Department of Mathematics

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Author: search.filters.author.Das, Dhiraj KumarSubject: search.filters.subject.Backward bifurcation

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    Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation
    (Elsevier, 2018-05) Das, Dhiraj Kumar
    We propose and analyze a mathematical model for tuberculosis (TB) transmission to study the role of exogenous reinfection and endogenous reactivation. The model exhibits two equilibria: a disease free and an endemic equilibria. We observe that the TB model exhibits transcritical bifurcation when basic reproduction number . Our results demonstrate that the disease transmission rate and exogenous reinfection rate plays an important role to change the qualitative dynamics of TB. The disease transmission rate give rises to the possibility of backward bifurcation for , and hence the existence of multiple endemic equilibria one of which is stable and another one is unstable. Our analysis suggests that may not be sufficient to completely eliminate the disease. We also investigate that our TB transmission model undergoes Hopf-bifurcation with respect to the contact rate and the exogenous reinfection rate . We conducted some numerical simulations to support our analytical findings.
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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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    Complex dynamics and fractional-order optimal control of an epidemic model with saturated treatment and incidence
    (World Scientific, 2023) Das, Dhiraj Kumar
    In this study, we have developed a novel SIR epidemic model by incorporating fractional-order differential equations and utilizing saturated-type functions to describe both disease incidence and treatment. The intricate dynamical characteristics of the proposed model, encompassing the determination of the conditions for the existence of all possible feasible equilibria with their local and global stability criteria, are investigated thoroughly. The model undergoes backward bifurcation with respect to the parameter representing the side effects due to treatment. This phenomenon emphasizes the critical role of treatment control parameters in shaping epidemic outcomes. In addition, to understand the optimal role of the treatment in mitigating the disease prevalence and minimizing the associated cost, we investigated a fractional-order optimal control problem. To further visualize the analytical results, we have conducted simulation works considering feasible parameter values for the model. Finally, we have employed local and global sensitivity analysis techniques to identify the factors that have the greatest potential to reduce the impact of the disease.