Narrow Results

The aim of this study is to analyze and investigate the COVID-19 transmission with effect of symptomatic and asymptomatic in the community. Mathematical model is converted into fractional order with the help of fractal fractional definition. The proposed fractional order system is investigated qualitatively as well as quantitatively to identify its stable position. Local stability of the COVID-19 system is verified and test the system is tested with flip bifurcation. Also the system is investigated for global stability using Lyapunov first and second derivative functions to see its rate of spread after recovery. The existence, boundedness and positivity of the COVID-19 are checked which are the key properties for such of type of epidemic problem to identify reliable findings. Effect of global derivative is demonstrated to verify its rate of effects according to their sub compartments to identify in which rate the symptomatic and asymptomatic transmission occurs. Solutions for fractional order system are derived with the help of advanced tool fractal fractional operator with generalized mittag-leffler kernel for different fractional values. Simulations are carried out to see symptomatic as well as asymptomatic effects of COVID-19 in the worldwide using MATLAB Coding. They show the actual behavior of COVID-19 especially for asymptomatic measures which will be helpful in early detection, also which will be helpful to understand the outbreak of COVID-19 as well as for future prediction and better control strategies.

This work examines a discrete Leslie–Gower predator–prey system with herd behavior, focusing on the influence of the slow–fast effect on predator dynamics. We analyze the presence and stability of fixed points, investigating period-doubling and Neimark–Sacker bifurcations at the positive fixed point. A hybrid control technique is implemented to successfully handle chaotic dynamics. Numerical simulations validate the theoretical findings, supporting our analysis. The parameter $𝜖$ in the predator–prey system has a significant impact on system dynamics due to the slow–fast effect it represents. A stable positive fixed point signifies a lasting and balanced coexistence of predator and prey populations within a certain $𝜖$ interval. Deviation from this range results in period-doubling and Neimark–Sacker bifurcations, increasing both unpredictability and instability in the system.

The SOS/ERK cascades are key signaling pathways that regulate cellular processes ranging from cellular proliferation, differentiation and apoptosis to tumor formation. However, the properties of these signaling pathways are not well understood. More importantly, how stochastic perturbations of internal and external cellular environment affect these pathways remains unanswered. To answer these questions, we, in this paper, propose a stochastic model according to the biochemical reaction processes of the SOS/ERK pathways, and, respectively, research the dynamical behaviors of this model under the four kinds of noises: Gaussian noise, colored noise, Lévy noise and fraction Brown noise. Some important results are found that Gaussian and colored noises have less effect on the stability of the system when the strength of the noise is small; Lévy and fractional Brownian noises significantly change the trajectories of the system. Power spectrum analysis shows that Lévy noise induces a system with quasi-periodic trajectories. Our results not only provide an understanding of the SOS/ERK pathway, but also show generalized rules for stochastic dynamical systems.

This study delves into the intricate interplay between social media platforms, interpersonal word-of-mouth communication, and the transmission dynamics associated with non-communicable diseases, with a particular emphasis on type 2 diabetes. Leveraging advanced mathematical modeling and epidemiological methodologies, our objective is to furnish a comprehensive understanding of how information dissemination through digital and interpersonal networks can impact the proliferation of such diseases within populations. We conduct sensitivity analysis to discern the pivotal model parameters that can wield a substantial influence on the dynamics of disease transmission and control. Moreover, we endeavor to explore the capacity of these model parameters to elicit stability or instability within the system. Our focus lies in the rigorous examination of Hopf and transcritical bifurcations within the system. Furthermore, we consider the influence of seasonal fluctuations in the growth rate of social media advertisements with an aim to discern its role in potentially instigating chaotic dynamics within the context of disease progression. In sum, this research seeks to offer a comprehensive and scientifically robust understanding of the patterns of type 2 diabetes and associated communicable diseases within the context of evolving digital communication landscapes.

The study addresses the global impact of COVID-19 by developing a mathematical model that combines within-host and between-host factors to better understand the disease’s dynamics. It begins by describing SARS-CoV-2 dynamics within individual human hosts using fractional-order differential equations. The model is shown to be Ulam–Hyers stable, ensuring reliable predictions. The research then investigates virus transmission from infected to susceptible individuals using agent-based modeling (ABM). This approach allows us to capture the diversity and heterogeneity among individuals, including variations in internal state of individuals, immune response and responses to interventions, making the model more realistic compared to aggregate models. The agent-based model places individuals on a square lattice, assigns health states (susceptible, infectious, or recovered), and relies on infected individuals’ viral load for transmission. Parameter values are stochastically generated via Latin hypercube sampling. The study further explores the impact of viral mutation and control measures. Simulations demonstrate that vaccination substantially reduces transmission but may not eliminate it entirely. The strategy is more effective when vaccinated individuals are evenly distributed across the population, as opposed to concentrated on one side. The research further reveals that while reducing transmission probability decreases infections by implementing prevention protocols, it does not proportionally correlate with the reduction magnitude. This discrepancy is attributed to the intervention primarily addressing inter-host transmission dynamics without directly influencing intra-host viral dynamics.

In this paper, a two-patch model with additive Allee effect, nonlinear dispersal and commensalism is proposed and studied. The stability of equilibria and the existence of saddle-node bifurcation, transcritical bifurcation are discussed. Through qualitative analysis of the model, we know that the persistence and the extinction of population are influenced by the Allee effect, dispersal and commensalism. Combining with numerical simulation, the study shows that the total population density will increase when the Allee effect constant $a$ increases or $m$ decreases. In addition to suppress the Allee effect, nonlinear dispersal and commensalism are crucial to the survival of the species in the two patches.

In an ecosystem, harvesting infected prey can assist in managing and containing the spread of the illness within the prey species. On the other hand, the harvesting of predators can be beneficial as it regulates their numbers, preventing them from over-consuming prey and subsequently preserving existence of the prey population. This study introduces a predator–prey model that encompasses prey infection, predator–prey interactions influenced by fear, switching and harvesting. We derive an analytic expression for the basic reproduction number, a critical determinant of disease spread. We investigate the global stability of disease-free and endemic equilibria contingent on the basic reproduction number’s value, highlighting the potential for disease eradication by maintaining it below unity. In-depth analysis of the deterministic model is undertaken, with a focus on Hopf bifurcations that delineate thresholds for disease-free and endemic states. Furthermore, the deterministic model is extended to incorporate environmental stochasticity. We obtain the conditions under which population extinction occurs. Our findings elucidate how the intensity of environmental noise influences population dynamics, providing valuable insights into extinction risks under varying noise levels.

In this paper, a single-species logistic model with both fear effect-type feedback control and additive Allee effect is developed and investigated using the new coronavirus as a feedback control variable. When the system introduces additive Allee effect and fear effect-type feedback control, more complicated dynamical behavior is obtained. The system can undergo transcritical bifurcation, saddle-node bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation. By numerical simulations, the system exhibits homoclinic bifurcation and saddle-node bifurcation of limit cycles as parameters are altered. Remarkably, it is the first time that two limit cycles have been discovered in a single-species logistic model with the Allee effect. Further, stronger Allee effect or stronger fear effect can lead to the extinction of the species population.

This paper investigates the dynamics of a new discrete form of the classical continuous Rosenzweig–MacArthur predator–prey model and the biological implications of the dynamics. The discretization method used here is to modify the continuous model to another with piecewise-constant arguments and then to integrate the modified model, which is quite different from the traditional Euler discretization method. First, the existence and local stability of fixed points have been thoroughly discussed. Then, all codimension-1 bifurcations have been studied, including the transcritical bifurcations at the trivial fixed point and the boundary fixed point, and the Neimark–Sacker bifurcation at the unique positive fixed point. Furthermore, it is proved that there are no codimension-2 bifurcations. Finally, the control of the bifurcations is studied. Regarding the biological significance, we have considered the following four aspects: When no harvesting effort is applied to both predators and prey, it is observed that providing more resources to the prey species leads to predator extinction, causing the ecosystem to lose stability. This implies that the paradox of enrichment occurs. When predators are harvested, the system exhibits the hydra effect, i.e. increasing the harvesting effort on predators will lead to an increase in the mean population density of predators. When the prey is harvested, numerical examples indicate that increasing the harvesting effort initially reduces prey density. Upon reaching the bifurcation point, prey density stabilizes while the predator density continues to decline. When both prey and predators are harvested, the biological significance is similar to the previous case. The outcomes of this paper have significant theoretical meaning in the study of population ecology. By MATLAB, the bifurcation diagrams, maximal Lyapunov exponent diagrams, phase portraits and mean population density curves are plotted. Numerical simulations not only show the correctness of the theoretical findings but also reveal many new and interesting dynamics.

This piece of work circles around the investigation of a predator–prey model that incorporates nonlinear harvesting and group defense, both of which are important phenomena occurring in the ecosystem. To analyze the proposed model, both analytical and numerical methods are employed. The model experiences a series of bifurcations, including saddle-node, Hopf and Bogdanov–Takens. The system exhibits bifurcations of codimension one and two but no bifurcation of codimension three emerged. Phase portrait diagrams are also presented to confirm the validity of both the analytical and numerical findings.

In this work, we study the Nanoparticles (NPs) impact on a phytoplankton–zooplankton interaction model with Ivlev-like and Holling type-II functional responses. We found that the growth rate of phytoplankton reduces due to NPs. In the non-spatial model, we investigated boundedness, stability, bifurcation and chaos. The stability criteria is determined using the Routh–Hurwitz criterion. Hopf bifurcation is demonstrated with parameter $K$, which represents the NPs carrying capacity while interacting with phytoplankton. The normal theory is used to examine the Hopf bifurcation direction and the stability of bifurcating periodic solutions. Moreover, the stability of non-hyperbolic equilibrium points have been determined using the Center Manifold theorem. Also, the parameter $\beta$, which represents the interaction rate between NPs and phytoplankton, exhibits chaotic behavior. Furthermore, we also investigated Hopf bifurcation and Turing instability in spatial model systems. This study demonstrates that NPs can influence the dynamics of the system in a balanced environment.

In this paper, a Michaelis–Menten-type predator harvesting predator–prey system with strong Allee effect on prey is proposed. The boundedness of the system is studied, and the existence of all possible equilibria is investigated. In particular, the system has at most four positive equilibria. The stability analysis of the coexistence positive equilibria is studied with the help of the graphical Jacobian method. Under some conditions, the bifurcations exhibited by the system are researched. With the support of Sotomayor’s theorem, saddle-node bifurcation and transcritical bifurcation are proven. The existence and the direction of the Hopf bifurcation have also been determined. The universal unfolding of the Bogdanov–Takens bifurcation of codimension 2 is calculated near the cusp. Our theoretical analysis is verified by numerical simulations. The results show that the strong Allee effect plays an important role in the stability of the system with respect to the bifurcation phenomenon, compared to the system without the strong Allee effect.

Analysis of a dynamic system helps scientists understand its properties and utilize it properly in different applications. This study analyzes the effects of various external excitements on a recently proposed mathematical neuron model derived from the original Fitzhugh–Nagumo model. Different bifurcation analyses on this system are conducted to detect chaotic behaviors that are common and of great importance in biological systems, considering the effects of different types of external excitements. Lyapunov exponents (LEs) confirm the existence of chaotic patterns. Furthermore, a bifurcation diagram that looks into the changes in the system dynamics caused by the simultaneous application of the external stimulants is represented. Neurons are bound to play a role in a network in which synchrony is an analytical quality. Therefore, the potential of a network of this model in showing synchronization is examined using the master stability function (MSF) technique. Ultimately, it is concluded that this neural model can produce chaotic behaviors and synchronous networks.

In this paper, we introduce and study an impulsive mutualistic model with variable coefficients. By using Floquent theorem, the invasion threshold and the stability of the boundary periodic solution are obtained. Furthermore, by using standard techniques of bifurcation theory, the existent condition of the positive periodic solution is obtained. Finally, numerical simulations are carried out to confirm the main theorems.

In this paper, we considered an SIR infectious disease model with two stages, immature and mature, with the time to maturity represented by a constant time delay. We obtain positivity and boundedness of solutions and analyze the equilibria and their stability properties.

Nowadays, biological resource in prey-predator ecosystem is commercially harvested and sold with aim of achieving economic interest. Furthermore, harvest effort is usually influenced by variation of economic interest of harvesting and spatial heterogeneous environment. In this paper, a delayed differential-algebraic bio-economic model is proposed, which is utilized to investigate interaction and coexistence mechanism of biological population in the harvested ecosystem due to the variation of economic interest of harvesting as well as the change of population spatial diffusion and gestation delay. Local stability analysis of the proposed model without gestation delay and diffusion reveals that there is a phenomenon of singularity induced bifurcation due to the variation of economic interest of harvesting, and state feedback controllers are designed to stabilize the proposed model at the interior equilibrium. Furthermore, local stability of the proposed model with gestation delay and diffusion is studied, which reveals that the interior equilibrium loses its stability at some critical values of gestation delay and corresponding cycle occurs. It is also shown that population spatial diffusion and harvesting have a stabilizing effect on population dynamics. Finally, numerical simulations are carried out to show consistency with theoretical analysis obtained in this paper.

An SIS epidemiological model in a population of varying size with two dissimilar groups of susceptible individuals has been analyzed. We prove that all the solutions tend to the equilibria of the system. Then we use the Poincaré Index theorem to determine the number of the rest points and their stability properties. It has been shown that bistability occurs for suitable values of the involved parameters. We use the perturbations of the pitchfork bifurcation points to give examples of all possible dynamics of the system. Some numerical examples of bistability and hysteresis behavior of the system has been also provided.

In this paper, a periodic Volterra model with mutual interference and impulsive effect is proposed and analyzed. By applying the Floquet theory of impulsive differential equation, some conditions are obtained for the linear stability of semi-trivial periodic solution. Some sufficient conditions are also given for the permanence of the system. Further, standard bifurcation theory is used to show the existence of coexistence state which arises near the semi-trivial periodic solution. Finally, theoretical results are confirmed by some special cases of the system.

This work deals with a three-dimensional system, which describes a food web model consisting of a prey, a specialist predator and a top predator which is generalist as it consumes the other two species. Using tools of dynamical systems we prove that the trajectories of system are bounded and that open subsets of parameters exist, such that the system in the first octant has at most two singularities. For an open subset of the parameters space, the system is shown to have an invariant compact set and this is a topologically transitive attractor set. Finally, we find another open set in the parameters space, such that the system has two limit cycles each contained in different invariant planes. The work is completed with a numeric simulation showing the attractor is a strange attractor.

Das et al. [Effect of disease-selective predation on prey infected by contact and external sources, Biosystems 95(3) (2009) 188–199] proposed an eco-epidemiological model where the prey species is infected through the external source of infection and contact of the species. In this present study we have modified their model by assuming that the predator consumes both the susceptible as well as the infected prey following the modified Holling type-II functional response. Our main focusing points of this study are the role of infection rate (both internal and external), alternative food, and half-saturation constant in the predator–prey dynamics with disease in the prey population. We have shown the local stability of the boundary as well as the interior equilibrium point under certain conditions. We have also worked out the permanence of the system. Our simulation results show that the system enters into limit cycle oscillations from stable position for higher values of the contact rate. But it is also shown that the external infection rate, enrichment of the alternative food of the predator population and the half-saturation constant can prevent limit cycle oscillations and stabilize the system. Thus external disease propagation, enrichment of the alternative food resource, and the half-saturation constant are the key factors for preventing the oscillatory behavior of the species.