Int J Biomath. 2019;12(07):1950073.
Revelle CS, Lynn WR, Feldmann F. Mathematical models for the economic allocation of tuberculosis control activities in developing nations. Am Rev Respir Dis. 1967;96(5):893-909.
Li MY, Muldowney JS. Global stability for the SEIR model in epidemiology. Math Biosci. 1995;125(2):155-164.
Newton EA, Reiter P. A model of the transmission of dengue fever with an evaluation of the impact of ultra-low volume (ULV) insecticide applications on dengue epidemics. Am J Trop Med Hyg. 1992;47(6):709-720.
Shokri A. The symmetric two-step P-stable nonlinear predictor-corrector methods for the numerical solution of second order initial value problems. Bull Iran Math Soc. 2015;41(1):201-215.
Shokri A. The symmetric P-stable hybrid Obrenchkoff methods for the numerical solution of second order IVPs. TWMS J Pure Appl Math. 2012;5:28-35.
Shokri A, Saadat HP. P-stability, TF and VSDPL technique in Obrechkoff methods for the numerical solution of the Schrödinger equation. Bull Iran Math Soc. 2016;42(3):687-706.
Sunday J, Shokri A, Marian D. Variable step hybrid block method for the approximation of Kepler problem. Fractal Fract. 2022;6(6):343.
Wang Y, Cao J. Global stability of general cholera models with nonlinear incidence and removal rates. J Franklin Inst. 2015;352(6):2464-2485.
Chennaf B, Abdelouahab MS, Lozi R. Analysis of the Dynamics of Tuberculosis in Algeria Using a Compartmental VSEIT Model with Evaluation of the Vaccination and Treatment Effects. Computation. 2023;11(7):146.
Diekmann O, Heesterbeek JAP, Metz JA. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J Math Biol. 1990;28:365-382.
Mickens RE. Nonstandard finite difference models of differential equations. World Scientific; 1994.
Mickens RE. Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations. J Difference Equ Appl. 2005;11(7):645-653.
Vieira A, Kailath T. On another approach to the Schur-Cohn criterion. IEEE Trans Circuits Syst. 1977;24(4):218-220.
Serban I, Najim M. A new multidimensional Schur-Cohn type stability criterion. In: 2007 American Control Conference. IEEE; 2007. p. 5533-5538.