Equilibrium Point Research Articles

Understanding population dynamics and management strategies for a newly emerging pest Carea sp. in Eucalyptus plantations in Indonesia through a mathematical model

In this current study, we investigate the population dynamics of Carea sp. (Lepidoptera: Nolidae), a newly emerging pest in Indonesian Eucalyptus plantations, utilizing the MELP-SI-B mathematical model based on the first reported outbreak in 2022. The larval infestation of Carea sp. has been observed to stunt plant growth and can even lead to plant mortality. The proposed model incorporates pest management interventions, including chemical control with pesticides and biological control through the introduction of natural predators. Analysis of the deterministic compartment model revealed four locally asymptotically stable equilibrium points, validated using eigenvalue approach and application of the center manifold theorem. Bifurcation diagrams are presented to elucidate the relationships between all equilibrium points. An optimal control problem is explored within this model, considering three scenarios: chemical control without predators, simultaneous chemical control with predators, and chemical control followed by predator management. The results of the optimal control strategy analysis highlight the critical importance of timing the initial chemical control application for effective pest management. Notably, this study reveals that a combined approach involving chemical control (without predators) followed by predator management during the post-chemical control period yields the most favorable outcomes among the examined strategies.

Investigation of an optimal control strategy for a cholera disease transmission model with programs

Cholera is a disease of poverty affecting people with inadequate access to safe water and basic sanitation. Conflict, unplanned urbanization and climate change all increase the risk of cholera. In this article, an optimal control deterministic mathematical model of cholera disease with cost-effectiveness analysis is developed and analyzed considering both direct and indirect contact transmission pathways. The model qualitative behaviors, such as the invariant region, the existence of a positive invariant solution, the two equilibrium points (disease-free and endemic equilibrium), and their stabilities (local as well as global stability) of the model are studied. Moreover, the basic reproduction number of the model is obtained. We also performed sensitivity analysis of the basic parameters of the model. Then an optimal control problem is designed with a control functional having five controls: vaccination, treatment, environment sanitation and personal hygiene, and water quality improvement program. We examined the existence and uniqueness of the optimal controls of the system. Through the implementation of Pontryagin's maximum principle, the characterization of the optimal controls optimality system is established. The numerical simulation results the integrated control strategies demonstrated that strategy 2, 7, and 12 are effective programs to combat cholera disease from the community. Based on the local circumstances, available funds, and resources, it is recommended to the government stakeholders and policymakers to execute any one of the three integrated intervention programs.

A conformable fractional finite difference method for modified mathematical modeling of SAR-CoV-2 (COVID-19) disease.

In this research, the ongoing COVID-19 disease by considering the vaccination strategies into mathematical models is discussed. A modified and comprehensive mathematical model that captures the complex relationships between various population compartments, including susceptible (Sα), exposed (Eα), infected (Uα), quarantined (Qα), vaccinated (Vα), and recovered (Rα) individuals. Using conformable derivatives, a system of equations that precisely captures the complex interconnections inside the COVID-19 transmission. The basic reproduction number (R0), which is an essential indicator of disease transmission, is the subject of investigation calculating using the next-generation matrix approach. We also compute the R0 sensitivity indices, which offer important information about the relative influence of various factors on the overall dynamics. Local stability and global stability of R0 have been proved at a disease-free equilibrium point. By designing the finite difference approach of the conformable fractional derivative using the Taylor series. The present methodology provides us highly accurate convergence of the obtained solution. Present research fills research addresses the understanding gap between conceptual frameworks and real-world implementations, demonstrating the vaccination therapy's significant possibilities in the struggle against the COVID-19 pandemic.

Chaos and feasibility analysis of a healthy lifestyle’s sustainability using fractional-order polycystic ovarian syndrome’s effect on infertility

In this study, we developed a fractional-order model for polycystic ovarian syndrome issue transmission in women through four compartments with the Mittag-Leffler kernel. For biological feasibility, we discussed the positivity and boundedness of solutions with fixed point mappings in Banach spaces. Sensitivity analysis is performed using different variables as response functions each time, employing Latin Hyper-cube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) with 200 iterations. Further, the reproductive number is determined for the model’s biological feasibility with an analysis of parameter impact through Matlab software. The solutions are examined for global stability using the Lyapunov function with a first derivative. Chaos Control will use the regulate for linear responses approach to bring the system to stabilize according to its points of equilibrium. Numerical solutions are derived by using a Newton polynomial in the generalized form of the Mittag-Leffler kernel, which illustrates the effect of the fractional operator and parameters involved in the system. Results are also analyzed with different assumptions of parameter values and initial conditions to observe the different impacts of treatment to overcome the risk of infertility in women. Such type of investigation will be useful to investigate the spread of disease as well as helpful in developing control strategies to overcome the risk of problems and best management care pre-detection of the issue and for treatment.

Stability and Bifurcations in Time-Delay Systems: A Novel and Efficient Blend of Methods

The goal of this paper is to introduce a hybrid technique, combining two existing methods conveniently, to analyze the stability and bifurcations of equilibrium points and periodic orbits in delay-differential equations of retarded type. One method is called the frequency-domain approach, an analytical tool based on control theory allowing to detect and represent periodic solutions emerging from the Hopf bifurcations, via Fourier series and harmonic balances. The second one, the so-called semi-discretization method, provides a numerical scheme to approximate the monodromy operator of a periodic solution, thus permitting to establish the stability and bifurcations of limit cycles as well as analyzing the equilibrium points. It is shown that a proper combination of these methods provides a straightforward strategy to study both local and global bifurcations in time-delay systems. The usefulness of the proposed approach is shown by three examples, where the results are compared with those given by the software DDE-BIFTOOL.

Analysis of Rumor Propagation Model Based on Coupling Interaction Between Official Government and Media Websites

The COVID-19 pandemic has not only brought a virus to the public, but also spawned a large number of rumors. The Internet has made it very convenient for media websites to record and spread rumors, while the official government, as the subject of rumor control, can release rumor-refutation information to reduce the harm of rumors. Therefore, this study took into account information-carrying variables, such as media websites and official governments, and expanded the classic ISR rumor propagation model into a five-dimensional, two-level rumor propagation model that interacts between the main body layer and the information layer. Based on the constructed model, the mean field equation was obtained. Through mathematical analysis, the equilibrium point and the basic reproduction number of rumors were calculated. At the same time, stability analysis was conducted using the Routh Hurwitz stability criterion. Finally, a numerical simulation verified that when the basic regeneration number was less than 1, rumors disappeared in the system; when the basic regeneration number was greater than 1, rumors continued to exist in the system and rumors erupted. The executive power of the official government to dispel rumors, that is, the effectiveness of the government, played a decisive role in suppressing the spread of rumors.

Chaos and attraction domain of fractional Φ6‐van der Pol with time delay velocity

This article investigates the chaotic analysis and attractive domain of a fractional‐order ‐van der Pol with time delay velocity under harmonic excitation. Firstly, eight different types of bifurcation states of the system under different parameters are calculated by using the undisturbed system. Secondly, the Melnikov method is used to explore the effect of time delay velocity on the threshold of chaos in the Smale horseshoe sense under the double‐well potential and three‐well potential of the system. Finally, through numerical analysis of the phase diagram, bifurcation diagram, and maximum Lyapunov exponent, the influence of time delay velocity on system chaos is studied. The results indicate that an increase in the delay velocity coefficient will lead to the system transitioning from a chaotic state to a periodic state, while an increase in the delay velocity term will lead to the system transitioning from a periodic state to a chaotic state. In the study of system bifurcation, it is found that the position of the equilibrium points of the system changes during periodic motion. Therefore, cell mapping is used to draw the attractive domain of the system is studying the influence of initial conditions on the equilibrium point of the system and the results show that there is a close relationship between the attraction domain and the process of chaos occurrence.

Stability and qualitative dynamics of a class of three‐dimensional difference equation system of exponential type

In this paper, we study the boundedness, persistence, and global stability of the equilibrium point and rate of convergence of positive solutions of the exponential form of a system of difference equations where are constants and are initial values. Furthermore, some numerical examples are also given, which support our theoretical results.

Fractional Einstein–Gauss–Bonnet Scalar Field Cosmology

Our paper introduces a new theoretical framework called the Fractional Einstein–Gauss–Bonnet scalar field cosmology, which has important physical implications. Using fractional calculus to modify the gravitational action integral, we derived a modified Friedmann equation and a modified Klein–Gordon equation. Our research reveals non-trivial solutions associated with exponential potential, exponential couplings to the Gauss–Bonnet term, and a logarithmic scalar field, which are dependent on two cosmological parameters, m and α0=t0H0 and the fractional derivative order μ. By employing linear stability theory, we reveal the phase space structure and analyze the dynamic effects of the Gauss–Bonnet couplings. The scaling behavior at some equilibrium points reveals that the geometric corrections in the coupling to the Gauss–Bonnet scalar can mimic the behavior of the dark sector in modified gravity. Using data from cosmic chronometers, type Ia supernovae, supermassive Black Hole Shadows, and strong gravitational lensing, we estimated the values of m and α0, indicating that the solution is consistent with an accelerated expansion at late times with the values α0=1.38±0.05, m=1.44±0.05, and μ=1.48±0.17 (consistent with Ωm,0=0.311±0.016 and h=0.712±0.007), resulting in an age of the Universe t0=19.0±0.7 [Gyr] at 1σ CL. Ultimately, we obtained late-time accelerating power-law solutions supported by the most recent cosmological data, and we proposed an alternative explanation for the origin of cosmic acceleration other than ΛCDM. Our results generalize and significantly improve previous achievements in the literature, highlighting the practical implications of fractional calculus in cosmology.

Modeling and analysis using piecewise hybrid fractional operator in time scale measure for ebola virus epidemics under Mittag–Leffler kernel

This emerging infectious disease poses one of the most severe threats to public health in these locations, but there are not many reliable therapies yet. In this work, we developed the Ebola virus dynamics and control factors epidemic model with a piecewise hybrid fractional Operator in time scale measure insight of Mittag–Leffler kernel. Patterns and structures that repeat at various scales are the focus of fractal analysis, which has applications in complex systems such as biological ones. Both qualitatively and statistically, a proposed model with the Lipschitz criteria and linear growth is examined, considering positive solutions, boundedness, and uniqueness at equilibrium points with Leray-Schauder results under time scale ideas. The regulation for linear responses approach will be used by Chaos Control to stabilize the system after its equilibrium points. A fractional-order framework with a controlled design will be considered, where solutions are bounded in the feasible domain of relations of different compartments. Ulam–Hyers stability results in the solution are treated when function (constant or rising) for the component of qualitative inquiry in generalized form. The dynamical behaviors of the suggested model are discussed with the Newton polynomial approach used to implement on model in the sense of classical piecewise and Mittag–Leffler kernel at different fractional order values. The model shows that solutions are stable and confined within a feasible range, ensuring reliability. Through detailed simulations, it effectively captures how different interventions and infection rates influence Ebola spread. This fractional-order model enhances understanding of Ebola transmission, providing a strong basis for predicting outbreaks and planning effective control measures, with practical applications for analyzing real-world data.

Dynamic modelling and equilibrium manifold of multi‐converter systems: A study on grid‐forming and grid‐following converters in renewable energy power plants

AbstractThis study explores the optimal balance between grid‐forming (GFM) and grid‐following (GFL) converter capacities within power stations to ensure stable operations. The investigation introduces a novel, generic modelling approach for analysing multiple converter systems in the wind and photovoltaic power plants. The method aims to elucidate the dynamic characteristics of the converters in power plants, particularly focusing on the continuity and existence of the equilibrium manifolds and their impact on system stability. Findings reveal a pronounced difference in the recovery capabilities of GFM and GFL following synchronization losses, highlighting an asymmetry in their abilities. Specifically, GFL converters exhibit more effectiveness in reinstating synchrony after synchronization losses caused by GFM. Conversely, GFM demonstrates a lesser capacity to recover from synchronization losses induced by GFL. Furthermore, analysis indicates that when the capacity ratio of GFL to the system's short‐circuit capacity significantly exceeds that of GFM (exceeding a 1:5 ratio), the system experiences an absence of a stable equilibrium point, thereby affecting the synchronization stability of GFM. These conclusions have been validated through joint controller hardware‐in‐the‐loop testing.

Mathematical Model for Antimicrobial Resistance Transmission Dynamics in Hospitals

In this study, we formulate and analyze a mathematical model to describe the possible transmission routes of Antimicrobial Resistance (A.M.R) in a hospital setting. We have examined the significant means of transmission of resistance in hospitals and found that the significant means of transmission is the use of antibiotics and through the contamination by health care workers. It has been shown that the resistance free equilibrium point is locally asymptotically stable. We have also shown that the model has a unique positive endemic equilibrium point which is locally asymptotically stable.

Titik Equilibrium Bebas dari Penyebaran Hoax Pemilu Lewat Media Sosial Tiktok

Election issues are sensitive information among the society because they relate to people's survival in terms of work, economy and education. Election hoaxes where information whose truth cannot be ascertained becomes a weapon used by certain parties to bring down political opponents. Social media such as TikTok is a media that is often used to spread hoaxes because TikTok social media users come from various ages and there is no distance or time between users. In this paper, a mathematical model is presented to see how the equilibrium point is to be free from hoax based on the basic reproduction number R0. The spread of hoaxes on social media occurs if R0 > 1 and does not occur if R0 < 1. The model approach used is the SEIR mathematical model because it is seen that there are people who receive hoax videos (exposed) and then spread them without finding out the truth (infected), and users with this condition it needs to be recovered. The method used in this research is a literature review. The results obtained show that the mathematical model presented shows that the basic reproduction number R0 is zero (R0 < 1). This means that there will be no spread of election hoaxes

Numerical Simulation of SEIR Model for COVID-19 Infection

One of the most basic models in epidemiology, the SEIR (Susceptible-Exposed-Infectious-Recovered) model explains how viral infections spread through communities. This study analyzes the SEIR model's differential equations to learn more about its dynamic behaviors. This study discusses the 2019 corona virus disease (COVID-19) epidemic model using numerical methods of susceptible exposed infected recovered (SEIR). A numerical description of the notion is provided using two numerical approaches, including forward Euler and RK-4 approaches. The subplots of SEIR model are drawn by ode 45 commands. The stability of disease free equilibrium and Endemic equilibrium points are depicted. The significance of comprehending the interaction between epidemiological variables and population dynamics in developing efficient public health treatments is brought to light in this study, which investigates the consequences of these equations on the transmission and management of infectious diseases. Insights into the dynamics of illness transmission and potential measures for mitigating its effects are presented through the use of mathematical analysis and computational simulations. Jagannath University Journal of Science, Volume 11, Number 1, June 2024, pp. 172−185

A Novel Method for Analyzing Global Bifurcation and Global Hysteresis of FGM Truncated Conical Shell Structure

In the dynamic bifurcation and dynamic hysteresis of a system, our first consideration is the tiny changes in time that can lead to variations in the vector field, potentially causing the occurrence of system dynamic bifurcation and dynamic hysteresis. This paper presents a novel method for analyzing the global bifurcation and hysteresis of Functionally Graded Material (FGM) truncated conical shell structure under aerodynamics and in-plane force along meridian near internal resonances. This method involves introducing the ratio of vector fields as a periodic perturbation parameter in the system’s nonlinear second-order ordinary differential governing equations. This allows for the analysis of the nonlinear ordinary differential equations as algebraic equations, yielding algebraic expressions that only involve parameters and do not contain state variables. Subsequently, the global bifurcation set and global hysteresis set of FGM truncated conical shell structure are obtained, providing the parameters interval ranges for the stability and instability of FGM truncated conical shell structure. By utilizing MAPLE software for numerical simulation, the images of the global bifurcation set and global hysteresis set about the amplitude of in-plane load are first simulated numerically. Subsequently, the amplitude graphs of the equilibrium points are numerically simulated for validation. The results demonstrate a perfect alignment between the images of the global bifurcation set and global hysteresis set and the data from the amplitude graphs of the equilibrium point. Due to the periodicity of the periodic perturbation parameter, it shuttles between different persistent regions as other parameters change, leading to the generation of chaotic solutions in the governing equations. The validity of the obtained results is confirmed through comparisons with existing literature.

Qualitative and optimal control analysis of a two-serotype dengue model with saturated incidence in co-infection

ABSTRACT The arboviral disease dengue is a life-threatening global public health problem due to its high prevalence rate, morbidity and mortality. Recent studies suggest that morbidity during secondary dengue infection is higher than during primary infection. To deepen our understanding, we have explored a two-serotype co-infected dengue model with bilinear incidence for primary infection and saturated incidence for secondary infection. In the model, we have also included four control measures: mosquito bite prevention control, treatment control, vaccine control and mosquito killing control. The existence and local stability of the disease free, serotype II free, serotype I free and endemic equilibrium points have been studied. The effect of inhibitory or awareness factors on disease dynamics has also been exhaustively examined. By using sensitivity analysis we have identified and quantified the most influential parameters to prevent as well as to control the dengue outbreak. To optimize our strategies, we have characterized the time-dependent optimal control to reduce dengue prevalence and the total implementation cost of the aforementioned controls. Efficiency analysis has been done to identify the best effective combination of the applied controls to reduce dengue infection. Mosquito bites with treatment control is the most effective combined strategy to control dengue infection.

Persistence and zero-Hopf equilibrium in the tritrophic food chain model with Holling functional response

In this paper, we analyze the persistence of three species in a three-level food chain model. We characterize when such a model exhibits a zero-Hopf equilibrium point and show that it is possible only if the functional responses in the model are of type Holling III or IV.

The dynamics and behavior of logarithmic type fuzzy difference equation of order two.

Fuzzy difference equations are becoming increasingly popular in fields like engineering, ecology, and social science. Difference equations find numerous applications in real-life problems. Our study demonstrates that the logarithmic-type fuzzy difference equation of order two possesses a nonnegative solution, and an equilibrium point, and exhibits asymptotic behavior. [Formula: see text] Where, xi represents the sequence of fuzzy numbers, and the parameters α, β, A, along with the initial conditions x-1 and x0, are positive fuzzy numbers. The characterization theorem is employed to convert each single logarithmic fuzzy difference equation into a set of two crisp logarithmic difference equations within a fuzzy environment. We evaluated the stability of the equilibrium point of the fuzzy system. Utilizing variational iteration techniques, the method of g-division, inequality skills, and a theory of comparison for logarithmic fuzzy difference equations, we investigated the governing equation dynamics, including its boundedness, existence, and both local and global stability analysis. Additionally, we provided some numerical solutions for the equation describing the system to verify our results.

Analisis Dinamik Model Predator-Prey dengan Efek Pemanenan Pada Populasi Ikan Nike (Awaous Melanocephalus) di Provinsi Gorontalo

Gorontalo Province is one of the areas directly adjacent to Tomini Bay, which is known for its rich natural resources, especially in the fisheries sector such as Nike fish (Awaous melanocephalus). Nike fish is the main food commodity for people in the Gorontalo region. However, intensive fishing activities can pose an extinction threat to this species. This study aims to understand the dynamics of the Nike fish population using a predator-prey model that integrates the effects of harvesting on the prey population. The research process includes formulation of assumptions, model building, model validation, and data analysis through mathematical analysis and numerical simulation. The analysis showed that the model has four equilibrium points, each with different stability conditions. Numerical simulations showed that an increase in harvesting rate and natural mortality of prey, as well as a decrease in birth rate, could threaten the viability of the Nike fish population. Thus, appropriate management efforts are needed to maintain ecosystem balance and the sustainability of the Nike fish population. The calculation results show that the Nike fish production threshold in Gorontalo Province is 8.470.858 kg per year. However, the amount of Nike fish production in Gorontalo Province in 2023 only reached 56,75% of the threshold. This indicates that the production of Nike fish in Gorontalo can still be increased as long as the harvest limit is observed to prevent extinction.

Enchanting dynamics: Juvenile and adult predators with prey model in an intuitionistic fuzzy environment

This study represents an interaction model including juvenile and adult predators along with prey, involving key characteristics like the inability of juvenile predators to reproduce, the impact of adult predators on juvenile growth rates and the transformation of some juveniles to adulthood. Furthermore, it is supposed that adult predators intake prey for sustenance, and the logistic growth of prey is assumed in the non-appearance of adult predators. Intra-species competition among adult predators, together with predation of juveniles and chasing, is conveyed. To tackle the imprecise biological parameters, most parametric values are considered as triangular intuitionistic fuzzy numbers. The model is further processed into an intuitionistic fuzzy model using the Hukuhara derivative and Yager’s ranking method is implemented to make the model crispified. Non-negativity and boundedness of the model are explored, after that, an investigation of possible equilibrium points is discussed. Local and global stability analyses close to these points are supervised. The study accentuates the significant influence of uncertain model parameters on the solution. Especially, uncertain estimates of intra-species competition among adult predators remarkably regulate the model’s stability. Extended uncertainty in intra-species competition for prey, although, is discovered to stabilize the system. In addition, increased consumption of juvenile predators by adult predators is observed to provide the system’s instability.