Stability Analysis Of Equilibria Research Articles

Effect of Fear in Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response Incorporating Prey Refuge

In the present paper, we study the effect of antipredator behavior due to fear of predation on a modified Leslie-Gower predator-prey model incorporating prey refuge which predation rate of predators follows Beddington-DeAngelis functional response. The biological justification of the model is demonstrated through non-negativity, boundedness, and permanence. Next, we perform the analysis of equilibrium and local stability. We obtain four equilibrium points where two points are locally asymptotically stable and other points are unstable. Besides, we show the effect of the fear in the model and obtain a conclusion that the increased rate of fear can decrease the density of both populations, and prey populations become extinct. Meanwhile, for the case with a constant rate of fear, the prey refuge helpful to the existence of both populations. However, for the case with the fear effect is large, prey refuge cannot cause the extinction of predators. Several numerical simulations are performed to support our analytical results.

Nonlinear stability and numerical simulations for a reaction–diffusion system modelling Allee effect on predators

Abstract A reaction–diffusion system governing the prey–predator interaction with Allee effect on the predators, already introduced by the authors in a previous work is reconsidered with the aim of showing destabilization mechanisms of the biologically meaning equilibrium and detecting some aspects for the eventual oscillatory pattern formation. Extensive numerical simulations, depicting such complex dynamics, are shown. In order to complete the stability analysis of the coexistence equilibrium, a nonlinear stability result is shown.

Mathematical model of COVID-19 in Nigeria with optimal control

The novel Coronavirus Disease 2019 (COVID-19) is a highly infectious disease caused by a new strain of severe acute respiratory syndrome of coronavirus 2 (SARS-CoV-2). In this work, we proposed a mathematical model of COVID-19. We carried out the qualitative analysis along with an epidemic indicator which is the basic reproduction number (R0) of this model, stability analysis of COVID-19 free equilibrium (CFE) and Endemic equilibrium (EE) using Lyaponuv function are considered. We extended the basic model into optimal control system by incorporating three control strategies. These are; use of face-mask and hand sanitizer along with social distancing; treatment of COVID-19 patients and active screening with testing and the third control is prevention against recurrence and reinfection of humans who have recovered from COVID-19. Daily data given by Nigeria Center for Disease Control (NCDC) in Nigeria is used for simulation of the proposed COVID-19 model to see the effects of the control measures. The biological interpretation of this findings is that, COVID-19 can be effectively managed or eliminated in Nigeria if the control measures implemented are capable of taking or sustaining the basic reproductive number R0 to a value below unity. If the three control strategies are well managed by the government namely; NCDC, Presidential Task Force (PTF) and Federal Ministry of Health (FMOH) or policymakers, then COVID-19 in Nigeria will be eradicated.

Volterra–Lyapunov Stability Analysis of the Solutions of Babesiosis Disease Model

This paper studies the global stability analysis of a mathematical model on Babesiosis transmission dynamics on bovines and ticks populations as proposed by Dang et al. First, the global stability analysis of disease-free equilibrium (DFE) is presented. Furthermore, using the properties of Volterra–Lyapunov matrices, we show that it is possible to prove the global stability of the endemic equilibrium. The property of symmetry in the structure of Volterra–Lyapunov matrices plays an important role in achieving this goal. Furthermore, numerical simulations are used to verify the result presented.

Stability analysis of the coexistence equilibrium of a balanced metapopulation model

We analyze the stability of a unique coexistence equilibrium point of a system of ordinary differential equations (ODE system) modelling the dynamics of a metapopulation, more specifically, a set of local populations inhabiting discrete habitat patches that are connected to one another through dispersal or migration. We assume that the inter-patch migrations are detailed balanced and that the patches are identical with intra-patch dynamics governed by a mean-field ODE system with a coexistence equilibrium. By making use of an appropriate Lyapunov function coupled with LaSalle’s invariance principle, we are able to show that the coexistence equilibrium point within each patch is locally asymptotically stable if the inter-patch dispersal network is heterogeneous, whereas it is neutrally stable in the case of a homogeneous network. These results provide a mathematical proof confirming the existing numerical simulations and broaden the range of networks for which they are valid.

Microbial insecticide model and homoclinic bifurcation of impulsive control system

A new microbial insecticide mathematical model with density dependent for pest is proposed in this paper. First, the system without impulsive state feedback control is considered. The existence and stability of equilibria are investigated and the properties of equilibria under different conditions are verified by using numerical simulation. Since the system without pulse has two positive equilibria under some additional assumptions, the system is not globally asymptotically stable. Based on the stability analysis of equilibria, limit cycle, outer boundary line and Sotomayor’s theorem, the existence of saddle-node bifurcation and global dynamics of the system are obtained. Second, we consider homoclinic bifurcation of the system with impulsive state feedback control. The existence of order-1 homoclinic orbit of the system is studied. When the impulsive function is slightly disturbed, the homoclinic orbit breaks and bifurcates order-1 periodic solution. The existence and stability of order-1 periodic solution are obtained by means of theory of semi-continuous dynamic system.

Stability Analysis of a Mathematical Model for the Use of Wolbachia to Stop the Spread of Zika Virus Disease

Summary The use of wolbachia-infected mosquitoes to stop the spread of zika virus disease is modeled and analyzed. The model consists of a system of 10 ordinary differential equations which describes the dynamics of the disease in the human population, a wolbachia-free Aedes aegypti population, and a wolbachia-infected Aedes aegypti population used for disease control. A stability analysis of the disease-free equilibrium is conducted, which shows that it is both locally and globally asymptotically stable when the reproduction number is less than one. The result of the stability analysis shows that the spread of zika virus disease can be stopped, irrespective of the initial sizes of the infected human and mosquito populations, when wolbachia-infected Aedes aegypti are introduced in the area where the disease is endemic.

Dynamics of a COVID-19 Model with a Nonlinear Incidence Rate, Quarantine, Media Effects, and Number of Hospital Beds

In many countries the COVID-19 pandemic seems to witness second and third waves with dire consequences on human lives and economies. Given this situation the modeling of the transmission of the disease is still the subject of research with the ultimate goal of understanding the dynamics of the disease and assessing the efficacy of different mitigation strategies undertaken by the affected countries. We propose a mathematical model for COVID-19 transmission. The model is structured upon five classes: an individual can be susceptible, exposed, infectious, quarantined or removed. The model is based on a nonlinear incidence rate, takes into account the influence of media on public behavior, and assumes the recovery rate to be dependent on the hospital-beds to population ratio. A detailed analysis of the proposed model is carried out, including the existence and uniqueness of solutions, stability analysis of the disease-free equilibrium (symmetry) and sensitivity analysis. We found that if the basic reproduction number is less than unity the system can exhibit Hopf and backward bifurcations for some range of parameters. Numerical simulations using parameter values fitted to Saudi Arabia are carried out to support the theoretical proofs and to analyze the effects of hospital-beds to population ratio, quarantine, and media effects on the predicted nonlinear behavior.

Role of immunotherapy in tumor-immune interaction: Perspectives from fractional-order modelling and sensitivity analysis

Immunotherapy plays a vital role in strengthening the immune system and enhancing its ability to fight cancer during the tumor-immune interaction. This interaction is a complex process and biological studies are still ongoing to explore the tumor microenvironment with the action of the immune system. The limitations associated with ethical consideration and the costs associated with the biological experiments on human samples, motivate researchers to use other available means, including mathematical modelling to find possible solutions to the problems. In this study, we use a fractional model for tumor-immune interaction incorporating the treatment of cytokine interleukin-2 (IL-2) to boost the immune system to fight cancer. The basic properties of the model such as positivity of the solutions and local stability analysis of the tumor free equilibrium are studied and the conditions for tumor removal highlighted. Furthermore, the existence and uniqueness of solution of the model is proved using the fixed point theory. Given the uncertainty in the selection of model parameters, sensitivity analysis was performed using the Latin Hypercube Sampling scheme to determine model parameters which describe the processes that significantly influence the changes in the model state variables. The model was numerically solved for different orders of the fractional derivative using the Adams–Bashforth–Moulton Method. Our results suggest that, satisfactory stable tumor control can be achieved by adoptive cellular immunotherapy (ACI) alone, or through a combination of ACI and IL-2. We further observed that, the processes affecting the tumor-immune system interaction influence the dynamics variably at different stages of the disease. This observation is vital in informing researchers about the essential processes of emphasis when implementing drug targeting intervention measures aimed at curtailing the progression of cancer.

Stability Analysis of Nash Equilibrium for Two-Agent Loss-Aversion-Based Noncooperative Switched Systems

The stability property of the loss-aversion-based noncooperative switched systems with quadratic payoffs is investigated. In this system, each agent adopts the lower sensitivity parameter in the myopic pseudo-gradient dynamics for the case of losing utility than gaining utility, and both systems' dynamics and switching events (conditions) are depending on agents' payoff functions. Sufficient conditions under which agents' state converges towards the Nash equilibrium are derived in accordance with the location of the Nash equilibrium. In the analysis, the mode transition sequence and interesting phenomena which we call flash switching instants are characterized. Finally, we present several numerical examples to illustrate the properties of our results.

System of Time Fractional Models for COVID-19: Modeling, Analysis and Solutions

The emergence of the COVID-19 outbreak has caused a pandemic situation in over 210 countries. Controlling the spread of this disease has proven difficult despite several resources employed. Millions of hospitalizations and deaths have been observed, with thousands of cases occurring daily with many measures in place. Due to the complex nature of COVID-19, we proposed a system of time-fractional equations to better understand the transmission of the disease. Non-locality in the model has made fractional differential equations appropriate for modeling. Solving these types of models is computationally demanding. Our proposed generalized compartmental COVID-19 model incorporates effective contact rate, transition rate, quarantine rate, disease-induced death rate, natural death rate, natural recovery rate, and recovery rate of quarantine infected for a holistic study of the coronavirus disease. A detailed analysis of the proposed model is carried out, including the existence and uniqueness of solutions, local and global stability analysis of the disease-free equilibrium (symmetry), and sensitivity analysis. Furthermore, numerical solutions of the proposed model are obtained with the generalized Adam–Bashforth–Moulton method developed for the fractional-order model. Our analysis and solutions profile show that each of these incorporated parameters is very important in controlling the spread of COVID-19. Based on the results with different fractional-order, we observe that there seems to be a third or even fourth wave of the spike in cases of COVID-19, which is currently occurring in many countries.

Necessary and sufficient conditions for the roots of a cubic polynomial and bifurcations of codimension-1, -2, -3 for 3D maps

We reconsider the well-known conditions which guarantee the roots of a third-degree polynomial to be inside the unit circle. These conditions are important in the stability analysis of equilibria and cycles of three-dimensional systems in discrete time. A simplified set of conditions determine the boundary of the stability region and we prove which kind of bifurcation occurs when the boundary is crossed at any of its points. These points correspond to the existence of one, two or three eigenvalues equal to 1 in modulus, real or complex conjugate. We give the explicit expressions of the eigenvalues at each point of the border of the stability region in the parameter space. The results are applied to a system representing a housing market model that gives rise to a Neimark–Sacker bifurcation, a flip bifurcation or a pitchfork bifurcation.

A dynamical systems approach to the interplay between tobacco smokers, electronic-cigarette smokers and smoking quitters

In this paper, the effect of e-cigarettes on smoking cessation is studied using the tools of dynamical systems theory. The purpose here is to examine this efficacy by representing and analyzing a non-linear ODE system modeling potential smokers, tobacco smokers, e-cigarette smokers and quitters. The transition from smoking class to e-cigarette smoking class is represented by the “peer pressure”. The model exhibits three possible equilibrium solutions which are the smoking-free equilibrium, e-cigarette smoking-free equilibrium and endemic equilibrium. It is shown that the smoking free equilibrium always exists. Moreover, this equilibrium is stable if the basic reproduction number R0 is less than unity. When R0>1, it is shown that the smoking free equilibrium is unstable, and of the other two equilibria is stable. The global stability analysis of the smoking-free equilibrium is also given. Some numerical simulations are plotted using the data obtained from the literature. The theoretical results are also confirmed by numerical results.

Dynamics of a Two Prey and One Predator System with Indirect Effect

We study a population model with two preys and one predator, considering a Holling type II functional response for the interaction between first prey and predator and taking into account indirect effect of predation. We perform the stability analysis of equilibria and study the possibility of Hopf bifurcation. We also include a detailed discussion on the problem of persistence. Several numerical simulations are provided in order to illustrate the theoretical results of the paper.

Mathematical modelling and analysis of COVID-19 epidemic and predicting its future situation in Ethiopia

The epidemic of the coronavirus disease 2019 (COVID-19) has been rising rapidly and life-threatening worldwide since its inception. The lack of an established vaccine for this disease has caused millions of illnesses and hundreds of thousands of deaths globally. Mathematical models have become crucial tools in determining the potential and seriousness of the disease and in helping the types of strategic intervention measures to be taken to prevent and control the intensity of the spread of the disease. In this study, a compartmental epidemic model of COVID-19 is proposed and analyzed to predict the transmission dynamics of the disease in Ethiopia. Analytically, the basic reproduction number is determined. To observe the dynamics of the system, a detailed stability analysis of the disease-free equilibrium (DFE) of the proposed model is carried out. Our result shows that the DFE is stable if the basic reproduction number is less than unity and unstable otherwise. Also, the parameters of the assumed model are estimated using the actual data of COVID-19 from Ethiopia reported for three months between March and June 2020. Furthermore, we performed a sensitivity analysis of the basic reproductive number and found that reducing the rate of transmission is the most important factor in achieving disease control. Numerical simulations demonstrate the suitability of the proposed model for the actual COVID-19 data in Ethiopia. In particular, the numerical simulation shows an increase in the rate of transmission leads to a significant increase in the infected individuals. Thus, results of the numerical simulations are in agreement with the sensitivity results of the system. The possible implication of this is that declining the rate of transmission to the desired level could enable us to combat the disease. Numerical simulations are also performed to forecast the disease prevalence in the community.

Qualitative analysis and optimal control of an SIR model with logistic growth, non-monotonic incidence and saturated treatment

This paper describes an SIR model with logistic growth rate of susceptible population, non-monotonic incidence rate and saturated treatment rate. The existence and stability analysis of equilibria have been investigated. It has been shown that the disease free equilibrium point (DFE) is globally asymptotically stable if the basic reproduction number is less than unity and the transmission rate of infection less than some threshold. The system exhibits the transcritical bifurcation at DFE with respect to the cure rate. We have also found the condition for occurring the backward bifurcation, which implies the value of basic reproduction number less than unity is not enough to eradicate the disease. Stability or instability of different endemic equilibria has been shown analytically. The system also experiences the saddle-node and Hopf bifurcation. The existence of Bogdanov-Takens bifurcation (BT) of co-dimension 2 has been investigated which has also been shown through numerical simulations. Here we have used two control functions, one is vaccination control and other is treatment control. We have solved the optimal control problem both analytically and numerically. Finally, the efficiency analysis has been used to determine the best control strategy among vaccination and treatment.

Farming awareness based optimum interventions for crop pest control.

We develop a mathematical model, based on a system of ordinary differential equations, to the upshot of farming alertness in crop pest administration, bearing in mind plant biomass, pest, and level of control. Main qualitative analysis of the proposed mathematical model, akin to both pest-free and coexistence equilibrium points and stability analysis, is investigated. We show that all solutions of the model are positive and bounded with initial conditions in a certain significant set. The local stability of pest-free and coexistence equilibria is shown using the Routh-Hurwitz criterion. Moreover, we prove that when a threshold value is less than one, then the pest-free equilibrium is locally asymptotically stable. To get optimum interventions for crop pests, that is, to decrease the number of pests in the crop field, we apply optimal control theory and find the corresponding optimal controls. We establish existence of optimal controls and characterize them using Pontryagin's minimum principle. Finally, we make use of numerical simulations to illustrate the theoretical analysis of the proposed model, with and without control measures.

Rock Landslides Induced by Earthquakes: A Study on Influence of Strength Criterion on Limit Equilibrium Stability Analysis

Rock Landslides Induced by Earthquakes: A Study on Influence of Strength Criterion on Limit Equilibrium Stability Analysis

The stability and duality of dynamic Cournot and Bertrand duopoly model with comprehensive preference

In this paper, a Cournot model and its dual Bertrand model where firms have comprehensive preferences are developed. The preference was proposed by Bowles (2004) based on the results of Rabin (1993) and Lebine (1998). Comparable static method is used to illustrate the impact of parameters on equilibrium. Three scenarios are classified by preference parameter: completely cooperative, hostile to each other and general Cournot or Bertrand model. In each situation, Cournot model and its dual model present different chaos phenomenon which have various and abundant strange attractors, while the shapes of the stability regions are similar in the same situation. In the static setting, the local stability analysis of equilibrium gives the bounded region of quantity or price. The one-dimensional Logistic mapping is applied to studying the dynamics of the models on invariant axes. The critical curves classify different regions according to the number of preimages (q1, q2) or (p1, p2). Besides, simulations give more intuitive results: the cycle attractor, chaotic attractor and the basin of attraction with “holes” are presented. The article also provides new findings that under the assumption of comprehensive preference, the Cournot model with substitutes (complements) and the Bertrand model with complements (substitutes) are still duality models in the dynamic settings.

Stability Analysis of the Disease Free Equilibrium of Malaria, Dengue and Typhoid Triple Infection Model

A Mathematical model of a system of non-linear differential equation is developed to study the transmission dynamics of malaria, dengue and typhoid triple infection. In this work, the basic reproduction number is derived using the Next Generation Matrix, also we computed the disease free equilibrium point. The disease free equilibrium (DFE) point is analyzed and was found that the DFE is locally stable but may be globally unstable when R0 < 1.