Department of Mathematics

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Analysis of a chronological age-structured epidemic model with a pair of optimal treatment controls
(IOP, 2024-11) Das, Dhiraj Kumar
Emerging infectious diseases are one of the core concerns in epidemiology, and they often lead to a global public health emergency and consequently affect socio-economic sectors. The risk of contracting an infection varies with different age groups in a population. In this investigation, we present a continuous age-structured epidemic model that incorporates a chronological age-dependent bilinear disease incidence rate. We establish the model's feasibility from the population perspective and derive the basic reproduction number, . The analysis reveals that the model consistently exhibits a disease-free equilibrium, and a unique endemic equilibrium emerges whenever . Furthermore, the value of determines the stability of the age-structured model. We further formulate an optimal control problem by introducing a pair of age-dependent control variables, namely, (i) pre-cautions or medical care to the latently infected individuals and (ii) treatments or hospitalization of the infected individuals. We aim to minimize the cost of implementing these two controls so that the severity of an epidemic can be mitigated. We derived the adjoint equations to the age-dependent optimal control problem using Gateaux derivatives, then proved the existence and uniqueness of optimal control solution pair using Ekeland's principle. Finally, numerical simulations are conducted to verify the analysis and visualize the solution profiles of the model. Our observations suggest that while both control measures are effective in reducing the impact of the disease, taking precautions proves to be significantly more effective in mitigating the spread of the epidemic.
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.
Complex dynamics of a fractional-order epidemic model with saturated media effect
(Springer, 2024-07) Das, Dhiraj Kumar
A four-compartmental fractional-order epidemic model has been investigated to understand the transmission mechanism of infectious diseases with the population’s memory effect. The existence and uniqueness criterion of the model solution of the proposed fractional-order model is verified. Utilizing the next-generation matrix method, a threshold quantity called, the basic reproduction number () is obtained. The model possesses two equilibrium points, infection-free and endemic. The asymptotic stability (local and global) of the proposed system at the equilibrium points has been analyzed thoroughly. It is observed that the total number of infections during the disease is influenced by the fractional-order of the model which represents the population’s memory. A transcritical bifurcation is exhibited around the infection-free equilibrium point when the basic reproduction number crosses unity. Additionally, a fractional-order optimal control problem has been studied by considering two disease interventions: media awareness and treatment. The policy containing infectious disease spread has been determined based on a cost-effectiveness analysis. Sensitivity indices are computed to determine which parameters significantly impact and hence may used in controlling the disease. Some numerical simulations have been performed to verify analytical results by using MATLAB2022a.
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.
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.
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.
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.
The effectiveness of contact tracing in mitigating COVID-19 outbreak: A model-based analysis in the context of India
(Elsevier, 2021-09) Das, Dhiraj Kumar
The ongoing pandemic situation due to COVID-19 originated from the Wuhan city, China affects the world in an unprecedented scale. Unavailability of totally effective vaccination and proper treatment regimen forces to employ a non-pharmaceutical way of disease mitigation. The world is in desperate demand of useful control intervention to combat the deadly virus. This manuscript introduces a new mathematical model that addresses two different diagnosis efforts and isolation of confirmed cases. The basic reproductive number, is inspected, and the model’s dynamical characteristics are also studied. We found that with the condition the disease can be eliminated from the system. Further, we fit our proposed model system with cumulative confirmed cases of six Indian states, namely, Maharashtra, Tamil Nadu, Andhra Pradesh, Karnataka, Delhi and West Bengal. Sensitivity analysis carried out to scale the impact of different parameters in determining the size of the epidemic threshold of . It reveals that unidentified symptomatic cases result in an underestimation of whereas, diagnosis based on new contact made by confirmed cases can gradually reduce the size of and hence helps to mitigate the ongoing disease. An optimal control problem is framed using a control variable projecting the effectiveness of diagnosis based on traced contacts made by a confirmed COVID patient. It is noticed that optimal contact tracing effort reduces effectively over time.
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.
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.
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
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.
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.
Modelling the risk of COVID-19 based on major clinical factors: A fuzzy rule approach
(IEEE, 2021) Das, Dhiraj Kumar
In this article, a Mamdani type fuzzy inference system is formulated in order to identify possible COVID-19 infected individuals based on three major clinical factors namely body-temperature, body-immunity level and vaccination efficacy. Measurements of the system's input and output parameters are considered as linguistic variable and assumed to follow trapezoidal type membership functions. The system based on total 27 fuzzy If-Then rules and called as Fuzzy Inference System (FIS) of Mamdani type. The system is analyzed using the Fuzzy Logic Toolbox of MATLAB. It has been found that with highly efficient vaccine a person with low body-immunity can escape the disease. On contrary, high body-temperature with high body-immunity power is not sufficient to exclude a person from the risk of having COVID-19.
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.
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.
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.
Spatially heterogeneous eco-epidemic model: Stabilizing role of non-local disease transmission
(Elsevier, 2024-07) Das, Dhiraj Kumar
Interaction between prey and predator in the presence of an infectious pathogen is the main focus of this article. A non-local transmission, that encompasses the possibility of acquiring infection from a distanced potential infected individual, is incorporated by utilizing a convolution of a spatial kernel function of compact support with the spatial distribution of the infected population. The spatial kernel function characterizes the likelihood of contracting an infection from an infected individual located within a certain range and its compact support indicates the extent of the non-local disease transmission. The model is governed by a system of semilinear parabolic integro-differential equations whose global-in-time classical solution has been established. The basic reproduction numbers of two distinct disease-free homogeneous steady states are derived for the non-local diffusive eco-epidemiological model. The associated non-spatial model undergoes a supercritical Hopf-bifurcation and possesses a stable limit cycle bifurcated from the stable endemic equilibrium point. The Turing instability conditions of the endemic homogeneous steady state are derived for both the local and non-local systems. A wide variety of spatio-temporal solutions over a one-dimensional spatial domain have been explored for both the local model and the non-local model with parabolic kernel function. It has been observed that an increase in the extent of the non-local disease transmission increases the parametric region of the stable spatially homogeneous solution. On the other hand, the parametric region for the spatially heterogeneous solutions, characterizing the patchy distributions of interacting populations, shrinks with an increase in the extent of the non-local disease transmission.
Spatio-temporal dynamics of an SIS model with nonlinear incidence and nonlocal disease transmission
(Springer, 2023-06) Das, Dhiraj Kumar
Investigation of the spatio-temporal patterns exhibited by infected communities sharing the same spatial region is the focus of many researchers. Typically, an individual’s susceptibility is substantially connected with the distance from nearby affected persons. Such a disease propagation mechanism is called the nonlocal infection which is primarily modeled with a kernel function K, whose support determines the range of the nonlocal infection area. In our current study, a susceptible–infected–susceptible-type epidemic model is analyzed considering the nonlinear disease incidence rate which is further extended to incorporate the nonlocal disease transmission and random movement of the individuals. Complete bifurcation characteristics of the associated temporal model include the saddle-node, subcritical Hopf, and homoclinic bifurcations. Our primary emphasis is to investigate the formation of a wide variety of spatio-temporal patterns that include stationary, quasi-periodic, periodic, and chaotic patterns, among others. Comparisons have been made between the spatio-temporal dynamics of the local and nonlocal disease transmission models. It is observed that the nonlocal disease transmission expands the parametric domain (referred to as Hopf and stable domains) on which the system possesses oscillatory and spatially homogeneous solutions. As a result, the spatially heterogeneous stationary solutions (referred to as Turing patterns) of the local system turn into irregular oscillatory solutions or spatially homogeneous solutions whenever the nonlocal extent of the disease transmission gradually increases. Also, the increased range of nonlocal infections reduces the number of stationary patches. In addition, the system exhibits “long transient” dynamics when the dispersal rate of the population tends to the Turing threshold. Exhaustive numerical simulations have been carried out to illustrate the wide range of spatio-temporal patterns displayed by the system in the presence and absence of nonlocal terms.
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.