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This paper establishes a new fractional frSIS model utilizing a continuous time random walk method. There are two main innovations in this paper. On the one hand, the model is analyzed from a mathematical perspective. First, unlike the classic SIS infectious disease model, this model presents the infection rate and cure rate in a fractional order. Then, we proved the basic regeneration number R0 of the model and studied the influence of orders a and b on R0. Second, we found that frSIS has a disease-free equilibrium point E0 and an endemic equilibrium point E∗. Moreover, we proved frSIS global stability of the model using R0. If R0<1, the model of E0 is globally asymptotically stable. If R0>1, the model of E∗ is globally asymptotically stable. On the other hand, from the perspective of infectious diseases, we discovered that appropriately increasing a and decreasing b are beneficial for controlling the spread of diseases and ultimately leading to their disappearance. This can help us provide some dynamic adjustments in prevention and control measures based on changes in the disease. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the tracking performance and antidisturbance ability of the widely used proportional-integral-derivative (PID) controllers in practice. Towards this end, we consider a basic class of second-order nonlinear stochastic control systems subject to model uncertainties and external disturbances, and focus on the ability of the classical PID controller to track time-varying reference signals. First, under some suitable conditions on the system nonlinear functions, reference signals, and external disturbances, we show that such control systems can be stabilized in the mean square sense, provided that the three PID gains are selected from a stability region constructed in the paper. Besides, it is shown that the steady-state tracking error has an upper bound proportional to the sum of the varying rates of reference signals, the varying rates of external disturbances, and the intensity of random noises. Meanwhile, its proportional coefficient depends on the selection of PID gains, which can be made arbitrarily small by choosing suitably large PID gains. Finally, by introducing a desired transient process which is shaped from the reference signal, a new PID tuning rule is presented, which can guarantee both the expected steady state and transient tracking performance. [ABSTRACT FROM AUTHOR]

In a previous paper (Feng et al., J. Differential Equations (2023)), we studied a seasonally interactive model between closed seasons and open seasons with Michaelis-Menten type harvesting, in which we assumed that the harvesting quantity was relatively large (0 < κ = l G c E < 1) and obtained a length threshold of the closed season T ¯ ⁎ depending on the harvesting parameter and the seasonal fluctuation period. It was shown that the origin is globally asymptotically stable if and only if T ¯ ≤ T ¯ ⁎ , and there exists a unique globally asymptotically stable T -periodic solution if and only if T ¯ > T ¯ ⁎. In this paper, we continue to investigate the periodic dynamics of this model when the harvesting quantity is relatively small (κ = l G c E ≥ 1). By finding another smaller length threshold T ¯ ⁎ ∈ (0 , T ¯ ⁎) , we determine the number of periodic solutions and study their stability, which imply the occurrence of bifurcation of periodic solutions. Our results demonstrate that designing the closed harvesting season properly can prevent the species from extinction. Moreover, by comparing with the continuous harvesting model and combining the results in our previous paper and this article, we provide a complete understanding on the global dynamics for the periodic switching model. Some numerical simulations are also carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

A significant proportion of malaria infections in humans exhibit no symptoms, but it is a reservoir for maintaining malaria transmission. A time periodic reaction-diffusion model for malaria spread is introduced in this paper, incorporating spatial heterogeneity, incubation periods, symptomatic and asymptomatic carriers. This paper introduces the concept of the basic reproduction number R 0 , which is defined as the spectral radius of the next generation operator, and we present some preliminary results by elementary analysis. The threshold dynamic behavior analysis shows that when R 0 < 1 , the disease is extinct, and when R 0 > 1 , the disease is persistent. We investigate the case of constant system parameters, focusing on the global asymptotic stability of the disease-free steady state when R 0 = 1 . In the numerical simulation section, we validate the theoretical results obtained, and then use elasticity analysis methods to explore the influence of parameters on the output solution. In addition, sensitivity analysis of the basic reproduction number under homogeneous conditions indicates direction of controlling malaria transmission. And several control measures are evaluated in the following steps. [ABSTRACT FROM AUTHOR]

The efficacy of vaccination, incomplete treatment and disease relapse are critical challenges that must be faced to prevent and control the spread of infectious diseases. Age heterogeneity is also a crucial factor for this study. In this paper, we investigate a new age-structured SVEIR epidemic model with the nonlinear incidence rate, waning immunity, incomplete treatment and relapse. Next, the asymptotic smoothness, the uniform persistence and the existence of interior global attractor of the solution semi-flow generated by the system are given. We define the basic reproduction number R 0 and prove the existence of the equilibria of the model. And we study the global asymptotic stability of the equilibria. Then the parameters of the model are estimated using tuberculosis data in China. The sensitivity analysis of R 0 is derived by the Partial Rank Correlation Coefficient method. These main theoretical results are applied to analyze and predict the trend of tuberculosis prevalence in China. Finally, the optimal control problem of the model is discussed. We choose to take strengthening treatment and controlling relapse as the control parameters. The necessary condition for optimal control is established. [ABSTRACT FROM AUTHOR]

In this paper, we consider a diffusive–advective predator–prey system in a spatially heterogeneous environment subject to a hostile boundary condition, where the interaction term is governed by a Holling type II functional response. We investigate the existence and global attractivity of both trivial and semi-trivial steady-state solutions and the existence and local stability of coexistence steady-state solutions, depending on the size of a key principal eigenvalue. In addition, we show that the effect of advection on the principal eigenvalue is monotonic for small advection rates, depending on the concavity of the resource distribution. For arbitrary advection rates, we consider two explicit resource distributions for which we can say precisely the behaviour of the principal eigenvalue as it depends on advection, highlighting that advection can either improve or impair a population's ability to persist, depending on the characteristics of the resource distribution. We present some numerical simulations to demonstrate the outcomes as they depend on the advection rates for the full predator–prey system. These insights highlight the intimate relationship between environmental heterogeneity, directed movement, and the hostile boundary. The methods employed include upper and lower solution techniques, bifurcation theory, spectral analysis, and the comparison principle. [ABSTRACT FROM AUTHOR]

Establishing a mapping between the emergent biological properties and the repository of network structures has been of great relevance in systems and synthetic biology. Adaptation is one such biological property of paramount importance that promotes regulation in the presence of environmental disturbances. This paper presents a nonlinear systems theory-driven framework to identify the design principles for perfect adaptation with respect to external disturbances of arbitrary magnitude. Based on the prior information about the network, we frame precise mathematical conditions for adaptation using nonlinear systems theory. We first deduce the mathematical conditions for perfect adaptation for constant input disturbances. Subsequently, we translate these conditions to specific necessary structural requirements for adaptation in networks of small size and then extend to argue that there exist only two classes of architectures for a network of any size that can provide local adaptation in the entire state space, namely, incoherent feed-forward (IFF) structure and negative feedback loop with buffer node (NFB). The additional positiveness constraints further narrow the admissible set of network structures. This also aids in establishing the global asymptotic stability for the steady state given a constant input disturbance. The proposed method does not assume any explicit knowledge of the underlying rate kinetics, barring some minimal assumptions. Finally, we also discuss the infeasibility of certain IFF networks in providing adaptation in the presence of downstream connections. Moreover, we propose a generic and novel algorithm based on non-linear systems theory to unravel the design principles for global adaptation. Detailed and extensive simulation studies corroborate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper is dedicated to Jim Cushing on the occasion of his 80th birthday. It is inspired by his work on evolutionary theory. We investigate the global dynamics of discrete-time phenotypic evolutionary models, both autonomous and periodic. We developed the theory of mixed monotone maps and applied it to show that the positive equilibrium of the autonomous evolutionary Ricker model of single and multi-species is globally asymptotically stable. Then we extend this result to the corresponding evolutionary Ricker model with periodic parameters. [ABSTRACT FROM AUTHOR]

In this paper, the author proposes and studies a mathematical model for the spread of coronavirus 2019. The effect of preventive programs on the control of disease is also considered. The basic reproduction number R0$$ {R}_0 $$ was computed and the results showed that forward bifurcation occured. Local and global stability of the equilibrium points of the model was studied by geometric stability method and Lyapunov function. Finally, the particular data of China which was reported for Covid‐19 were used for numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, we study a network-based SIR epidemic model with a saturated treatment function in which a parameter α is introduced to measure the extent of the effect of the infected being delayed for treatment. Our aim is to present a global analysis and to investigate how the parameter α affects the spreading of diseases. Our main results are as follows: (1) In the case of the threshold value R 0 < 1 , there exist two values of α : α c and α 0 , such that the disease-free equilibrium is globally asymptotically stable when α ≤ α c and multiple endemic equilibria exist when α ≥ α 0 . This means that the parameter α has an essential influence on the spreading of the disease. (2) In the case of the threshold value R 0 > 1 , if the model has only one endemic equilibrium, then the unique endemic equilibrium is globally attractive. In this case, it is also proved that if α ≤ α c , then the endemic equilibrium has only one, so is globally attractive. In addition, numerical simulation is performed to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Contracting cancer typically induces a state of terror among the individuals who are affected. Exploring how chemotherapy and anxiety work together to affect the speed at which cancer cells multiply and the immune system's response model is necessary to come up with ways to stop the spread of cancer. This paper proposes a mathematical model to investigate the impact of psychological scare and chemotherapy on the interaction of cancer and immunity. The proposed model is accurately described. The focus of the model's dynamic analysis is to identify the potential equilibrium locations. According to the analysis, it is possible to establish three equilibrium positions. The stability analysis reveals that all equilibrium points consistently exhibit stability under the defined conditions. The bifurcations occurring at the equilibrium sites are derived. Specifically, we obtained transcritical, pitchfork, and saddle-node bifurcation. Numerical simulations are employed to validate the theoretical study and ascertain the minimum therapy dosage necessary for eradicating cancer in the presence of psychological distress, thereby mitigating harm to patients. Fear could be a significant contributor to the spread of tumors and weakness of immune functionality. [ABSTRACT FROM AUTHOR]

This paper investigates both the local and global stability of a system of rational difference equations and its connection to co-balancing numbers. The study delves into the intricate dynamics of mathematical models and their stability properties, emphasizing the broader implications of global stability. Additionally, the investigation extends to the role of co-balancing numbers, elucidating their significance in achieving equilibrium within the solutions of the rational difference equations. The interplay between global stability and co-balancing numbers forms a foundational aspect of the analysis. The findings contribute to a deeper understanding of the mathematical structures underlying dynamic systems and offer insights into the factors influencing their stability and equilibrium. This article serves as a valuable resource for mathematicians, researchers, and scholars interested in the intersection of global stability and co-balancing sequences in the realm of rational difference equations. Moreover, the presented examples and figures consistently demonstrate the global asymptotic stability of the equilibrium point throughout the paper. [ABSTRACT FROM AUTHOR]

This paper focuses on the S-asymptotically Periodic oscillation for a type of fractional-order fuzzy Cohen-Grossberg neural networks (CGNNs) by employing some properties of Mittag-Leffler mappings and fixed point theorems. Further, the global asymptotic stability of CGNNs is received. For CGNNs, our works in this paper not only enrich its theoretical achievements, but also expand its application scope. [ABSTRACT FROM AUTHOR]

In this paper, we develop discrete models of Tuberculosis (TB). This includes SEI endogenous and exogenous models without treatment. These models are then extended to a SEIT model with treatment. We develop two types of net reproduction numbers, one is the traditional R0 which is based on the disease-free equilibrium, and a new net reproduction number R0 (E*) based on the endemic equilibrium. It is shown that the disease-free equilibrium is globally asymptotically stable if R0 ≤ 1 and unstable if R0 > 1. Moreover, the endemic equilibrium is locally asymptotically stable if R0(E*) < 1 < R0. [ABSTRACT FROM AUTHOR]

In this paper, we present and assess a predator–prey Leslie–Gower model including disease, refuge and treatment in prey population. There are two groups of prey: those who are susceptible and infected. It is hypothesized that prey population is affected by diseases and refuge, and grows logistically in the absence of predators. Infected prey population receives treatment. The predators’ growth rate is governed by the modified Leslie–Gower dynamics. The dynamical attributes of the resulting system are boundedness, positivity of solutions, extinction criteria, existence and (local and global) stability. Biology uses mathematical analysis to identify the possible attributes of equilibrium points. The focus of this study is to assess how treatment and refuge affect the populations of ill prey, susceptible prey, predators and treated prey. The numerical simulation indicates that the influence of treatment, and refuge change the dynamics of the system (2.1). Extensive numerical simulations were performed to validate our analytical findings by using the Mathematica and MATLAB software. [ABSTRACT FROM AUTHOR]

The application of mathematical biology and dynamical systems has proven to be an effective approach for studying viral infection models. To contribute to this research, our paper proposes a new CHIKV model that takes into account an adaptive immune response and distributed time delays, which accurately reflects the time lag between initial viral contacts and the production of new active CHIKV particles. By analyzing the model’s qualitative behavior, we establish a biological threshold number that can predict whether CHIKV will be cleared from or persist in the body. We demonstrate the global stability of both CHIKV-present and CHIKV-free steady states using the Lyapunov functional method and LaSalle’s invariance principle. In addition, we conduct numerical simulations to examine how time delays can affect the stability of the steady states. Through these simulations, we gain insights into how varying time delays can influence the persistence or clearance of CHIKV within the host. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop and investigate a novel mathematical model of the dynamical behaviors of chronic hepatitis B virus infection. The model includes exposed infected hepatocytes, intracellular HBV DNA-containing capsids, uses a general incidence function for viral infection covering a variety of special cases available in the literature, and describes the interaction of cytotoxic T lymphocytes that kill the infected hepatocytes and the magnitude of B-cells that send antibody immune defense to neutralize free virions. Further, one time delay is incorporated to account for actual capsids production. The other time delays are used to account for maturation of capsids and free viruses. We start with the analysis of the proposed model by establishing the local and global existence, uniqueness, non-negativity and boundedness of solutions. After defined the threshold parameters, we discuss the stability properties of all possible steady state constants by using the crafty Lyapunov functionals, the LaSalle’s invariance principle and linearization methods. The impacts of the three time delays on the HBV infection transmission are discussed through local and global sensitivity analysis of the basic reproduction number and of the classes of infected states. Finally, an application is provided and numerical simulations are performed to illustrate and interpret the theoretical results obtained. It is suggested that, a good strategy to eradicate or to control HBV infection within a host should concentrate on any drugs that may prolong the values of the three delays. [ABSTRACT FROM AUTHOR]

This paper presents a study on a Lotka-Volterra ammensalism model that incorporates the fear effect, which can potentially decrease the birth rate and raise the mortality rate of the species. For the autonomous case, equilibrium points' local and global stability are discussed. For the nonautonomous case, sufficient conditions which ensure the persistence and extinction, and global asymptotic stability of the positive solutions are obtained, respectively. The study has shown that with the increase of the fear effect, the final density of the affected population will decrease, and when the fear effect is large enough, it will cause population's extinction. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the global dynamics of a Lotka-Volterra competition-diffusion-advection system with nonlinear boundary conditions, including the existence, nonexistence and global stability of coexistence steady states. We start with the investigation of the principal eigenvalue of linearized system to get the local stability of steady states and then discuss the global dynamics in terms of competition coefficients. [ABSTRACT FROM AUTHOR]

Mathematical models play a crucial role in controlling and preventing the spread of diseases. Based on the communication characteristics of diseases, it is necessary to take into account some essential epidemiological factors such as the time delay that takes an individual to progress from being latent to become infectious, the infectious age which refers to the duration since the initial infection and the occurrence of reinfection after a period of improvement known as relapse, etc. Moreover, age-structured models serve as a powerful tool that allows us to incorporate age variables into the modeling process to better understand the effect of these factors on the transmission mechanism of diseases. In this paper, motivated by the above fact, we reformulate an SEIR model with relapse and age structure in both latent and infected classes. Then, we investigate the asymptotic behavior of the model by using the stability theory of differential equations. For this purpose, we introduce the basic reproduction number R 0 of the model and show that this threshold parameter completely governs the stability of each equilibrium of the model. Our approach to show global attractivity is based on the fluctuation lemma and Lyapunov functionals method with some results on the persistence theory. The conclusion is that the system has a disease-free equilibrium which is globally asymptotically stable if R 0 < 1 , while it has only a unique positive endemic equilibrium which is globally asymptotically stable whenever R 0 > 1 . Our results imply that early diagnosis of latent infection with decrease in both transmission and relapse rates may lead to control and restrict the spread of disease. The theoretical results are illustrated with numerical simulations, which indicate that the age variable is an essential factor affecting the spread of the epidemic. [ABSTRACT FROM AUTHOR]

In this paper, we continue the research in Li et al. (J Differ Equ 371:353–395, 2023) and study the linear and global stability of a class of reaction-diffusion systems with general degenerate diffusion. The establishment of these systems is based on the acid-mediated invasion hypothesis, which is a candidate explanation for the Warburg effect. Our theoretical results characterize the effects of acid resistance and mutual competition between healthy cells and tumor cells on local and long-term tumor development, i.e., whether the healthy cells and tumor cells coexist or the tumor cells prevail after tumor invasion. We first consider the linear stability of the steady states and give a complete characterization by transforming the linearized analysis into an algebraic problem. In discussing global stability, the main difficulty of this model arises from density-limited diffusion terms, which can lead to degeneracy in the parabolic equations. We find that the method established in Li et al. (J Differ Equ 371:353–395, 2023) works well to overcome the degenerate problem. This method combines the Lyapunov functionals and upper/lower solutions, and it can be applied to a broader range of reaction-diffusion systems even if the diffusion terms degenerate and have very poor properties. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the global asymptotic behaviors of a mathematical susceptibleinfected(SI) age-infection-structured human immunodeficiency virus(HIV) model with heterogeneous transmission. Mathematical analysis shows that the local and global dynamics are completely determined by the basic reproductive number R 0. If R 0 < 1, disease-free equilibrium is globally asymptotically stable. If R 0 >1, it shows that diseasefree equilibrium is unstable and the unique endemic equilibrium is globally asymptotically stable. The proofs of global stability utilize Lyapunov functions. Besides, the numerical simulations are illustrated to support these theoretical results and sensitivity analysis of each parameter for R 0 is performed by the method of partial rank correlation coefficient(PRCC). [ABSTRACT FROM AUTHOR]

In this paper, we propose a diffusive prey-predator model with saturated hunting cooperation and predator-taxis. We first establish the global classical solvability and boundedness, and provide some sufficient conditions to assure the existence of a unique positive homogeneous steady state and the global uniform asymptotic stability of the predator-free homogeneous steady state. Secondly, we study the pattern formation mechanism and reveal that pattern formation is driven by the joint effect of predator-taxis, hunting cooperation, and slow diffusivity of predators. Moreover, we find that a strong predator-taxis can annihilate the spatiotemporal patterns, but a weak predator-taxis supports the pattern formation when diffusion-driven instability is present in the model without predator-taxis. However, if diffusion-driven instability is absent, predator-taxis cannot destabilize the unique positive spatially homogeneous steady state. Additionally, we highlight that spatially heterogeneous steady states do not exist when the diffusion coefficient ratio of predators to prey is sufficiently large under specific parametric conditions. To explore the various types of spatially heterogeneous steady states, we derive amplitude equations based on the weakly nonlinear analysis theory. Finally, numerical simulations, including the hexagonal pattern, stripe pattern, a mixed pattern combining hexagons and stripes, and the square pattern, are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study an age‐structured epidemiological compartment model and its respective problem, applied but not limited to the COVID‐19 pandemic, in order to investigate the role of the age of the individuals in the evolution of epidemiological phenomena. We investigate the well‐posedness of the model, as well as the global dynamics of it in the sense of basic reproduction number, via constructing Lyapunov functions. [ABSTRACT FROM AUTHOR]

In this paper, SIR model with feedback control has been considered. By constructing the Lyapunov function, using the Routh–Hurwitz criterion and the positive definite quadratic form theory, some sufficient conditions have been obtained, and some well-known results have been generalized. It is worth mentioning that if the feedback control variables have an inhibitory effect on the susceptible population and the infected population, the number of basic regeneration of the disease will be reduced. The reduction depends only on the control of the susceptible population, but not on the control of the infected population. This is a very interesting conclusion. [ABSTRACT FROM AUTHOR]

In this paper, we present a survey about the latest results in global stability concerning the discrete-time evolutionary Ricker competition model with n species, in both, autonomous and periodic models. The main purpose is to convey some arguments and new ideas concerning the techniques for showing global asymptotic stability of fixed points or periodic cycles in these kind of discrete-time models. In order to achieve this, some open problems and conjectures related to the evolutionary Ricker competition model are presented, which may be a starting point to study global stability, not only in other competition models, but in predator–prey models and Leslie–Gower-type models as well. [ABSTRACT FROM AUTHOR]

The sigmoid functional response in the predator–prey model was posed in 1977. But its dynamics has not been completely characterized. This paper completes the classification of the global dynamics for the classical predator–prey model with the sigmoid functional response, whose denominator has two different zeros. The dynamical phenomena we obtain here include global stability, the existence of the heteroclinic and homoclinic loops, the consecutive canard explosions via relaxation oscillation, and the canard explosion to a homoclinic loop among others. As we know, the last one is a new dynamical phenomenon, which has never been reported previously. In addition, with the help of geometric singular perturbation theory, we solve the problem of connection between stable and unstable manifolds from different singularities, which has not been well settled in the published literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the global dynamics of a general reaction-diffusion model based on acid-mediated invasion hypothesis, which is a candidate explanation for the Warburg effect. Our theoretical results characterize the effects of acid resistance and mutual competition of healthy cells and tumor cells on tumor progression in the long term, i.e., whether the healthy cells and tumor cells coexist or the tumor cells prevail after tumor invasion. A key feature of this model is the density-limited tumor diffusion term for tumor cells, which might give rise to the degeneracy of the parabolic equation. To overcome this difficulty, we combine the construction of suitable Lyapunov functionals and upper/lower solutions. This paper continues and improves the work begun in [15]. [ABSTRACT FROM AUTHOR]

This paper considers the dynamical properties of a space and time discrete fractional reaction–diffusion epidemic model, introducing a novel generalized incidence rate. The linear stability of the equilibrium solutions of the considered discrete fractional reaction–diffusion model has been carried out, and a global asymptotic stability analysis has been undertaken. We conducted a global stability analysis using a specialized Lyapunov function that captures the system's historical data, distinguishing it from the integer-order version. This approach significantly advanced our comprehension of the complex stability properties within discrete fractional reaction–diffusion epidemic models. To substantiate the theoretical underpinnings, this paper is accompanied by numerical examples. These examples serve a dual purpose: not only do they validate the theoretical findings, but they also provide illustrations of the practical implications of the proposed discrete fractional system. [ABSTRACT FROM AUTHOR]

This paper analyzes an allelopathic phytoplankton competition model, which was proposed by Bandyopadhyay [Dynamical analysis of a allelopathic phytoplankton model, J Biol Syst14(02):205–217, 2006]. Our study refines the previous results and finds at most three positive equilibria for the system. The existence conditions of all positive equilibria and the corresponding stability cases are given in the paper. Interesting dynamical phenomena such as bistability, saddle-node bifurcation, and cusp bifurcation are found. It is shown that the rate of toxin releases heavily influences the positive equilibria of the system under certain conditions. Numerical simulations verify the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of traveling wavefronts for a three-component Lotka–Volterra system with nonlocal dispersal. This system arises in the study of three-species competition model in which there is no competition between two of these three species. It has been shown that this system admits a bistable traveling wavefront. In this paper, we further investigate the stability of bistable traveling wavefronts. By constructing suitable super- and sub-solutions and using a dynamical system approach, we obtain the globally asymptotic stability of the bistable traveling wavefronts. [ABSTRACT FROM AUTHOR]

In this paper, we consider a food chain model with ratio-dependent functional response and prey-taxis. We first investigate the global existence and boundedness of the unique positive classical solutions of the system in a bounded domain with smooth boundary and Neumann boundary conditions. Then, we analyze the local stability of the system and the existence of Hopf bifurcation. In addition, we prove the global asymptotic stability of steady states under some conditions by constructing a Lyapunov functional, and investigate convergence rates. Finally, we present several numerical simulations to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we study an SSvEIQR model with nonlinear contact rate, isolation rate and vaccination rate driven by media coverage. First, the basic reproduction number R0 is derived. Then, the threshold dynamics of the disease are obtained in terms of R0: when R0 ≤ 1, the global stability of the disease-free equilibrium is obtained by constructing an appropriate Lyapunov function; when R0 > 1, the sufficient conditions to prove the globally stability of endemic equilibrium are obtained by applying the geometric method into the four-dimensional system, which needs to estimate the Lozinskiǐ measure of a 6 × 6 matrix. Further, we conduct some numerical simulations to validate our theoretical results, and analyze the impact of media coverage on disease transmission, the results show that media coverage could effectively suppress the spread of the disease and reduce the number of infected individuals. Finally, through the sensitivity analysis of R0, we obtain some measures to control the spread of the disease, such as reducing contact, strengthening isolation and vaccination. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of population dynamics of an EIS epidemic reaction-diffusion model in spatially heterogeneous environments, where the rates of disease-induced mortality and disease transmission are space dependent. For such a class of systems, we provide clear pictures on the global dynamics in terms of the basic reproduction number R 0 , which improves the local dynamics proposed in Wang and Zhao (2012) [20]. When R 0 ≤ 1 , the disease-free steady state is globally asymptotically stable. Different from earlier works for many epidemic models, we obtain the positive steady states by using bifurcation theory for R 0 > 1 , which can be easily generalized to the case where the reaction-diffusion system involves partial diffusion coefficients of zero. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Zika transmission model which considers human-to-human sexual transmission, the extrinsic incubation period of mosquitoes, and the vector-bias effect. Firstly, the explicit expression of the basic reproduction number R0 is given by using the next-generation operator method, and the global dynamics of the model are established by taking R0 as the threshold condition, that is, if R0 ≤ 1, the disease-free equilibrium is globally asymptotically stable, if R0 > 1, the model has a unique endemic equilibrium that is locally asymptotically stable and the disease persists. And when we ignore the vector-bias effect, the global asymptotic stability of the endemic equilibrium is proved by constructing a Lyapunov function. Then, we select the reported epidemic data from Brazil for fitting, which verifies the obtained theoretical results. Meanwhile, we study the impact of human-to-human sexual transmission rate and mosquito-to-human transmission rate on the spread and prevalence of Zika. In addition, we calculate the sensitivity indices of R0 to the model parameters and provide effective measures to control Zika transmission. The simulation results indicate that extending the extrinsic incubation period of mosquitoes is beneficial for disease control while ignoring the vector-bias effect will underestimate the risk of Zika transmission. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a two-species competition model with seasonal succession and different harvesting strategies. We aim to comprehend the impact of the proportion of good season on species competition outcomes and harvesting practices. For the case of weak harvest, we establish proportion thresholds for the good season for the species. Based on these threshold settings, we determine conditions for the existence and uniqueness of two semi-trivial periodic solutions and the existence, uniqueness and stability of positive periodic solutions under certain conditions. Numerical simulations and brief discussions are presented to validate our theoretical findings. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive study of disease spreading dynamics through the application of a nonlinear fractional order epidemic SEIRS model. By incorporating the Crowley–Martin type functional response and a saturated treatment function, the model effectively captures the intricacies of real-world epidemics. Our research establishes the existence, uniqueness, non-negativity and boundedness of the solution, while also investigating the model’s fundamental reproduction number. Additionally, we conduct a thorough analysis of the specific conditions governing the local and global stability of the model’s equilibriums. A notable observation is the variation of the reproduction number with the fractional-order α, which represents a memory effect on individuals’ dynamic behavior and reveals the influence of interactions between compartments. To validate these theoretical findings, we employ numerical simulations using Matlab, demonstrating that inhibition measures for susceptibles and the saturated treatment parameters play a pivotal role in determining the disease state. Specifically, we observe that as these parameter values increase, the transition from endemic equilibrium to disease-free equilibrium occurs. [ABSTRACT FROM AUTHOR]

Heterogeneity in susceptibility and infectivity is a central issue in epidemiology. Although the latter has received some attention recently, the former is often neglected in modeling of epidemic systems. Moreover, very few studies consider both of these heterogeneities. This paper is concerned with the characterization of epidemic models with differential susceptibility and differential infectivity under a general setup. Specifically, we investigate the global asymptotic behavior of equilibria of these systems when the network configuration of the Susceptible-Infectious interactions is strongly connected. These results prove two conjectures by Bonzi et al. (J Math Biol 62:39–64, 2011) and Hyman and Li (Math Biosci Eng 3:89–100, 2006). Moreover, we consider the scenario in which the strong connectivity hypothesis is dropped. In this case, the model exhibits a wider range of dynamical behavior, including the rise of boundary and interior equilibria, all based on the topology of network connectivity. Finally, a model with multidirectional transitions between infectious classes is presented and completely analyzed. [ABSTRACT FROM AUTHOR]

This paper considers stability analysis of a Susceptible-Exposed-Infected-Recovered-Virus (SEIRV) model with nonlinear incidence rates and indicates the severity and weakness of control factors for disease transmission. The Lyapunov function using Volterra–Lyapunov matrices makes it possible to study the global stability of the endemic equilibrium point. An optimal control strategy is proposed to prevent the spread of coronavirus, in addition to governmental intervention. The objective is to minimize together with the quantity of infected and exposed individuals while minimizing the total costs of treatment. A numerical study of the model is also carried out to investigate the analytical results. [ABSTRACT FROM AUTHOR]

Free-floating space robotic manipulators (FSRMs) are robotic arms mounted on space platforms, such as spacecraft or satellites which are used for the repair of space vehicles or the removal of noncooperating targets such as inactive material remaining in orbit. In this paper, a novel nonlinear optimal control method is applied to the dynamic model of FSRMs. First, the state-space model of a 3-DOF free-floating space robot is formulated and its differential flatness properties are proven. This model undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the free-floating space robot a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution of the nonlinear optimal control problem under model uncertainty and external perturbations. To compute the controller's feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed nonlinear optimal control approach achieves fast and accurate tracking of setpoints under moderate variations of the control inputs and a minimum dispersion of energy by the actuators of the free-floating space robot. [ABSTRACT FROM AUTHOR]

This paper deals with the global boundedness and asymptotic stability of the solution of the two-predator and one-prey systems with density-dependent motion in a n-dimensional bounded domain with Neumann boundary conditions. In a previous paper, Qiu et al. (J Dyn Differ Equ, 1–25, 2021) proved the global existence and uniform boundedness of classical solution by limiting the conditions on motility functions and the coefficients of logistic source. By contrast, we relax the limitation conditions in Qiu et al. (2021) by constructing the weight function. Moreover, under diverse competition circumstances, the global stabilities of nonnegative spatially homogeneous equilibria for the special model are established. [ABSTRACT FROM AUTHOR]

In this paper, the population dynamics of a novel coronavirus are studied. An extended SEIR model with quarantine, hospitalization, and the environment compartment is proposed to simulate the novel coronavirus epidemic. The model considers eight distinct epidemiological classes: susceptible, exposed, asymptomatic, symptomatic, quarantined, hospitalized, recovered, and viruses in the environment. The basic reproduction numbers are determined by using a method called the next-generation matrix. The model has two equilibria: a disease-free equilibrium and an endemic equilibrium. The Lyapunov function and the LaSalle invariance principle are used to analyze the global asymptotical stability of the equilibria of the proposed model. The disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity, and the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than unity. To study cost-effectiveness assessments for the four optimum control strategies, we ran numerical simulations. When the four control strategies were compared, it was discovered that Strategy A (public health education and intensive medical treatment) was the most economical and efficient control intervention in the absence of vaccination. However, we observe that strategies A, B, and D are similarly effective at containing COVID-19 in terms of infection prevention. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a population suppression model and a population replacement model with periodic impulsive releases of Nilaparvata lugens infected with wStri. The conditions for the stability of wild- $ N.\,lugens $ N. l u g e n s -eradication periodic solution of two systems are obtained by applying the Floquet theorem and comparison theorem. And the sufficient conditions for the persistence in the mean of wild $ N.\,lugens $ N. l u g e n s are also given. In addition, the sufficient conditions for the extinction and persistence of the wild $ N.\,lugens $ N. l u g e n s in the subsystem without wLug are also obtained. Finally, we give numerical analysis which shows that increasing the release amount or decreasing the release period are beneficial for controlling the wild $ N.\,lugens $ N. l u g e n s , and the efficiency of population replacement strategy in controlling wild populations is higher than that of population suppression strategy under the same release conditions. [ABSTRACT FROM AUTHOR]

In this paper. we propose a mathematical model that describes the dynamics of transmission of COVID-19 between potential people and infected ones, as well as between bats as virus carriers and people. This article deals with optimal control applied to vaccination, treatment strategies, and the hunt for a SIR-B epidemic model with logistic growth. The global stability of the disease-free and endemic equilibrium is verified. The existence of optimal control is demonstrated. The Pontryagin maximum principle is employed to describe these optimal controls. To validate our previous theoretical results. the optimality system is numerically resolved using a Matlab algorithm based on the Runge Kutta approximation 4. [ABSTRACT FROM AUTHOR]

In this paper we consider the ultimate dynamics of one 4D cancer model which was created for studying the immune response to the two-phenotype tumors. Our approach is based on the localization method of compact invariant sets. The existence of a positively invariant polytope is shown and its size is calculated depending on the parameters of this cancer model. Various convergence conditions to the tumor free equilibrium point were proposed. This property has the biological meaning of global asymptotic tumor eradication (GATE). Further, the case in which local asymptotic tumor eradication (LATE) conditions entail GATE conditions was found. Our theoretical studies of ultimate dynamics are complemented by numerical simulation results. [ABSTRACT FROM AUTHOR]

In this paper, a novel pest-natural enemy model with additional food source and Holling-( p +1) type functional response is put forward for plant pest management by considering multiple food sources for predators. The dynamical properties of the model are investigated, including existence and local asymptotic stability of equilibria, as well as the existence of limit cycles. The inhibition of natural enemy on pest dispersal and the impact of additional food sources on system dynamics are elucidated. In view of the fact that the inhibitory effect of the natural enemy on pest dispersal is slow and in general deviated from the expected target, an integrated pest management model is established by regularly releasing natural enemies and spraying insecticide to improve the control effect. The influence of the control period on the global stability and system persistence of the pest extinction periodic solution is discussed. It is shown that there exists a time threshold, and as long as the control period does not exceed that threshold, pests can be completely eliminated. When the control period exceeds that threshold, the system can bifurcate the supercritical coexistence periodic solution from the pest extinction one. To illustrate the main results and verify the effectiveness of the control method, numerical simulations are implemented in MATLAB programs. This study not only enriched the related content of population dynamics, but also provided certain reference for the management of plant pest. [ABSTRACT FROM AUTHOR]

In this paper, an attempt has been made to explore a new delayed epidemiological model assuming that the disease is transmitted among the susceptible population and possessing nonlinear incidence function along with a saturated treatment rate. Due attention is paid to the positivity and boundedness of the solutions and the bifurcation of the dynamical system as well. Basic reproduction number is being calculated, and considering the latent period as a bifurcation parameter, it has been examined that a Hopf-bifurcation occurs near the endemic equilibrium point while the parameter attains critical values. We have also discussed the stability and direction of Hopf-bifurcation near the endemic equilibrium point, the global stability analysis and the optimal control theory. We conclude that the system reveals chaotic dynamics through a specific time-delay value. Numerical simulations are being performed in order to explain the accuracy and effectiveness of the acquired theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a predator–prey model that takes into account the interactions between three species of fish, zooplankton, and phytoplankton has been developed. The logistic growth of phytoplankton has been considered in the absence of zooplankton. It is assumed that zooplankton eats phytoplankton. It is considered that the growth rate of zooplankton decreases due to the fear of fish. The release of toxins by phytoplankton and fish consumption are two further factors that are assumed to decrease the growth rate of zooplankton. The harvesting of fish has also been considered in this study. Local and global stability of the system around the equilibrium points has been investigated. The existence condition of oscillatory behaviour of the system has been studied through Hopf bifurcation. It is found that the increasing rate of minimum cost of fear η may stabilize the system. It is obtained that the rate of increase of release of toxin substance ρ by phytoplankton may be responsible for the stability of the system. It is also found that the system may become unstable for the increase of level of fear ξ of zooplankton. Some numerical simulation results are presented to test the theoretical findings. [ABSTRACT FROM AUTHOR]

The anti-predator element because of fear from predator population in prey–predator models has an excellent significance and it is able to lessen the duplicate of prey population to a positive degree. In this paper, a delayed prey–predator model including fear effect and predator-taxis sensitivity is studied. Local stability and occurrence of Hopf bifurcation are analyzed by fixing the time delay as the bifurcating parameter. Then, a series of sufficient criteria for global stability is established by designing a suitable Lyapunov function. Furthermore, direction and stability of the Hopf bifurcation are studied. Lastly, numerical simulation outcomes are performed to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]