Mathematical modelling of TB with the effects of case detection and treatment

Abstract

In this paper a non-linear mathematical model of tuberculosis with case detection and treatment is proposed and analyzed. The whole population under consideration is divided into four compartments e.g. susceptible, exposed, infected and Recovered to study the transmission dynamics of the tuberculosis. Based on the immunity level, susceptible individuals move to exposed class or directly to infected class once they come into contact with an infective. This has been incorporated through progression rate which could be fast or slow. The equilibria of the model and the basic reproduction number R 0 are computed. It is observed that the disease free equilibrium of the model is locally asymptotically stable when R 0 < 1. The model exhibits backward bifurcation under certain restriction on parameters, which gives rise to existence of multiple endemic equilibria for R 0 < 1. This suggests that an accurate estimation of parameters and the level of control measures are required to reduce the infection prevalence of TB in endemic region and just R 0 < 1 is not enough to eliminate the disease from the population. R 0 needs to be lowered much below one to confirm the global stability of the disease free equilibrium. Numerical simulation is performed to demonstrate the analytical results. It is found that the increase in the rate of case detection shifts the bacward bifurcation diagram towards right which leads to increase in the threshold value of R 0. It is also shown that the treatment is reducing the equilibrium level of infective

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

“…The analysis focused on the applications of dynamical systems methods and optimal control theory to the epidemiology of TB. In agreement with the earlier studies on TB models with exogenous re‐infection property (see, for examples, References 3,4,6,9, and 13), it was shown that the autonomous deterministic TB model exhibits backward bifurcation whether incomplete treatment property is present or not. The occurrence of backward bifurcation in TB transmission dynamics makes the disease control more taxing.…”

“…The existence of two endemic equilibria in item (iii) of Theorem 2 shows that the TB model () has the potential to exhibit backward bifurcation property (a situation where the stable DFE and stable EE exist simultaneously when ), which can be caused by exogenous re‐infection of latently‐infected individuals. This result has been well‐established in literature on TB dynamics (see References 3‐5, and 9).…”

“…A number of mathematical models have been developed to analyze the dynamical spread of TB in human population (see, eg, References 3‐11 and some of the references therein). In Reference 7, the authors presented a mathematical model for TB with exogenous re‐infection, which allows for the possibility of a subcritical bifurcation at a certain critical value of the basic reproduction number.…”

“…Adewale et al 3 rigorously analyzed a deterministic TB model by considering the detection and treatment of individuals with active TB. Similarly, Athithan and Ghosh 4 studied mathematical modeling of TB transmission with exogenous re‐infection by assessing the effects of case detection and treatment. It should be noted that all the aforementioned studies did not consider time‐dependent control strategies to suggest or design optimal programs for preventing and controlling TB transmission in human population.…”

Modeling the Effects of Vaccination and Treatment on Tuberculosis Transmission Dynamics

“…ese models apply to di erent types of populations such as a city or country, a school, prison, or refugee camp, see for instance, [8,9] Also, di erent models would focus on various factors such as progression rate, treatment, vaccination, immigration, etc. see [10][11][12].…”