Department of Mathematics

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Subject: search.filters.subject.Basic reproduction number

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Now showing 1 - 6 of 6
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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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    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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    Global dynamics of a fractional-order HFMD model incorporating optimal treatment and stochastic stability
    (Elsevier, 2022-08) Das, Dhiraj Kumar
    Hand, foot and mouth disease (HFMD) is highly contagious and occurs primarily among children under the age of five. Analysis of transmission dynamics of infectious diseases is essential to prevent the adverse effects caused by the diseases. The current study presents a fractional-order SEIR-type epidemic model to investigate the dynamics of HFMD transmission. The biological feasibility of the proposed model system is demonstrated from an epidemiological perspective. The basic reproduction number, R0, is obtained through the next-generation matrix approach. Around the feasible equilibrium points, the asymptotic dynamics of the proposed model system are examined, both at local and global levels. It is found that, the model undergoes transcritical bifurcation at R0 = 1. As a result, R0 plays the role of threshold in determining the future course of the disease. The optimal treatment control of fractional epidemic models are less explored. Here, an optimal control problem is formulated considering a time-dependent treatment measure u(t) both in exposed and infected classes. The findings are also visualized and verified by simulating the model using some feasible parameter values, from which it can be concluded that fractional-order gives better results. A gradual decrease in the total cases as well as in the peak is observed in the presence of treatment control. Further, we extend the study in a stochastic environment with the help of white noise and investigate the stochastic stability of the endemic equilibrium point. Finally, parameters defining the threshold quantity R0 are scaled with the normalized forward sensitivity index.
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    Modeling and analysis of Caputo-type fractional-order SEIQR epidemic model
    (Springer, 2023-11) Das, Dhiraj Kumar
    In this research, a susceptible-exposed-infected-quarantine-recovered-type epidemic model containing fractional-order differential equations is suggested and examined in order to better understand the dynamical behavior of the infectious illness in the presence of vaccination and treatments. The non-negative and bounded solutions of our proposed model are examined for existence and uniqueness. We investigate the explicit formulation of a threshold , often known as the basic reproduction number, using the next-generation matrix technique. Depending on the value of , one endemic equilibrium exists and is stable for , and one disease-free equilibrium (exist for all values of ) is stable for . This article has also noticed the emergence of a transcritical bifurcation. The relevance of using vaccination and treatments as controls has been met by formulating a fractional-order optimal control problem. The resulting theoretical conclusions are supported by a few numerical simulations. Ultimately, a global sensitivity analysis is carried out to identify the parameters that have the greatest influence.
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    Threshold dynamics of an age-structured vaccinated epidemic model with both direct and indirect routes of infections
    (Elsevier, 2024-04) Das, Dhiraj Kumar
    Age-structured epidemic models play a crucial role in the study of epidemiological modeling. Motivated by this, in this article, we propose an epidemic model with age since infection and vaccination age of individuals, a coupled system of PDE and integro-differential equations (IDE). Here, two contagious routes, (a) direct human-to-human contact and (b) indirect environmentally contaminated surfaces or objects are considered. First, we establish the well-posedness of the model, followed by the basic reproduction number () and the role of the threshold value of in the asymptotic profile of the solution semi-flow is established. We observe the global stability of the disease-free steady state for , the uniform persistence of the disease and the existence of the endemic steady state for . This endemic steady state is also globally asymptotically stable for . We have further analyzed the influence of vaccination age and age since infection in the threshold parameter . Our analysis shows that the threshold parameter does not depend explicitly on vaccination age, but it strictly decreases with the natural depletion rate of the contaminated environment. Finally, the model is discretized using the finite difference method to illustrate our theoretical results numerically.
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    Stability and bifurcation analysis of an infectious disease model with different optimal control strategies
    (Elsevier, 2023-11) Dubey, Balram; Dubey, Uma S.
    This paper deals with the non-linear Susceptible–Infected–Hospitalized–Recovered model with Holling type II incidence rate, treatment with saturated type functional response for the prevention and control of disease with limited healthcare facilities. The well-posedness of the model is ensured with the help of the non-negativity and boundedness of the solution of the system. The feasibility of the model with DFE (Disease-free equilibrium) and EE (endemic equilibrium) is analysed. The local and global stability are discussed with the help of the computed basic reproduction number . At , we use the Centre manifold theory to analyse the transcritical bifurcation exhibited by the system. It is found that the disease is not eradicated even if due to the occurrence of backward bifurcation. The occurrence condition of Hopf bifurcation is obtained. The optimal control theory is used to analyse the effects of the minimum possible medical facilities, hospital beds, and awareness creation on the population dynamics. The Hamiltonian function is constructed with the extended optimal control model and solved by Pontryagin’s maximum principle to get the minimum possible expenditure. Different types of control strategies are shown by numerical simulation. The sensitivity analysis is discussed with the help of a crucial parameter that depends on the reproduction number. Further, the model is simulated numerically to support the theoretical studies. This paper emphasizes the significance of treatment intensity, the total number of hospital bed available and their occupancy rate as vital parameters for prevention of disease prevalence.