Mathematical Model Of Tuberculosis Research Articles

MATHEMATICAL ANALYSIS OF A TUBERCULOSIS MODEL WITH TWO DIFFERENT STAGES OF INFECTION

Tuberculosis is an infectious disease. This disease causes death and the world notes that Tuberculosis has a high mortality rate. A mathematical model of Tuberculosis with two infection stages of individuals, pre infected and actively infected, is studied in this paper. The rate of treatment considered in this model. The stability analysis of the equilibrium is determined by the basic reproduction ratio. Routh Hurwitz linearization is used for investigate the local stability of uninfected equilibrium. While the global stability of endemic equilibrium is investigated by construct Lyapunov function. The effect of treatment in pre infected and actively infected stages can reduce the spread rate of Tuberculosis as shown in numerical simulation.

A nonlinear epidemic model for tuberculosis with Caputo operator and fixed point theory

Tuberculosis (TB) is one of the most dangerous infectious diseases which spreads from one person to another through the air and often attacks the lungs. A mathematical model of TB with control measures is considered in this paper. The integer order time derivative is modeled via the Caputo fractional order operator. A reliable numerical technique was applied to discretize the fractional derivative. The existence and uniqueness of solutions are examined. The effect of the control parameters was investigated through some numerical experiments. The effect of variation of the fractional-order parameters on the susceptible, early latent, infected, persistent latent, and recovered population with respect to time are given via figures and discussed. It was observed that a model with a fractional order parameter could serve as a good control measure to the spread of TB.

A mathematical model of tuberculosis and COVID-19 coinfection with the effect of isolation and treatment

In this research, we developed a coinfection model of tuberculosis and COVID-19 with the effect of isolation and treatment. We obtained two equilibria, namely, disease-free equilibrium and endemic equilibrium. Disease-free equilibrium is a state in which no infection of tuberculosis and COVID-19 occurs. Endemic equilibrium is a state in which there occurs not only the infection of tuberculosis and COVID-19 but also the coinfection of tuberculosis and COVID-19. We assumed that the parameters follow the uniform distribution, and then, we took 1,000 samples of each parameter using Latin hypercube sampling (LHS). Next, the samples were sorted by ranking. Finally, we used the partial rank correlation coefficient (PRCC) to find the correlation between the parameters with compartments. We analyzed the PRCC for three compartments, namely, individuals infected with COVID-19, individuals infected with tuberculosis, and individuals coinfected with COVID-19 and tuberculosis. The most sensitive parameters are the recovery rate and the infection rate of each COVID-19 and tuberculosis. We performed the optimal control in the form of prevention for COVID-19 and tuberculosis. The numerical simulation shows that these controls effectively reduce the infected population. We also concluded that the effect of isolation has an immediate impact on reducing the number of COVID-19 infections, while the effect of treatment has an impact that tends to take a longer time.

Asymptotic behavior of Tuberculosis between smokers and non-smokers

Tuberculosis (TB) is an ailment caused by Mycobacterium tuberculosis. So, by the continuous spread of TB, this study presents a new mathematical model of TB that investigates the impact of smoking and contact rate on the transmission dynamics of the disease in a population. The basic reproduction number R 0 , of the model, is computed by employing the next generation matrix approach and the dynamical behavior of the model is explored in detail. Mathematical analysis reveals that the disease-free equilibrium solution is globally asymptotically stable when the associated basic reproduction number is less than unity. It is further shown that the model has a unique endemic equilibrium point, which is proved to exist when R 0 > 1 . The global asymptotic stability of the unique endemic equilibrium, when the associated basic reproduction number exceeds unity, is investigated through numerical simulations. Sensitivity analysis is carried out to identify key parameters that have the greatest influence on the transmission dynamics of TB in the population. The sensitivity results show that the top four parameters of the model, that have the most influence on R 0 of the model are the recruitment rate of smokers into the population, contact rate, progression rate to latent TB stage, and recovery rate of infectious individuals, with other key parameters influencing the outcome of the other output responses. Numerical experiments are performed to support the analytical findings. Numerical simulation of the model shows that, if the contact rate α = 0 . 001 , then R 0 is estimated to be 0.4611, which shows that TB dies out of the population. It further shows that, if the contact rate α = 0 . 005 , R 0 is estimated to be 2.3058, which indicates that TB establishes itself in the population. We observed from the above results that the smoking habit should be discouraged in society to reduce the prevalence of TB in the population.

Analysis of the Epidemic Biological Model of Tuberculosis (TB) via Numerical Schemes

Tuberculosis (TB) is caused by bacillus Mycobacterium tuberculosis (MTB). In this study, a mathematical model of tuberculosis (TB) is analyzed. The numerical behaviour of the considered model is analyzed including basic reproduction number and stability. We applied three numerical techniques to this model, i.e., nonstandard finite difference (NSFD) scheme, Runge–Kutta method of order 4(RK-4), and forward Euler (FD) scheme. NSFD scheme preserves all the essential properties of the model. Acquired results corroborate that NSFD scheme converges for each step size. While the other two schemes failed to preserve some properties of the model such as positivity and convergence. A graphical comparison presented in this study confirms the numerical stability of the NSFD technique shown here is maintained over a large area.

Drug-sensitivity and passive immunity mathematical epidemiological model for tuberculosis

Regardless of many decades of research, the widespread availability of a vaccine and more recently highly visible WHO efforts to promote a unified global control strategy, Tuberculosis remains a leading cause of infectious mortality. In this paper, a Mathematical Model for Tuberculosis Epidemic with Passive Immunity and Drug-Sensitivity is presented. We carried out analytical studies of the model where the population comprises of eight compartments: passively immune infants, susceptible, latently infected with DS-TB. The Disease Free Equilibrium (DFE) and the Endemic Equilibrium (EE) points were established. The next generation matrix method was used to obtain the reproduction number for drug sensitive (𝑅𝑜𝑠) Tuberculosis. We obtained the disease-free equilibrium for drug sensitive TB which is locally asymptotically stable when 𝑅𝑜𝑠 < 1 indicating that tuberculosis eradication is possible within the population. We also obtained the global stability of the disease-free equilibrium and results showed that the disease-free equilibrium point is globally asymptotically stable when 𝑅𝑜𝑠 ≤ 1 which indicates that tuberculosis naturally dies out.

Mathematical model of tuberculosis considering observed treatment and vaccination interventions

In this study, a mathematical model of tuberculosis (TB) spread is constructed, considering vaccination and observed treatment. Slow-fast infection and treatment failures are also considered in the model. Mathematical analysis show that the TB-free equilibrium is locally stable if the basic reproduction number is smaller than unity, and it becomes unstable when this number exceeds unity. Furthermore, the TB endemic equilibrium exists only (and uniquely) if the basic reproduction number is larger than unity. Our results show that even though the basic reproduction number is more sensitive to vaccination than observed treatment interventions, both interventions exhibit potential for controlling the spread of TB in the population.

The accuracy comparison of the RK-4 and RK-5 method of SEIR model for tuberculosis cases in South Sulawesi

This research discusses about the accuracy of the 4th and 5th order Runge-Kutta methods (RK-4 and RK-5) as a numerical solution in the Susceptible-Exposed-Infected-Recovered (SEIR) mathematical model of Tuberculosis (TB) transmission in South Sulawesi. The data used are secondary data on the number of TB patients in South Sulawesi sourced from the South Sulawesi Health Office. The research begins by further examining the SEIR model on TB transmission, then looking for general solutions of the SEIR model with the 4th-order and 5th-order Runge-Kutta methods, parameter determination, simulation and results analysis. In this research, the movement graph is obtained from the SEIR model with real data. After analysing the numerical simulation, the accuracy of the 4th-order and 5th-order Runge-Kutta methods is compared, then the two methods are compared to observe a more accurate method. The result of this research shown that the simulation of SEIR model is accurate to predict TB cases in South Sulawesi. The Runge-Kutta method can used to observe trends in the spread of TB in South Sulawesi. The numerical simulation shown that the 5th-order more accurate than the 4th-order Runge-Kutta methods. The results of the modelling are simulated using Maple can predict the number of TB cases which can be used by the government as a consideration to prevent the spread of TB in South Sulawesi.

Optimal control analysis of a tuberculosis model with exogenous re‐infection and incomplete treatment

SummaryA new mathematical model of tuberculosis (TB) featuring exogenous re‐infection and incomplete treatment is presented and analyzed. The model divides total population into susceptible, latently infected, actively infected (uninformed and enlightened), and treatment classes. The model with or without incomplete treatment exhibits backward bifurcation phenomenon, which is caused by the presence of exogenous re‐infection. However, further investigation reveals that the absence of incomplete treatment has the potential to reduce the backward bifurcation range significantly. The global dynamics of the TB model without exogenous re‐infection is completely determined by the basic reproduction number under certain modifications on parameters. Furthermore, the model is extended to include three time‐dependent control functions, such as public awareness campaign, treatment effort, and preventive control against incomplete treatment. The existence of the optimal control for the nonautonomous model is proven and the three controls are characterized using Pontryagin's maximum principle. Numerical simulations are performed to show the significance of singular implementation of each of the controls and combination of the three controls in minimizing the TB burden in the population.

A mathematical model of tuberculosis (TB) transmission with children and adults groups: A fractional model

We consider a novel fractional model to investigates the (tuberculosis) TB model dynamics with two age groups of human, that is, the children and the adults. First, we formulate the model and present the basic results associated to the model. Then, using the fractional operators, Caputo and the Atangana-Baleanu and obtain a generalized model. Further, we give a novel numerical approach for the solution of the fractional model and obtain their approximate solution. We show graphical results with various values of the fractional order. A comparison of the two operators are shown graphically. The results obtained through Atangana-Baleanu operator is flexible than that of Caputo derivative. The infection in tuberculosis (TB) infected people decreases fast when decreasing the fractional order.

Modeling the Effects of Vaccination and Treatment on Tuberculosis Transmission Dynamics

In this study, we develop and analyze a deterministic mathematical model for tuberculosis (TB) transmission dynamics. The model includes vaccination for newborns and treatment for both high-risk latent and active TB patients. The stability of disease-free equilibrium point is discussed in detail. In the numerical simulation, the model parameters are estimated using reported TB incidence data in Ethiopia from the years 2003 to 2017, and R0 is calculated as R0≈2.13. Finally, the sensitivity indices of R0 with respect to the model parameters are performed, and their corresponding graphical results are presented. Our results quantify the positive influence of vaccination and the treatment for high-risk latent and active TB patients on the control of tuberculosis.

GLOBAL DYNAMICS OF A MATHEMATICAL MODEL OF TUBERCULOSIS WITH AGE-DEPENDENT LATENCY AND ACTIVE INFECTION

In this paper, mathematical analysis is carried out for a mathematical model of Tuberculosis (TB) with age-dependent latency and active infection. The model divides latent TB infection into two stages: an early stage of high risk of developing active TB and a late stage of lower risk for developing active TB. Infected persons initially progress through the early latent TB stage and then can either progress to active TB infection or progress to late latent TB infection. The model is formulated by incorporating the duration that an individual has spent in the stages of the early latent TB, the late latent TB and the active TB infection as variables. By constructing suitable Lyapunov functionals and using LaSalle’s invariance principle, it is shown that the global dynamics of the disease is completely determined by the basic reproduction number: if the basic reproduction number is less than unity, the TB always dies out; if the basic reproduction number is greater than unity, a unique endemic steady state exists and is globally asymptotically stable in the interior of the feasible region and therefore the TB becomes endemic. Numerical simulations are carried out to illustrate the theoretical results.

A Mathematical Model of Tuberculosis with Drug Resistance Effects

Despite the enormous progress in prevention and treatment, tuberculosis disease remains a leading cause of death worldwide and one of the major sources of concern is the drug resistant strain, MDR-TB (multidrug resistant tuberculosis) and XDR-TB (extensively drug resistant tuberculosis). In this work, we extend the standard SEIRS epidemiology model of tuberculosis to include MDR-TB. For that, we considered compartments of susceptible, exposed, infected, resistant to a first line of treatment and recovered humans and we modeled the natural growth, the interactions between these populations and the effects of treatments. We calculate the basic reproduction number, , using the next generation method. The DFE and the EE are established and their stability analysis done to show that they are locally and globally asymptotically stable. Numerical analysis for the model with and without delay is done and demonstrated that in the case of patients with both active tuberculosis and MDR tuberculosis, both strains will still persist due to lack of permanent immunity to tuberculosis while the recovered can still lose their immunity to become susceptible again.

Practical aspects of backward bifurcation in a mathematical model for tuberculosis

In this work, we examine practical aspects of backward bifurcation for a data-based model of tuberculosis that incorporates multiple features which have previously been shown to produce backward bifurcation (e.g. exogenous reinfection and imperfect vaccination) and new considerations such as the treatment of latent TB infection (LTBI) and the BCG vaccine׳s interference with detecting LTBI. Understanding the interplay between these multiple factors and backward bifurcation is particularly timely given that new diagnostic tests for LTBI detection could dramatically increase rates of both LTBI detection and vaccination in the coming decades.By establishing analytic thresholds for the existence of backward bifurcation, we identify those aspects of TB׳s complicated pathology that make backward bifurcation more or less likely to occur. We also examine the magnitude of the backward bifurcation produced by the model and its sensitivity to various model parameters. We find that backward bifurcation is unlikely to occur. While increased vaccine coverage and/or increased detection and treatment of LTBI can push the threshold for backward bifurcation into the region of biological plausibility, the resulting bifurcations may still be too small to have any noticeable epidemiological impact.

Evaluating the potential impact of enhancing HIV treatment and tuberculosis control programmes on the burden of tuberculosis

HIV has fuelled increasing tuberculosis (TB) incidence in sub-Saharan Africa. Better control of TB in this region may be achieved directly through TB programme improvements and indirectly through expanded use of antiretroviral therapy (ART) among those with HIV. We used a mathematical model of TB and HIV in South Africa to examine the potential epidemiological impact in scenarios involving improvements in three dimensions of TB programmes: coverage, diagnosis and treatment effectiveness, as well as expanded ART use through broadened eligibility. We projected the effect of alternative scenarios on TB prevalence, incidence and TB-related mortality over 20 years. Of the three dimensions of TB programme improvement, expanding coverage would produce the greatest reduction in TB burden. Compared with current performance, combined TB programme improvements were projected to decrease TB incidence by 30% over 5 years and 46% over 20 years, and decrease TB-related mortality by 45% over 5 years and 69% over 20 years. Expanded ART eligibility was projected to decrease TB incidence by 22% over 5 years and 45% over 20 years, and TB-related mortality by 22% over 5 years and 50% over 20 years. We found that over a 20-year horizon, TB-specific and HIV-specific programme changes contribute equally to incidence reductions, whereas the TB-specific changes produce a majority of the mortality benefits. An aggressive expansion of ART alongside traditional TB-specific control measures has the potential to greatly reduce TB burden, with the different elements of a combined approach having a synergistic effect in reducing long-term TB incidence and mortality.

Cost-Effectiveness Analysis of Optimal Control Measures for Tuberculosis

We propose and analyze an optimal control problem where the control system is a mathematical model for tuberculosis that considers reinfection. The control functions represent the fraction of early latent and persistent latent individuals that are treated. Our aim was to study how these control measures should be implemented, for a certain time period, in order to reduce the number of active infected individuals, while minimizing the interventions implementation costs. The optimal intervention is compared along different epidemiological scenarios, by varying the transmission coefficient. The impact of variation of the risk of reinfection, as a result of acquired immunity to a previous infection for treated individuals on the optimal controls and associated solutions, is analyzed. A cost-effectiveness analysis is done, to compare the application of each one of the control measures, separately or in combination.