Department of Mathematics

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Author: search.filters.author.Das, Dhiraj Kumar

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    Optimizing pharmaceutical and non-pharmaceutical interventions in an age-structured epidemic model
    (Elsevier, 2025-09) Das, Dhiraj Kumar
    Besides being transmitted through direct contact with infected individuals, infectious diseases are often transmitted through contact with contaminated inanimate objects, known as fomites. Therefore, in addition to conventional pharmaceutical interventions, environmental sanitation plays a crucial role in minimizing disease transmission. This study examines two different ways of disease transmission by incorporating an additional compartment into an age-dependent SIRS model framework, which represents the concentration of pathogens in fomites. The basic reproduction number, is obtained for the model. It is found that is a monotonic decreasing function with respect to the half-saturation level of environmental contamination. It possesses two steady states, disease-free and endemic. Their stability and uniform persistence criterion are obtained with respect to the threshold value , as it crosses the unit value. The model is further extended to an optimal control problem (OCP), which includes two control interventions (i) the age-dependent treatment of infected individuals and (ii) environmental sanitation. The existence and uniqueness of the OCP are examined utilizing the concept of Gâteaux derivative. The model is numerically simulated considering feasible age-dependent parameters to visualize the analytical results. Moreover, several solutions to the OCP are plotted considering both the presence or absence of the single or both control variables. It is found that each of the control interventions is capable of containing the disease spread. However, a combined control strategy with low-cost controls is most effective in minimizing disease transmission while the two routes of infection are considered.
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    Dynamical behaviour of tuberculosis transmission
    (Biomath Communications, 2018) Das, Dhiraj Kumar
    Tuberculosis(TB) is a contagious disease in human caused by infection withВ Mycobacterium tuberculosis(Mtb). Most infections results a clinically asymptotic state termed as latent TB infection(LTBI) whereas a smaller portion ofВ infected individuals grow symptomatic active pulmonary TB. The main difference between TB and other infectious diseases is that, the disease progressionВ from primary infection(LTBI) to active pulmonary TB is signicantly time-consuming. We proposed and study an SEIR type mathematical model for TB transmission incorporating roles of both exogenous re-infection and endogenous reactivation. Our model possesses two kinds of steady states: infection free andВ endemic. The epidemiological threshold key that is, basic reproduction numberВ R0 has been obtained by using next-generation matrix. We observe that theВ disease transmission rate and exogenous re-infection level plays a signicantВ role in order to determine the qualitative behaviour of our proposed model system. Our results demonstrate that when exogenous re-infection level crosses aВ critical value our system undergoes backward bifurcation and hence a stable endemic equilibrium exists in spite of the fact R0 < 1. Therefore, reducing R0 lessВ than unity is not sucient to eradicate TB completely. We further investigateВ that proposed model experience stable periodic solutions as increases throughВ a critical value. Various numerical simulations have been conducted coveringВ the breadth of feasible parameter space to support analytical establishments.
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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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    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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    Dynamical behaviour of infected predator–prey eco-epidemics with harvesting effort
    (Springer, 2021-04) Das, Dhiraj Kumar
    This investigation accounts for a predator–prey system where the predator community is affected by infectious disease and also subjected to harvest. The model considers the behavioural change in susceptible predators due to the crowding effect of infected predators. The dynamical characteristics are studied encompassing asymptotic stability of the existing equilibrium points and bifurcation analysis. A sufficient parametric condition for global stability of the interior equilibrium point is investigated using a geometric approach. The system undergoes a Hopf-bifurcation around interior equilibrium point considering disease transmission rate as a bifurcation parameter. An optimal control problem is formulated by considering a time-dependent fishing effort as a control variable. The objective of this optimal control problem is to maximize the present value of the economic revenue obtained by fishing. Finally, several numerical simulations are conducted to visualize our analytical results.
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    Optimal control strategy for adherence to different treatment regimen in various stages of tuberculosis infection
    (EPJ Plus, 2021-08) Das, Dhiraj Kumar
    In this article, we propose a new mathematical model for tuberculosis considering the infectivity of both smear-positive and smear-negative individuals, searching for an efficient control strategy that may be followed to curtail the disease. We have employed different treatment regimens in various stages of tuberculosis infection. The fundamental epidemic threshold quantity R0 is inspected by the next-generation matrix method. The forward normalized sensitivity indices of the model parameters connected with R0 are computed to scale their impacts on the basic reproduction number. An optimal control problem is constructed considering three different treatment regimens in different possible stages of TB, and the control problem is solved analytically. The simulation results suggest that the combined implementation of all the controls optimally is the best policy to minimize the tuberculosis prevalence with the least interventions implementations costs.