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This article focuses on the qualitative analysis of complex dynamics arising in a few mathematical models in neuroscience context. We first discuss the dynamics arising in the three-dimensional FitzHugh-Rinzel (FHR) model and then illustrate those arising in a class of non-homogeneous FitzHugh-Nagumo (Nh-FHN) reaction-diffusion systems. FHR and Nh-FHN models can be used to generate relevant complex dynamics and wave-propagation phenomena in neuroscience context. Such complex dynamics include canards, mixed-mode oscillations (MMOs), Hopf-bifurcations and their spatially extended counterpart. Our article highlights original methods to characterize these complex dynamics and how they emerge in ordinary differential equations and spatially extended models.

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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 perspective. Varying the perturbation and coupling parameters, we observe the propagation of activities with various amplitude modulations and transition phases of different wave profiles that affect the speed of pulses in certain parameter regimes. We observe different types of traveling pulses, such as envelope solitons and multi-bump solutions, and show how system parameters and coupling play a major role in the formation of different traveling pulses. Finally, we obtain the conditions for stable and unstable plane waves.

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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.

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Pandemic is an unprecedented public health situation, especially for human beings with comorbidity. Vaccination and non-pharmaceutical interventions only remain extensive measures carrying a significant socioeconomic impact to defeating pandemic. Here, we formulate a mathematical model with comorbidity to study the transmission dynamics as well as an optimal control-based framework to diminish COVID-19. This encompasses modeling the dynamics of invaded population, parameter estimation of the model, study of qualitative dynamics, and optimal control problem for non-pharmaceutical interventions (NPIs) and vaccination events such that the cost of the combined measure is minimized. The investigation reveals that disease persists with the increase in exposed individuals having comorbidity in society. The extensive computational efforts show that mean fluctuations in the force of infection increase with corresponding entropy. This is a piece of evidence that the outbreak has reached a significant portion of the population. However, optimal control strategies with combined measures provide an assurance of effectively protecting our population from COVID-19 by minimizing social and economic costs.

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The analysis of dynamical complexity in nonlinear phenomena is an effective tool to quantify the level of their structural disorder. In particular, a mathematical model of tumor-immune interactions can provide insight into cancer biology. Here, we present and explore the aspects of dynamical complexity, exhibited by a time-delayed tumor-immune model that describes the proliferation and survival of tumor cells under immune surveillance, governed by activated immune-effector cells, host cells, and concentrated interleukin-2. We show that by employing bifurcation analyses in different parametric regimes and the 0-1 test for chaoticity, the onset of chaos in the system can be predicted and also manifested by the emergence of multi-periodicity. This is further verified by studying one- and two-parameter bifurcation diagrams for different dynamical regimes of the system. Furthermore, we quantify the asymptotic behavior of the system by means of weighted recurrence entropy. This helps us to identify a resemblance between its dynamics and emergence of complexity. We find that the complexity in the model might indicate the phenomena of long-term cancer relapse, which provides evidence that incorporating time-delay in the effect of interleukin in the tumor model enhances remarkably the dynamical complexity of the tumor-immune interplay.

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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 R 0 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, with emphasis on testing and identifying the hot spots of highly infected populations. It has also been suggested that a complete lockdown, followed by lockdowns in selected regions, is more effective than the reverse.

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Excitable cells often produce different oscillatory activities that help us to understand the transmitting and processing of signals in the neural system. The diverse excitabilities of an individual neuron can be reproduced by a fractional-order biophysical model that preserves several previous memory effects. However, it is not completely clear to what extent the fractional-order dynamics changes the firing properties of excitable cells. In this article, we investigate the alternation of spiking and bursting phenomena of an uncoupled and coupled fractional leech-heart (L-H) neurons. We show that a complete graph of heterogeneous de-synchronized neurons in the backdrop of diverse memory settings (a mixture of integer and fractional exponents) can eventually lead to bursting with the formation of cluster synchronization over a certain threshold of coupling strength, however, the uncoupled L-H neurons cannot reveal bursting dynamics. Using the stability analysis in fractional domain, we demarcate the parameter space where the quiescent or steady-state emerges in uncoupled L-H neuron. Finally, a reduced-order model is introduced to capture the activities of the large network of fractional-order model neurons.

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Fractional-order dynamics of excitable systems can be physically described as a memory dependent phenomenon. It can produce diverse and fascinating oscillatory patterns for certain types of neuron models. To address these characteristics, we consider a nonlinear fast-slow FitzHugh-Rinzel (FH-R) model that exhibits elliptic bursting at a fixed set of parameters with a constant input current. The generalization of this classical order model provides a wide range of neuronal responses (regular spiking, fast-spiking, bursting, mixed-mode oscillations, etc.) in understanding the single neuron dynamics. So far, it is not completely understood to what extent the fractional-order dynamics may redesign the firing properties of excitable systems. We investigate how the classical order system changes its complex dynamics and how the bursting changes to different oscillations with stability and bifurcation analysis depending on the fractional exponent (0 < α ≤ 1). This occurs due to the memory trace of the fractional-order dynamics. The firing frequency of the fractional-order FH-R model is less than the classical order model, although the first spike latency exists there. Further, we investigate the responses of coupled FH-R neurons with small coupling strengths that synchronize at specific fractional-orders. The interesting dynamical characteristics suggest various neurocomputational features that can be induced in this fractional-order system which enriches the functional neuronal mechanisms.

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This work is mainly focused on the series of dynamical analysis of tritrophic food chain model with Sokol-Howell functional response, incorporating the multiple gestation time delays for more realistic formulation. Basic properties of the proposed model are studied with the help of boundedness, stability analysis, and Hopf-bifurcation theory. By choosing the fixed parameter set and varying the value of time delay, the stability of the model has been studied. There is a critical value for delay parameter. Steady state is stable when the value of delay is less than the critical value and further increase the value of delay beyond the critical value makes the system oscillatory through Hopf-bifurcation. Whereas, another delay parameter has a stabilizing effect on the system dynamics. Chaotic dynamics has been explored in the model with the help of phase portrait and sensitivity on initial condition test. Numerical simulations are performed to validate the effectiveness of the derived theoretical results and to explore the various dynamical structures such as Hopf-bifurcation, periodic solutions and chaotic dynamics.

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Electrical activities of excitable cells produce diverse spiking-bursting patterns. The dynamics of the neuronal responses can be changed due to the variations of ionic concentrations between outside and inside the cell membrane. We investigate such type of spiking-bursting patterns under the effect of an electromagnetic induction on an excitable neuron model. The effect of electromagnetic induction across the membrane potential can be considered to analyze the collective behavior for signal processing. The paper addresses the issue of the electromagnetic flow on a modified Hindmarsh-Rose model (H-R) which preserves biophysical neurocomputational properties of a class of neuron models. The different types of firing activities such as square wave bursting, chattering, fast spiking, periodic spiking, mixed-mode oscillations etc. can be observed using different injected current stimulus. The improved version of the model includes more parameter sets and the multiple electrical activities are exhibited in different parameter regimes. We perform the bifurcation analysis analytically and numerically with respect to the key parameters which reveals the properties of the fast-slow system for neuronal responses. The firing activities can be suppressed/enhanced using the different external stimulus current and by allowing a noise induced current. To study the electrical activities of neural computation, the improved neuron model is suitable for further investigation.

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We study the spatiotemporal dynamics of a conductance-based neuronal cable. The processes of one-dimensional (1D) and 2D diffusion are considered for a single variable, which is the membrane voltage. A 2D Morris-Lecar (ML) model is introduced to investigate the nonlinear responses of an excitable conductance-based neuronal cable. We explore the parameter space of the uncoupled ML model and, based on the bifurcation diagram (as a function of stimulus current), we analyze the 1D diffusion dynamics in three regimes: phasic spiking, coexistence states (tonic spiking and phasic spiking exist together), and a quiescent state. We show (depending on parameters) that the diffusive system may generate regular and irregular bursting or spiking behavior. Further, we explore a 2D diffusion acting on the membrane voltage, where striped and hexagonlike patterns can be observed. To validate our numerical results and check the stability of the existing patterns generated by 2D diffusion, we use amplitude equations based on multiple-scale analysis. We incorporate 1D diffusion in an extended 3D version of the ML model, in which irregular bursting emerges for a certain diffusion strength. The generated patterns may have potential applications in nonlinear neuronal responses and signal transmission.

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We examine the dynamics of a spatially extended excitable neuron model between phase state and stable/unstable equilibrium point depending on the parameter regimes. The solitary wave profiles in the excitable medium are characterized by an improved Hindmarsh-Rose (H-R) spiking-bursting neuron model with an injected decaying current function. Linear stability and the nature of deterministic system dynamics are analyzed. Further investigation for the existence of wave using the reaction-diffusion H-R system and the criteria for diffusion-driven instabilities are performed. An approximation method is introduced to analyze traveling wave profiles for the oscillatory neuron model that allows the explicit analytical treatment of both the speed equations and shape of the traveling wave solution. The solitary wave profiles exhibited by the system are explored. The analytical expression for the solution scheme is validated with good accuracy in a wide range of the biophysical parameters of the system. The traveling wave fronts and speed equations control the variations of the information transmission, and the speed of signal transmission may be affected by the injection of certain drugs.

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Fear can influence the overall population size of an ecosystem and an important drive for change in nature. It evokes a vast array of responses spanning the physiology, morphology, ontogeny and the behavior of scared organisms. To explore the effect of fear and its dynamic consequences, we have formulated a predator-prey model with the cost of fear in prey reproduction term. Spatial movement of species in one and two dimensions have been considered for the better understanding of the model system dynamics. Stability analysis, Hopf-bifurcation, direction and stability of bifurcating periodic solutions have been studied. Conditions for Turing pattern formation have been established through diffusion-driven instability. The existence of both supercritical and subcritical Hopf-bifurcations have been investigated by numerical simulations. Various Turing patterns are presented and found that the change in the level of fear and diffusion coefficients alter these structures significantly. Holes and holes-stripes mixed type of ecologically realistic patterns are observed for small values of fear and relative increase in the level of fear may reduce the overall population size.

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Pyramidal neurons produce different spiking patterns to process information, communicate with each other and transform information. These spiking patterns have complex and multiple time scale dynamics that have been described with the fractional-order leaky integrate-and-Fire (FLIF) model. Models with fractional (non-integer) order differentiation that generalize power law dynamics can be used to describe complex temporal voltage dynamics. The main characteristic of FLIF model is that it depends on all past values of the voltage that causes long-term memory. The model produces spikes with high interspike interval variability and displays several spiking properties such as upward spike-frequency adaptation and long spike latency in response to a constant stimulus. We show that the subthreshold voltage and the firing rate of the fractional-order model make transitions from exponential to power law dynamics when the fractional order α decreases from 1 to smaller values. The firing rate displays different types of spike timing adaptation caused by changes on initial values. We also show that the voltage-memory trace and fractional coefficient are the causes of these different types of spiking properties. The voltage-memory trace that represents the long-term memory has a feedback regulatory mechanism and affects spiking activity. The results suggest that fractional-order models might be appropriate for understanding multiple time scale neuronal dynamics. Overall, a neuron with fractional dynamics displays history dependent activities that might be very useful and powerful for effective information processing.

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In the present paper, we propose and analyze an eco-epidemiological model with diffusion to study the dynamics of rabbit populations which are consumed by lynx populations. Existence, boundedness, stability and bifurcation analyses of solutions for the proposed rabbit-lynx model are performed. Results show that in the presence of diffusion the model has the potential of exhibiting Turing instability. Numerical results (finite difference and finite element methods) reveal the existence of the wave of chaos and this appears to be a dominant mode of disease dispersal. We also show the mechanism of spatiotemporal pattern formation resulting from the Hopf bifurcation analysis, which can be a potential candidate for understanding the complex spatiotemporal dynamics of eco-epidemiological systems. Implications of the asymptotic transmission rate on disease eradication among rabbit population which in turn enhances the survival of Iberian lynx are discussed.

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In this paper, we have proposed and analysed a mathematical model to figure out possible ways to rescue a damaged eco-epidemiological system. Our strategy of rescue is based on the realization of the fact that chaotic dynamics often associated with excursions of system dynamics to extinction-sized densities. Chaotic dynamics of the model is depicted by 2D scans, bifurcation analysis, largest Lyapunov exponent and basin boundary calculations. 2D scan results show that µ, the total death rate of infected prey should be brought down in order to avoid chaotic dynamics. We have carried out linear and nonlinear stability analysis and obtained Hopf-bifurcation and persistence criteria of the proposed model system. The other outcome of this study is a suggestion which involves removal of infected fishes at regular interval of time. The estimation of timing and periodicity of the removal exercises would be decided by the nature of infection more than anything else. If this suggestion is carefully worked out and implemented, it would be most effective in restoring the health of the ecosystem which has immense ecological, economic and aesthetic potential. We discuss the implications of this result to Salton Sea, California, USA. The restoration of the Salton Sea provides a perspective for conservation and management strategy.

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The mathematical models proposed and studied in the present paper provide a unified framework to understand complex dynamical patterns in vole populations in Europe and North America. We have extended the well-known model provided by Hanski and Turchin by incorporating the diffusion term and spatial heterogeneity and performed several mathematical and numerical analyses to explore the dynamics in space and time of the model. These models successfully predicted the observed rodent dynamics in these regions. An attempt has been made to bridge the gap between the field and theoretical studies carried out by Turchin and Hanski (1997) and Turchin and Ellner (2000). Simulation experiments, mainly two-dimensional parameter scans, show the importance of spatial heterogeneity in order to understand the poorly understood fluctuations in population densities of voles in Fennoscandia and Northern America. This study shed new light upon the dynamics of voles in these regions. The nonlinear analysis of vole data suggests that the dynamical shift is from stability to chaos. Diffusion driven model systems predict a new type of dynamics not yet observed in the field studies of vole populations carried out so far. This has been termed as chaotic in time and regular in space (CTRS). We observed CTRS dynamics in several simulation experiments. This directs us to expect that dynamics of this animal would be de-correlated in time and simultaneously mass extinctions might be possible at many spatial locations.

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Avian influenza, commonly known as bird flu, is an epidemic caused by H5N1 virus that primarily affects birds like chickens, wild water birds, etc. On rare occasions, these can infect other species including pigs and humans. In the span of less than a year, the lethal strain of bird flu is spreading very fast across the globe mainly in South East Asia, parts of Central Asia, Africa and Europe. In order to study the patterns of spread of epidemic, we made an investigation of outbreaks of the epidemic in one week, that is from February 13-18, 2006, when the deadly virus surfaced in India. We have designed a statistical transmission model of bird flu taking into account the factors that affect the epidemic transmission such as source of infection, social and natural factors and various control measures are suggested. For modeling the general intensity coefficient f ( r ) , we have implemented the recent ideas given in the article Fitting the Bill, Nature [R. Howlett, Fitting the bill, Nature 439 (2006) 402], which describes the geographical spread of epidemics due to transportation of poultry products. Our aim is to study the spread of avian influenza, both in time and space, to gain a better understanding of transmission mechanism. Our model yields satisfactory results as evidenced by the simulations and may be used for the prediction of future situations of epidemic for longer periods. We utilize real data at these various scales and our model allows one to generalize our predictions and make better suggestions for the control of this epidemic.