Local Stability Analysis Research Articles

Stability properties of a Cournot-type oligopoly model with time delay

Abstract The paper studies a Cournot game involving n private firms, where the firms’ interactions are analyzed in a discrete-time setting with time delay. The past production levels of other private firms have an impact on the production levels of each private firm. We performed a local stability analysis on the nonlinear system associated with the model and found that it has only one equilibrium point, which is asymptotically stable if and only if some conditions involving the number of firms, the degree of product di erentiation and the time delay are fulfilled. In addition, numerical examples are presented in this paper, which illustrate the theoretical results obtained.

Within-host models of dengue virus transmission with immune response

AbstractDengue fever is an infectious viral fever. The complex behavior of the virus within the body can be explained through mathematical models to understand the virus’s dynamics. We propose two different with-in host models of dengue virus transmission with humoral immune response. The proposed models differ from one another because one of the models assumes that newly formed viruses infect healthy cells again. To understand the dynamics of the proposed models, we perform a comparative study of stability analysis, numerical simulation, and sensitivity analysis. The basic reproduction number (BRN) of the two models is computed using next-generation matrix method. The local stability (l.s) analysis is discussed using the linearization method. The Lyapunov’s direct method is used to check the global stability (g.s) of the models. It has been found that both the equilibrium states for both the models, namely, virus-free equilibrium state and endemic equilibrium state, are globally stable, based on the value of BRN. Results show the influence of immune response on the cell dynamics and virus particles. The virus neutralization rate by antibodies and rate that affects the antibody growth are highly sensitive for the two models. Optimal control is applied to explore the possible control strategies to prevent virus spread in the host system. It is evident from the results that the strategy to administrate antibiotic drugs and home remedies slow down the virus spread in the host.

Evolutionary game model based on cumulative prospect theory for information management mechanism in SIoT

As nodes in Social Internet of Things (SIoT) become more intelligent, malicious information occurs more frequently and spreads more widely. This problem can severely affect the trustworthiness of services and applications in SIoT. Methods to effectively control malicious information spreading in SIoT are essential and necessary. Reputation mechanism provides a powerful tool to tackle this challenge. In this paper, we propose a reputation-based mechanism to activate the self-purification capacity of the SIoT network by balancing information conflicts triggered by reporters and supporters. In order to find the best rewarding and punishment strategy, a bilateral cumulative-prospect-based evolutionary game model of SIoT network information conflict is constructed. Using local stability analysis and numerical simulation, the evolutionary trends of the proposed game model under different theoretical application scenarios are analyzed. The findings indicate that the basic income and deposit of both sides, the popularity of information as well as the importance of the conformity effect all have a significant impact on the system's steady state and evolutionary path. The specific conditions that both participating sides of the game tend to treat conflicts relatively rationally are analyzed. Dynamic evolution analysis and sensitivity analysis of selected parameters show that basic income is positively related to smart object's feedback strategies, while deposit is negatively related to that. While weight of conformity effect or the information popularity goes up, the rising of feedback probability is observed. Based on the above results, suggestions on dynamic reward and punishment strategies are given. The proposed model is a helpful attempt to model the evolution of information spreading in SIoT networks, with the ability to simulate several well-known regularities of message dissemination. Proposed model and suggested quantitative strategies can be helpful to build feasible malicious information control facilities in SIoT networks.

Stability and bifurcations in scalar differential equations with a general distributed delay

For a differential equation involving a general distributed time delay, a local stability and bifurcation analysis is performed, relying on fundamental properties of the characteristic function of the random variable whose probability density function is the delay distribution. Based on the root locus method, the bifurcation curves are determined in the considered parameter plane, also providing the number of unstable roots of the analyzed characteristic equation in each of the open connected regions delimited by these curves. This leads to a characterisation of the stability region of the considered equilibrium in the corresponding parameter plane. A Hopf bifurcation analysis is also completed in the general setting, and the criticality is analyzed by employing the method of multiple times scales. In contrast with some previously reported results from the literature, our analysis is accomplished in a general context and only then exemplified for particular types of delay distributions (e.g. Dirac, Gamma, uniform and triangular). The theoretical results are showcased in the framework of a simple neural model.

Dynamical behaviors of a delayed SIR information propagation model with forced silence function and control measures in complex networks.

Due to the advanced network technology, there is almost no barrier to information dissemination, which has led to the breeding of rumors. Intended to clarify the dynamic mechanism of rumor propagation, we formulate a SIR model with time delay, forced silence function and forgetting mechanism in both homogeneous and heterogeneous networks. In the homogeneous network model, we first prove the nonnegativity of the solutions. Based on the next-generation matrix, we calculate the basic reproduction number . Besides, we discuss the existence of equilibrium points. Next, by linearizing the system and constructing a Lyapunov function, the local and global asymptotically stability of the equilibrium points are found. In the heterogeneous network model, we derive the basic reproduction number through the analysis of a rumor-prevailing equilibrium point . Moreover, we conduct the local and global asymptotic stability analysis for the equilibrium points according to the LaSalle's Invariance Principle and stability theorem. As long as the maximum spread rate is large enough, the rumor-prevailing point is locally asymptotically stable when . Additionally, it hits that the system exists bifurcation behavior at due to the newly added forced silence function. Later, after adding two controllers to the system, we research the problem of optimal control. Finally, aimed at authenticating the above theoretical results, a serious of numerical simulation experiments are carried out.

Differential and Time-Discrete SEIRS Models with Vaccination: Local Stability, Validation and Sensitivity Analysis Using Bulgarian COVID-19 Data

Bulgaria has the lowest COVID-19 vaccination rate in the European Union and the second-highest COVID-19 mortality rate in the world. That is why we think it is important better to understand the reason for this situation and to analyse the development of the disease over time. In this paper, an extended time-dependent SEIRS model SEIRS-VB is used to investigate the long-term behaviour of the COVID-19 epidemic. This model includes vaccination and vital dynamics. To apply the SEIRS-VB model some numerical simulation tools have been developed and for this reason a family of time-discrete variants are introduced. Suitable inverse problems for the identification of parameters in discrete models are solved. A methodology is proposed for selecting a discrete model from the constructed family, which has the closest parameter values to these in the differential SEIRS-VB model. To validate the studied models, Bulgarian COVID-19 data are used. To obtain all these results for the discrete models a mathematical analysis is carried out to illustrate some biological properties of the differential model SEIRS-VB, such as the non-negativity, boundedness, existence, and uniqueness. Using the next-generation method, the basic reproduction number associated with the model in the autonomous case is defined. The local stability of the disease-free equilibrium point is studied. Finally, a sensitivity analysis of the basic reproduction number is performed.

Analysis of fractional order model on higher institution students’ anxiety towards mathematics with optimal control theory

Anxiety towards mathematics is the most common problem throughout nations in the world. In this study, we have mainly formulated and analyzed a Caputo fractional order mathematical model with optimal control strategies on higher institution students’ anxiety towards mathematics. The non-negativity and boundedness of the fractional order dynamical system solutions have been analysed. Both the anxiety-free and anxiety endemic equilibrium points of the Caputo fractional order model are found, and the local stability analysis of the anxiety-free and anxiety endemic equilibrium points are examined. Conditions for Caputo fractional order model backward bifurcation are analyzed whenever the anxiety effective reproduction number is less than one. We have shown the global asymptotic stability of the endemic equilibrium point. Moreover, we have carried out the optimal control strategy analysis of the fractional order model. Eventually, we have established the analytical results through numerical simulations to investigate the memory effect of the fractional order derivative approach, the behavior of the model solutions and the effects of parameters on the students anxiety towards mathematics in the community. Protection and treatment of anxiety infectious students have fundamental roles to minimize and possibly to eradicate mathematics anxiety from the higher institutions.

Modeling and analysis of sustainable approach for dynamics of infections in plant virus with fractal fractional operator

In this paper, studying a sustainable approach to see the dynamics of infection in the plant by using a fractal fractional derivative. To depict a time-fractional order plant virus model including disease consequences, we suggested a set of fractional differential equations. For the fractional order system, both qualitative and quantitative studies are carried out. To prove the existence and uniqueness of the proposed model with the effect of global derivative, Linear growth and Lipschitz conditions are used. Positiveness and boundedness of solutions of the fractional order model are verified. Local stability analysis and global stability is verified by using the Lyapunov function with the first and second derivative tests. Sensitivity analysis is performed to see the influence of parameters on the fractional order model. To study the effect of the fractional operator, which demonstrates the impact of the illness on plants, solutions are generated with a two-step Lagrange polynomial in the generalized form of the Mittag-Leffler kernel. To observe the behavior of the fractional order plant virus model, numerical simulation is used. Such an inquiry will help to comprehend how the virus behaves and to create defences against infected plants.

On the Solution of a Nonlinear Fractional-Order Glucose-Insulin System Incorporating β -cells Compartment

In this work, we are interested in studying variations in plasma glucose and insulin levels over time using a fractional-order version of a mathematical model. Applying the fractional-order Caputo derivative, we can investigate different concentration rates among insulin, glucose, and healthy β-cells. The main aim is to obtain the numerical solution of the proposed model in order to show variations in plasma glucose and insulin levels over time, by applying the generalized Euler method. The local stability analysis of the proposed (discretization) Caputo fractional-order model was discussed. To check the feasibility of our analysis, we have investigated some numerical simulations for various fractional orders by varying values of the parameters with help of Mathematica. Numerical simulations were in good agreement with the theoretical findings. Three specific numerical examples are given as applications of the main results.

The Qualitative Analysis of Host–Parasitoid Model with Inclusion of Spatial Refuge Effect

The objective of this work was to investigate the dynamics of host–parasitoid model with spatial refuge effect. For this, two discrete host–parasitoid models were considered under spatial refuge effect. Suppose that a constant population of hosts may seek refuge and protection from an attack of parasitoids. We found the parametric factors affecting the existence of the equilibrium points and uniqueness of equilibrium points. A local stability analysis of host–parasitoid models was also carried out. Bifurcation theory was used to observe that the host–parasitoid models undergo Neimark–Sacker bifurcation. The effect of the existence of constant refuge effect on the local stability and bifurcation of models was also explored. Hybrid chaos control methodology was used to control the chaotic behavior of model. In addition, numerical simulations, bifurcation diagrams, and phase portraits of the models are also presented.

Modelling and Analysis of a Measles Epidemic Model with the Constant Proportional Caputo Operator

Despite the existence of a secure and reliable immunization, measles, also known as rubeola, continues to be a leading cause of fatalities globally, especially in underdeveloped nations. For investigation and observation of the dynamical transmission of the disease with the influence of vaccination, we proposed a novel fractional order measles model with a constant proportional (CP) Caputo operator. We analysed the proposed model’s positivity, boundedness, well-posedness, and biological viability. Reproductive and strength numbers were also verified to examine how the illness dynamically behaves in society. For local and global stability analysis, we introduced the Lyapunov function with first and second derivatives. In order to evaluate the fractional integral operator, we used different techniques to invert the PC and CPC operators. We also used our suggested model’s fractional differential equations to derive the eigenfunctions of the CPC operator. There is a detailed discussion of additional analysis on the CPC and Hilfer generalised proportional operators. Employing the Laplace with the Adomian decomposition technique, we simulated a system of fractional differential equations numerically. Finally, numerical results and simulations were derived with the proposed measles model. The intricate and vital study of systems with symmetry is one of the many applications of contemporary fractional mathematical control. A strong tool that makes it possible to create numerical answers to a given fractional differential equation methodically is symmetry analysis. It is discovered that the proposed fractional order model provides a more realistic way of understanding the dynamics of a measles epidemic.

Switching effect on a two prey–one predator system with strong Allee effect incorporating prey refuge

In an environment, the food chains are balanced by the prey–predator interactions. When a predator species is provided with more than one prey population, it avails the option of prey switching between prey species according to their availability. So, prey switching of predators mainly helps to increase the overall growth rate of a predator species. In this work, we have proposed a two prey–one predator system where the predator population adopts switching behavior between two prey species at the time of consumption. Both the prey population exhibit a strong Allee effect and the predator population is considered to be a generalist one. The proposed system is biologically well-defined as the system variables are positive and do not increase abruptly with time. The local stability analysis reveals that all the predator-free equilibria are saddle points whereas the prey-free equilibrium is always stable. The intrinsic growth rates of prey, the strong Allee parameters, and the prey refuge parameters are chosen to be the controlling parameters here. The numerical simulation reveals that in absence of one prey, the other prey refuge parameter can change the system dynamics by forming a stable or unstable limit cycle. Moreover, a situation of bi-stability, tri-stability, or even multi-stability of equilibrium points occurs in this system. As in presence of the switching effect, the predator chooses prey according to their abundance, so, increasing refuge in one prey population decreases the count of the second prey population. It is also observed that the count of predator population reaches a comparatively higher value even if they get one prey population at its fullest quantity and only a portion of other prey species. So, in the scarcity of one prey species, switching to the other prey is beneficial for the growth of the predator population.

Impacts of Predation-Driven Allee Effect in a Predator–Prey Model

In the present work, we introduce one of the most significant biological factors — predator-driven Allee effect in a modified Leslie–Gower (LG) model. Our mathematical analyses include local stability analysis, Hopf bifurcation analysis, direction of Hopf bifurcation and Bogdanov–Tankens bifurcation around the coexisting equilibrium point. We also perform numerical simulations to validate our analytical results and to explore rich dynamics. Our results indicate that introducing predator-driven Allee effect not only destabilizes the system but also changes the characteristics of Hopf bifurcation from supercritical to subcritical. The system exhibits two types of bistability behavior between prey-free equilibrium and coexisting equilibrium, and prey-free equilibrium and limit cycle oscillation. Further, we explore an interesting result that the probability of extinction risk of prey population of modified LG system with predator-driven Allee effect is much higher in comparison to modified LG system when Allee term is absent. We have also demonstrated that predation-driven Allee effect increases extinction risk of prey population when the initial predator population size is large.

STABILITY ANALYSIS OF DYNAMICAL MODEL INTERACTIONS TEA PLANTS, PESTS, AND DISEASES WITH FUNGICIDES AND INSECTICIDES CONTROLS

Tea plants are one of the exports commodities in Indonesia. In their development, the plantation ecosystem is heavily influenced by several factors, both internal and external factors. In the field of applied mathematics, mathematical modelling can be used to analyze the development of tea plant growth and their interactions each other in their ecosystem. The mathematical model in this research is combining three main models, there are logistic model, epidemiological model, and predator prey model by adding fungicide and insecticide controls. Furthermore, local and global stability analysis is carried out and the optimal control problem is solved by Pontryagin maximum principle. The results of the analysis obtained five equilibrium points. Local stability analysis was carried out using the Routh Hurwitz criteria which showed the fifth equilibrium point is locally asymptotically stable. The basic reproduction number in the model is 0.99. Because 16R0<1" style="width: 28pt; height: 11pt;">, it can be concluded that there is no spread of disease in the tea plantation ecosystem after a period of 5 years. The control provided can reduce pest and disease attacks. After being given control, the population of infected tea plants decreased by 93.21%, Empoasca pests decreased by 99.47%, and leaf roller caterpillars decreased by 99.31% compared to the model that was not given control.

Energetic optimization and local stability of heliothermal plant models under three thermo-economic performance regimes

Solar power plants have the advantage of not producing greenhouse gases into the atmosphere. In their operation, some design and construction parameters can be identified for improving their performance. In this work within the context of Finite-Time Thermodynamics (FTT), we analyze a Müser–Curzon–Ahlborn heat engine model by maximizing economic-objective functions which not only provide optimal operation modes, but also they can reduce the energy dissipated in the form of heat towards the atmosphere. In our thermo-economic study we take into account both investment as well as operation and maintenance (O&M) costs. From operation data of three commercial tower-power plants: Eurelios, Gemasolar and Ivanpah, we find optimal values of solar concentration (Ci) for each economic-operation regime, accounting for the effect of internal irreversibilities (through a lumped parameter R). The Ci-values that we found are in the range of current reported values for the three power plants aforementioned. Besides, we present a local stability analysis for the three aforementioned solar plants and we show that for large size plants, disturbed isothermal branches return faster to their respective steady states, in less dissipative regimes, (e.g. the ecological regime).

Dynamic Behavior of a Predator–Prey Model with Double Delays and Beddington–DeAngelis Functional Response

In the predator–prey system, predators can affect the prey population by direct killing and predation fear. In the present study, we consider a delayed predator–prey model with fear and Beddington–DeAngelis functional response. The model incorporates not only the fear of predator on prey with an intraspecific competition relationship, but also fear delay and pregnancy delay. Apart from the local stability analysis of the equilibrium points of the model, we find that time delay can change the stability of the system and cause Hopf bifurcation. Taking time delay as the bifurcation parameter, the critical values of delays in several cases are derived. In addition, we extend it to the random environment and study the stochastic ultimate boundedness of the stochastic process. Finally, our theoretical results are validated by numerical simulation.

Modelling, Analysis, and Simulation of Measles Disease Transmission Dynamics

Measles is one of the top communicable diseases, which is still responsible for 2.6 million deaths every year. Due to this reason, the paper focuses on measles transmission dynamics concerning the impact of indirect contact rate (transmitted from the host of the virus to the healthy individual) and improving the SEVIR model into the SVIRP model. From the model, we first estimated the disease-free equilibrium, calculated the effective reproduction numberREff, and established the stability analysis. The Castillo–Chavez stability criterion is used to demonstrate the global stability of the disease-free equilibrium point, while the linearization method is used to justify its local stability analysis and gives a result REff<1. The stability analysis of endemic equilibrium point is explained by defining a Lyapunov function, and its global stability exists whenREff> 1. To identify the effect of parameters on the transmission dynamics, we performed sensitivity index and numerical simulation. From the result, we obtained that the indirect contact rate has the highest impact in maximizing the transmission dynamics of measles. Also, we found that working on prevention and treatment strategies brings a significant contribution in reducing the disease effect in the community.

Study of A Delayed SIVA Within-Host Model of Dengue Virus Transmission

During the process of immune response to the infection caused by dengue virus, antibodies are generated by plasma cells which are produced by B-cells. In some cases, it is observed that there is a delay in the production of plasma cells from B-cells which causes a delay in the immune response. We propose a SIVA within-host model of the virus transmission with delayed immune response to articulate the dynamics of the cell and virus population. The stability analysis of different equilibrium states is also studied. The basic reproduction number (BRN) of the model is computed using next generation matrix (NGM) method. The local stability analysis is discussed using the method of linearisation. The stability conditions of the equilibrium states are validated using the Li´enard - Chipart criterion. Hopf bifurcation analysis is carried out as the system has time lag in the immune response. Three equilibrium states, namely, virus free equilibrium state, endemic equilibrium state with and without immune response, have been observed. It has been found that the virus free equilibrium state is locally asymptotically stable if BRN is less than or equal to 1. Additionally, the conditions for the stability of the endemic equilibrium points are derived and elaborated. Numerical simulations for different values of time delay parameter τ are presented and illustrated using graphs. A Hopf bifurcation is observed if the delay parameter τ crosses a threshold value and then the system becomes unstable with periodic solution. To determine the relative importance of the model parameters to the virus transmission and prevalence, sensitivity analysis of the parameters is illustrated using graphs. Due to the time lag in the immune response, an increase in the virus growth is observed in large quantity. As a result, the infection spreads more quickly within the host.

Diffusion Mixture Minimum Total Error Entropy Adaptive Filtering Algorithm and Its Performance Analysis

The distributed adaptive filtering algorithms applied in the error-in-variables (EIV) model consider that both input and output signals suffer from noise disturbance for practical applications. However, the traditional distributed adaptive filtering algorithms applied in the EIV model cannot combat non-Gaussian noises well. To address this issue, this paper develops a mixture minimum total error entropy (MMTE) method and its diffusion mixture minimum total error entropy (DMMTE) algorithm. The proposed DMMTE has the significant performance improvement in non-Gaussian noises, and is more general than the diffusion minimum total error entropy (DMTE) algorithm. To select mixture coefficient effectively, a variable mixture coefficient method is further proposed to generate a variable DMMTE (VDMMTE) algorithm. Moreover, the local mean stability analysis and steady-state performance analysis of DMMTE are presented. Simulations verify the correctness of the theoretical analysis and illustrate the performance superiority of the proposed DMMTE.

MULTICOMPARTMENTAL ANALYSIS OF MIDDLE EASTERN RESPIRATORY SYNDROME CORONAVIRUS MODEL UNDER FRACTIONAL OPERATOR WITH NEXT-GENERATION MATRIX METHODS

Deterministic and fractional properties of an epidemic model for the dynamics of the Middle Eastern respiratory syndrome coronavirus (MERS-COV) with various infection stages are proposed in this study whose aim is to show via a mathematical model the transmission of MERS-COV between humans and camels, which are suspected to be the primary source of infection. The mathematical aspects together with biological feasibility of MERS-COV model are provided. The basic reproduction number [Formula: see text] has been calculated by using the next-generation matrix. With the help of [Formula: see text], we show the local and global stability analysis of the proposed model. Analysis of sensitivity for the threshold number is performed to understand the most sensitive parameter. Moreover, the concepts of strength number and second-order derivative of the Lyapunov function have been utilized for the waves detection. By using the concept of fixed point approach for the model, concerning if it really exists or not, we prove the existence and uniqueness results for the proposed model. The numerical solutions are obtained with the help of well-known fractional Adams–Bashforth technique. For the approximate solution, with the help of Runge–Kutta technique of order four, we accomplish the numerical simulations to support our analytical outcomes which are believed to have an effective impact on developing preventive measures for MERS-CoV, including disease control as well as prevention of spread and transmission in related populations.