Nontrivial Equilibrium Point Research Articles

Study of chronic myeloid leukemia with T-cell under fractal-fractional order model

Abstract This research work is devoted to investigate myeloid leukemia mathematical model. We give some details about the existence of trivial and nontrivial equilibrium points and their stability. Also, local asymptotical stability of disease-free and endemic equilibrium points is discussed. Also, positivity of the solution has been discussed. Some sufficient results are achieved to study the local existence and uniqueness of solution to the considered model for Mittag–Leffler kernel using the Banach contraction theorem. Three numerical algorithms are derived in obtaining the numerical solution of suggested model under three different kernels using Adams–Basforth technique. Numerical results have been presented for different fractals and fractional orders to show the behavior of the proposed model.

A mathematical model with uncertainty quantification for allelopathy with applications to real-world data

We revisit a deterministic model for studying the dynamics of allelopathy. The model is formulated in terms of a non-homogeneous linear system of differential equations whose forcing or source term is a piecewise constant function (square wave). To account for the inherent uncertainties present in this natural phenomenon, we reformulate the model as a system of random differential equations where all model parameters and the initial condition are assumed to be random variables, while the forcing term is a stochastic process. Taking extensive advantage of the so-called Random Variable Transformation (RVT) method, we obtain the solution of the randomized model by providing explicit expressions of the first probability density function of the solution under very general assumptions on the model data. We also determine the joint probability density function of the non-trivial equilibrium point, which is a random vector. If the source term is a time-dependent stochastic process, the RVT method might not be applicable since no explicit solution of the model is available. We then show an alternative approach to overcome this drawback by applying the Liouville–Gibbs partial differential equation. All the theoretical findings are illustrated through several examples, including the application of the randomized model to real-world data on alkaloid contents from leaching thornapple seed.

Modelling the adverse impacts of urbanization on human health

Abstract Urbanization has been a growing trend for decades, but the Coronavirus disease 2019 (COVID-19) pandemic has accelerated this trend in several ways. In many countries, the pandemic has prompted significant investments in urban health infrastructure, including hospitals, clinics, and public health facilities. Also, there are many studies suggesting that COVID-19, urbanization, and chronic diseases are interconnected in several ways, and addressing one aspect requires a comprehensive approach that considers the interconnectedness of these factors. To date, there are no studies depicting COVID-19 as a cause of emerging chronic diseases within the urban population. To address this sensitive issue, we propose a mathematical model in which we divide the total human population into three compartments: susceptible, diseased, and recovered. We have obtained the equilibrium points of the model and the conditions for their local stabilities. A suitable Lyapunov function has been used to derive the condition for the global stability of the nontrivial equilibrium point. A detailed numerical simulation of the model has been carried out, which is followed by sensitivity analysis of the model system.

Modeling and stability analysis of substance abuse in women with control policies

Substance abuse is considered to be predominantly a problem for all in society, and much substance abuse research is done with a male majority. Recent drug addiction studies, however, reveal that there are substantial gender variations in substance-related epidemiology. The epidemiology of women’s substance misuse raises issues distinct from those addressed by men’s substance abuse. Women with addictions are more likely than males to experience several impediments to accessing and entering substance misuse treatment. In this regard, we framed a mathematical model by considering the whole population as a women population. For this system, we found pair of equilibrium points namely the Addiction-free equilibrium point and the Non-trivial equilibrium point. Moreover, the addicts generation number (threshold parameter) RA has been found to investigate the spread of addiction. Also, the local and global stability analysis for both the equilibrium points have investigated. Further, to analyze the decrease and increase of the infected population and recovery population respectively, an optimum control analysis is done with two control parameters using Pontryagin’s maximum principle. The influence of addict generation numbers on significant parameters involved in RA has been shown. An optimum control study projected that social media awareness is substantially more efficient than the fixed control in optimizing drug addiction among women. Through numerical simulations we have exhibited, the effect of with and without control on each population. Finally, it produced positive efficacy by reducing the number of infected and increasing the rehabilitation population.

Bifurcación de Hopf en un sistema autónomo presa-predador con crecimiento logístico y respuesta funcional de Holling tipo II

The autonomous prey-predator system with logistic growth and Holling type II functional response, which describes the population dynamics of two species. In this study, the equilibrium points of the system (1.1) were identified. Two saddle-node points and one non-trivial P3 equilibrium point were found. For this latter point, the conditions were determined for the Jacobian matrix of (1.1), evaluated at P3, to have a pair of purely complex eigenvalues (necessary condition for the Hopf bifurcation). Through this analysis, values c0, δ0, k0 were found that satisfied these conditions. Subsequently, each of these values is considered as a bifurcation parameter value, and the remaining two are considered as control parameters, under the assumptions of the normal form theorem for the Hopf bifurcation, it’s concluded that by varying these values slightly, the system undergoes the Hopf bifurcation. Finally, the first Lyapunov coefficient was calculated to determine the conditions under which the system exhibits supercritical, subcritical, and degenerate Hopf bifurcation. The analysis was supported by using MAPLE and MATLAB software, which enabled graphical visualization of the obtained results.

Time delays and economic profit in fractional differential‐algebraic predator–prey model

In this paper, we formulate a fractional differential‐algebraic model with two discrete delays and double linear species harvesting. One of the delays denotes the time taken for digestion of the prey, and the other reflects the negative feedback of the predator's density. Taking into account of the economic profit, linear predator harvesting and prey harvesting have been incorporated into the proposed predator–prey system. From the biological viewpoint, we discuss the sufficient conditions for the existence of nontrivial positive equilibrium point. By jointly using the stability theory of fractional differential‐algebraic equations and the bifurcation theory, we study the delay‐induced instability and Hopf bifurcations. Finally, the correctness of the theoretical analysis is verified by numerical simulations, and the effects of delays, economic profit, and fractional exponents on the stability of the system are explored.

Memory effect on prey–predator dynamics: Exploring the role of fear effect, additional food and anti-predator behaviour of prey

Memory over past events has great importance on the survival of species in an ecological system. It is helpful in searching favourite food, escaping from predator attacks, group defence, use of environmental protection in the future time for the species. To explore the role of different ecological effects like fear effect, additional food and anti-predator behaviour of prey in the presence memory on an ecological system, in this paper we have proposed a two-dimensional prey–predator model and formulated mathematically using a system of two fractional order differential equations. Here the Caputo sense fractional derivative is used to construct the fractional order differential equations. The existence–uniqueness, non-negativity and boundedness of system solutions have been established. The detailed mathematical and graphical analysis for the feasibility of different equilibrium points and their analytical conditions for local asymptotic stability has been studied. The sufficient parametric conditions for the global stability of non-trivial equilibrium points are investigated. The Hopf bifurcation analysis of the system around the coexistence equilibrium point is derived in terms of system parameters. Using numerical simulations we have established that the memory effect can stabilize the system from an unstable periodic situation and makes the above-considered ecological effects more positively effective.

Stability Analysis of a Prey-Predator Model with Prey Refuge and Fear of Adult Predator

This paper is concerned with a Holling-II stage-structured predator-prey system in which predators are divided into an immature and mature predators. The aim is to explore the impact of the prey's fear caused by the dread of mature predators in a prey-predator model including intraspecific competitions and prey shelters. The theoretical study includes the local and global stability analysis for the three equilibrium points of the system and shows the prey's fear may lead to improving the stability at the positive equilibrium point. A numerical analysis is given to ensure the accuracy of the theoretical outcomes and to testify the conditions of stability of the system near the non-trivial equilibrium points.

Controlling of periodicity and chaos in a three dimensional prey predator model introducing the memory effect

In an ecological system, omnivores are often significantly important as they feed upon more than one trophic level in a food chain model. Intraguild predation is a special kind of omnivory that is ubiquitous in many ecological communities. A lot of field experiments suggest that the experience over a time duration in the recent past affects the growth rate of all species at the present time. Here, we have considered the Caputo type fractional ordered three-species food chain model. The memory effect is explained in terms of the order of the fractional derivative. Intermediate, top predators feed upon prey by Holling type I functional response and the top predator consumes intermediate predator by Holling type II functional response. Here, we established the existence and uniqueness of the solution of the considered system. Also, we prove the positivity and boundedness of the system solution. All possible feasible equilibrium points with required parametric conditions are discussed in the text. The local stability of all feasible equilibrium points and the parametric conditions of global stability of all non-trivial equilibrium points are discussed. The memory effect can stabilize the system from the periodic oscillatory situation which has been proved through the Hopf bifurcation analysis. Finally, some numerical simulation has been carried out to make significant comments on the dynamics of the considered system about the memory effect and other system parameters. It is clear from the numerical simulations that the order of fractional derivative can be taken as the indicator of chaos control. The biological interpretations make the article most significant from the ecological point of view.

Habitat Destruction and Restoration in Relation to Extinction and Survival of Species in Competitive Communities

We studied the N-species competitive coexistence model with direct effect on habitat destruction to analyze the behaviors of abundant and extinct species in the system caused by habitat loss. The nontrivial equilibrium points of the system are determined for a general habitat destruction function. For the trivial equilibrium, species that survived the habitat destruction are identified using eigenvalues of the Jacobian matrix. Solutions of the system are also presented using the recursive method. Three special cases of habitat destruction functions are addressed: continuous destruction, which is a typical habitat destruction; sudden habitat destruction, which is similar to natural phenomena such as earthquakes or floods; and sudden habitat destruction with aftershocks. The proportional abundances of 50 species are numerically portrayed in each case. We found that the survival of a species is guaranteed if its corresponding eigenvalue is positive. However, the fact that a species has negative corresponding eigenvalue does not guarantee its extinction, as this also depends on the initial number of that species.

Bifurkasi Pada Model Penyebaran Penyakit MERS-CoV di Dua Wilayah dengan Populasi Konstan

Middle East Respiratory Syndrome Coronavirus (MERS-CoV) is caused by a novel coronavirus and it can be a human-to-human transmission disease. World Health Organization (WHO) reported the disease outbreak first happened in Saudi Arabia in 2012 and the last case is reported in 2019. In 2018, MERS-CoV outbreaks were reported in the Republic of Korea, United Kingdom of Great Britain, Northern Ireland, Saudi Arabia, Uni Arab Emirates, Oman, and Malaysia. Cases that are identified outside the Middle East are usually caused by traveling people who were infected in the Middle East and then traveled back to their country. The previous research had constructed a mathematical model for the transmission of MERS-CoV in two areas by separating the human population into susceptible and infectious groups. It focused on the basic reproductive number and sensitivity analysis. In this paper, we simplify the model with the assumption that the total population of each area is constant. Using Lagrange Multiplier Method, we find some co-dimension one and co-dimension two bifurcations i.e.fold bifurcation and cusp bifurcation, respectively. We get the domain of parameters where three, two and one non-trivial equilibrium point occurs. We also find a transcritical bifurcation point such that the disease-free equilibrium point is stable on some parameter domains.

Food and environmental degradation as causative agents of honey bee colonies decline: Mathematical model approach

In this research, a new compartment model of honey bee population is developed to study the effects of gradual change of food availability and environmental degradation on bee population growth and development. The model is proved to be mathematical well posed and a non-trivial equilibrium point is shown to exist and asymptotically stable under certain conditions. The model predicts a critical threshold environmental degradation rate above which the population size of bees decline and subsequently collapse. Low environmental degradation and high food availability leads to stable bee population. Global sensitivity analysis is conducted to determine the most sensitive parameters of the model that can lead to colony collapse disorder. Numerical simulations are conducted to illustrate all the results.

SEIR Mathematical Model of Convalescent Plasma Transfusion to Reduce COVID-19 Disease Transmission

In some diseases, due to the restrictive availability of vaccines on the market (e.g., during the early emergence of a new disease that may cause a pandemic such as COVID-19), the use of plasma transfusion is among the available options for handling such a disease. In this study, we developed an SEIR mathematical model of disease transmission dynamics, considering the use of convalescent plasma transfusion (CPT). In this model, we assumed that the effect of CPT increases patient survival or, equivalently, leads to a reduction in the length of stay during an infectious period. We attempted to answer the question of what the effects are of different rates of CPT applications in decreasing the number of infectives at the population level. Herein, we analyzed the model using standard procedures in mathematical epidemiology, i.e., finding the trivial and non-trivial equilibrium points of the system including their stability and their relation to basic and effective reproduction numbers. We showed that, in general, the effects of the application of CPT resulted in a lower peak of infection cases and other epidemiological measures. As a consequence, in the presence of CPT, lowering the height of an infective peak can be regarded as an increase in the number of remaining healthy individuals; thus, the use of CPT may decrease the burden of COVID-19 transmission.

Qualitative analysis of a mathematical model about population of green turtles on the Galapagos island

According to the IUCN, most sea turtles fall into one of the endangered categories. Since, sea turtles, like many other reptiles, present an unusual developmental process, marked by the determination of the sex of the offspring by environmental factors, more specifically by temperature. In the temperature sex determination (TSD) system the temperature of an embryo's environment during incubation period will dictate the embryo's sex development. This developmental process, together with the complex mating and nesting behavior and the vulnerability of sea turtles to threats of a natural or anthropogenic nature, naturally lead to the study of the population dynamics of the species.? For this reason, in this paper, we have developed a continuous model given by a system of three ordinary differential equations to study the dynamics of the green sea turtle population long-term, focusing the mathematical simulations on the data obtained for the nesting species of Galapagos Islands. Through the qualitative analysis of the model, the following is demonstrated: 1) The flow induced by the system is positively invariant on the region of biological interest and 2). The given condition on is necessary and sufficient for the unique nontrivial equilibrium point to be globally asymptotically stable in that region. When implementing the estimated values for our parameters in the numerical simulations, it was observed that indeed the population of Galapagos green sea turtles complies with the condition for which the nontrivial critical point is globally asymptotically stable.

Bifurcation and Chaos Control of a System of Rational Difference Equations

We study a system of rational difference equations in this article. For equilibrium points, we present the stability conditions. In addition, we show that the system encounters period-doubling bifurcation at the trivial equilibrium point O and Neimark-Sacker bifurcation at the non-trivial equilibrium point E. To control the chaotic behavior of the system, we use the hybrid control approach. We also verify our theoretical outcomes at the end with some numerical applications

The Modified Palomba Economic Model by Difference Equations and its Stability Analysis

In this study, Palomba economic model was analyzed through difference equations. For these products given as capital and consumer goods in the model, their non-marketable loss rates were also taken into account. Tirivial and non-tirivial equilibrium points were found and local asymptotic stability (LAS) conditions of these equilibrium points were investigated. Although the non-trivial equilibrium point is always unstable, the conditions for the stability of the trivial equilibrium point were found. In this way, in addition to the thought put forward by Palomba, the absence of periodicity was expressed too. The findings were supported by numerical studies.

Multistability and circuit implementation of tabu learning two-neuron model: application to secure biomedical images in IoMT.

In this paper, the dynamics of a non-autonomous tabu learning two-neuron model is investigated. The model is obtained by building a tabu learning two-neuron (TLTN) model with a composite hyperbolic tangent function consisting of three hyperbolic tangent functions with different offsets. The possibility to adjust the compound activation function is exploited to report the sensitivity of non-trivial equilibrium points with respect to the parameters. Analysis tools like bifurcation diagram, Lyapunov exponents, phase portraits, and basin of attraction are used to explore various windows in which the neuron model under the consideration displays the uncovered phenomenon of the coexistence of up to six disconnected stable states for the same set of system parameters in a TLTN. In addition to the multistability, nonlinear phenomena such as period-doubling bifurcation, hysteretic dynamics, and parallel bifurcation branches are found when the control parameter is tuned. The analog circuit is built in PSPICE environment, and simulations are performed to validate the obtained results as well as the correctness of the numerical methods. Finally, an encryption/decryption algorithm is designed based on a modified Julia set and confusion–diffusion operations with the sequences of the proposed TLTN model. The security performances of the built cryptosystem are analyzed in terms of computational time (CT = 1.82), encryption throughput (ET = 151.82 MBps), number of cycles (NC = 15.80), NPCR = 99.6256, UACI = 33.6512, χ2-values = 243.7786, global entropy = 7.9992, and local entropy = 7.9083. Note that the presented values are the optimal results. These results demonstrate that the algorithm is highly secured compared to some fastest neuron chaos-based cryptosystems and is suitable for a sensitive field like IoMT security.

Student Group Dynamic Model Based on Understanding in Mathematics Subjects

This study discusses the interaction of students with a mathematical modeling point of view. This interaction involves students who understand and do not understand mathematics subject matter. The interaction process between groups is modeled in a two-dimensional system of differential equations. Variable A is the percentage of students who understand the material, and variable B is the percentage of students who do not understand the material. The dynamic analysis results obtained by one trivial equilibrium point and three non-trivial equilibrium points exist with several conditions. Based on the stability analysis of the non-trivial equilibrium point, it is found that the conditions without students do not understand mathematics subject matter. This condition is the goal of this study, which involves interaction between students; it can increase the learning process's success.

Stability analysis of nontrivial stationary solution and constant equilibrium point of reaction–diffusion neural networks with time delays under Dirichlet zero boundary value

In this paper, Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem and variational methods are employed to derive the existence and the unique existence results of globally exponentially stable (positive) stationary solution of delayed reaction–diffusion cell neural networks under Dirichlet zero boundary value, including the global stability criteria in the classical meaning. Next, sufficient conditions are proposed to guarantee the global stability invariance of ordinary differential systems under the influence of diffusions. New theorems show that the diffusion is a double-edged sword in judging the stability of diffusion systems. Besides, an example is constructed to illuminate that any non-zero constant equilibrium point must be not in the phase plane of dynamic system under Dirichlet zero boundary value, or it must lead to a contradiction. Next, under Lipschitz assumptions on active function, another example is designed to prove that the small diffusion effect will cause the essential change of the phase plane structure of the dynamic behavior of the delayed neural networks via a Saddle point theorem. Finally, a numerical example illustrates the feasibility of the proposed methods. It is worth mentioning that some interesting mathematical problems are originally put forward in the last chapter.

Complexity in time-delay networks of multiple interacting neural groups

<p style='text-indent:20px;'>Coupled networks are common in diverse real-world systems and the dynamical properties are crucial for their function and application. This paper focuses on the behaviors of a network consisting of mutually coupled neural groups and time-delayed interactions. These interacting groups can include different sets of nodes and topological architecture, respectively. The local and global stability of the system are analyzed and the stable regions and bifurcation curves in parameter planes are obtained. Different patterns of bifurcated solutions arising from trivial and non-trivial equilibrium points are given, such as the coexistence of non-trivial equilibrium points and periodic responses and multiple coexisting periodic orbits. The bifurcation diagrams are shown and plenty of complex dynamic phenomena are observed, such as multi-period oscillations and multiple coexisting attractors.</p>