Papers by Ranjit Kumar Upadhyay
Nonlinear Dynamics, 2019
The control of highly contagious diseases is very important today. In this paper, we proposed an ... more The control of highly contagious diseases is very important today. In this paper, we proposed an SEIR model with Crowley-Martin-type incidence rate and Holling type II and III treatment rates. Dynamics of the spread of infection and its control are performed for both the cases of treatment functions. We have performed the stability and bifurcation analyses of the model system. The sensitivity analysis of all the parameters with respect to the basic reproduction number has been performed. Furthermore, we discussed the optimal control strategy using Pontryagin's maximum principle and determined the effect of control parameter u on the model dynamics. Moreover, we validate the theoretical results using numerical simulations. Between both the treatment functions, we observe that the implementation of Holling type II treatment is most effective to prevent the spread of diseases. Thus, we conclude that the pervasive effect of treatment not only
International Journal of Bifurcation and Chaos, 2016
Recently, the 2014 Ebola virus (EBOV) outbreak in West Africa was the largest outbreak to date. I... more Recently, the 2014 Ebola virus (EBOV) outbreak in West Africa was the largest outbreak to date. In this paper, an attempt has been made for modeling the virus dynamics using an SEIR model to better understand and characterize the transmission trajectories of the Ebola outbreak. We compare the simulated results with the most recent reported data of Ebola infected cases in the three most affected countries Guinea, Liberia and Sierra Leone. The epidemic model exhibits two equilibria, namely, the disease-free and unique endemic equilibria. Existence and local stability of these equilibria are explored. Using central manifold theory, it is established that the transcritical bifurcation occurs when basic reproduction number passes through unity. The proposed Ebola epidemic model provides an estimate to the potential number of future cases. The model indicates that the disease will decline after peaking if multisectorial and multinational efforts to control the spread of infection are main...
arXiv (Cornell University), Jan 9, 2023
The concept of superfluous prey, or an excess of prey in certain areas within a patchy ecosystem,... more The concept of superfluous prey, or an excess of prey in certain areas within a patchy ecosystem, has significant implications for the synchronization of the predator population. These areas, known as "hotspots," have a higher density of prey compared to other areas and attract a higher concentration of predators. As a result, the predator population becomes more stable and predictable, as they are less likely to migrate to other areas in search of food. This phenomenon can have important consequences for both the predators and their prey, as well as the overall functioning of the ecosystem. This work investigates the synchronization between two chaotic food webs using the generalized Lotka-Volterra (GLV) model consisting of one prey and two predator populations. We, first, examine the impact of three functional responses (linear, Holling type II, and Holling type III) on system dynamics For the study, we consider the model with a linear functional response consisting of chaotic oscillations and apply controllers to stabilize its unstable fixed points. This research contributes to the understanding of how to apply chaotic ecological models to predict the population of competing species in one habitat using information about similar populations in another system. To do this, we configure a drive-response system where prey acts as the driving variable and both predators depend only on the prey. We use active and adaptive control methods to synchronize two coupled GLV models and verify the analytical results through numerical simulations.
Spatial Dynamics and Pattern Formation in Biological Populations, 2021
Spatial Dynamics and Pattern Formation in Biological Populations, 2021
arXiv (Cornell University), Feb 9, 2023
In this study, we investigate the dynamics of a spatial and non spatial prey-predator interaction... more In this study, we investigate the dynamics of a spatial and non spatial prey-predator interaction model that includes the following: (i) fear effect incorporated in prey birth rate; (ii) group defence of prey against predators; and (iii) prey refuge. We provide comprehensive mathematical analysis of extinction and persistence scenarios for both prey and predator species. We investigate how the prey and predator equilibrium densities are influenced by the prey birth rate and fear level. To better explore the dynamics of the system, a thorough investigation of bifurcation analysis has been performed using fear level, prey birth rate, and prey's death rate caused by intra-prey competition as bifurcation parameter. All potential occurrences of bi-stability dynamics have also been investigated for some relevant sets of parametric values. Our numerical evaluations show that high levels of fear can stabilize the prey-predator system by ruling out the possibility of periodic solutions. Also, our model's Hopf bifurcation is subcritical in contrast to traditional prey-predator models, which ignore the cost of fear and have supercritical Hopf bifurcations in general. In contrast to the general trend, predator species go extinct at higher values of prey birth rates. We have also found that, contrary to the typical tendency for prey species to go extinct, both prey and predator populations may coexist in the system as intra-prey competition level grows noticeably. We have also been obtained that both prey and predator equilibrium densities increase (decrease) as the prey birth rate (fear level in prey) increases. The stability and Turing instability of associated spatial model have also been investigated analytically. We also perform the numerical simulation to observe the effect of different parameters on the density distribution of species. Different types of spatiotemporal patterns like spot, mixture of spots and stripes have been observed via variation of time evolution, diffusion coefficient of predator population, level of fear factor and prey refuge. The fear level parameter (k) has a great impact on the spatial dynamics of model system.
arXiv (Cornell University), Jan 24, 2022
The transmission dynamics of an infectious disease are most sensitive to the social contact patte... more The transmission dynamics of an infectious disease are most sensitive to the social contact patterns in a population of a particular community and to analyze the precautions people use to reduce the transmission of the disease. The social contact pattern depends on the age distribution of the specific community via different location such as work, school and recreation etc. Therefore, knowing the age-specific prevalence and incidence of the infectious disease is essential for modeling the future burden of the disease and the effectiveness of interventions such as vaccination. In the present study, we consider an SEIR age-structured multi-group epidemic model to understand the impact of social contact patterns in controlling the disease. To observe how fluctuations in social mixing have affected the spreading of the emerging infectious disease, we used synthetic location-specific social contact matrices in the community. For mathematical analysis, we computed the basic reproduction number (R 0) for the system and also illustrated the global behavior of the system in terms of basic reproduction number. Further, the existence of optimal control for the associated problem has been established and computed mathematically. The transmission rate for the proposed model using the real data of COVID-19 for India from September 1, 2020, to December 31, 2020, has also been estimated. We simulated lifting of different non-pharmaceutical interventions by permitting the people to go back for their works in a phased-manners and investigated the effects of returning to work at different stages, accordingly. Our results suggest that awareness of symptomatic infected individuals of age groups 20 − 49 years is beneficial to reduce the number of infected individuals when all schools are closed. However, awareness of symptomatic infected individuals of school children age groups also plays a significant role in reducing disease cases when some schools are partially opened. The simulation results also recommend that the number of cases could be reduced in large numbers by controlling the contacts at school and other gathering places. Interestingly, it has been investigated that the time-dependent transmission rate is more realistic rather than the constant spread rate to COVID-19 for India via estimating transmission rate using the least square method. Our study suggests that the early and sudden lifting of control measures could lead to other peaks and a high COVID-19 burden, which could be flattened and reduced by relaxing the interventions gradually. We hope that our results would help health policymakers in deciding appropriate and timely age-based vaccination distribution strategies and, therefore, control the disease.
Journal of Applied Mathematics and Computing, 2022
We employ an age-structured susceptible-infected-quarantined-recovered model to simulate the prog... more We employ an age-structured susceptible-infected-quarantined-recovered model to simulate the progression of COVID-19 in France, Spain, and Germany. In the absence of a vaccine or conventional treatment, non-pharmaceutical interventions become more valuable, so our model takes into account the efficacy of official social distancing and lockdown measures. Using data from February to July 2020, we make useful predictions for the upcoming months, and further simulate the effect of lifting the lockdown at a later stage. A control model is also proposed and conditions for optimality are also obtained using optimal control theory. Motivated by the recent surge in cases in France and Spain, we also examine the possibility of a second wave of the pandemic. We conclude that further measures need to be taken in these two countries, while Germany is on its way to mitigating the disease.
Differential Equations and Dynamical Systems, 2021
Robust testing and tracing are key to fighting the menace of coronavirus disease 2019 (COVID-19).... more Robust testing and tracing are key to fighting the menace of coronavirus disease 2019 (COVID-19). This outbreak has progressed with tremendous impact on human life, society and economy. In this paper, we propose an age-structured SIQR model to track the progression of the pandemic in India, Italy and USA, taking into account the different age-structures of these countries. We have made predictions about the disease dynamics, identified the most infected age-groups and analysed the effectiveness of social distancing measures taken in the early stages of infection. The basic reproductive ratio R0 has been numerically calculated for each country.We propose a strategy of age-targeted testing, with increased testing in the most proportionally infected age-groups. We observe a marked flattening of the infection curve upon simulating increased testing in the 15-40 year age-groups in India. Thus, we conclude that social distancing and widespread testing are effective methods of control, wit...
Journal of Biological Systems, 2017
In this paper, we have formulated a compartmental epidemic model with exponentially decaying tran... more In this paper, we have formulated a compartmental epidemic model with exponentially decaying transmission rates to understand the Ebola transmission dynamics and study the impact of control measures to basic public health. The epidemic model exhibits two equilibria, namely, the disease-free and unique endemic equilibria. We have calculated the basic reproduction number through next generation matrix and investigated the spatial spread of the epidemic via reaction–diffusion modeling. Instead of fitting the model to the observed pattern of spread, we have used previously estimated parameter values and examined the efficacy of predictions of the designed model vis-à-vis the pattern of spread observed in Sierra Leone, West Africa. Further, we conducted a sensitivity analysis to determine the extent to which improvement in predictions is achievable through better parameterization. We performed numerical simulations with and without control measure for the designed model system. A signifi...
Asian-European Journal of Mathematics, 2017
An attempt has been made to understand the complex dynamics of a spatial predator–prey system wit... more An attempt has been made to understand the complex dynamics of a spatial predator–prey system with Beddington–DeAngelis type functional response in the presence of prey-taxis and subjected to homogenous Neumann boundary condition. To describe the active movement of predators to the regions of high prey density or if the predator is following some sort of odor to find the prey, the prey-taxis phenomenon is included in a general reaction–diffusion equation. We have studied the linear stability analysis of both spatial and non-spatial models. We have performed extensive simulations to identify the conditions to generate spatiotemporal patterns in the presence of prey-taxis. It has been observed that the increasing predator active movement from the bifurcation value, the system shows chaotic behavior whereas increasing value of random movement brings the system back to order from the disordered state.
International Journal of Bifurcation and Chaos, 2016
In this paper, an eco-epidemiological model in which both species diffuse along a spatial gradien... more In this paper, an eco-epidemiological model in which both species diffuse along a spatial gradient has been shown to exhibit temporal chaos at a fixed point in space. The proposed model is a modification of the model recently presented by Upadhyay and Roy [2014]. The spatial interactions among the species have been represented in the form of reaction–diffusion equations. The model incorporates the intrinsic growth rate of fish population which varies linearly with the depth of water. Numerical results show that diffusion can drive otherwise stable system into aperiodic behavior with sensitivity to initial conditions. We show that spatially induced chaos plays an important role in spatial pattern formation in heterogeneous environment. Spatiotemporal distributions of species have been simulated using the diffusivity assumptions realistic for natural eco-epidemic systems. We found that in heterogeneous environment, the temporal dynamics of both the species are drastically different an...
In this paper, we investigate a Rosenzweig-McAurthur model and its variant for phytoplankton, zoo... more In this paper, we investigate a Rosenzweig-McAurthur model and its variant for phytoplankton, zooplankton and fish population dynamics with Holling type II and III functional responses. We present the theoretical analysis of processes of pattern formation that involves organism distribution and their interaction of spatially distributed population with local diffusion. The choice of parameter values is important to study the effect of diffusion, also it depends more on the nonlinearity of the system. With the help of numerical simulations, we observe the formation of spatiotemporal patterns both inside and outside the Turing space.
International Journal of Bifurcation and Chaos, May 1, 2017
The modified three-dimensional (3D) Morris–Lecar (M–L) model is very useful to understand the spi... more The modified three-dimensional (3D) Morris–Lecar (M–L) model is very useful to understand the spiking activities of neurons. The present article addresses the random dynamical behavior of a modified M–L model driven by a white Gaussian noise with mean zero and unit spectral density. The applied stimulus can be expressed as a random term. Such random perturbations are represented by a white Gaussian noise current added through the electrical potential of membrane of the excitatory principal cells. The properties of the stochastic system (perturbed one) and noise induced mixed mode oscillation are analyzed. The Lyapunov spectrum is computed to present the nature of the system dynamics. The noise intensity is varied while keeping fixed the predominant parameters of the model in their ranges and also observed the changes in the dynamical behavior of the system. The dynamical synchronization is studied in the coupled M–L systems interconnected by excitatory and inhibitory neurons with noisy electrical coupling and verified with similarity functions. This result suggests the potential benefits of noise and noise induced oscillations which have been observed in real neurons and how that affects the dynamics of the neural model as well as the coupled systems. The analysis reports that the modified M–L system which has the limit cycle behavior can show a type of phase locking behavior which follows either period adding (i.e. 1:1, 2:1, 3:1, 4:1) sequences or Farey sequences. For the coupled neural systems, complete synchronization is shown for sufficient noisy coupling strength.
Algorithms
Synchronization of food chain models is an intensely investigated area in dynamical systems. Two ... more Synchronization of food chain models is an intensely investigated area in dynamical systems. Two pioneering models in three species food chain systems exhibiting chaotic dynamics are the Hastings and Powell (HP) model and the Upadhyay and Rai (UR) model. These are known to synchronize, even though the top predators in the two models behave differently. In the current manuscript, we show that although the HP and UR models synchronize for certain initial conditions, they do not synchronize for arbitrarily large initial conditions due to the blow-up dynamics present in the UR model. Thus, the synchronization of these model systems is purely a local (in initial data) phenomenon. Interestingly, we find that a similar result holds for the modified UR model as well, which has global in-time solutions for any positive initial condition. Thus, the lack of synchrony could also be attributed to the difference in the top predator’s feeding preferences in the model systems. Our results have larg...
Fractals
In this study, we present a delayed predator–prey model with cannibalism including a prey, a juve... more In this study, we present a delayed predator–prey model with cannibalism including a prey, a juvenile predator and an adult predator. The adult predator hunts not only the prey but also the juvenile predator. The alternative resource is considered to reduce cannibalism in predator species. The main results are demonstrated by local stability and Hopf bifurcation. Sufficient criteria for local stability of the interior equilibrium point and the exhibition of a Hopf bifurcation are derived. Global stability is analyzed via designing a suitable Lyapunov function. Additionally, stability of the Hopf bifurcation is explored. Finally, numerical simulations are executed to validate the obtained results.
Chaos: An Interdisciplinary Journal of Nonlinear Science
In this article, we report on the generation and propagation of traveling pulses in a homogeneous... more In this article, we report on the generation and propagation of traveling pulses in a homogeneous network of diffusively coupled, excitable, slow-fast dynamical neurons. The spatially extended system is modeled using the nearest neighbor coupling theory, in which the diffusion part measures the spatial distribution of coupling topology. We derive analytically the conditions for traveling wave profiles that allow the construction of the shape of traveling nerve impulses. The analytical and numerical results are used to explore the nature of propagating pulses. The symmetric or asymmetric nature of traveling pulses is characterized, and the wave velocity is derived as a function of system parameters. Moreover, we present our results for an extended excitable medium by considering a slow-fast biophysical model with a homogeneous, diffusive coupling that can exhibit various traveling pulses. The appearance of series of pulses is an interesting phenomenon from biophysical and dynamical p...
Mathematics and Computers in Simulation
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Papers by Ranjit Kumar Upadhyay