Local Stability Of Equilibrium Points Research Articles

Mathematical simulations and sensitivity visualization of fractional order disease model describing human immunodeficiency

In this study, we explore a mathematical model of human immunodeficiency virus type 1 (HIV-1) infection in CD4+ T-cells, considering fractional-order dynamics in the Caputo sense. Fractional models are of great significance due to their capacity to predict disease outbreaks while incorporating memory and non-local characteristics. To establish the existence and uniqueness of solutions for the fractional model, a fixed-point theorem and an iterative technique are employed. Furthermore, it is confirmed that the solutions of the model are both positive and bounded. The basic reproduction number, denoted as R0, is calculated using the next-generation matrix approach, and the equilibrium points of the model, including disease-free and endemic equilibrium points, are presented. A sensitivity analysis of R0 is performed by varying different parameters. To analyze the local stability of equilibrium points, the Routh-Hurwitz technique is employed. The global stability of equilibrium points is demonstrated using the Lyapunov function and LaSalle's principle, as well as through UlamHyers (UH) stability and the generalized UH stability conditions. The proposed numerical scheme Adams-Bashforth predictor-corrector is validated by comparing it with the fourth-order Runge-Kutta (RK4) technique. We examine the influence of the fractional order (ξ) by conducting numerical simulations for different values of ξ using the Adams-Bashforth predictor-corrector approach. Furthermore, graphical presentations illustrate how different crucial model parameters impact disease dynamics. The results obtained from this study demonstrate that the adopted strategies significantly improve prediction accuracy.

A stochastic co-infection model for HIV-1 and HIV-2 epidemic incorporating drug resistance and dual saturated incidence rates

This study investigates a mathematical model encompassing both HIV-1 and HIV-2 epidemics, with a focus on drug resistance and dual saturated incidence rates. We begin by analyzing the local stability of equilibrium points in the deterministic system using Routh-Hurwitz and Jacobian matrix approaches. Furthermore, we establish the existence of a global positive solution for our model. We identify specific parameter conditions that lead to disease extinction. Additionally, we explore the persistence of HIV-1 and HIV-2 infections by examining their mean values. To assess the practical implications of our theoretical analysis, we conduct graphical evaluations under various noise disturbances, employing Milstein's Higher Order Method. This research contributes to a deeper understanding of the dynamics of HIV-1 and HIV-2 co-infection, with potential implications for treatment strategies and disease management.

Bifurcation and Stability Analysis of a New Fractional-Order Prey–Predator Model with Fear Effects in Toxic Injections

This paper proposes a prey–predator model affected by fear effects and toxic substances. We used the Lipschitz condition to prove the uniqueness of the model solution and Laplace transform to prove the boundedness of the model solution. We used the fractional-order stability theorem to provide sufficient conditions for the local stability of equilibrium points, and selected fractional-order derivatives as parameters to perform Hopf bifurcation analysis on the system. Finally, the theoretical results are verified via numerical simulation. The results show that a value of α will affect the stability of the system and that the population size and the effect of toxic substances have a huge impact on the stability of the system.

Dynamics and control of an SITR COVID-19 model with awareness and hospital bed dependency

In this article, the complex dynamics of susceptible-infected-treated-recovered (SITR) model describing the COVID-19 transmission is explored. Two types of transmission are included here, such as bi-linear with unaware susceptible individuals and nonlinear with aware susceptible individuals. Modified saturation recovery in treated individuals and a difference self-quarantined parameter in susceptible individuals have been considered. Sensitivity indices of the parameters associated with the reproduction number (R0) have been derived using the normalized forward sensitivity index to recognize the significant parameters in changing the disease dynamics. The criteria of existence and local stability of equilibrium points are discussed. The backward bifurcation, forward bifurcation and Hopf bifurcation are demonstrated analytically and numerically, indicating rich dynamics of the proposed model. Chaotic dynamics and supercritical Hopf bifurcation also have been investigated by computing the Lyapunov exponent and the first Lyapunov coefficient respectively. It is observed that, difference self-quarantined rate and psychological or awareness effect ensure the existence of only disease-free equilibrium for R0<1. Optimal control theory and Pontryagin’s maximum principle are applied to compute the efficiency of vaccination and isolation controls in lowering infection while minimizing the associated costs. According to the simulation results, applying both controls together is the most efficient way to prevent the spread of the disease, compared to the situation where a single control is used to achieve the same.

Kinetic Behavior and Optimal Control of a Fractional-Order Hepatitis B Model

The fractional-order calculus model is suitable for describing real-world problems that contain non-local effects and have memory genetic effects. Based on the definition of the Caputo derivative, the article proposes a class of fractional hepatitis B epidemic model with a general incidence rate. Firstly, the existence, uniqueness, positivity and boundedness of model solutions, basic reproduction number, equilibrium points, and local stability of equilibrium points are studied employing fractional differential equation theory, stability theory, and infectious disease dynamics theory. Secondly, the fractional necessary optimality conditions for fractional optimal control problems are derived by applying the Pontryagin maximum principle. Finally, the optimization simulation results of fractional optimal control problem are discussed. To control the spread of the hepatitis B virus, three control variables (isolation, treatment, and vaccination) are applied, and the optimal control theory is used to formulate the optimal control strategy. Specifically, by isolating infected and non-infected people, treating patients, and vaccinating susceptible people at the same time, the number of hepatitis B patients can be minimized, the number of recovered people can be increased, and the purpose of ultimately eliminating the transmission of hepatitis B virus can be achieved.

A delayed deterministic and stochastic [formula omitted] model: Hopf bifurcation and stochastic analysis

In this paper, we present a delayed deterministic and stochastic SIRICV models to investigate the effects of the white noise intensities and the waning immunity of vaccinated individuals in the evolution of the disease. For the deterministic SIRICV model, the basic reproduction number R0 and the equilibrium points are calculated. The local stability of equilibrium points is analyzed. Particularly, when R0<1 the disease-free equilibrium is locally stable for any positive value of τ. Furthermore, when R0>1, the local stability and sufficient conditions to ensure the occurrence of Hopf bifurcation for the endemic equilibrium point are established by considering the time delay τ as a bifurcation parameter. For the stochastic SIRICV model, the conditions of the extinction and persistence of the disease are given by using the stochastic basic reproduction numbers R0s and R0s∗. Numerical simulations are presented to enhance our analytical results and contrast the deterministic and stochastic models.

Local Stability of Equilibrium Point of the Tuberculosis Transmission Model

A crucial part of illness prevention over the past few decades has been played by mathematical models. The dynamic system can be used to characterize the TB infection process. For the purpose of developing future prevention strategies, it is crucial to comprehend the effect of vaccination approach on the control of TB. We investigated the impact of vaccination strategies on TB disease transmission through a dynamic model. The model discussed involves logistical population growth. The purpose of this discussion is to analyze the local stability of the equilibrium point of the TB disease transmission model. Numerical simulations are provided to illustrate the theoretical results. The existence and local stability of the model equilibrium point depends on the basic reproduction number analytically. Based on secondary data, the basic reproduction number values are 0.98 and 4.12, respectively. Numerical simulations for these two values support the analysis results obtained. If the basic reproduction number is less than one, then the transmission of TB disease can be eradicated. However, if the basic reproduction number is more than one, the vaccination strategy is not sufficient to control TB transmission.

Co-infection dynamics of two short term diseases with effect of recovery delay

A co-infection model with two short-term diseases with delay in recovery is proposed. Here, we consider the simultaneous transmission of infection does not happen but of simultaneous recovery from both illnesses. The system consists of four epidemiological classes populations, namely: susceptible (\(S\)), an infected class with the first disease (\(I_1\)), an infected class with the second disease (\(I_2\)), co-infected class (\(I_{12}\)). We have found all possible equilibrium states, and the basic reproduction number also examined their stability without and with delay. Analytically, we have established that the local stability of equilibrium points depends on the basic reproduction number in the absence of recovery delay. But with delay, it requires some additional conditions. We have also checked the effect of delay on stability of endemic steady state numerically and showed that beyond a critical threshold value of delay parameter, the system loses its stability, and Hopf bifurcation occurs. Finally, a numerical simulation presented supports the analytical findings.

Dynamical Behaviours of Stage-Structured Fractional-Order Prey-Predator Model with Crowley–Martin Functional Response

In this paper, the dynamic behaviour of the stage-structure prey-predator fractional-order derivative system is considered and discussed. In this model, the Crowley–Martin functional response describes the interaction between mature preys with a predator. The existence, uniqueness, non-negativity, and the boundedness of solutions are proved. All possible equilibrium points of this system are investigated. The sufficient conditions of local stability of equilibrium points for the considered system are determined. Finally, numerical simulation results are carried out to confirm the theoretical results.

THE EFFECT OF ALLEE FACTOR ON A NONLINEAR DELAYED POPULATION MODEL WITH HARVESTING

The Allee effect may have a stabilizing or destabilizing effect on population dynamics, or reaching stability may be delayed. Also this effect can push populations to extinction, especially under sharp harvesting effect. Compensatory or overcompensatory population maps are situations that change the effect of the Allee factor on population dynamics. The harvesting factor has also the effect of changing the equilibrium levels differently in theoretical population models. This study includes the local stability of the equilibrium point of the delayed discrete-time population model exposed to the harvest effect with and without Allee effect. We also review the effect of Allee factor on the local stability of equilibrium point of the discrete-time population model involving delay [1]. The dynamics of this population models are given together with the enriched dynamics.

On the Dynamics of a Discrete Fractional-Order Cournot–Bertrand Competition Duopoly Game

A discreet fractional-order Cournot–Bertrand competition duopoly game is introduced based on the fractional-order difference calculus of the Caputo operator. The model is designed when players can make long memory decisions. The local stability of equilibrium points is discussed for the proposed model. Some numerical simulations explore the model’s bifurcation and chaos by employing bifurcation diagrams, phase portraits, maximal Lyapunov exponents, and time series. According to our findings, the fractional-order parameter has an effect on the game’s stability and dynamics.

Optimal control for dengue eradication program under the media awareness effect

Abstract In this article, a mathematical model is proposed to assess the effects of media awareness on dengue eradication programs. First, the existence and local stability of equilibrium points are discussed using the concept of the basic reproduction number. Using the center-manifold theorem, it is shown that the proposed model always undergoes a forward bifurcation at the basic reproduction number equal to unity. It is observed that the high-intensity media awareness could reduce the size of the endemic equilibrium. Based on local sensitivity analysis, we identify the three most sensitive parameters, namely the natural death rate of mosquito (μ v ), infection rates (β h1, β v1), and hospitalization rate (η). Hence, control variables need to be introduced to increase/reduce these parameters. In this article, we use three different control variables, namely the media campaign, (u 1(t)), to reduce infection rates, additional hospitalization rate, (u 2(t)), and fumigation rate, (u 3(t)), to increase mosquitoes death rate. Pontryagin’s maximum principle is used to determine the optimal conditions. Some numerical simulations are performed to describe a possible scenario in the field. Cost effectiveness analysis is then conducted to determine the best strategy for the dengue eradication program. We conclude that a combination of media campaigns and fumigation is the most effective strategy to prevent a significant increase in the number of infected individuals.

Bifurcations, chaos, and multistability in a nonautonomous predator-prey model with fear.

Classical predator-prey models usually emphasize direct predation as the primary means of interaction between predators and prey. However, several field studies and experiments suggest that the mere presence of predators nearby can reduce prey density by forcing them to adopt costly defensive strategies. Adoption of such kind would cause a substantial change in prey demography. The present paper investigates a predator-prey model in which the predator's consumption rate (described by a functional response) is affected by both prey and predator densities. Perceived fear of predators leads to a drop in prey's birth rate. We also consider both constant and time-varying (seasonal) forms of prey's birth rate and investigate the model system's respective autonomous and nonautonomous implementations. Our analytical studies include finding conditions for the local stability of equilibrium points, the existence, direction of Hopf bifurcation, etc. Numerical illustrations include bifurcation diagrams assisted by phase portraits, construction of isospike and Lyapunov exponent diagrams in bi-parametric space that reveal the rich and complex dynamics embedded in the system. We observe different organized periodic structures within the chaotic regime, multistability between multiple pairs of coexisting attractors with intriguing basins of attractions. Our results show that even relatively slight changes in system parameters, perturbations, or environmental fluctuations may have drastic consequences on population oscillations. Our observations indicate that the fear effect alters the system dynamics significantly and drives an otherwise irregular system toward regularity.

Investigating the Impact of Social Awareness and Rapid Test on A COVID-19 Transmission Model

In this article, we propose and analyze a mathematical model of COVID-19 transmission among a closed population, with social awareness and rapid test intervention as the control variables. For this, we have constructed the model using a compartmental system of the ordinary differential equations. Dynamical analysis regarding the existence and local stability of equilibrium points is conducted rigorously. Our analysis shows that COVID-19 will disappear from the population if the basic reproduction number is less than one, and persist if the basic reproduction number is greater than one. In addition, we have shown a trans-critical bifurcation phenomenon based on our proposed model when the basic reproduction number equals one. From the elasticity analysis, we have observed that rapid testing is more promising in reducing the basic reproduction number as compared to a media campaign to improve social awareness on COVID-19. Using the Pontryagin Maximum Principle (PMP), the characterization of our optimal control problem is derived analytically and solved numerically using the forward-backward iterative algorithm. Our cost-effectiveness analysis shows that using rapid test and media campaigns partially are the best intervention strategy to reduce the number of infected humans with the minimum cost of intervention. If the intervention is to be implemented as a single intervention, then using solely the rapid test is a more promising and low-cost option in reducing the number of infected individuals vis-a-vis a media campaign to increase social awareness as a single intervention.

Stability, Multistability, and Complexity Analysis in a Dynamic Duopoly Game with Exponential Demand Function

In this paper, a discrete-time dynamic duopoly model, with nonlinear demand and cost functions, is established. The properties of existence and local stability of equilibrium points have been verified and analyzed. The stability conditions are also given with the help of the Jury criterion. With changing of the values of parameters, the system shows some new and interesting phenomena in terms to stability and multistability, such as V-shaped stable structures (also called Isoperiodic Stable Structures) and different shape basins of attraction of coexisting attractors. The eye-shaped structures appear where the period-doubling and period-halving bifurcations occur. Finally, by utilizing critical curves, the changes in the topological structure of basin of attraction and the reason of “holes” formation are analyzed. As a result, the generation of global bifurcation, such as contact bifurcation or final bifurcation, is usually accompanied by the contact of critical curves and boundary.

Stability analysis of SEIR model related to efficiency of vaccines for COVID-19 situation

This work is aimed to formulate and analyze a mathematical modeling, SEIR model, for COVID-19 with the main parameters of vaccination rate, effectiveness of prophylactic and therapeutic vaccines. Global and local stability of the model are investigated and also numerical simulation. Local stability of equilibrium points are classified. A Lyapunov function is constructed to analyze global stability of the disease-free equilibrium. The simulation part is based on two situations, the US and India. In the US circumstance, the result shows that with the rate of vaccination 0.1% per day of the US population and at least 20% effectiveness of both prophylactic and therapeutic vaccines, the reproductive numbers R0 are reduced from 2.99 (no vaccine) to less than 1. The same result happens in India case where the maximum reproductive number R0 in this case is 3.38. To achieve the same infected level of both countries, the simulation shows that with the same vaccine's efficiency the US needs a higher vaccination rate per day. Without vaccines for this pandemic, the model shows that a few percentages of the populations will suffering from the disease in the long term.

Energy mechanism analysis for chaotic dynamics of gyrostat system and simulation of displacement orbit using COMSOL

Most of the existing research on gyrostat systems uses the Melnikov method to analyze chaotic dynamic characteristics. The mechanism of chaotic motion from the energy perspective of gyrostat systems has rarely been analyzed. The gyrostat system with positive damping disturbance is modeled in this paper. Casimir energy and power are given. The mechanism of the orbital expansion and contraction of the gyrostat is revealed through the sign change of Casimir power and the ellipsoid of Casimir energy. The bifurcations of the Casimir power corresponding to the state bifurcation and Lyapunov exponent spectrum reflect the evolution and transition of gyrostat's dynamics. The energy-level leap expressed by the variance of Casimir energy is found to determine the bifurcation of dynamics for the first time. The coexistence of the gyrostat is explained using Casimir power and local stability of equilibrium point. The COMSOL finite element simulation platform is built to study the state changes of the displacement orbit of the frame gyrostat model. A comparison is performed for the displacement results conducted by the COMSOL and the numerical simulation for the gyrostat system kinematic model. The different displacement states of the gyrostat system under free motion and in the presence of damping disturbance and external torques are obtained.

A widespread interaction between generalist and specialist enemies: The role of intraguild predation and Allee effect

A natural food web system involves a diverse community of natural enemies and one of the widespread and common phenomena of such systems is intraguild predation (IGP) (i.e., presence of eating and killing among potential competitors). Here, we investigate the dynamics of a food web system with Allee effect and intraguild predation. In the present study, generalist and specialist natural enemies competing for a shared resource (prey) (intraguild predation) have been introduced. The Allee effect has also been incorporated in the shared prey growth rate. We investigate how parameters defining Allee effect and intraguild predation affect the long-term persistence, extinction and co-existence regions of such species. We outline the conditions under which different types of interior and non-interior equilibria exist and are locally stable. Bi-stable dynamics has also been investigated for the proposed model system for a suitable range of parametric values. A threshold condition on the strength of Allee effect has been obtained assuring the absence of IG predator population along with extinction region of shared prey. To understand the dynamics of the system, a comprehensive study of bifurcation analysis has also been provided taking Allee effect and fertility rate of intraguild predator as bifurcation parameters. These two parameters generate various interesting bifurcations like saddle-node bifurcation and Hopf-bifurcation. We have obtained different parametric regions of Allee parameter for the existence of different boundary and interior equilibria. All the analytical results related with local stability of equilibrium points and all possible successive bifurcations have been supported by different numerical examples, one and two parameter bifurcation diagrams, bi-stability diagram and stability regions of all possible equilibrium points. The impacts of Allee effect on co-existence, stability, extinction of species, their persistence, bistability and bifurcations have been explicitly discussed and the whole dynamics has also been successfully compared with the dynamics of food web without Allee effect. It is observed that the introduction of Allee effect and IG predator induce more rich dynamics and compel the system to be more sensitive to initial population densities.

Hopf bifurcation and stability analysis of the Rosenzweig-MacArthur predator-prey model with stage-structure in prey.

We consider a stage-structure Rosenzweig-MacArthur model describing the predator-prey interaction. Here, the prey population is divided into two sub-populations namely immature prey and mature prey. We assume that predator only consumes immature prey, where the predation follows the Holling type II functional response. We perform dynamical analysis including existence and uniqueness, the positivity and the boundedness of the solutions of the proposed model, as well as the existence and the local stability of equilibrium points. It is shown that the model has three equilibrium points. Our analysis shows that the predator extinction equilibrium exists if the intrinsic growth rate of immature prey is greater than the death rate of mature prey. Furthermore, if the predation rate is larger than the death rate of predator, then the coexistence equilibrium exists. It means that the predation process on the prey determines the growing effects of the predator population. Furthermore, we also show the existence of forward and Hopf bifurcations. The dynamics of our system are confirmed by our numerical simulations.

Optimal Investments for Burning and Recycling of Plastics

Now a day’s plastic has become necessity of human being. In this model, we have deliberated to budget in health policy and recycling of plastic garbage instead of plastic usage; using the system of non-linear differential equations. This study has been elaborated by optimizing investment policy. Health policy for the community and recycling policy in advantages of plastic usage have been invested with a limited budget. Existences of equilibrium points are computed as it play an important role for the decision maker. Local stability of equilibrium points has been performed to scrutinize this dynamical model. Global dynamical behavior is established using graph theory results. Invested optimality system is numerically simulated.