Department of Mathematics
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Item The impact of radio-chemotherapy on tumour cells interaction with optimal control and sensitivity analysis(Elsevier, 2024-03) Dubey, Balram; Dubey, Uma S.Oncologists and applied mathematicians are interested in understanding the dynamics of cancer-immune interactions, mainly due to the unpredictable nature of tumour cell proliferation. In this regard, mathematical modelling offers a promising approach to comprehend this potentially harmful aspect of cancer biology. This paper presents a novel dynamical model that incorporates the interactions between tumour cells, healthy tissue cells, and immune-stimulated cells when subjected to simultaneous chemotherapy and radiotherapy for treatment. We analysed the equilibria and investigated their local stability behaviour. We also study transcritical, saddle–node, and Hopf bifurcations analytically and numerically. We derive the stability and direction conditions for periodic solutions. We identify conditions that lead to chaotic dynamics and rigorously demonstrate the existence of chaos. Furthermore, we formulated an optimal control problem that describes the dynamics of tumour-immune interactions, considering treatments such as radiotherapy and chemotherapy as control parameters. Our goal is to utilize optimal control theory to reduce the cost of radiotherapy and chemotherapy, minimize the harmful effects of medications on the body, and mitigate the burden of cancer cells by maintaining a sufficient population of healthy cells. Cost-effectiveness analysis is employed to identify the most economical strategy for reducing the disease burden. Additionally, we conduct a Latin hypercube sampling-based uncertainty analysis to observe the impact of parameter uncertainties on tumour growth, followed by a sensitivity analysis. Numerical simulations are presented to elucidate how dynamic behaviour of model is influenced by changes in system parameters. The numerical results validate the analytical findings and illustrate that a multi-therapeutic treatment plan can effectively reduce tumour burden within a given time frame of therapeutic intervention.Item The impact of social media advertisements and treatments on the dynamics of infectious diseases with optimal control strategies(Elsevier, 2024-05) Dubey, Uma S.; Dubey, BalramThe dissemination of public health information through television and social media posts is essential for informing the public about the transmission of contagious diseases, which is crucial in preventing the spread of various infectious diseases. In this paper, we propose a non-linear mathematical model to assess the effect of advertisements through social media in creating awareness and limiting treatment on spreading infectious diseases. These initiatives may alter population behaviour and divide the susceptible population into subgroups. In addition, to comprehend these dynamics better, we use half-saturation constant rates for media coverage and treatment. The model’s well-posedness and feasibility are evaluated. The possible biological equilibrium points are calculated. Local and global stability are carried out. The objective of our study is to produce the model’s bifurcation. Transcritical, Saddle–node, Hopf bifurcation of codimension 1 and Cusp, Generalized-Hopf (Bautin), and Bogdanov–Takens (BT) bifurcation of codimension 2 are studied for this purpose. Due to the limited medical resources and supply efficiency, the model exhibits backward bifurcation, resulting in bistability. Moreover, the occurrence condition for stability and direction of Hopf bifurcation is discussed. This model study demonstrates that the system is significantly influenced by the pace with which awareness programmes are implemented and that raising this value above a threshold may result in continuous oscillation. Sensitivity analysis employs the normalized forward sensitivity index of the basic reproduction number to provide a comprehensive understanding of the effect of various parameters on accelerating and limiting disease spread. Further, the minimum possible cost is determined by formulating an optimal control system based on sensitivity analysis and applying Pontryagin’s maximum principle. Methods of cost-effectiveness, such as ACER and ICER, are used to determine the most cost-effective control intervention strategy among all the strategies. Numerical simulations have been done to support all theoretical findings.Item Study of a cannibalistic prey–predator model with Allee effect in prey under the presence of diffusion(Elsevier, 2024-05) Dubey, BalramIn this study, we have investigated the temporal and spatio-temporal behavior of a prey–predator model with weak Allee effect in prey and the quality of being cannibalistic in a specialist predator. The parameters responsible for the Allee effect and cannibalism impact both the existence and stability of coexistence steady states of the temporal system. The temporal system exhibits various kinds of local bifurcations such as saddle–node, Hopf, Generalized Hopf (Bautin), Bogdanov–Takens, and global bifurcation like homoclinic, saddle–node bifurcation of limit cycles. For the model with self-diffusion, we establish the non-negativity and prior bounds of the solution. Subsequently, we derive the theoretical conditions in which self-diffusion leads to the destabilization of the interior equilibrium. Additionally, we explore the conditions under which cross-diffusion induces the Turing-instability where self-diffusion fails to do so. Further, we present different kinds of stationary and dynamic patterns on varying the values of diffusion coefficients to depict the spatio-temporal model’s rich dynamics. It has been found that the addition of self and cross-diffusion in a prey–predator model with the Allee effect in prey and cannibalistic predator play essential roles in comprehending the pattern formation of a distributed population model. It is expected that the comprehensive mathematical analysis and extensive numerical simulations used in investigating the global dynamics of the proposed model can facilitate researchers in studying the temporal and spatial aspects of prey–predator models in more significant detail.Item Stability Switching in a Cooperative Prey-Predator Model with Transcritical and Hopf-bifurcations(Springer, 2022-10) Dubey, BalramIn nature organisms attempt to adopt new techniques to diminish the possibilities of being falling prey. Interspecies cooperation is one of these approaches which two different types of prey can use against a common predator. Inspired by this, we purpose a prey-predator model having two prey who cooperate with each other while interacting with a predator. For making the model more general and realistic, the interactions between prey and predator are handled through general Holling type-IV and Crowley-Martin functional responses. For well-posedness of the proposed model, firstly, its boundedness is investigated which is followed by the vigorous proofs for the existence of equilibrium points, their stability analysis, evaluation of conditions for occurrence of transcritical and Hopf-bifurcations. Numerically, we observe that as the inverse measure of predator’s immunity from first prey and coefficient of cooperation from first prey to second prey crosses some respective critical values, there is occurrence of Hopf-bifurcation.Transcritical bifurcation is also depicted numerically for the intrinsic growth rate of first prey and the death rate of predator species. Several phase portraits, bifurcation diagrams are drawn to support our analytical findings. We also endorse the attribute of bistability, and basins of attraction for both stable equilibrium points are also drawn.Item Impact of Cooperative Hunting and Fear-Induced in a Prey-Predator System with Crowley-Martin Functional Response(Springer, 2022-10) Dubey, BalramCooperative hunting among predators and the fear-induced growth rate reduction in prey populations is an ecologically significant phenomenon. Many researchers have studied the effects of hunting cooperation and fear independently, but there has not been much research on the combined effect. This study analyzed a classical predator-prey system incorporating hunting cooperation and fear effect with Crowley-Martin functional response. We have done the basic analysis, including positivity, boundedness of solutions, existence and stability analysis of equilibria, Hopf-bifurcation, saddle-node bifurcation. We analyzed that incorporating cooperative hunting among predators may destabilize the system dynamics by producing limit cycles via Hopf-bifurcation. Furthermore, we noticed that the system shows bi-stability behavior between predator-free equilibrium and the coexistence equilibrium. Also, analysis shows that the system becomes unstable for a fixed hunting cooperation parameter on increasing the strength of fear. To validate the analytical conclusions, numerical simulations are conducted.