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# A Discrete Model of TB Dynamics in Khyber Pakhtunkhwa-Pakistan

1 Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, South Africa
2 Department of Mathematics, City University of Science and Information Technology, Peshawar, Pakistan
3 Department of Mathematics, The Hashemite university, Zarqa, Jordan
4 Department of Basic Sciences, Princess Sumaya University for Technology, Amman, Jordan
* Corresponding Author: Muhammad Altaf Khan. Email:
(This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)

Computer Modeling in Engineering & Sciences 2020, 123(2), 777-795. https://doi.org/10.32604/cmes.2020.08208

Received 06 August 2019; Accepted 29 November 2019; Issue published 01 May 2020

### Abstract

The present paper investigates the theoretical analysis of the tuberculosis (TB) model in the discrete-time case. The model is parameterized by the TB infection cases in the Pakistani province of Khyber Pakhtunkhwa between 2002 and 2017. The model is parameterized and the basic reproduction number is obtained and it is found R0 ¼ 1:5853. The stability analysis for the model is presented and it is shown that the discrete-time tuberculosis model is stable at the disease-free equilibrium whenever R0 < 1 and further we establish the results for the endemic equilibria and prove that the model is globally asymptotically stable if R0 > 1. A discrete fractional model in the sense of Caputo derivative is presented. The numerical results of the model with various parameters and their effect on the model are presented. A comparison of discrete-time method with continuous-time model is presented graphically. A discrete fractional approach is compared with the existing method in literature and some reasonable results are achieved. Finally, a summary of results and conclusion are presented.

### Keywords

Discrete TB model; real data; equilibria; Lyapunov; stability; Caputo derivative

Khan, M. A., Khan, K., Safi, M. A., DarAssi, M. H. (2020). A Discrete Model of TB Dynamics in Khyber Pakhtunkhwa-Pakistan. CMES-Computer Modeling in Engineering & Sciences, 123(2), 777–795.

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