Department of Mathematics

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Subject: search.filters.subject.Tuberculosis model

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    Influence of multiple re-infections in tuberculosis transmission dynamics: A Mathematical Approach
    (IEEE, 2019) Das, Dhiraj Kumar
    This investigation accounts a TB transmission model with the possibility of both exogenous re-infections and recurrent TB. The qualitative characteristic of the model system has been analyzed covering stability of existing equilibrium points and bifurcation criteria. The basic reproduction number is obtained by using the next-generation matrix method. It has been observed that the system performs a backward bifurcation at Ro = 1 and hence Ro <; 1 can not guaranty the disease elimination. Several numerical simulations have been performed to support the analytical findings.
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    Transmission dynamics of tuberculosis with multiple re-infections
    (Elsevier, 2020-01) Das, Dhiraj Kumar
    We propose and analyze an epidemic model describing the transmission dynamics of tuberculosis (TB) with the possibilities of re-infections and fast progression of the disease. The qualitative behavior of the system is studied, covering several distinct aspects of disease transmission. The epidemiological threshold, known as the basic reproduction number, R0, is determined using the next-generation matrix approach. It is observed that the present epidemic system may exhibit a backward bifurcation for R0 < 1. Therefore, we may conclude that reducing R0 to less than unity is not sufficient for eradication of tuberculosis. However, reducing R0 to less than the sub-threshold obtained in the absence of recurrent TB, it is possible to eradicate the disease. We notice that a sufficient proportion of newly infected individuals developing a direct progression to the active stage can overcome the possibility of backward bifurcation. We also insight the qualitative nature of backward bifurcation with variation in re-infection level. It is found that increasing the level of re-infections makes the disease eradication more challenging. The theoretical investigations are being supplemented by numerical simulations whenever necessary.
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    The impact of the media awareness and optimal strategy on the prevalence of tuberculosis
    (Elsevier, 2020-02) Das, Dhiraj Kumar
    In this present study, we propose and analyze a mathematical model of tuberculosis (TB) transmission considering social awareness effects during an epidemic. Possible equilibrium points of the model are investigated, and their stability criterion is discussed. Basic reproduction number R0 of the model is obtained through the next-generation matrix method. It has been shown that the infection-free equilibrium is locally stable when R0 < 1 and unstable for R0 > 1. The global asymptotic stability of the endemic equilibrium P* is verified by constructing a suitable Lyapunov function. The possibility of two endemic equilibria when R0 < 1 urges the system through backward bifurcation at also verified using center manifold theory. The media awareness parameters influence the occurrence of backward bifurcation. An optimal control problem is framed considering a media intervention parameter u(t) as a control variable. The existence and characterization of the optimal solution to the problem solved analytically. Optimal media control strategy with accessible media intervention cost gradually reduce the prevalence of the disease. In addition to our analytical results, several numerical simulations are also performed to make the analysis more significant. A short discussion on the media guided transmission characteristic of the disease, obtained from our investigation is conducted at last
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    Dynamical analysis of an age-structured tuberculosis mathematical model with LTBI detectivity
    (Elsevier, 2020-12) Das, Dhiraj Kumar
    The age-dependent heterogeneity observed in tuberculosis (TB) epidemiology includes susceptibility, infectiousness, contact preferences of an individual. Also, the chance of finding a direct route to infectious pulmonary TB (PTB) of certain vulnerable risk-group and the diagnosis effort to detect latent TB individual (LTBI) are critical factors in TB epidemiology. The current investigation proposes a mathematical model based on a set of coupled partial differential equations (PDE) to encounter these vital characteristics of TB transmission. The analytical study mainly encompasses well-posedness of the PDE system, the asymptotic behavior of the model around the disease-free equilibrium point and existence criterion of endemic equilibrium point ⁎. A threshold quantity , called basic reproductive number provides the average size of infected population due to a single infectious individual introduced in the naive community. The current expression of offers a notable refinement in basic reproduction number compared to previous estimations. Also, theoretically we observe, detectivity of LTBI cases can both increase and decrease the size of depending upon a parametric condition.
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    Global dynamics of a tuberculosis model with sensitivity of the smear microscopy
    (Elsevier, 2021-05) Das, Dhiraj Kumar
    Sputum smear microscopy and chest X-ray are the key TB diagnosis methods available in resource-constrained health settings of many developing countries worldwide. The test has moderate sensitivity towards the detection of pulmonary tuberculosis (PTB) cases. However, the undetected cases are also capable of transmitting the disease with a reduced transmission possibility. In this work, a five-dimensional compartmental model is formulated considering the infectivity of both smear-positive and negative individuals. The next-generation matrix method yields the expression of basic reproduction number . The global asymptotic stability of the disease-free equilibrium point for and that of endemic equilibrium point for are established with suitably constructed Lyapunov functions. The sensitivity indices of the associated parameters of are obtained with a suitable choice of parameter values. It has been found that neglecting the transmission capacity of smear-negative individuals underestimates the value of whereas ignoring the smear-negative compartment overestimates the same quantity.
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    Qualitative analysis of TB transmission dynamics considering both the age since latency and relapse
    (Elsevier, 2024-11) Das, Dhiraj Kumar
    Since the beginning of time, tuberculosis (TB) has been a fatal illness that predominantly affects the human lungs before spreading to other organs including the brain, spine, etc. The main elements of TB mitigation are age-dependent heterogeneity, identifying those who are latently infected, and treating them using the right diagnostic strategy. In this present work, the complex transmission mechanism of this disease in a population is described by a coupled system of integro-partial differential equations (IDE-PDE). The system’s well-posedness requirement is confirmed. The proposed system’s basic reproduction number () is obtained. This work provides a complete analysis of the qualitative properties of the model, including steady state existence, asymptotic smoothness of the solution semi-flow, uniform persistence of the endemic equilibrium, and the global asymptotic stability criterion. It is observed that in assessing the severity of the pandemic, the value of is crucial. Additionally, the stability results are visually illustrated by solving the model equations numerically while assuming two hypothetical cases. The current work also suggests several methods for reducing the value of the basic reproductive number () by manipulating a few parameter values, which may help to lessen the prevalence of TB in a community.