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Few existing studies establish the fact that although human pathogen exists in the environmental reservoir, infectious disease outbreak is rare. In this article, we investigate the dynamics of cholera disease where the transmission occurs via contact with a contaminated reservoir. A deterministic iSIR compartmental epidemic model is formulated and explored the realistic situation, in which the quantity of ingested pathogens must exceed a critical threshold, the minimal infectious dose (MID), in order to infect healthy population. Bacteria growth rate is considered to follow the Allee effect. This model system shows multi-stability and undergoes transcritical and saddle-node bifurcations with respect to various parameters. Thus, external environmental noise can give rise to observable dynamics for the stochastic counterpart. One can see a noise-induced frequent transitions between two separate interior attractors, in particular for the cases where the deterministic model displays bistability. The global sensitivity using PRCC method is performed. Also, the optimal control for the developed deterministic cholera epidemic model is carried out with numerical illustrations. Our model primarily designed to explain and predict the indirectly transmitted cholera epidemic model incorporating the idea of MID. This study gives a different outlook for the transmission of disease from the human-centered perspective to socioeconomic context. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a mathematical model of plankton dynamics in an aquatic system. We consider the fact that phytoplankton take refuge and release toxic chemicals to reduce their predation by the zooplankton who is assumed to be a generalist predator. We make the system more realistic with the inclusion of seasonal variations of the model parameters representing input of nutrients, refuge taken by phytoplankton, toxins released by phytoplankton and supply of additional foods for zooplankton. The autonomous and nonautonomous systems are analyzed mathematically, and numerical simulations explore various dynamics of both the systems. In the absence of seasonal forcing, we observe that the system destabilizes on a gradual increase in the input rate of nutrients and recycling of dead biomass of phytoplankton into nutrients concentration. In contrast, refuge taken by phytoplankton, toxins liberation by phytoplankton and intraspecific competition among the species of phytoplankton have potentials to terminate the persistent oscillations, and bring stability in the ecosystem. Supply of additional foods to zooplankton in low (high) quantity creates instability (stability) in the planktonic system. Moreover, in the presence of seasonal forcing, we observe different dynamical features including periodic solution, higher periodic solutions, bursting patterns and boundary periodic solution. Overall, our findings not only provide insights about the sustainability of biodiversity by potentially explaining the emergence of seasonally recurring planktonic blooms, but also could be employed to assist develop management strategies to preserve and restore the integrity of an aquatic habitat. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of a stage-structured SI epidemic system with a saturated incidence rate and a saturated treatment function. Two discrete time delays are incorporated to represent the time required for immature to be mature, and the infected individuals to move into recover class, respectively. A thorough investigation of endogenous equilibrium states of the SI system has been conducted, and the characteristics of the dynamical system around these states, including local stability and Hopf bifurcation, have been studied. Using sensitivity analysis, the model is evaluated to determine which parameters play greater roles in the model results and subsequently which of them may be used to control the disease. Finally, some numerical simulations have been conducted to confirm the analytical results. [ABSTRACT FROM AUTHOR]

An e-epidemic SIRS (susceptible-infected-recovered-susceptible) model is proposed with a generalized non-monotone incidence rate characterizing the psychological effect of some devastating malware when the number of infected nodes are getting larger. Existence of unique and multiple equilibria, linear and non-linear stability with the help of basic reproduction number, and the nature of temporal system dynamics are analyzed. Bifurcation analyses (backward and forward transcritical bifurcation, Hopf bifurcation) are performed that are exhibited by the temporal system. It has been shown that saturation recovery of infected nodes along with the malware transmission rate lead to vital dynamics such as monostable and bistable, when basic reproduction number ( R 0 ) is less than unity. Further, state feedback controller is introduced along with the time delay for extending the region of stability and delaying the occurrence of Hopf bifurcation in the network. All the analytical results are validated through numerical simulation experiments. Numerically, it reflects that a larger value of inhibition factor, a and smaller value of γ can accelerate the malware removal and reduces the infection level in the network. Our study helps to understand how the psychological effect impacts on the network malware propagation process. The analysis and simulation results are useful for the policy-making insights of the anti-malware practices in the information sharing networks. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This article reports a method for the estimation of parameters of a 3D Hindmarsh Rose neural model under a noisy measurement. Estimation procedure is based on reparametrization of the 3D model in a linear form. Method of least square has been used to estimate the biophysical parameters of the model. It is known that the presence of unavoidable noise effects the estimation procedure. To reduce the influence of noise to a certain extent, a denoising algorithm based on local projection is considered. Estimation procedure has been derived both in noisy and denoised condition to present the effectiveness of the algorithm. The denoising technique has been applied to reduce the influence of noisy stimuli in an experimentally collected EEG data set and the results are presented in terms of reduction in variance level. The effectiveness of the method is presented using analytical/mathematical and simulation results. [ABSTRACT FROM AUTHOR]

Fear of top predators may be as or more important than direct killing in causing trophic cascades. The mere presence of top predators gives rise to a "landscape of fear", buffering lower trophic levels from overconsumption by herbivores. In the present work, we have proposed a three-species food chain model of wolf spiders (top predator), insect pests (herbivores), and plant community interactions incorporating the cost of fear of wolf spiders in the predation rate of insect pests. It is assumed that some of the insect pests migrate due to the fear of wolf spiders. Positivity, boundedness, and parameter regimes for an ecologically realistic parameter set is identified. We have derived the local and global stability conditions for the proposed system. The existence of Hopf-bifurcation, direction and stability conditions of the bifurcating periodic solutions have been discussed. It is observed that the fear of wolf spiders has a stabilizing impact on system dynamics. It contributes positively to the role of wolf spiders as a biocontrol agent. One and two-parameter bifurcation diagrams have been plotted. It is observed that fear level decreases with an increase in the migration rate and for relatively large values of conversion rate, insect pests become less sensitive toward the perceived predation risk. [ABSTRACT FROM AUTHOR]

The article focuses on the issue of a spatiotemporal excitable biophysical model that describes the propagation of electrical potential called spikes to model the diffusion-induced dynamics based on an analytical development of amplitude equations. Considering the Izhikevich neuron model consisting of coupled systems of ODEs, we demonstrate various results of spatiotemporal architecture (PDEs) using a suitable parameter regime. We analytically perform the saddle-node bifurcation and Hopf bifurcation analysis with bifurcating periodic solutions that show the transition phases in the system dynamics. We study different firing patterns both analytically and numerically by the formation of Riccati differential equation. To examine the characteristics of diffusive instabilities, we use Turing amplitude equations by multiscaling method. The instabilities and Turing bifurcation are established using theoretical analysis and numerical simulations. The spatial dynamics has potential effects on the deterministic system as a result of the diffusive matrices with various couplings, and the coupled oscillators with this nearest-neighbor coupling show synchronization measured by the synchronization factor analysis. Our results qualitatively reproduce different phenomena of the extended excitable system based with an efficient analytical scheme. [ABSTRACT FROM AUTHOR]

In this paper, we have proposed a model system of two preys that mutually contribute to each other with restricted growth rates upon arrival of a generalist predator (who can sexually reproduce) whose predation is followed by Holling type II functional response where three species are self-diffusing for their best of the survival. In spatial case, we investigate the conditions of the blow-up phenomena of sexually reproductive predator of generalist category at finite time. The stability of the system is studied, and we discuss about the conditions of Turing instability. Near bifurcation threshold, major slow down in the dynamical system has been noticed. Variation in the dynamical behaviour of the model system makes structural changes which suggests emergence of patterns. Lastly via numerical simulation, we illustrate different significant patterns such as spots, spots-stripes and stripes which are generated by variation of diffusion parameters and investigate the transition from chaotic to stability properties of the system. Finally, we explain our obtained results where prey exhibits mutual relation among themselves and investigate the populations of the alien/invasive species being fulminated in the ecosystem. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The objective of this paper is to propose a delayed susceptible-infectious-recovered (SIR) model for the transmission of porcine reproductive respiratory syndrome virus (PRRSV) among a swine population, including the latent period delay of the virus and the time delay due to the period the infectious swines need to recover. By taking different combinations of the two delays as the bifurcation parameter, local stability of the disease-present equilibrium and the existence of Hopf bifurcation are analyzed. Sufficient conditions for global stability of the disease-present equilibrium are derived by constructing a suitable Lyapunov function. Directly afterwards, properties of the Hopf bifurcation such as direction and stability are studied with the aid of the normal form theory and center manifold theorem. Finally, numerical simulations are presented to justify the validity of the derived theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a delayed tobacco smoking model containing users in the form of snuffing. Its dynamics is studied in terms of local stability and Hopf bifurcation by regarding the time delay as a bifurcation parameter and analyzing the associated characteristic transcendental equation. Specially, specific formulas determining the stability and direction of the Hopf bifurcation are derived with the aid of the normal form theory and the center manifold theorem. Using LMI techniques, global exponential stability results for smoking present equilibrium have been presented. Computer simulations are implemented to explain the obtained analytical results. [ABSTRACT FROM AUTHOR]

Many prey organisms have developed defense mechanisms against predation. If the predator can encounter in a limited spatial domain, the feeding rates reflect interference between predators and the corresponding functional response depends on both predator and prey densities. In this manuscript, an attempt has been made to understand the role of top predator interference and gestation delays on the dynamics of a three-species food chain model involving intermediate and top predators population. Interaction between the prey and an intermediate predator follows the Monod–Haldane functional response, while that between the top predator and its prey depends on Beddington–DeAngelis-type functional response. Analytically, we study the essential mathematical features such as boundedness, stability and direction of bifurcating periodic solution around the coexisting equilibrium for the model system. Numerically, we study the Hopf and transcritical bifurcations scenarios with respect to inhibitory effect of phytoplankton against zooplankton and death rate of fish population for the non-delayed system. Further, we study the stability behavior of the delayed model system. Model system exhibits irregular behavior when the interference is high or gestation period is larger than its critical value. Further, the system shows extinction of predators with the increase in inhibitory effect. Stability domain plots with respect to different important system parameters for non-delayed system and delay parameters for delayed system give a significant impact to study the stability of the different equilibrium points and bifurcation scenarios of both non-delayed and delayed systems. Our results point out the complexity of three-way interactions between phytoplankton, zooplankton and fish population in marine environment and highlight the role of predator interference and gestation delays in exhibiting the chaotic dynamics and extinction of predator population. [ABSTRACT FROM AUTHOR]

This paper considers a plankton–fish interaction model by comparing the dynamics between non-spatial and spatial systems. In the model, both the fish and phytoplankton populations are assumed to be growing logistically, the fish population is nonlinearly harvested, and the interaction between phytoplankton and zooplankton is described by the Crowley–Martin functional response. Stability analyses for the plankton–fish interaction model have been carried out for both non-spatial and spatial systems. The theoretical results are then supported by numerical simulations. In the case of non-spatial system, single-parameter bifurcation diagrams are used, while in the spatial system numerical simulations for one- and two-dimensional cases are performed. In the case of a non-spatial system, the conditions for the existence of a positive equilibrium point, as well as their local and global stability analyses, have been obtained. In the spatial system, the stability and existence of Hopf bifurcation have been discussed, and different conditions for Turing pattern information have been established through diffusion-driven instability analysis. Taken together, these results show that spatial heterogeneity, the mortality rate of phytoplankton, and the constant harvesting of the fish population play important roles in the dynamical behavior of the marine system. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, an attempt is made to understand the transmission dynamics of Ebola virus disease (EVD) incorporating fuzziness in all biological parameters due to its natural variability. To characterize the transmission trajectories of Ebola outbreak, we propose and analyze two SEIR and SEIRHD type transmission models. Using triangular fuzzy numbers for the imprecise parameters, we first study the existence of the equilibria and their stability. Both of the model have two equilibria, namely the disease-free and endemic. Stability of the disease-free and endemic equilibria is related with basic reproduction number that has been calculated from next generation matrix. Stability analysis of the system shows that the disease free equilibrium is locally as well as globally asymptotically stable when the basic reproduction number is less than unity. Under some additional conditions, the model system becomes locally asymptotically stable at unique endemic equilibrium when basic reproduction number is greater than unity. Finally, we perform some numerical experiments to justify the theoretical estimate. [ABSTRACT FROM AUTHOR]

Predator–prey interactions with stochastic forcing have been extensively investigated in the literature. However there are not many investigations of such models, that include prey defense. The goal of the current manuscript is to investigate a stochastic predator–prey model with mutual interference, and various Holling type functional responses, where the prey is able to release toxins as defense against a predator. This can also be generalized to include group or herd defense, toxin production and mimicry. We establish local and global existence for the stochastic model, and perform various numerical simulations to support our theoretical results. Our key result is that we have globally existing solutions independent of the magnitude of the toxin release parameter, or the predation rates. We also show that large enough noise intensity in solely the prey, can lead to extinction in the noisy model, for both species, whilst there is persistence in the deterministic model. [ABSTRACT FROM AUTHOR]

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 reduces the basic reproduction number as the control parameter u increases with nonlinear treatment, h(I) but also controls the spread of disease infection among the population. [ABSTRACT FROM AUTHOR]

The firing activities of multiple timescale dynamics for single neurons can be treated with fractional-order derivative. It has been shown in previous studies (theoretical and experimental) that the passive properties of membranes may be considered by fractional-order dynamics as it can produce various types of memory-dependent dynamics. Spiking and bursting play major role in information processing for cortical neurons. However, it is not completely clear to what extent the dynamics of fractional-order excitable systems may redesign the properties of excitable cells. It is demonstrated that the fractional dynamics of Izhikevich neuron model is capable to exhibit various oscillations for cortical neurons such as regular spiking and various bursting patterns, mixed mode oscillations, chattering, fast spiking. It characterizes various firing modes (hence the firing frequency) better than the classical-order model. It shows the dynamical differences between the integer-order and fractional-order system. The firing frequency is increased with the decrease of fractional exponents. Further, the responses of a network of excitatory and inhibitory Izhikevich neurons with a controlled parameter set show different dynamical behavior at various fractional orders and it controls the long-term interactions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We propose an e-epidemic time-delay Susceptible-Latent-Breaking out-Susceptible (S L B S ) model to study delay dynamics appearing due to antivirus software, which takes time to clean the viruses from latent and breaking-out computers. We perform nonlinear stability analysis, Hopf bifurcation analysis, and its direction and stability. Numerical simulation results (time series analysis and bifurcation diagram) give useful insights for delay dynamics. We investigate the effect of the control parameters like rate of infection of all the classes and cure rates on the model system. Our results suggest that time delay is responsible for destabilizing the system dynamics. For smooth functionality of a computer system, our results suggest the minimum use of removable storage devices like smart phones, optical discs, memory cards, external hard disk, digital cameras, and so on and use of effective antivirus softwares. [ABSTRACT FROM AUTHOR]

We are concerned with the Hopf bifurcation of an SVEIR computer virus model with time delay and nonlinear incident rate. First of all, by analyzing the associated characteristic equation we obtain sufficient conditions for its local stability and the existence of a Hopf bifurcation. Directly afterward, by means of the normal form theory and the center manifold theorem we derive explicit formulas that determine the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions. Finally, we carry out numerical simulations to illustrate and verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this work, an attempt is made to understand the dynamics of a modified Leslie-Gower model with nonlinear harvesting and Holling type-IV functional response. We study the model system using qualitative analysis, bifurcation theory and singular optimal control. We show that the interior equilibrium point is locally asymptotically stable and the system under goes a Hopf bifurcation with respect to the ratio of intrinsic growth of the predator and prey population as bifurcation parameter. The existence of bionomic equilibria is analyzed and the singular optimal control strategy is characterized using Pontryagin’s maximum principle. The existence of limit cycles appearing through local Hopf bifurcation and its stability is also examined and validated numerically by computing the first Lyapunov number. Optimal singular equilibrium points are obtained numerically for various discount rates. [ABSTRACT FROM AUTHOR]