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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]

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]

This paper is concerned with the global dynamics of a two-species Grindrod clustering model with Lotka–Volterra competition. The model takes the advective flux to depend directly upon local population densities without requiring intermediate signals like attractants or repellents to form the aggregation so as to increase the chances of survival of individuals like human populations forming small nucleated settlements. By imposing appropriate boundary conditions, we establish the global boundedness of solutions in two-dimensional bounded domains. Moreover, we prove the global stability of spatially homogeneous steady states under appropriate conditions on system parameters, and show that the rate of convergence to the coexistence steady state is exponential while the rate of convergence to the competitive exclusion steady state is algebraic. [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]

This paper is the first to propose an allelopathic phytoplankton competition ODE model influenced by the fear effect based on natural biological phenomena. It is shown that the interplay of this fear effect and the allelopathic term cause rich dynamics in the proposed competition model, such as global stability, transcritical bifurcation, pitchfork bifurcation, and saddle-node bifurcation. We also consider the spatially explicit version of the model and prove analogous results. Numerical simulations verify the feasibility of the theoretical analysis. The results demonstrate that the primary cause of the extinction of non-toxic species is the fear of toxic species compared to toxins. Allelopathy only affects the density of non-toxic species. The discussion guides the conservation of species and the maintenance of biodiversity. [ABSTRACT FROM AUTHOR]

In this paper, we consider an SEIS epidemic model with infectious force in latent and infected period, which incorporates by nonlinear incidence rates. The local stability of the equilibria is discussed. By means of Lyapunov functionals and LaSalle's invariance principle, we proved the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium. An application is given and numerical simulation results based on real data of COVID-19 in Morocco are performed to justify theoretical findings. [ABSTRACT FROM AUTHOR]

This paper delves into the study of a diffusive SIR epidemic model characterized by reaction–diffusion equations enriched with a fractional derivative within the Caputo framework. Within this model, we incorporate a general incidence function and meticulously analyze how the adoption of mask-wearing and adherence to physical distancing protocols intricately shape the dynamics of susceptible and infected individuals. Our exploration commences by establishing the existence and uniqueness of a positively bounded solution for the model, employing powerful Banach's fixed point theorem. Moreover, we showcase that this solution demonstrates distinctive global mild attributes. Subsequently, we elucidate the two equilibrium points inherent in the system: the disease-free and endemic points. Employing the LaSalle–Lyapunov theorem, we establish that the global stability of these equilibrium points is predominantly contingent upon the basic reproduction number of the system. This stability assertion holds true across various values of the non-integer order derivative. Lastly, we substantiate our findings with a series of numerical simulations that provide tangible support for the preceding analytical results. [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]

PID (proportional-integral-derivative) control is recognized to be the most widely and successfully employed control strategy by far. However, there are limited theoretical investigations explaining the rationale why PID can work so well when dealing with nonlinear uncertain systems. This paper continues the previous researches towards establishing a theoretical foundation of PID control, by studying the regulation problem of PID control for nonaffine uncertain nonlinear stochastic systems. To be specific, a three dimensional parameter set will be constructed explicitly based on some prior knowledge on bounds of partial derivatives of both the drift and diffusion terms. It will be shown that the closed-loop control system will achieve exponential stability in the mean square sense under PID control, whenever the controller parameters are chosen from the constructed parameter set. Moreover, similar results can also be obtained for PD (PI) control in some special cases. A numerical example will be provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, Lotka–Volterra recurrent neural networks with time-varying delays on time scales are considered. Using Banach's fixed-point principle, the theory of calculus on time scales and suitable Lyapunov functional, some sufficient conditions for the existence, uniqueness and Stepanov-exponential stability of positive weighted Stepanov-like pseudo almost periodic solution on time scales to the recurrent neural networks are established. Finally, an illustrative example and simulations are presented to demonstrate the effectiveness of the theoretical findings of the paper. The results of this paper are new and generalize some previously-reported results in the literature. [ABSTRACT FROM AUTHOR]

This paper explores the effect of immigration in a generalized model that considers susceptibles, infected, chronic carriers, and recovered, the incidence rate is considered as a general function and the immigration as a constant in all its populations. This model has the characteristic that carriers and infected can transmit the disease, besides it has not a disease-free equilibrium point and no basic reproductive number when the immigration is considered. Using an appropriate Lyapunov function and with suitable conditions on the functions involved in the general incidence, we show that the endemic equilibrium point is globally asymptotically stable. When the immigration is not considered the model has a disease-free equilibrium point, endemic equilibrium and basic reproductive number which are globally asymptotically stable depending on the magnitude of this threshold. Through numerical simulations we show that even having a good vaccination rate, recovery rate, diagnosis rate cannot stop the transmission of the disease if migration or immigration is considered. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamic of a multi-strain SEIR model with both saturated incidence and treatment functions. Two basic reproduction numbers are extracted from the epidemic model, noted R 0 , 1 and R 0 , 2 . Using the Lyapunov method, we investigate the global stability of the disease free equilibrium and prove that it is globally asymptotically stable when R 0 , 1 and R 0 , 2 are less than one. Moreover, we formulate the optimal control problem, solve it, and perform some numerical simulations, to support the analytical results and test how well the proposed model may be applied in practice. [ABSTRACT FROM AUTHOR]

In this paper, we study a classical two species stage structure Lotka–Volterra diffusion–advection system. By employing the Krein–Rutman theorem, analyzing the principal eigenvalue, and combining with the theory of monotone dynamical systems, a classification of the global dynamics is given for the weak competition case. [ABSTRACT FROM AUTHOR]

This paper aims to the investigation of the global threshold dynamics of an infection age-space structured HIV infection model. The model is formulated in a bounded domain involving two infection routes (virus-to-cell and cell-to-cell) and Neumann boundary conditions. We first transform the original model to a hybrid system containing two partial differential equations and a Volterra integral equation. By appealing to the theory of fixed point problem together with Picard sequences, the well-posedness of the model is shown by verifying that the solution exists globally and the solution is ultimately bounded. Under the Neumann boundary condition, we establish the explicit expression of the basic reproduction number. By analyzing the distribution of characteristic roots of the associated characteristic equation in terms of the basic reproduction number, we achieve the local asymptotic stability of the steady states. The global asymptotic stability of the steady states is established by the technique of Lyapunov functionals, respectively. Numerical simulations are performed to validate our theoretical results. [ABSTRACT FROM AUTHOR]

Age and spatial heterogeneities play a significant role in the prediction of transmission patterns of the cholera prevalence. In this paper, we propose a diffusive cholera epidemic model with chronological age and infection age structures. First, we adopt some embedding tricks and transform the model into a non-densely Cauchy problem. Then we establish the existence, uniqueness and dissipativity of the model by the integral semigroup and the Volterra integral equation theory. Finally, we show that the global dynamics of each feasible equilibrium determined by the basic reproduction number R 0 . If R 0 < 1 , the disease-free equilibrium is globally asymptotically stable; if R 0 > 1 , the endemic equilibrium E ∗ is globally attractive as long as the shedding rate of an infected individual is a function of chronological age or infection age. [ABSTRACT FROM AUTHOR]

A hepatitis B or C virus (HBV or HCV) epidemic model with intra- and extra-hepatic coinfection, immune delay and saturation incidence, as well as antiviral therapy is proposed in this paper. The existence of equilibria (infection-free, immune-free and immune-activated), the basic reproduction numbers, i.e., R 0 , R 1 , are given respectively, by which the criteria on (local and global) stability of above equilibria are established. Furthermore, if the immune delay τ > τ 0 , both the existence of subcritical (supercritical) Hopf bifurcation on the immune-activated equilibrium E ∗ , and the stability of bifurcating periodic solutions are obtained. Finally, the theoretical results are demonstrated by numerical simulations. We derive that the immune delay and intra- and extra-hepatic coinfection have significant influence on the transmission of HBV/HCV, could cause more complicated dynamics at E ∗ from stability to unstablity untill bifurcation, which greatly increases the difficulty of disease control. While effective antiviral therapy could evidently decrease the spread of HBV/HCV. [ABSTRACT FROM AUTHOR]

In this paper we study a modified SEIR model with general incidence function of the form f (s) [ g (I 1) + h (I 2) ] where I 1 and I 2 are two infection categories different and the migration is constant in all compartments. The model admits neither a disease-free equilibrium point nor a basic reproduction number. Using a suitable Lyapunov function and under sufficient conditions on the functions f, g and h we show that the endemic equilibrium point is globally asymptotically stable. The considered model without migration has a disease-free equilibrium, at least one endemic equilibrium and a basic reproduction number. We show according to the values of R 0 that these equilibria are globally asymptotically stable. To illustrate the results obtained we use a non-linear incidence function given by β S I 1 1 + α 1 I 1 + η I 2 1 + α 2 I 2 where I 1 modeling uneducated infected individuals and I 2 modeling educated infected individuals. Next, we performed sensitivity analysis to determine how each parameter of the model may affect disease transmission. Finally, using reasonably chosen numerical data, we confirm our analytical results. [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]

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]

This paper discusses autonomous and nonautonomous epidemic models with nonlinear incidence rate of saturated mass action and feedback controls. The global asymptotic stability of disease-free equilibrium and the endemic equilibrium of the autonomous system is established using suitable Lyapunov functional. It is shown that by choosing suitable values of feedback control variables, one can make the disease endemic or extinct as time evolves. Moreover, the effect of coefficient of inhibition on the persistence of disease is also discussed. We discuss the permanence, existence, uniqueness and asymptotic stability of an almost periodic solution of the model. The analytical results obtained in this paper are illustrated with the help of numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of an infectious disease is studied by considering age-structured models; a stage structure and an age-structured epidemic model. The respective basic reproduction numbers for the proposed models are calculated, and the local analyses of the equilibria of the models are investigated by using the method of linearization. The global dynamics of the two models are analyzed by using the wave lemma and the Lyapunov function theory. This study establishes a solid theoretical framework and a rigorous mathematical formulation for the prevention and control of pseudorabies. [ABSTRACT FROM AUTHOR]

In this paper, the tools of dynamical systems theory are applied to examine the streamline patterns and their local and global bifurcations for ciliary-induced peristalsis of non-Newtonian fluid (blood) with the suspension of hybrid nanoparticles (Cu–Ag/Phan–Thien–Tanner based fluid) in a tube with heat source effect. The thermodynamics of this model are recently described by Ali et al. (Biomech Model Mechanobiol 20:2393–2412, 2021), where the fluid flows through a tube whose inner walls are considered to be ciliated with small hair-like structures. However, our novel approach allows us to create a complete picture of the model's overall dynamic behavior in terms of bifurcation point analysis exhibiting qualitatively different flow modes. Special attention is paid to the computing, analysis and simulation of equilibrium points in terms of capturing the global dynamics, such as evaluating the heteroclinic bifurcation, which is used to identify trapping phenomena in response to biological characteristics such as wave amplitude, Weissenberg and wave numbers. The main novelty here is the ability to control the position of the equilibrium points in the domain of interest, allowing one to identify global bifurcations that reflect key dynamic properties of the model. Based on the advantages of this technique, the maximum trapping volume and symmetric trapping zones adjacent to the walls are determined as a novel result. We also show that as the solid volume fraction of copper and the Brinkman number increases, the isotherm patterns become more distorted. Our findings highlight a novel class of complex behavior that governs transitions between qualitatively different modes and trapping phenomena. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to explore the dynamical behavior of a three-species prey-predator interaction model with sexually reproductive generalized type top predator under the consideration of environmental fluctuations. Our discussion has involved a continuous tritrophic food chain model with Crowley-Martin senses and Holling type II functional responses. For the deterministic model, the existence of equilibria as well as boundedness of the solution has been established here. The feasibility and local stability of the interior and non-interior equilibrium points has been investigated. The global dynamics of the co-existence of all the three species are shown at the interior equilibrium point. To determine the direction of Hopf bifurcations under the non-hyperbolic case, the first Lyapunov number is computed using the center manifold theorem. Several bifurcation analyses are performed at the interior equilibrium point. The effect of environmental fluctuation on some of model parameters are studied here through the verification of existence of unique positive global solution, existence and persistence of stationary solution. Numerical simulation has been carried out to illustrate the theoretical findings by using the MATLAB and Maple software packages and finally some concluding remarks are given. [ABSTRACT FROM AUTHOR]

This paper presents a modal and non-modal stability analysis of the boundary layer developed on a hot plate. A liquid-type temperature-dependent viscosity model has been considered to account for the viscosity variation in the boundary layer region. The base flow is uniform and parallel to the surface at the leading edge. The base flow solution is obtained using an open-source finite volume source code. The Reynolds number (Re) is defined based on the displacement thickness (δ*) at the inlet of the computation domain. The spectral collocation method is used for spatial discretization of governing stability equations. The formulated generalized eigenvalue problem (EVP) is solved using Arnoldi's iterative algorithm with the shift and invert strategy. The global temporal eigenmodes are calculated for the sensitivity parameter β from 1 to 7, Re = 135, 270, and 405, and the span wise wave-number N from 0 to 1. The modal and non-modal stability analysis have been performed to study the least stable eigenmodes and the optimal initial conditions and perturbations (using mode superposition), respectively. The global temporal eigenmodes are found more stable for β > 0 at a given value of N. Thus, heating the boundary layer within the considered range of β (0 < β ≤ 7) leads to the stabilization of flow. The optimal energy growth increases with the β due to reducing the perturbation energy loss. Tilted elongated structures of the optimal perturbations are found near the outflow boundary. However, the length scale of the elongated cellular mode structure reduces with increase in β. The same qualitative structure of the optimal perturbations has been found at a given value of N. [ABSTRACT FROM AUTHOR]

In this paper, we first formulate a system of ODEs–PDE to model diseases with latency-age and differential infectivity. Then, based on the ways how latent individuals leave the latent stage, one ODE and two DDE models are derived. We only focus on the global stability of the models. All the models have some similarities in the existence of equilibria. Each model has a threshold dynamics for global stability, which is completely characterized by the basic reproduction number. The approach is the Lyapunov direct method. We propose an idea on constructing Lyapunov functionals for the two DDE and the original ODEs–PDE models. During verifying the negative (semi-)definiteness of derivatives of the Lyapunov functionals along solutions, a novel positive definite function and a new inequality are used. The idea here is also helpful in applying the Lyapunov direct method to prove the global stability of some epidemic models with age structure or delays. [ABSTRACT FROM AUTHOR]

A one-dimensional discrete-time dynamical system can be also seen as a recurrence, a difference equation, or an iteration scheme; and sometimes theoretical results come from different contexts. In this paper, I present a short survey about a particular family of one-dimensional maps that I have found in different situations. First, I introduce and explain the various motivations for the equation, and then I state some relevant results, with suitable references. Finally, I include some open problems and some words of caution about a series of recent poor-quality papers that, pretending to rediscover this equation, provide trivial results. [ABSTRACT FROM AUTHOR]

The stability of a new two-species discrete ratio-dependent predator-prey system is considered. By using the linearization method, we obtain some sufficient conditions for the local stability of the positive equilibria. We also obtain a new sufficient condition to ensure that the positive equilibrium is globally asymptotically stable by using an iteration scheme and the comparison principle of difference equations, which generalizes what paper (Chen and Zhou in J Math Anal Appl 27:7358-7366, 2003) has done. The method given in this paper is new and very resultful comparing with articles (Damgaard in J Theor Biol 227:197-203, 2004; Edmunds in Theor Popul Biol 72:379-388, 2007; Fan and Wang in Math Comput Model 35:951-961, 2002; Muroya in J Math Anal Appl 330:24-33, 2007; Huo and Li in Appl Math Comput 153:337-351, 2004; Liao et al. in Appl Math Comput 190:500-509, 2007) and it can also be applied to study other global asymptotic stability for general multiple species discrete population systems. At the end of this paper, we present two open questions. [ABSTRACT FROM AUTHOR]

Considering that HBV belongs to the DNA virus family and is hepatotropic, we model the HBV DNA-containing capsids as a compartment. In this paper, a delayed HBV infection model is established, where the general incidence function and two infection routes including cell-virus infection and cell-cell infection are introduced. According to some preliminaries, including well-posedness, basic reproduction number and existence of two equilibria, we obtain the threshold dynamics for the model. We illustrate numerical simulations to verify the above theoretical results, and furthermore explore the impacts of intracellular delay and cell-cell infection on the global dynamics of the model. [ABSTRACT FROM AUTHOR]

The flexibility and programmability of SDN enable dynamic and automated network configuration and traffic routing. However, this also provides more avenues for malicious code propagation, leading to serious risks such as service disruptions and privacy breaches. To address this problem, we first designed three modules to suppress malicious code propagation: the abnormal traffic detection module, the malicious code analysis module, and the abnormal traffic tracing module. Then, the sharing mechanism is introduced. In order to analyze the process of malicious code propagation more clearly, based on the above strategy, this paper introduces the warning node into the classical SIR model, which can be exploited for studying how to control malicious code propagation to prevent large-scale outbreaks. The propagation threshold and equilibrium point of the proposed model are obtained through calculations. By constructing a Lyapunov function, the equilibrium point is proven stable. Finally, numerical simulation results indicate that when the detection rate reaches 90%, approximately 86.3% fewer nodes are infected at the peak point. Through comparative analysis, our system demonstrates optimal performance, validating the effectiveness of the analytical results. [ABSTRACT FROM AUTHOR]

Several diseases are transmitted from the infected individual to the uninfected one by different ways, like, pests, vectors, bacteria, viruses, etc. The living materials generally suffer from diseases some of which are contagious and the others are not infectious. There are lots of infectious diseases which are carried by the pests to the new individual to get infected after coming in touch of those particular pests. In this paper, a non-linear mathematical model is proposed to study the dynamics of disease transmission via pest among both components of susceptible prey and infected prey. The crisp model so formulated is converted to fuzzy model in which the control parameters are taken as triangular fuzzy numbers. The fuzzy model is reduced to the defuzzified model by using graded mean integration technique for defuzzification for triangular fuzzy number for the ease to find the solution. Also, the model is compared with fully stochastic model using a Markov process. The existence, uniqueness and boundedness of the solution of the proposed model are discussed. Also, all the feasible equilibrium points are determined and local stability, global stability analysis and Hopf bifurcations are investigated. Further, a numerical experiment is done with the help of MATLAB packages. Finally, the sensitivities of the control parameters are studied and presented graphically from which the control of the disease transmission can be handled. [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate a mathematical model describing the interactions between hepatitis B virus with DNA-containing capsids, liver cells (hepatocytes) and the adaptive immune response. This adaptive immunity will be represented by cytotoxic T-lymphocytes and antibody immune responses. The positivity and boundedness of solutions for non negative initial data are proved which is consistent with the biological studies. The local stability of the equilibria is established. In addition to this, the global stability of the disease-free equilibrium and the endemic equilibria is fulfilled by using appropriate Lyapunov functions. Finally, numerical simulations are performed to support our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider the population models with resource-dependent dispersal for single-species and two-species with competition. For the single-species model, it is well-known that the total population supported by the environment is always greater than the environmental carrying capacity if the dispersal is simply random diffusion. However, we find that the total population supported can be equal or smaller than the environmental carrying capacity when the dispersal depends on the resource distribution. This analytical finding not only well agrees with the yeast experiment observation of Zhang et al. (Ecol Lett 20(9):1118–1128, 2017), but also indicates that resource-dependent dispersal may produce different outcomes compared to the random dispersal. For the two-species competition model, when two competing species use the same dispersal strategy up to a multiplicative constant (i.e. their dispersal strategies are proportional) or both diffusion coefficients are large, we give a classification of global dynamics. We also show, along with numerical simulations, that if the dispersal strategies are resource-dependent, even one species has slower diffusion, coexistence is possible though competitive exclusion may occur under different conditions. This is distinct from the prominent result that with random dispersal the slower diffuser always wipes out its fast competitor. Our analytical results, supported by the numerical simulations, show that the resource-dependent dispersal strategy has profound impact on the population dynamics and evolutionary processes. [ABSTRACT FROM AUTHOR]

Delay-differential-algebraic equations have been widely used to model some important phenomena in science and engineering. Since, in general, such equations do not admit a closed-form solution, it is necessary to solve them numerically by introducing suitable integrators. The present paper extends the class of block boundary value methods (BBVMs) to approximate the solutions of nonlinear delay-differential equations with algebraic constraint and piecewise continuous arguments. Under the classical Lipschitz conditions, convergence and stability criteria of the extended BBVMs are derived. Moreover, a couple of numerical examples are provided to illustrate computational effectiveness and accuracy of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stability problem on perturbation near a physically steady state solution of the 3D generalized incompressible magnetohydrodynamic system in Lei-Lin space. The global stability and analytic estimates for small perturbation are established by the semigroup method in the critical space χ 1 - 2 α (R 3) with 1 2 ≤ α ≤ 1 , where linear terms from perturbation incur much difficulty. By introducing a diagonalization process we successfully eliminate the linear terms. Then, by virtue of the analytic estimates for a solution, the temporal decay rate (1 + t) - (5 4 α - 1) of the global solution is obtained. [ABSTRACT FROM AUTHOR]

This paper deals with a deterministic mathematical model of dengue based on a system of fractional-order differential equations (FODEs). In this study, we consider dengue control strategies that are relevant to the current situation in Malaysia. They are the use of adulticides, larvicides, destruction of the breeding sites, and individual protection. The global stability of the disease-free equilibrium and the endemic equilibrium is constructed using the Lyapunov function theory. The relations between the order of the operator and control parameters are briefly analysed. Numerical simulations are performed to verify theoretical results and examine the significance of each intervention strategy in controlling the spread of dengue in the community. The model shows that vector control tools are the most efficient method to combat the spread of the dengue virus, and when combined with individual protection, make it more effective. In fact, the massive use of personal protection alone can significantly reduce the number of dengue cases. Inversely, mechanical control alone cannot suppress the excessive number of infections in the population, although it can reduce the Aedes mosquito population. The result of the real-data fitting revealed that the FODE model slightly outperformed the integer-order model. Thus, we suggest that the FODE approach is worth to be considered in modelling an infectious disease like dengue. [ABSTRACT FROM AUTHOR]

In this paper, an attempt has been made to study and investigate a non-linear, non-integer SIR epidemic model for COVID-19 by incorporating Beddington–De Angelis incidence rate and Holling type II saturated cure rate. Beddington–De Angelis incidence rate has been chosen to observe the effects of measure of inhibition taken by both: susceptible and infective. This includes measure of inhibition taken by susceptibles as wearing proper mask, personal hygiene and maintaining social distance and the measure of inhibition taken by infectives may be quarantine or any other available treatment facility. Holling type II treatment rate has been considered for the present model for its ability to capture the effects of available limited treatment facilities in case of Covid 19. To include the neglected effect of memory property in integer order system, Caputo form of non-integer derivative has been considered, which exists in most biological systems. It has been observed that the model is well posed i.e., the solution with a positive initial value is reviewed for non-negativity and boundedness. Basic reproduction number R 0 is determined by next generation matrix method. Routh Hurwitz criteria has been used to determine the presence and stability of equilibrium points and then stability analyses have been conducted. It has been observed that the disease-free equilibrium Q d is stable for R 0 < 1 i.e., there will be no infection in the population and the system tends towards the disease-free equilibrium Q d and for R 0 > 1 , it becomes unstable, and the system will tend towards endemic equilibrium Q e . Further, global stability analysis is carried out for both the equilibria using R 0 . Lastly numerical simulations to assess the effects of various parameters on the dynamics of disease has been performed. [ABSTRACT FROM AUTHOR]

We consider in this paper an ecological model, in a predator-prey interaction with the presence of a herd behavior. For the analysis of the model, the existence of positive solution and also the existence Hopf bifurcation, Turing driven instability, and Turing-Hopf bifurcation point have bee proved. Then by calculating the normal form, on the center of the manifold associated to the Hopf bifurcation points, the stability of the periodic solution has been proved. In the last part of the paper, numerical simulations has been given to illustrate our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we have mathematically analyzed a within-host model of SARS-CoV-2 which is used by Li et al. in the paper "The within-host viral kinetics of SARS-CoV-2" published in (Math. Biosci. Eng. 17(4):2853–2861, 2020). Important properties of the model, like nonnegativity of solutions and their boundedness, are established. Also, we have calculated the basic reproduction number which is an important parameter in the infection models. From stability analysis of the model, it is found that stability of the biologically feasible steady states are determined by the basic reproduction number (χ 0) . Numerical simulations are done in order to substantiate analytical results. A biological implication from this study is that a COVID-19 patient with less than one basic reproduction ratio can automatically recover from the infection. [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]

In this paper, the velocity tracking control problem of heavy haul trains (HHTs) is focused on multibody and control-oriented dynamic models. A composite learning algorithm is developed by combining a neural network with a high-order disturbance observer to approximate unknown nonlinearities and compounded disturbances collaboratively, where a prediction error is introduced to assess and improve the approximation accuracy. Since the neural network approximation ability holds only over a compact set, neural network-based control schemes can only ensure semi-globally uniform ultimate boundedness. Thus, a global tracking control scheme for HHTs is proposed that can switch between the composite learning controller and an additional robust controller via a switching mechanism. Finally, the globally uniform ultimate boundedness of closed-loop system signals is proven through Lyapunov theory. Simulation experiments are carried out based on an HXD1-type HHT running on the Da-Qin Line in China, and the results demonstrate the effectiveness of the proposed models and control technique. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss the dynamic behavior of a fuzzy Clifford-valued system with distributed delays involving the Equi-Weyl almost periodic coefficients. In particular, we will be looking at the conditions for the existence and uniqueness of the EW ap p solution and the system's stability by means of Lyapunov functions, as well as its efficiency through the utilization of numerical examples with simulations. The difficulties of such a system will be discussed, as it deals with fuzzy operations on a Clifford algebra. Therefore, a novel approach will be proposed in order to overcome them (do not break down the model into real-valued submodels), opening the door to the analysis of more realistic fuzzy Clifford-valued models with time delays. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an SVEIR epidemic model with reaction–diffusion and nonlinear incidence. We establish the well-posedness of the solutions and the basic reproduction number R 0 . Moreover, we show that the disease-free steady state is globally asymptotically stable when R 0 < 1 , whereas the disease will be persistent when R 0 > 1 . Furthermore, using the method of Lyapunov functional, we prove the global stability of the positive steady state for the spatial homogeneous model. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the global dynamics of a multiscale hepatitis C virus model. The model takes into account the evolution of the virus in cells and RNA. For the model, we establish the globally asymptotical stability of both infection-free and infected equilibria. We first give the basic reproduction number R 0 of the model, and then find that the system holds infected equilibrium when R 0 > 1 . Using eigenvalue analysis, Lyapunov functional, persistence theory and so on, it is proved that infection-free and infected equilibria are globally asymptotically stable when R 0 < 1 and R 0 > 1 , respectively. Thus, extinction and persistence of viruses in cells are theoretically judged. Finally, we show our theoretical results by means of numerical simulation. [ABSTRACT FROM AUTHOR]

As is known to us all, global solvability and nonnegativity need to be proved in the process of considering spatiotemporal dynamics of some biological models because a local or negative solution is meaningless in reality. In this paper, we investigate two categories of chemotactic reaction–diffusion intraguild predation models with different functional responses and chemotactic sensitivities in a bounded high-dimensional domain with smooth boundary: one category possesses saturating functional responses and general chemotactic sensitivities; the other category possesses Lotka–Volterra functional responses and constant chemotactic sensitivities. We first show global classical solvability for spatial dimension n ≥ 1 by Moser–Alikakos iteration technique for the former category. Then we prove global classical solvability for spatial dimension 1 ≤ n ≤ 2 by energy identity method for the latter category. Finally, we study a kind of specific reaction–diffusion–chemotaxis intraguild predation model with logistic growth rate and obtain global stability for all possible nonnegative constant steady states by constructing proper Lyapunov functionals, which especially implies that chemotaxis cannot destabilize the unique positive steady state for small chemotaxis coefficient at certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, an epidemic dynamical model with vaccination is proposed. Vaccination of both newborn and susceptible is included in the present model. The impact of the vaccination strategy with the vaccine efficacy is explored. In particular, the model exhibits backward bifurcations under the vaccination level, and bistability occurrence can be observed. Mathematically, a bifurcation analysis is performed, and the conditions ensuring that the system exhibits backward bifurcation are provided. The global dynamics of the equilibrium in the model are also investigated. Numerical simulations are also conducted to confirm and extend the analytic results. [ABSTRACT FROM AUTHOR]

In this paper, PID (proportional-integral-derivative) controllers will be designed to solve the tracking problem for a class of coupled multi-agent systems, where each agent is described by a second-order high-dimensional nonlinear uncertain dynamical system, which only has access to its own tracking error information and does not need to communicate with others. This paper will show that a 3-dimensional manifold can be constructed based on the information about the Lipschitz constants of the system nonlinear dynamics, such that whenever the three parameters of each PID controller are chosen from the manifold, the whole multi-agent system can be stabilized globally and the tracking error of each agent approaches to zero asymptotically. For a class of coupled first-order multi-agent nonlinear uncertain systems, a PI controller will be designed to stabilize the whole system. [ABSTRACT FROM AUTHOR]

In this paper, we study the global properties of a computer virus propagation model. It is, interesting to note that the classical method of Lyapunov functions combined with the Volterra-Lyapunov matrix properties, can lead to the proof of the endemic global stability of the dynamical model characterizing the spread of computer viruses over the Internet. The analysis and results presented in this paper make building blocks towards a comprehensive study and deeper understanding of the fundamental mechanism in computer virus propagation model. A numerical study of the model is also carried out to investigate the analytical results. [ABSTRACT FROM AUTHOR]

Plant population response to herbivore attack by releasing volatile organic compounds (VOCs) in a highly regulated fashion. The natural enemies of herbivores are attracted by such induced volatile. This type of mechanism is known as "indirect defense mechanism" for plants. A tritrophic interaction among the plants, herbivores and their carnivorous enemies is considered along with "indirect defense mechanism" of plants in recruitment rate of carnivorous enemies. A systematic analysis of the dynamics of plant–herbivore–carnivore system with Holling type II functional response is done in this paper. The conditions for existence and feasibility of boundary and interior equilibrium points of the system are also determined. The conditions for the global stability of positive equilibrium point is derived. Persistence of the system is studied. Here, different types of bifurcations are analyzed with respect to many parameters. A key role is played by immigration rate of carnivores in determining the eventual state of the ecosystem. Finally, numerical simulations are performed to validate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper a mathematical model is proposed and analyzed to study the dynamics of tri-trophic food chain model with refugia and allee effect in prey species. Criteria for local stability, instability and global stability of the non-negative equlibria are obtained. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system and global stability of the positive interior equilibrium solution are discussed. Further we pay attention to the chaotic dynamics which is produced by half-saturation constant. Our numerical simulations reveal that the three species food chain model without refugia and allee effect induced chaos from stable focus for increasing the value of half-saturation constant. We conclude that chaotic dynamics can be controlled by the Allee and refugia parameters. [ABSTRACT FROM AUTHOR]