Abstract
This article addresses the dynamics of the bacterial disease tuberculosis in Khyber Pakhtunkhwa, Pakistan, through a mathematical model. In this work, the latent compartment is divided into slow and fast kinds of progresses. The model is parameterized based on the reported tuberculosis-infected cases in Khyber Pakhtunkhwa for the period 2002–2017. We obtain the basic reproduction number [Formula: see text] of the model using the next-generation method. The estimated value of [Formula: see text] for the given period is approximately 1.38. Furthermore, it is shown that the model has two types of equilibria: disease-free and endemic equilibriums. The global stability analysis of the model equilibria is shown via Lyapunov functions. We also perform the sensitivity analysis of [Formula: see text] and present their corresponding graphical results to examine the relative importance of various model parameters to tuberculosis transmission and prevalence. Finally, some numerical simulations are done for the estimated parameters and the key parameters effects are considered on the curtailing tuberculosis disease. From the numerical results and model sensitivity analysis, it is found that the spread of tuberculosis can be minimized by increasing the treatment rate [Formula: see text] of infected people and decreasing the effective disease transmission rate [Formula: see text] and the rate [Formula: see text] at which the individuals leave treated class reenter infected classes.
Highlights
Tuberculosis (TB) is a contagious bacterial infection that is caused due to bacillus mycobacterium TB (MTB)
Keeping the above discussion and the previous literature in view, in this article, we develop a mathematical model with standard incidence rate for the transmission dynamics of TB
This study is focused on the formulation, analysis, and simulations of a deterministic model for assessing the TB infection dynamics in Khyber Pakhtunkhwa
Summary
Tuberculosis (TB) is a contagious bacterial infection that is caused due to bacillus mycobacterium TB (MTB). In order to ensure that the TB infection model described in equation (2) is meaningful from the epidemiological point of view, it is necessary to prove that the solutions of system (2) with nonnegative initial conditions will remain nonnegative for any value of t such that t.0 For this purpose we present the following lemma. The qualitative dynamics of such models are completely examined by a threshold parameter, known as the basic reproduction number and is usually denoted by R0 This important threshold quantity is defined as ‘‘an average number of tuberculosis infections is inserted to purely susceptible population that generate further secondary infection cases of tuberculosis by a typical infected person.’’16 With the help of R0 one can determine that either the infectious disease will remain or die out in a population.
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