Stability Of Equilibrium Point Research Articles

Stability Analysis for the Spread of Mumps Based on SIQR Model

Mumps is an infectious disease caused by a virus that is quite dangerous because vaccine strategies often fail. In severe cases, mumps can lead to infection of the covering of the brain (15%), pancreatitis (4%), permanent deafness, and painful swelling of the testes which rarely causes infertility. In addition, genetic mutations that are continuously carried out by the virus and the movement of individuals from one area to another can result in the spread of causing an epidemic in an area. This study discusses the spread of mumps by assuming there is a population of individuals who are quarantined. In addition, it is possible to die from mumps in the presence of complications from other diseases. The purpose of this research is to look at the behavior of the system solution analytically depending on the basic reproduction number ( ) accompanied by numerical simulations on initial values with certain parameters. The SIQR model is used by assumes that individuals who have recovered will get permanent immunity against Mumps. The analysis was carried out around the disease-free equilibrium point and the endemic equilibrium point. It was found that the disease-free equilibrium point is locally asymptotically stable with conditions . This means that if the birth rate is less than the natural death rate and the contact rate between individuals infected with mumps and susceptible individuals is less than the sum of the recovery rate, death rate and quarantine rate, then the mumps outbreak will eventually disappear from the population. On the other hand, the local asymptotically stable endemic equilibrium point if . This means that if the mumps disease is about to become epidemic and infected individuals are still in the population. This can be interpreted as if the birth rate is greater than the natural death rate and the contact rate between individuals infected with mumps and susceptible individuals is greater than the sum of the recovery rate, death rate and quarantine rate then the mumps outbreak will still exist in the population. However, if the value of the quarantine rate is increased, the value will decrease and the quarantined individual population approaches the infected individual population. This means that the proposed quarantine scenario with the addition of class Q makes the system solution better. Furthermore, a simulation was carried out with the Maple 18 program.

Mathematical modeling and analysis of a novel monkeypox virus spread integrating imperfect vaccination and nonlinear incidence rates

In this paper, we introduce a novel model that simulates the spread of the monkeypox virus. The new model takes into account the effect of the interaction between the human and rodent population along with some realistic factors that have not been introduced before such as imperfect vaccination and nonlinear incidence rates. Moreover, the human population is further divided into low-risk and high-risk groups to better reflect recent observations. To understand the dynamics of the new model, the existence, uniqueness, and continuous dependence on initial conditions, as well as its positivity and boundedness of the model are investigated in detail. The obtained solution is proved to be positive and has bounded behavior. Furthermore, the reproduction number R0 for the proposed system is computed and detailed bifurcation analysis is performed to reveal major qualitative changes in the model's dynamics. In addition, we investigate the influence of key parameters on the basic reproduction number. We also explore and analyze the stability of equilibrium points in the space of parameters. The new findings reveal that the virus is more frequently witnessed among the high-risk group, which may provide some restrictions to stop the spread through social limitations. To validate our theoretical results, we conduct numerical simulations, which provide insights into the behavior of the model under different conditions. The new findings aim in developing preventive control measures to suppress the spread of the virus and to develop effective strategies for controlling and preventing outbreaks, ultimately protecting public health and minimizing the impact of the disease.

The Volterra-Lyapunov matrix theory and nonstandard finite difference scheme to study a dynamical system

A compartmental model is considered to study the transmission dynamics of COVID-19. The proposed model is investigated for different results by using Volterra-Lyapunov (V-L) matrix theory. In this regard, first we presented a modified form of SEIR model by incorporating three new compartments C (protected), D (death due to corona) and Q (quarantined). Both equilibrium points are computed together with basic reproductive number. In addition, local stability of both equilibrium points for our proposed model is examined by assuming that wearing of mask, testing of the unaware infected individuals and medical care of the individuals that got infected should constantly be maintained. Hence, subsequently by combining the V-L stable matrix theory with the traditional methodology of constructing the Lyapunov functions, a procedure for the global stability analysis of COVID-19 is presented. Furthermore, based on LaSalle and Lipschitz invariance principle, the global stability of disease free equilibrium point is also examined. The technique we introduced in this paper will provide the more profound comprehension to understand the basic structure of COVID-19. Moreover, for numerical interpretation of our proposed model non-standard finite difference (NSFD) scheme is utilized for simulations. Different graphical illustrations are provided to understand the transmission dynamics.

Modeling and Analysis of Malware Propagation for IoT Heterogeneous Devices

With the rapid development and application of Internet of Things (IoT) technology, a large number of various types of IoT devices with security vulnerabilities have been connected to the public network. Attackers only need to use malware to successfully infect a few types of IoT devices to generate large-scale propagation in the network and capture a large number of broilers, which in turn lays the foundation for subsequent attacks. In this study, an innovative malware propagation model [i.e., IoT-susceptible-latent-propagated-recovered (IoT-SLPR)] based on epidemiological theory is developed for malware propagation in multiple IoT heterogeneous devices. The proposed model incorporates the propagation mechanism of malware in real IoT networks, and considers heterogeneous devices with different infection rates and recovery rates. This study calculates and proves the basic reproductive rate of malware and the stability of equilibrium points for different types of devices. In addition, this study designs a dynamic recovery rate optimal defense strategy, which comprehensively considers the cost and the overall infection rate of the device. The experimental results show that the proposed propagation model can effectively reflect the propagation dynamics of malware in multiple heterogeneous devices, and the achieved dynamic recovery rate defense strategy performs better than the static strategy regarding the overall results.

Modeling the dynamics of information propagation in the temporal and spatial environment

In this paper, we try to establish a non-smooth susceptible–infected–recovered (SIR) rumor propagation model based on time and space dimensions. First of all, we prove the existence and uniqueness of the solution. Secondly, we divide the system into two parts and discuss the existence of equilibrium points for each of them. For the left part, we define R 0 to study the relationship between R 0 and the existence of equilibrium points. For the right part, we classify many different cases by discussing the coefficients of the equilibrium point equation. Then, on this basis, we perform a bifurcation analysis of the non-spatial system and find conditions that lead to the existence of saddle-node bifurcation. Further, we consider the effect of diffusion. We specifically analyze the stability of equilibrium points. In addition, we analyze the Turing instability and Hopf bifurcation occurring at some equilibrium points. According to the Lyapunov number, we also determine the direction of the bifurcation. When I = I c , we discuss conditions for the existence of discontinuous Hopf bifurcation. Finally, through numerical simulations and combined with the practical meaning of the parameters, we prove the correctness of the previous theoretical theorem.

Hopf Bifurcation Analysis of a Continuous Investment Update Project Model

Some investment projects aim not only to produce goods but more importantly, to update and efficiently supply products in accordance with market demand. A double time delay differential dynamics model is formulated for continuous renewal investment projects based on the flowchart of the capital appreciation process and the assumed transfer functions. By analyzing the mathematical model, it can be determined that a unique local asymptotically stable positive equilibrium point exists for the continuous investment project. In accordance with the Hopf branching theorem, the model displays periodic behavior in proximity to its positive equilibrium point under certain conditions. The simulation results are compared under various conditions, and the validity of the relevant conclusions is confirmed.

ANALISIS KESTABILAN MODEL SVIQR PADA PENYEBARAN PENYAKIT DIFTERI DENGAN PENGARUH VAKSINASI DAN KARANTINA

Diphtheria is an acute disease caused by bacteria (Corynebacterium Diphtheriae). This disease is transmitted through the air and droplets (very small drops of fluid) from infected individuals. Vaccination can be carried out as a preventive measure so as not to be infected with bacteria (Corynebacterium Diphtheriae) and quarantine is carried out as a healing process because this disease is one type of disease included in the hospital-based STP (Integrated Disease Surveillance) data source. This study aims to compile and analyze a model of the spread of diphtheria using the SVIQR model. This model contains five subpopulations, namely susceptible (S), infected (I), cured (R), quarantined (Q) and vaccinated (V). Then determine the numerical simulation by estimating the parameters. From the results of numerical simulations of the mathematical model of diphtheria transmission under the influence of vaccination and quarantine, it shows that by selecting the vaccination fade rate parameter, =0.3795491181<1, which means that the disease-free equilibrium point is stable. The asymptotically stable endemic equilibrium point obtained is =1.207495030>1. The smaller the value of the vaccination fading rate parameter ε, the more asymptotically stable the disease-free point means that the spread of disease or endemic disease can be prevented if the vaccine given does not fade easily.
 Keywords: Stability Analysis, Mathematical Models, Diphtheria, Quarantine, Vaccination.

Emergency Volunteer Participation in the Evolutionary Game of Public Security Governance under Community Incentives

The outbreak of the COVID-19 epidemic has brought profound changes to all aspects of our society and also reflects the importance of community emergency volunteers actively participating in epidemic prevention and control in the face of unexpected public security events. As a bridge between the implementation of government policies and the masses of the community, community emergency volunteers have the characteristics of high efficiency and low cost, which has a great impact on the advancement of modern social governance. In order to motivate volunteers, the community will introduce incentive mechanisms. How does the evolutionary process of a dynamic game between volunteer engagement and community motivation change? How should communities maximize the service investment of volunteers in the game process? However, the current research rarely focuses on the role of community volunteers in the modernization of Community Governance. In order to clarify this game process, this article constructs a public safety governance incentive game model consisting of communities and emergency volunteers. Based on evolutionary game theory, we obtain the evolutionary stable equilibrium point by solving the replicator dynamic equations of all parties in the dynamic system under different constraints. Finally, some numerical examples were provided to simulate the selection of agents. The research results show that the degree of community public security risk, the degree of active involvement of volunteers, the degree of inactive involvement of volunteers, and the level of community incentives have an important impact on the enthusiasm of volunteer community service investment decision-making behavior. In addition, the choice of community incentive-volunteer service investment strategy is a dynamic process, which can converge to the ideal state under certain conditions after continuous adjustment and optimization. In addition, this study puts forward suggestions and measures conducive to the game between both sides, which can provide valuable guidance for the practice of community public security governance and the improvement of government efficiency in China.

Modeling the Impact of Pollution on Sea Turtles

Human activities are currently threatening sea turtles at all life stages, both on nesting beaches and at sea. The debris and toxic waste dumped on the coast or at sea pollutes the sea and puts marine life in danger. In recent years, the number of global turtle population have noticeably decreased, and this is largely due to plastic pollution. However, how the sea pollution affects the sea turtles’ populations is not fully understood. Therefore, in this study, using the mathematical model, we will investigate the impact of pollution on sea turtle population. The model system is analyzed using standard mathematical techniques, including positivity of solutions and stability analysis. Our findings showed that there are two possible equilibrium points (i.e. steady-state solutions) for the model proposed, in which the stability analysis showed that only one of the solutions is asymptotically stable. Thus, the conditions of stability for both equilibrium points were also derived analytically based on their eigenvalues. As for the numerical simulations, the parameter of contamination rate is varied to investigate the effect of pollution on the population of sea turtles. The results suggested that if the contamination rate is high, then the population of sea turtles are expected to decrease and extinct approximately within 10 years. The comparison of survival and extinction of sea turtles are shown using the time series plots.

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.

Modeling self-propagating malware with epidemiological models

Self-propagating malware (SPM) is responsible for large financial losses and major data breaches with devastating social impacts that cannot be understated. Well-known campaigns such as WannaCry and Colonial Pipeline have been able to propagate rapidly on the Internet and cause widespread service disruptions. To date, the propagation behavior of SPM is still not well understood. As result, our ability to defend against these cyber threats is still limited. Here, we address this gap by performing a comprehensive analysis of a newly proposed epidemiological-inspired model for SPM propagation, the Susceptible-Infected-Infected Dormant-Recovered (SIIDR) model. We perform a theoretical analysis of the SIIDR model by deriving its basic reproduction number and studying the stability of its disease-free equilibrium points in a homogeneous mixed system. We also characterize the SIIDR model on arbitrary graphs and discuss the conditions for stability of disease-free equilibrium points. We obtain access to 15 WannaCry attack traces generated under various conditions, derive the model’s transition rates, and show that SIIDR fits the real data well. We find that the SIIDR model outperforms more established compartmental models from epidemiology, such as SI, SIS, and SIR, at modeling SPM propagation.

COMBINED IMPACT OF CANNIBALISM AND ALLEE EFFECT ON THE DYNAMICS OF A PREY–PREDATOR MODEL

In ecological environment, Allee effect is one of the important factors which cause significant changes to the system dynamics. In this paper, using the theory of dynamical systems, we analyze a variation of a standard cannibalistic two-dimensional prey–predator model with Holling type-II functional response in the presence of both weak and strong Allee effects. We have analyzed the impact of strong and weak Allee effects on the dynamics of a cannibalistic system, knowing the dynamics of the cannibalistic model without Allee effect. We have deduced that in the presence of cannibalism, both strong and weak Allee effects generate bistability between equilibrium points. For strong Allee effect, bistability occurs between trivial equilibrium point and predator-free equilibrium point as well as between trivial and coexistence equilibrium points. But for weak Allee effect, bistability occurs only between coexistence equilibrium points. We also pointed out that the cannibalistic system without Allee effect exhibits tristability among the trivial equilibrium point, coexistence equilibrium point having low prey concentration and coexistence equilibrium point having comparatively high prey concentration. But in the presence of strong Allee effect, cannibalistic system experiences tristability among trivial and two other stable coexistence equilibrium points. By a comprehensive bifurcation analysis, we have observed that Allee effect enriches both the local and global dynamics of the system. Here, we have reported all possible codimension-one and codimension-two bifurcations extensively by choosing cannibalism, Allee effect and predator natural death rate as the bifurcation parameters. In the analysis of bifurcations, we have explored the existence of transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and Bautin bifurcation. Our analytical findings are validated through exhaustive numerical simulations. Finally, we have reported a comparative study between the impacts of strong and weak Allee effects on the dynamics of the cannibalistic system.

Quasi-periodic structure in chaotic bursting attractor for a controlled jerk oscillator

The main purpose of the paper is to show that parts of the trajectory for chaotic bursting oscillations may oscillate according to different limit cycles, which implies orderly movement exists in chaos. For a typical jerk oscillator with a cubic nonlinearity, when a slow-varying controlling term is introduced, bursting oscillations may occur, appearing in the combination of large-amplitude and small-amplitude oscillations. Two coexisting stable equilibrium points may evolve to chaos via different cascades of period-doubling bifurcation, respectively, which means different types of stable attractors may coexist with the variation of slow-varying parameter, while several periodic windows exist in the chaotic region. It is found that the spiking oscillations of the full system may change according to different cycles not only for 2-D torus bursting oscillations but also for the chaotic bursting attractor. Furthermore, whether the influence of the short periodic window of the fast subsystem on the bursting oscillations appears or not depends on the size of the slow scale, since the inertia of the movement may cause the delay effect of the bifurcations at both two edges of the window. The findings suggest that coupling of multiple scales may be one of key factors for the diversity of real world.

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.

Analyzing the stability and sensitivity of the deterministic SVEQ1IQ2R model for the spread of COVID-19

Covid-19 has become a pandemic in various countries since 2019. The spread of Covid-19 is easy and fast through droplets that come out when coughing, sneezing, or talking. Several ways to overcome the spread are conducting home quarantine, medical quarantine, carrying out health protocols, and vaccinating. This study proposed a deterministic model called SVEQ1IQ2R, which considers home and medical quarantine and vaccination. The model divides the population into seven compartments, namely susceptible (S), exposed (E), vaccinated (V), home quarantine (Q1), infected (I), medical quarantine (Q2), and recovered (R). This study aims to analyze the stability of equilibrium points and the sensitivity of the proposed model. The stability analysis results show that disease-free and endemic equilibrium points are locally asymptotically stable using Routh-Hurwitz criteria and Castillo-Chavez and Song theorems, respectively. Moreover, the sensitivity analysis shows that reducing the contact rate and increasing vaccine effectiveness are the most appropriate steps to take to reduce the spread of Covid-19. Based on numerical simulations, increasing the vaccine effectiveness and the vaccination rate as well as reducing the contact rate can make the transmission of Covid-19 resolved more quickly.

Bifurcation analysis and simulations of a modified Leslie–Gower predator–prey model with constant‐type prey harvesting

The paper discusses the effects of constant prey harvesting on the dynamics of a modified Leslie–Gower predator–prey model. Prey harvesting will lead to enriched dynamics to help us realize ecological phenomena such as the extinction of some species with a positive initial density when the harvesting rate is larger than a critical level. All these may be very useful for biological management. Based on these reasons, firstly, we analyze the existence conditions and stability of different equilibrium points to predict the eventual state of the given system. In particular, the existence of cusps of Codimensions 2 and 3 is proved. Secondly, we successfully demonstrate the presence of Hopf bifurcation and saddle‐node bifurcation for specified parameter values. To prove the limit cycle stability of Hopf bifurcation, we use the first Lyapunov coefficient for illustration. As well as calculating the universal expansion near the cusp, the Bogdanov–Takens bifurcations of Codimensions 2 and 3 are investigated. If the parameters are chosen fittingly, there will be a stable or an unstable limit cycle, a homoclinic loop, two limit cycles, or both a limit cycle and a homoclinic loop simultaneously in the system. Finally, numerical simulations are performed using MATLAB to illustrate the theoretical results.

Fractional dynamics of a Chikungunya transmission model

In this paper, we introduce a Caputo-sense fractional derivative to an existing model for the Chikungunya transmission dynamics, replacing integer derivative. After reviewing previous work on the integer order model, we suggest a change whereby the integer derivative is replaced by the Caputo derivative operator, and then we obtain results about the asymptotic stability of equilibrium points in this new fractional model. We prove the existence and uniqueness of the solution as well as the global stability of the fractional system. We then construct a numerical scheme for the fractional model, and use parameter values from the Chikungunya epidemic in Chad to perform numerical simulations. This allows us to validate our theoretical results and compare the dynamics behaviour of both models. We find from numerical simulations, that for the fractional-order parameter η in the range (0.85;1), daily detected cases are closer to those the model predict. Thus in the case of Chikungunya epidemic in Chad, the model with fractional derivative produces better results than which with integer derivative.

Dinamika dan Stabilitas Populasi pada Model Penyebaran COVID-19 dengan Vaksinasi dan Migrasi Penduduk

This article discusses the mathematical model of the spread of COVID-19 by considering vaccination and population migration. The former model is analyzed by determining the equilibrium point, basic reproduction number, analyzing the stability of the equilibrium point, sensitivity analysis, and accompanied by numerical simulation. Analysis of the stability of disease-free and endemic equilibrium points using the Routh-Hurwitz Criteria and the Castillo-Chaves and Song theorems. The results of the analysis show that there are two equilibrium points, namely a disease-free equilibrium point (T1), which is locally asymptotically stable when R0 1, and an endemic equilibrium point (T2), which is locally asymptotically stable when R0 1. Furthermore, the sensitivity analysis showed that the most sensitive parameters to changes in the basic reproduction number were the emigration rate parameter (m2) and the infection probability parameter after contact between infected and susceptible individuals without vaccination (h). In addition, the numerical simulation results show that the sensitive parameter values, namely m2, h, zse, g, and # have a significant effect on the basic reproduction numbers. Suppressing the chance of infection in susceptible individuals and the rate of contact between susceptible and exposed individuals, as well as increasing the number of individuals who emigrate and who are vaccinated, can reduce the transmission of COVID-19.

Complex Dynamics and Chaos Control of a Discrete-Time Predator-Prey Model

The objective of this study is to investigate the complexity of a discrete predator-prey system. The discretization is achieved using the piecewise constant argument method. The existence and stability of equilibrium points, as well as transcritical and Neimark–Sacker bifurcations, are all explored. Feedback and hybrid control methods are used to control the discrete system’s bifurcating and fluctuating behavior. To validate the theoretical conclusions, numerical simulations are performed. The findings of the study suggested that the discretization technique employed in this investigation preserves bifurcation and displays more effective dynamic consistency in comparison to the Euler method.

Impact of Cooperation and Intra-Specific Competition of Prey on the Stability of Prey–Predator Models with Refuge

The main objective of this study is to find out the influences of cooperation and intra-specific competition in the prey population on escaping predation through refuge and the effect of the two intra-specific interactions on the dynamics of prey–predator systems. For this purpose, two mathematical models with Holling type II functional response functions were proposed and analyzed. The first model includes cooperation among prey populations, whereas the second one incorporates intra-specific competition. The existence conditions and stability of different equilibrium points for both models were analyzed to determine the qualitative behaviors of the systems. Refuge through intra-specific competition has a stabilizing role, whereas cooperation has a destabilizing role on the system dynamics. Periodic oscillations were observed in both systems through Hopf bifurcation. From the analytical and numerical findings, we conclude that intra-specific competition affects the prey population and continuously controls the refuge class under a critical value, and thus, it never becomes too large to cause predator extinction due to food scarcity. Conversely, cooperation leads the maximal number of individuals to escape predation through the refuge so that predators suffer from low predation success.