Global Stability Of Equilibria Research Articles

A mathematical model for HIV dynamics with multiple infections: implications for immune escape.

Multiple infections enable the recombination of different strains, which may contribute to viral diversity. How multiple infections affect the competition dynamics between the two types of strains, the wild and the immune escape mutant, remains poorly understood. This study develops a novel mathematical model that includes the two strains, two modes of viral infection, and multiple infections. For the representative double-infection case, the reproductive numbers are derived and global stabilities of equilibria are obtained via the Lyapunov direct method and theory of limiting systems. Numerical simulations indicate similar viral dynamics regardless of multiplicities of infections though the competition between the two strains would be the fiercest in the case of quadruple infections. Through sensitivity analysis, we evaluate the effect of parameters on the set-point viral loads in the presence and absence of multiple infections. The model with multiple infections predict that there exists a threshold for cytotoxic T lymphocytes (CTLs) to minimize the overall viral load. Weak or strong CTLs immune response can result in high overall viral load. If the strength of CTLs maintains at an intermediate level, the fitness cost of the mutant is likely to have a significant impact on the evolutionary dynamics of mutant viruses. We further investigate how multiple infections alter the viral dynamics during the combination antiretroviral therapy (cART). The results show that viral loads may be underestimated during cART if multiple-infection is not taken into account.

Global properties of SARS‐CoV‐2 and IAV coinfection model with distributed‐time delays and humoral immunity

Severe acute respiratory syndrome coronavirus 2 (SARS‐CoV‐2) and influenza A virus (IAV) are two respiratory viruses and are similar concerning seasonal occurrence, transmission routes, clinical manifestations, and related immune responses. Recent studies showed that a substantial number of patients infected by SARS‐CoV‐2 were also coinfected with IAV. In this paper, we study the global properties of SARS‐CoV‐2 and IAV coinfection model in vivo. The role of humoral (antibody) immunity in controlling the coinfection is modeled. The model tracks uninfected epithelial cells, latent SARS‐CoV‐2‐infected epithelial cells, latent IAV‐infected epithelial cells, active SARS‐CoV‐2‐infected epithelial cells, active IAV‐infected epithelial cells, free SARS‐CoV‐2 particles, free IAV particles, SARS‐CoV‐2‐specific antibodies and IAV‐specific antibodies. The model includes six distributed‐time delays, two delays for the formation of latent SARS‐CoV‐2‐infected and latent IAV‐infected cells; two delays for the reactivation of latent SARS‐CoV‐2‐infected cells and latent IAV‐infected cells; and two delays for the maturation of new released SARS‐CoV‐2 and IAV virions. The regeneration and death of the uninfected epithelial cells are considered. We first examine the basic qualitative properties of the model, then we calculate all equilibria and prove their global stability. The global stability of equilibria is established using the Lyapunov method. The theoretical findings are demonstrated via numerical simulations. The importance of considering humoral immunity in the coinfection dynamics model is discussed. It is found that without modeling the humoral immunity, the case of IAV and SARS‐CoV‐2 coexistence will not occur. We discuss the effect of time delays on the dynamics of SARS‐CoV‐2 and IAV coinfection. It is noted that increasing the time delay length has similar impact as antiviral therapies.

Global Stability of Equilibria and Lyapunov Functions in a Vaccinating Behavioural SIR Model with Lagged Information

Global Stability of Equilibria and Lyapunov Functions in a Vaccinating Behavioural SIR Model with Lagged Information

Dynamical analysis of a predator-prey system with prey vigilance and hunting cooperation in predators.

In this work, we propose a predator-prey system with a Holling type Ⅱ functional response and study its dynamics when the prey exhibits vigilance behavior to avoid predation and predators exhibit cooperative hunting. We provide conditions for existence and the local and global stability of equilibria. We carry out detailed bifurcation analysis and find the system to experience Hopf, saddle-node, and transcritical bifurcations. Our results show that increased prey vigilance can stabilize the system, but when vigilance levels are too high, it causes a decrease in the population density of prey and leads to extinction. When hunting cooperation is intensive, it can destabilize the system, and can also induce bi-stability phenomenon. Furthermore, it can reduce the population density of both prey and predators and also change the stability of a coexistence state. We provide numerical experiments to validate our theoretical results and discuss ecological implications.

Forward hysteresis and Hopf bifurcation in an Npzd model with application to harmful algal blooms.

Nutrient-Phytoplankton-Zooplankton-Detritus (NPZD) models, describing the interactions between phytoplankton, zooplankton systems, and their ecosystem, are used to predict their ecological and evolutionary population dynamics. These organisms form the base two trophic levels of aquatic ecosystems. Hence understanding their population dynamics and how disturbances can affect these systems is crucial. Here, starting from a base NPZ modeling framework, we incorporate the harmful effects of phytoplankton overpopulation on zooplankton-representing a crucial next step in harmful algal bloom (HAB) modeling-and split the nutrient compartment to formulate an NPZD model. We then mathematically analyze the NPZ system upon which this new model is based, including local and global stability of equilibria, Hopf bifurcation condition, and forward hysteresis, where the bi-stability occurs with multiple attractors. Finally, we extend the threshold analysis to the NPZD model, which displays both forward hysteresis with bi-stability and Hopf bifurcation under different parameter regimes. We also examine ecological implications after incorporating seasonality and ecological disturbances. Ultimately, we quantify ecosystem health in terms of the relative values of the robust persistence thresholds for phytoplankton and zooplankton and find (i) ecosystems sufficiently favoring phytoplankton, as quantified by the relative values of the plankton persistence numbers, are vulnerable to both HABs and (local) zooplankton extinction (ii) even healthy ecosystems are extremely sensitive to nutrient depletion over relatively short time scales.

Mathematical model and optimal control analysis of coffee berry borer with temperature and rainfall variability

This study focuses on a nonlinear deterministic mathematical model for coffee berry borer (Hypothenemus hampei) with temperature and rainfall variability. In the model analysis, CBB free and coexistence equilibria are computed. The basic reproduction numbers at a minimum and maximum temperature and rainfall are derived. The qualitative analysis of the model revealed the scenario for equilibria together with basic reproduction numbers. The local stability of equilibria is established through the Jacobian matrix and the Routh–Hurwitz criteria, while the global stability of equilibria is demonstrated using an appropriate Lyapunov function. The normalized sensitivity analysis has also been performed to observe the impact of different parameters on basic reproduction numbers. The proposed model is extended into an optimal control problem by incorporating two control variables, namely, the preventive measure variable based on the separation of susceptible coffee berries from contacting the pests based on biological control and an increase in the death rate of colonizing females of CBB based on chemical control. Optimal disease control analysis is examined using Pontryagin’s minimum principle. Finally, the numerical simulations are performed based on analytical results and are discussed quantitatively. Furthermore, the cost-effectiveness of control strategies to determine the best approach to minimize the CBB burden was studied. The study is significant in providing reliable information on how one can use mathematical modeling to improve the roles of control strategies and prevention in CBB transmission in a coffee farm. The outcome of the study may guide public agriculture policymakers on optimal control strategies to control the pests. In particular, using chemical pesticides is very effective to combat pests with minimum costs.

Dynamics of a non-smooth pest-natural enemy model with the threshold control strategy

Pest issues have always been the focus of attention in agriculture. The Integrated Pest Management(IPM) method is currently the most popular way to be applied for pest control. In this study, according to the IPM strategy, we regard pest quantity as a threshold index and extend the Leslie-Gower model into a non-smooth Filippov system through combining chemical and biological control. To maintain the pest population at or below the given economic threshold(ET), we investigate the global dynamics of the proposed model, including the existence of sliding mode and various equilibria, sliding dynamics and bifurcations, and global stability of equilibria. The result shows that desired equilibria can be globally stable under some conditions, meaning that our control tactics work. In particular, the case where our strategy fails to be effective arouses interest. In the end, the biological implications of the results are discussed and given in detail.

Spatiotemporal dynamics of a gene expression model with positive feedback loop

Developing appropriate spatiotemporal models can help us understand at the molecular level how organisms function, when and where certain genes are expressed, and to what extent. In this paper, we establish a reaction-diffusion model with positive feedback loop in the process of gene expression, which describes the expression of genes according to probability and random diffusion. We focus on the long-term behavior of the system, including the determination of invariant regions and the global stability of equilibria. In particular, in the absence of a globally stable equilibrium, we derive some results on the existence and nonexistence of nonconstant steady state solutions, and clarify the dependence of the existence of nonconstant solutions on the diffusion rate of reactants and the size of the reaction environment. These results contribute to understanding the heterogeneity of cell characterization. We also carry out numerical simulations to support our theoretical results.

On a two-strain epidemic mathematical model with vaccination

In this paper, we study mathematically a two strains epidemic model taking into account non-monotonic incidence rates and vaccination strategy. The model contains seven ordinary differential equations that illustrate the interaction between the susceptible, the vaccinated, the exposed, the infected and the removed individuals. The model has four equilibrium points, namely, disease free equilibrium, endemic equilibrium with respect to the first strain, endemic equilibrium with respect to the second strain and the endemic equilibrium with respect to both strains. The global stability of the equilibria has been demonstrated using some suitable Lyapunov functions. The basic reproduction number is found depending on the first strain reproduction number R 0 1 and the second reproduction number R 0 2 . We have shown that the disease dies out when the basic reproduction number is less than unity. It was remarked that the global stability of the endemic equilibria depends, on the strain basic reproduction number and on the strain inhibitory effect reproduction number. We have also observed that the strain with high basic reproduction number will dominate the other strain. Finally, the numerical simulations are presented in the last part of this work to support our theoretical results. We notice that our suggested model has some limitations and does not predicting the long-term dynamics for some reproduction numbers cases.

A nonlinear mathematical model on the Covid-19 transmission pattern among diabetic and non-diabetic population

In this paper, a three tier mathematical model describing the interactions between susceptible population, Covid-19 infected, diabetic population and Covid-19 infected, non diabetic population is proposed. Basic properties of such a dynamic model, namely, non negativity, boundedness of solutions, existence of disease-free and disease equilibria are studied and sufficient conditions are obtained. Basic reproduction number for the system is derived. Sufficient conditions on functionals and parameters of the system are obtained for the local as well as global stability of equilibria, thus, establishing the conditions for eventual prevalence of disease free or disease environment, as the case may be. The stability aspects are discussed in the context of basic reproduction number and vice versa. An important contribution of this article is that a novel technique is presented to estimate some key, influencing parameters of the system so that a pre-specified, assumed equilibrium state is approached eventually. This enables the society to prepare itself with the help of these key, influencing parameters so estimated. Several examples are provided to illustrate the results established and simulations are provided to visualize the examples.

Global stability of a Lotka-Volterra piecewise-smooth system with harvesting actions and two predators competing for one prey

The global dynamics of a three-dimensional Lotka-Volterra system described by two predator species competing for one prey and with human harvesting action on the predator species is studied in this paper. The harvesting action is introduced by means of two switching control actions defined on the predator species. A well-known result in the study of ecosystem modeling is that there are two states of coexistence of one of the predatory species with the prey species assuming that the principle of competitive exclusion or coexistence of competing species is fulfilled. In this sense, the three species cannot coexist in this class of system. In this work, it is proved that there is a global stable equilibrium point where the three species can coexist due to the proposed harvesting action.

Bifurcation and overexploitation in Rosenzweig-MacArthur model

<p style='text-indent:20px;'>In this paper, we propose a Rosenzweig–MacArthur predator-prey model with strong Allee effect and trigonometric functional response. The local and global stability of equilibria is studied, and the existence of bifurcation is determined in terms of the carrying capacity of the prey, the death rate of the predator and the Allee effect. An analytic expression is employed to determine the criticality and codimension of Hopf bifurcation. The existence of supercritical Hopf bifurcation and the non-existence of Bogdanov–Takens bifurcation at the positive equilibrium are proved. A point-to-point heteroclinic cycle is also found. Biologically speaking, such a heteroclinic cycle always indicates the collapse of the system after the invasion of the predator, i.e., overexploitation occurs. It is worth pointing out that heteroclinic relaxation cycles are driven by either the strong Allee effect or the high per capita death rate. In addition, numerical simulations are given to demonstrate the theoretical results.</p>

Modelling HIV dynamics with cell‐to‐cell transmission and CTL response

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell‐to‐cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so‐called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov–Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle‐node bifurcation, Bogdanov–Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell‐to‐cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so‐called new type bifurcation persist. Finally, some conclusions and discussions are also given.

Modeling and Stability Analysis of Within-Host IAV/SARS-CoV-2 Coinfection with Antibody Immunity

Studies have reported several cases with respiratory viruses coinfection in hospitalized patients. Influenza A virus (IAV) mimics the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) with respect to seasonal occurrence, transmission routes, clinical manifestations and related immune responses. The present paper aimed to develop and investigate a mathematical model to study the dynamics of IAV/SARS-CoV-2 coinfection within the host. The influence of SARS-CoV-2-specific and IAV-specific antibody immunities is incorporated. The model simulates the interaction between seven compartments, uninfected epithelial cells, SARS-CoV-2-infected cells, IAV-infected cells, free SARS-CoV-2 particles, free IAV particles, SARS-CoV-2-specific antibodies and IAV-specific antibodies. The regrowth and death of the uninfected epithelial cells are considered. We study the basic qualitative properties of the model, calculate all equilibria and investigate the global stability of all equilibria. The global stability of equilibria is established using the Lyapunov method. We perform numerical simulations and demonstrate that they are in good agreement with the theoretical results. The importance of including the antibody immunity into the coinfection dynamics model is discussed. We have found that without modeling the antibody immunity, the case of IAV and SARS-CoV-2 coexistence is not observed. Finally, we discuss the influence of IAV infection on the dynamics of SARS-CoV-2 single-infection and vice versa.

The stability of a dynamic duopoly Cournot–Bertrand game model

The paper is devoted to exploring the complex dynamics of a Cournot–Bertrand model, where one agent chooses quantity and the other chooses price. A nonlinear discrete system is built to illustrate the game model with bounded rationality. Theoretically, the local and global stability of equilibrium are investigated. The simulation reveals that Flip and Neimark–Sacker​ bifurcation phenomena occur when the adjustment speed of one firm increases through different boundary curves. Therefore three different chaotic attractors are presented. The standard Logistic mapping is applied to analyze the dynamics of the system on invariant axes. The critical curves classify the number of preimage of the quantity or price. Besides, simulations give more intuitive results: the cycle attractor, chaotic attractor and the basin of attraction with “holes” are given.

Global dynamics of IAV/SARS-CoV-2 coinfection model with eclipse phase and antibody immunity.

Coronavirus disease 2019 (COVID-19) and influenza are two respiratory infectious diseases of high importance widely studied around the world. COVID-19 is caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), while influenza is caused by one of the influenza viruses, A, B, C, and D. Influenza A virus (IAV) can infect a wide range of species. Studies have reported several cases of respiratory virus coinfection in hospitalized patients. IAV mimics the SARS-CoV-2 with respect to the seasonal occurrence, transmission routes, clinical manifestations and related immune responses. The present paper aimed to develop and investigate a mathematical model to study the within-host dynamics of IAV/SARS-CoV-2 coinfection with the eclipse (or latent) phase. The eclipse phase is the period of time that elapses between the viral entry into the target cell and the release of virions produced by that newly infected cell. The role of the immune system in controlling and clearing the coinfection is modeled. The model simulates the interaction between nine compartments, uninfected epithelial cells, latent/active SARS-CoV-2-infected cells, latent/active IAV-infected cells, free SARS-CoV-2 particles, free IAV particles, SARS-CoV-2-specific antibodies and IAV-specific antibodies. The regrowth and death of the uninfected epithelial cells are considered. We study the basic qualitative properties of the model, calculate all equilibria, and prove the global stability of all equilibria. The global stability of equilibria is established using the Lyapunov method. The theoretical findings are demonstrated via numerical simulations. The importance of considering the antibody immunity in the coinfection dynamics model is discussed. It is found that without modeling the antibody immunity, the case of IAV and SARS-CoV-2 coexistence will not occur. Further, we discuss the effect of IAV infection on the dynamics of SARS-CoV-2 single infection and vice versa.

Global Dynamics of a New Huanglongbing Transmission Model with Quarantine Measures

An epidemic model which describes Huanglongbing transmission is proposed with the goal of investigating the effect of quarantine measures on the spread of diseases. First of all, the analytical formula for the basic reproduction number R0 is obtained by the means of next generation matrix, and the existence of disease-free equilibrium and endemic equilibrium is discussed. Then, the local stability and the global stability of equilibria are investigated by using Routh-Hurwitz criterion and Lyapunov function, respectively. Numerical simulations indicate that comprehensive quarantine measures can effectively control the spread of Huanglongbing. It provides a reliable tactic basis for preventing the epidemic outbreak.

Dynamics of an immunological viral infection model with lytic and non-lytic immune response in presence of cell-to-cell transmission and cure of infected cells

In this paper, we propose a mathematical model that describes the dynamics of viral infection with both modes of transmission, virus-to-cell and cell-to-cell by taking into account the non-cytolytic cure of infected cells, the lytic and non-lytic humoral immune response. We first prove the non-negativity and boundedness of the solutions of the proposed model. Furthermore, the dynamical behaviors of the model including the local and global stability of equilibria are rigorously investigated.

Research on travelers’ transportation mode choice between carsharing and private cars based on the logit dynamic evolutionary game model

The rapid increase in private cars has exacerbated urban traffic and environmental problems. As an alternative to private cars, carsharing is more economical and environmentally friendly than private cars and is becoming internationally recognized as sustainable transportation. However, the benefits of carsharing in alleviating traffic and environmental problems would only be significant when many travelers switch from private cars to carsharing. Thus, understanding travelers' dynamic selection processes when choosing between carsharing and private cars is important in promoting carsharing. Assuming that travelers' behavior regarding travel mode choice conforms to the principle of maximizing random utility, a logit evolutionary game model based on logit dynamics and evolutionary game theory was constructed to explain and analyze the dynamic selection processes and mechanisms by which travelers choose between carsharing and private cars. In addition to the traditional time and monetary cost, the psychological benefit of the driving experience was also considered in the travelers’ payment functions. The unique global stable equilibrium point of the logit dynamic evolution model was derived through mathematical analysis. Data computed from a simple interview survey in Qingdao City were then used to perform an experimental numerical analysis on the evolutionary model of this study to further explain the dynamic evolution. Moreover, the relationships between the evolutionarily stable points and the time cost, monetary cost, and driving experience benefit differences between carsharing and private cars are discussed. The relationships between the benefits of the negative externalities of carsharing and private cars and the evolutionarily stable points are also examined. Furthermore, four scenarios are established, and their corresponding evolutionarily stable points are discussed to explain the influence of adjusting the external circumstances on the evolutionarily stable point. Finally, based on practical application, suggestions for encouraging travelers to use carsharing rather than private cars are provided to help carsharing organizations, government departments, and other related institutions.

Global Stabilization of a Single-Species Ecosystem with Markovian Jumping under Neumann Boundary Value via Laplacian Semigroup

By applying impulsive control, this work investigated the global stabilization of a single-species ecosystem with Markovian jumping, a time delay and a Neumann boundary condition. Variational methods, a fixed-point theorem, and Laplacian semigroup theory were employed to derive the unique existence of the global stable equilibrium point, which is a positive number. Numerical examples illuminate the feasibility of the proposed methods.