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This paper studies an impact of media epidemic system with diffusion and linear source. We first derive the uniform bounds of solutions to impact on media reaction diffusion system. Then, the basic reproduction number is calculated and the threshold dynamics of impact media reaction diffusion system is also given and the Kuratowski measure κ of non-compactness is also considered. In addition, assume the spatial environment is homogeneous, it is shown that the unique endemic equilibrium of the system is global stability by constructing suitable Lyapunov function. Finally, we discuss the asymptotic profile of the system when the diffusion rate of the susceptible (infected) individuals for the system tends to zero or infinity. The main results show that the activities of infected individuals can only be at low risk, and then the virus eventually will be extinct, that is, to control the entry of viruses from abroad and increase the detection of domestic viruses. Finally, some numerical simulations are worked out to confirm the results obtained in this paper. [ABSTRACT FROM AUTHOR]

A stochastic three species predator–prey model with intermediate predator and anti-predator behavior is established and studied in this paper. By constructing suitable Lyapunov functions and combining knowledge of stochastic differential equations, the behavior of global positive solutions to the model are investigated. Firstly, we prove that there is unique global positive solution to the model, and establish the stochastic boundedness of the positive solution as well. Secondly, the sufficient condition for the existence of a unique ergodic stationary distribution is provided. Thirdly, the persistence and extinction of the populations are discussed. Finally, some numerical simulations demonstrate that the results obtained in this paper are true, moreover, the influence of white noise on the populations is revealed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the sufficient conditions that guarantee the stability, continuity, and boundedness of solutions for a type of second-order stochastic delay integro-differential equation (SDIDE). To demonstrate the main results, we employ a Lyapunov functional. An example is provided to demonstrate the applicability of the obtained result, which includes the results of this paper and obtains better results than those available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we incorporate stoichiometry into a competition model in chemostat culture. We first discuss the dynamics of this stoichiometric chemostat model for single algae species. We obtain the uniform persistence and find that an increase of phosphorus input or a slow dilution rate facilitates the algal persistence. Then, we consider a stoichiometric chemostat model in which two species competing for single nutrient, investigate how stoichiometry, dilution rate and concentration of phosphorus input affect the result of competition between algae species. Competitive exclusion and coexistence of two competing algae species are explored by discussing the existence, local stability of all feasible equilibria. Previous studies suggested that competitive exclusion holds in general chemostat models with two species competing for one limiting nutrient. Our theoretical and numerical results demonstrate that stoichiometry brings the coexistence of two competing algae species. Moreover, under low phosphorus input or fast dilution rate, competitive exclusion still holds. High phosphorus input or slow dilution rate facilitates the coexistence of multiple species. [ABSTRACT FROM AUTHOR]

This paper is concerned with the bounded solutions for a nonlinear second-order differential equation with asymptotic conditions and boundary condition which arise from the study of Arctic gyres. In the case of Lipschitz continuous nonlinearities, we prove the existence, uniqueness and stability of the bounded solution. An existence result for the general nonlinear vorticity term is also obtained. [ABSTRACT FROM AUTHOR]

This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results. [ABSTRACT FROM AUTHOR]

Fermentation is an indispensable link in wine brewing, and mathematical modeling is an effective means to study fermentation process, which can reveal the characteristics of state variables and help to optimize the control of fermentation process. In this paper, a new model with fractional derivative of the wine fermentation is proposed. The basic properties of the solution and the stability at the equilibrium point of the new model are proved. Then the numerical simulation of the fractional wine fermentation model is given by the generalized Euler method. Compared with the classical integer order wine fermentation model, the new fractional wine fermentation model proposed in the article is more responsive and reflects more comprehensive trends through qualitative and quantitative analysis. We expect that this fractional wine fermentation model can be applied to wine production in real world, which is beneficial for oenologists to grasp all kinds of data more accurately, thus improving the quality of wine. [ABSTRACT FROM AUTHOR]

Hepatitis is inflammation of the liver, and one of its types, hepatitis B, is a contagious infection. Using mathematical models, the nature of the spread of the Hepatitis B virus can be predicted. In the present paper, a hepatitis B epidemic model with a Beddington–DeAngelis type incidence rate and a constant vaccination rate is considered. Some dynamical properties of this model, such as non-negativity, boundedness character, the basic reproduction number R 0 , stability nature, and the bifurcation phenomenon, are investigated. By the Bendixson theorem, it is demonstrated that the disease-free equilibrium is globally asymptotically stable. It is shown that a transcritical bifurcation phenomenon occurs when R 0 = 1 . It is concluded that the endemic equilibrium is globally asymptotically stable when R 0 > 1 , by utilizing Dulac's criteria. Also, a discrete system of difference equations is obtained by constructing a non-standard finite difference (NSFD) scheme for the continuous model. It is shown that the solutions of this discrete system are dynamically consistent for all finite step sizes. The theoretical results obtained are also supported and visualized by numerical simulations. These simulations also demonstrate that the NSFD scheme produces much more efficient results than the Euler or RK4 schemes, as shown in the theoretical results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we provide a unified approach to simulate diffusion bridges. The proposed method covers a wide range of processes including univariate and multivariate diffusions, and the diffusions can be either time-homogeneous or time-inhomogeneous. We provide a theoretical framework for the proposed method. In particular, using the parametrix representations we show that the approximated probability transition density function converges to that of the true diffusion, which in turn implies the convergence of the approximation. Unlike most of the methods proposed in the literature, our approach does not involve acceptance-rejection mechanics. That is, it is acceptance-rejection free. Extensive numerical examples are provided for illustration and demonstrate the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on using piecewise derivatives to simulate the dynamic behavior and investigate the crossover effect within the coupled fractional system with delays by dividing the study interval into two subintervals. We establish and prove significant lemmas concerning piecewise derivatives. Furthermore, we extend and develop the necessary conditions for the existence and uniqueness of solutions, while also investigating the Hyers–Ulam stability results of the proposed system. The results are derived using the Banach contraction principle and the Leary–Schauder alternative fixed-point theorem. Additionally, we employ a numerical method based on Newton’s interpolation polynomials to compute approximate solutions for the considered system. Finally, we provide an illustrative example demonstrating our theoretical conclusions’ practical application. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Basener–Ross system with time-dependent coefficients. Based on a stability criterion of damped equations combined with the quantitative information obtained by the method of upper and lower solutions on the reversed order, some results on the existence and Lyapunov stability of the periodic solutions are obtained. [ABSTRACT FROM AUTHOR]

In the present paper, we study stochastic stability and stochastic boundedness for the stochastic differential equation (SDE) with multi-delay of third order. The derived results extend and improve some earlier results in the relevant literature, which are related to the qualitative properties of solutions to third-order delay differential equations (DDEs) and SDEs with multi-delay. Two examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

Assessment of the stability/boundedness of solutions to nonlinear systems with variable coefficients brings long-standing and challenging problems which emerge in various application domains. These problems naturally evolved into more arduous and largely open problems concerned with the estimation of the corresponding stability/boundedness regions. This paper develops a novel approach furnishing computationally tractable boundedness/stability criteria which underscores a methodology providing recursive estimation of the boundaries of the trapping/stability regions for a broad class of multidimensional and nonlinear systems with variable nonperiodic coefficients. Furthermore, our approach naturally conveys the bilateral bounds for the norms of solutions to the corresponding systems via the application of successive approximations which are introduced in this paper. The developed techniques are validated in inclusive simulations which endorse their applications to systems with large and complex nonlinear components and bounded in-norm time-varying disturbances. [ABSTRACT FROM AUTHOR]

This paper studies the pth moment exponential stability of stochastic delayed differential systems with impulsive control involving delays. By using the Razumikhin method, average impulsive delay and average dwell-time, sufficient conditions for pth moment exponential stability of stochastic delayed differential systems with impulsive control involving delays are obtained. In addition, we apply the derived theoretical results to nonlinear impulsive stochastic delayed systems. Finally, the validity and generality of the obtained results are demonstrated by an example. [ABSTRACT FROM AUTHOR]

Ideally, the state variable follows a constant motion law over time. However, due to the finiteness of motion speed, almost all systems have time delay. In this paper, we investigate a new class of piecewise conformable fractional impulsive differential system with delay under two point inhomogeneous boundary condition. In the system, the motion laws of state variable vary at different time periods, and they interact with each other through time delay " τ " and time leading " - τ ", so as to be more realistic. By employing the well-know fixed point theorems, the sufficient conditions for the existence and uniqueness of solutions to the system are established. Under the conditions of ensuring the existence of the system's solutions, we conclude further that the system has Ulam–Hyers stability and Ulam–Hyers–Rassias stability by means of nonlinear functional analysis method. Finally, we give a feasible example to explain our result. [ABSTRACT FROM AUTHOR]

Temperature-dependent sex determination (TSD) is an environmental phenomenon in which the temperature during the embryonic or larval stages influences the determination of sex. It is mainly observed in teleost fish, where high water temperatures can induce a female-to-male transition. The exact mechanism by which environmental changes affect TSD is poorly understood, but cortisol, a stress hormone, is considered a potential factor. Although excessive secretion of cortisol is known to cause side effects, it can lead to a switch in sex hormones, potentially resulting in TSD. In this paper, we investigate the mechanism of TSD in teleost fish. To assess the impact of stress caused by changes in water temperature on TSD, we established a mathematical model for hormonal dynamics that incorporates cortisol. First, we conducted a stability analysis to qualitatively examine the sex determination. The temperature dependence was modeled using the Eyring–Polanyi equation, and we examined corresponding hormonal changes with water temperature. Furthermore, we theoretically investigated the role of a cortisol inhibitor in preventing side effects during TSD. [ABSTRACT FROM AUTHOR]

Recent research indicates the need for improved models of physical phenomena with multiple shocks. One of the newest methods is to use differential inclusions instead of differential equations. In this work, we intend to investigate the existence of solutions for an m-dimensional system of quantum differential inclusions. To ensure the existence of the solution of inclusions, researchers typically rely on the Arzela–Ascoli and Nadler's fixed point theorems. However, we have taken a different approach and utilized the endpoint technique of the fixed point theory to guarantee the solution's existence. This sets us apart from other researchers who have used different methods. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables, and some figures. The paper ends with an example. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a diffusive prey–predator system with strong Allee effect in prey, spatial memory, and gestation delay of predator. Turing instability and Hopf bifurcation are investigated extensively. The system more easily creates Turing patterns, when the predator owns sluggish memory-induced diffusion. If the predator owns rapid memory-induced diffusion, then the system may present abundant dynamics. Predator with long memory displays a spatial nonhomogeneous periodic spread once the system only contains spatial memory delay. Populations are in a homogeneous or nonhomogeneous periodic spread with gestation period. However, under the combined influences of memory delay and gestation delay, the stability switches appear. Strong Allee effect has obvious influences on the system. The positive equilibrium is linearly stable if there is no strong Allee effect; once strong Allee effect is reasonable, it is unstable. But if the Allee effect is huge enough, then two populations can die out. Finally, we investigate the results by numerical simulations, which demonstrate that spatial memory, gestation delay, and strong Allee effect are conductive to the dynamics of the system. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a five-dimensional tuberculosis model incorporating slow-fast progression, endogenous reactivation, exogenous reinfection, and the assumption that infectives pass the smear microscopy test. This study disseminates information regarding how the presence and infectiousness of smear-negative patients in a tuberculosis epidemic significantly affect the threshold epidemic quantity R 0 (basic reproduction number). Stability and bifurcation analysis is carried out, and we find that the exogenous reinfection causes backward bifurcation and multiple endemic steady states in the system. We analytically and numerically explored the case of the periodic oscillations in the population via Hopf bifurcation for R 0 < 1 as well as R 0 > 1. [ABSTRACT FROM AUTHOR]

In this paper, we extend the classical theory of normal forms for continuous and difference dynamical systems to dynamic equations on discrete time scales. As consequences of the well known results from the theory of analytic differential equations, we obtain some versions of the Poincaré and Siegel theorems for dynamic equations on discrete time scales. Using these results and known results on the stability of dynamic equations on time scales, we obtain some stability results for the nonlinear dynamic equations. We also prove some results on generic properties of bifurcation curves and the saddle-node bifurcation for one-parameter families of dynamic equations on arbitrary time scales. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present an example of an Ordinary Differential Equation x ′ = F (x) in the infinite-dimensional Hilbert space ℓ 2 with F being of class C 1 in the Fréchet sense, such that the origin is an asymptotically stable equilibrium point but the spectrum of the linearized operator DF(0) intersects the half-plane R (z) > 0 . The possible existence or not of an example of this kind has been an open question until now, to our knowledge. An analogous example, but of a non-invertible map instead of a flow defined by an ODE was recently constructed by the authors in Rodrigues and Solà-Morales (J. Differ. Equ. 269:1349–1359, 2020). The two examples use different techniques, but both are based on a classical example in Operator Theory due to S. Kakutani. [ABSTRACT FROM AUTHOR]

In this paper, we study the solvability and generalized Ulam–Hyers (UH) stability of a nonlinear Atangana–Baleanu–Caputo (ABC) fractional coupled system with a Laplacian operator and impulses. First, this system becomes a nonimpulsive system by applying an appropriate transformation. Secondly, the existence and uniqueness of the solution are obtained by an F-contractive operator and a fixed-point theorem on metric space. Simultaneously, the generalized UH-stability is established based on nonlinear analysis methods. Thirdly, a novel numerical simulation algorithm is provided. Finally, an example is used to illustrate the correctness and availability of the main results. Our study is a beneficial exploration of the dynamic properties of viscoelastic turbulence problems. [ABSTRACT FROM AUTHOR]

This paper studies a two-age structured SIRS epidemic model with logistic growth of susceptible population and two-time delays. We simultaneously introduce two-time delays, i.e., the immunity and incubation periods, into this dynamic system and investigate their impact on different dynamic behaviors for the model. By means of the C 0 -semigroup theory, the model is transformed into a non-densely defined abstract Cauchy problem, and the condition of the existence and uniqueness of the endemic equilibrium is obtained. Following the spectral analysis, the characteristic equation technique, and the Hopf bifurcation theorem, we show that different combinations of the two delays perform a vital role in the instability/stability as well as the Hopf bifurcation results of equilibrium solutions. We numerically provide some graphical representations to check the main theoretical results and show the rich dynamics by varying the two delay parameters. [ABSTRACT FROM AUTHOR]

In this paper, a fractional model of a special structure of bidirectional associative memory (BAM) neural networks called tri-diagonal BAM neural networks (TdBAMNNs) is considered. The Hopf bifurcation analysis is made for the proposed fractional system in the presence of leakage and communication delays. The feasibility of the obtained theoretical results is verified by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we establish an Ebola epidemic model incorporating limited medical resources and immunity loss, and also consider the tracking and quarantining of susceptible individuals. Meanwhile the stability of the disease-free equilibrium is proved. Furthermore, we deduce the existence of multiple endemic equilibria and the occurrence of bifurcation. And then, using the partial ranking correlation coefficient (PRCC), it is found that the control reproduction number is most correlated with the parameter ρ (incineration or burial rate of dead bodies). Additionally, based on the data of the Ebola epidemic in Uganda in 2022, the fitted curve of accumulated infectious cases stabilizes at 143, which is eventually consistent with the actual data. Finally, numerical simulation results indicate that the peak value of infectious individuals decreases by 52.3% when the maximum treatment recovery rate changes from 20% to 40%. And while the hospital bed-population ratio increases by 10%, the peak value of the infectious decreases by 29.1%. Simultaneously, when the rate of immune loss decreasing from 60% to 20%, the peak value of the infectious decreases by 13.2%. While enhancing the tracking and quarantining of susceptible individuals have significant impact on the prevention and control of Ebola. [ABSTRACT FROM AUTHOR]

In this paper, we aim to prove the existence and uniqueness of the mean square solution to the Caputo fractional random boundary value problem by using the fixed point theorems. Moreover, we introduce the Ulam–Hyers stability and generalized Ulam–Hyers stability for this problem. Finally, we give an example to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the robust stability of implicit dynamic equations with causal operators on time scales. First, we investigate the solvability of these dynamic equations and then consider the preservation of stability under small perturbations. An L p version of Bohl–Perron principle for implicit dynamic equations is also studied. [ABSTRACT FROM AUTHOR]

The anthropogenic carbon dioxide (C O 2) emission from the burning of fossil fuels is the prime cause behind the menace of global warming. Over the past few decades, fossil fuel consumption has increased drastically to fulfill the energy demand of the growing population and economy. The population pressure has not only contributed to the increase in fossil fuel consumption but also accelerated the deforestation for industrial, agricultural, and infrastructure expansion. This paper presents a nonlinear mathematical model to study the effect of an increase in fossil fuel use and deforestation due to population pressure on atmospheric carbon dioxide concentration. Further, the effect of economic efforts applied to reduce the population pressure over the control of atmospheric C O 2 levels is explored. The model analysis shows that an increase in the fossil fuel consumption rate causes an increase in the equilibrium level of carbon dioxide. Further, it is found that an increase in the deforestation rate coefficient has a destabilizing effect on the stability of positive state of the system. If the deforestation rate crosses a critical threshold, the positive state of the system loses stability and the periodic solutions arise via Hopf-bifurcation. It is shown that at high deforestation rates, an increase in the implementation rate of economic efforts applied to reduce the population pressure may cause reduction in the amplitude of periodic oscillations. The periodic oscillations may disappear if the implementation rate of economic effort increased beyond a critical threshold and the concentration of carbon dioxide gets stabilized. [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, a nonlinear integro-dynamic equation on time scales with local initial condition is considered. The purpose of this paper is to prove existence and uniqueness of solutions and to investigate qualitative properties of solutions of this equation such as boundedness, dependence of solutions on initial conditions, functions, and parameters, and Ulam stability. The analysis is based on the Krasnoselskiĭ fixed point theorem and Gronwall-type dynamic inequalities. For the illustrative purpose of our main results, examples on a nonstandard time scale domain are provided. [ABSTRACT FROM AUTHOR]

This paper focuses on the global asymptotic stability of fractional-order complex-valued fuzzy cellular neural networks (CVFCNNs) with impulse effects and time-varying delays. In this paper, the new Lyapunov–Krasovskii functional is designed to derive the global asymptotic stability criterion for the CVFCNNs by employing the fractional Barbalat's lemma and some inequality techniques. In order to derive the existence and uniqueness for considered fractional-order systems, the contraction mapping principle is used to obtain the constraints. A numerical example is carried out to validate the efficacy of the proposed method and fractional-order derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of practically susceptible population, which is a fraction of the biologically susceptible population. Assuming that the fraction depends on the severity of the epidemic and the public's level of precaution (as a response of the public to the epidemic), we propose a general framework model with the response level evolving with the epidemic. We firstly verify the well-posedness and confirm the disease's eventual vanishing for the framework model under the assumption that the basic reproduction number R 0 < 1 . For R 0 > 1 , we study how the behavioural response evolves with epidemics and how such an evolution impacts the disease dynamics. More specifically, when the precaution level is taken to be the instantaneous best response function in literature, we show that the endemic dynamic is convergence to the endemic equilibrium; while when the precaution level is the delayed best response, the endemic dynamic can be either convergence to the endemic equilibrium, or convergence to a positive periodic solution. Our derivation offers a justification/explanation for the best response used in some literature. By replacing "adopting the best response" with "adapting toward the best response", we also explore the adaptive long-term dynamics. [ABSTRACT FROM AUTHOR]

Infectious diseases have a devastating impact on individual health and social development. Different external environmental factors will affect the scale and the speed of disease outbreaks, such as sanitary conditions and policy interventions. In this paper, we attempt to establish a dual-system susceptible-infectious-quarantine-recovered model with different environmental impacts to explore it. For the deterministic model, we calculate the basic reproduction number, simultaneously, and investigate the local asymptotic stability of the disease-free equilibrium and the endemic equilibrium. Numerical simulations and theoretical analyses verify the conclusion of this paper. It is a surprise that there is a great probability of finding a quarantine inflection point, which can effectively control the scale of infection outbreak when two different systems of infection rate and recovery rate are determined. [ABSTRACT FROM AUTHOR]

The time-delayed Moran-Ricker population model is investigated in this paper with an aim to identify some of its unknown features. In this model, the decline of the essential resources arising from the previous generation emerges as a delay in the density dependency of the population. The random fluctuations in population size may cause the model’s dynamics to change. In this study, we aim to scrutinize the model thoroughly and reveal more properties of the model. A discussion about the fixed points and their stability is presented in a brief way. By studying the normal form of the model through the reduction of the model to the associated center manifold, we show that the model will experience flip (period-doubling), Neimark–Sacker, strong resonances, and period-doubling-Neimark Sacker bifurcations. The bifurcation conditions are extracted with their critical coefficients. Numerical bifurcation analysis confirms the validity of theoretical findings. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of Ulam stability of first-order nonlinear impulsive dynamic equations on finite-time scale intervals. Our main objective is to formulate sufficient conditions under which the class of first-order nonlinear impulsive dynamic equations on time scales we consider exhibits Ulam stability. Our methods rely on the extended integral inequality on time scales for piecewise-continuous functions. We provide an example to support the validity of the results obtained. [ABSTRACT FROM AUTHOR]

The classical p -Laplace equation is one of the special and significant second-order ODEs. The fractional-order p -Laplace ODE is an important generalization. In this paper, we mainly treat with a nonlinear coupling (p 1 , p 2) -Laplacian systems involving the nonsingular Atangana–Baleanu (AB) fractional derivative. In accordance with the value range of parameters p 1 and p 2 , we obtain sufficient criteria for the existence and uniqueness of solution in four cases. By using some inequality techniques we further establish the generalized UH-stability for this system. Finally, we test the validity and practicality of the main results by an example. [ABSTRACT FROM AUTHOR]

Epidemic diseases and media campaigns are closely associated with each other. Considering most epidemics have multiple pathogenic strains, in this paper, we take the lead in proposing two multi-strain SIS epidemic models in heterogeneous networks incorporating awareness programs due to media. For the first model, we assume that the transmission rates for strain 1 and strain 2 depend on the level of awareness campaigns. For the second one, we further suppose that awareness divides susceptible population into two different subclasses. After defining the basic reproductive numbers for the whole model and each strain, we obtain the analytical conditions that determine the extinction, coexistence and absolute dominance of two strains. Moreover, we also formulate its optimal control problem and identify an optimal implementation pair of awareness campaigns using optimal control theory. Given the complexity of the second model, we use the numerical simulations to visualize its different types of dynamical behaviors. Through theoretical and numerical analysis of these two models, we discover some new phenomena. For example, during the persistence analysis of the first model, we find that the characteristic polynomials of two boundary equilibria may have a pair of pure imaginary roots, implying that Hopf bifurcation and periodic solutions may appear. Most strikingly, multistability occurs in the second model and the growth rate of awareness programs (triggered by the infection prevalence) has a multistage impact on the final size of two strains. The numerical results suggest that the spread of a two-strain epidemic can be controlled (even be eradicated) by taking the measures of enhancing awareness transmission, reducing memory fading of aware individuals and ensuring high-level influx and rapid growth of awareness programs appropriately. [ABSTRACT FROM AUTHOR]

Getting an efficient neural network can be a very difficult task for engineers and researchers because of the huge number of hyperparameters to tune and their interconnections. To make the tuning step easier and more understandable, this work focuses on probably one of the most important leverage to improve Neural Networks efficiency: the optimizer. These recent years, a great number of algorithms have been developed but they need an accurate tuning to be efficient. To get rid of this long and experimental step, we are looking for generic and desirable properties for non-convex optimization. For this purpose, the optimizers are reinterpreted or analyzed as a discretization of a continuous dynamical system. This continuous framework offers many mathematical tools in order to interpret the sensitivity of the optimizer with respect to the initial guess such as Lyapunov stability. By enforcing the discrete decrease of Lyapunov functionals, new robust and efficient optimizers are designed. They also considerably simplify the tuning of hyperparameters (learning rate, momentum etc.). These Lyapunov based algorithms outperform several state of the art optimizers on different benchmarks of the literature. Drawing its inspiration from the numerical analysis of PDEs, this paper emphasizes the essential role of some hidden energy/entropy quantities for machine learning tasks. [ABSTRACT FROM AUTHOR]

It is well known that electromagnetic dispersive structures such as metamaterials can be modelled by generalized Drude–Lorentz models. The present paper is the first of two articles dedicated to dissipative generalized Drude–Lorentz open structures. We wish to quantify the loss in such media in terms of the long-time decay rate of the electromagnetic energy for the corresponding Cauchy problem. By using an approach based on frequency-dependent Lyapunov estimates, we show that this decay is polynomial in time. These results extend to an unbounded structure the ones obtained for bounded media in Nicaise and Pignotti (Z Angew Math Phys 71(3):76, 2020) via a quite different method based on the notion of cumulated past history and semi-group theory. A great advantage of the approach developed here is to be less abstract and directly connected to the physics of the system via energy balances. [ABSTRACT FROM AUTHOR]

In this paper, we study the practical exponential stability of nonlinear nonautonomous differential equations under nonlinear perturbations. By introducing a new method, we obtain some explicit criteria for the practical exponential stability of these equations. Furthermore, several characterizations for the exponential stability of a class of nonlinear differential equations are also presented. The obtained results generalize some existing results in the literature. Applications to neutral networks are investigated. Some examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The concept of a face of population states arises naturally in evolutionary games. This paper studies faces of profiles in asymmetric evolutionary games with infinite strategy space. The concepts of strong immovable and immutable faces of profiles are introduced and stability results for these faces are discussed. [ABSTRACT FROM AUTHOR]

In this paper, a deterministic mathematical model for Dengue Fever (DF) and Zika virus (ZIKV) co-infection transmission dynamics is formulated and analyzed. Two sub-models, namely ZIKV-only and DF-only sub-models, are considered first of all. Rigorous qualitative analysis of the sub-models reveals that each of the diseases undergoes the phenomenon of backward bifurcation when the associated reproduction number of both ZIKV-only and DF-only sub-models (denoted by R 0 z + and R 0 d respectively) is less than unity. It is shown, using the centre manifold theory that the full ZIKV-DF co-infection model undergoes a backward bifurcation phenomenon. In addition, simulation of the full ZIKV-DF model shows the clear picturization of disease transmission and also we provide the solution of model numerically in detail. [ABSTRACT FROM AUTHOR]

In this paper, a two-DoF (degree-of-freedom) semi-passive compass-like biped model with nonlinear impulse thrust is considered and investigated. Conditions for the existence and stability of period gaits of the linearized system are obtained. The conditions for flip bifurcation of periodic gait are derived, and bifurcation control is rigorously studied. The results show that impulse thrust and the angle between two legs at the impact can modify the stability of the semi-passive compass-like biped model and lead to chaos through flip bifurcation. It is possible to improve the stability of compass-like biped model by bifurcation control. Period - n gait (n > 2) and chaotic gait are stabilized to period - 1 gait or period - 2 gait by reducing the angle between the two legs at the impact. Numerical results for periodic gaits, bifurcation diagrams and bifurcation control diagrams are illustrated with an example, and they are in good agreement with the theoretical analysis. The obtained results about period gaits and bifurcation of the linearized system can be used to predict the nonlinear walking pattern. [ABSTRACT FROM AUTHOR]

An SVEIR SARS-CoV-2 Omicron variant model is proposed to provide some insights to coordinate non-pharmaceutical interventions (NPIs) and vaccination. Mathematically, we define the basic reproduction number R 0 and the effective reproduction number R e to measure the infection potential of Omicron variant and formulate an optimal disease control strategy. Our inversion results imply that the sick period of Omicron variant in the United States is longer than that of Delta variant in India. The decrease in the infectious period of the infection with infectiousness implies that the risk of hospitalization is reduced; but the increasing period of the infection with non-infectiousness signifies that Omicron variant lengthens the period of nucleic acid test being negative. Optimistically, Omicron's death rate is only a quarter of Delta's. Moreover, we forecast that the cumulative cases will exceed 100 million in the United States on February 28, 2022, and the daily confirmed cases will reach a peak on February 2, 2022. The results of parameters sensitivity analysis imply that NPIs are helpful to reduce the number of confirmed cases. In particular, NPIs are indispensable even if all the people were vaccinated when the efficiency of vaccine is relatively low. By simulating the relationships of the effective reproduction number R e , the vaccination rate and the efficacy of vaccine, we find that it is impossible to achieve the herd immunity without NPIs while the efficiency of vaccine is lower than 88.7 % . Therefore, the herd immunity area is defined by the evolution of relationships between the vaccination rate and the efficacy of vaccine. Finally, we present that the disease-induced mortality rate demonstrates the periodic oscillation and an almost periodic function is deduced to match the curve. A discussion completes the paper. [ABSTRACT FROM AUTHOR]

In previous articles, we formalized the problem of optimal allocation strategies for a (perfect) vaccine in an infinite-dimensional metapopulation model. The aim of the current paper is to illustrate this theoretical framework with multiple examples where one can derive the analytic expression of the optimal strategies. We discuss in particular the following points: whether or not it is possible to vaccinate optimally when the vaccine doses are given one at a time (greedy vaccination strategies); the effect of assortativity (that is, the tendency to have more contacts with similar individuals) on the shape of optimal vaccination strategies; the particular case where everybody has the same number of neighbors. [ABSTRACT FROM AUTHOR]

In this paper, we provide a simple ODEs model with a generic nonlinear incidence rate function and incorporate two treatments, blocking the virus binding and inhibiting the virus replication to investigate the impact of calibration on model predictions for the SARS-CoV-2 infection dynamics. We derive conditions of the infection eradication for the long-term dynamics using the basic reproduction number, and complement the characterization of the dynamics at short-time using the resilience and reactivity of the virus-free equilibrium are considered to inform on the average time of recovery and sensitivity to perturbations in the initial virus free stage. Then, we calibrate the treatment model to clinical datasets for viral load in mild and severe cases and immune cells in severe cases. Based on the analysis, the model calibrated to these different datasets predicts distinct scenarios: eradication with a non reactive virus-free equilibrium, eradication with a reactive virus-free equilibrium, and failure of infection eradication. Moreover, severe cases generate richer dynamics and different outcomes with the same treatment. Calibration to different datasets can lead to diverse model predictions, but combining long- and short-term dynamics indicators allows the categorization of model predictions and determination of infection severity. [ABSTRACT FROM AUTHOR]

In the present paper, a spatially discretized diffusive Brusselator model with zero-flux boundary conditions is considered. Firstly, the global existence and uniqueness of the positive solution are proved. Then the local stability of the unique spatially homogeneous steady state is considered by analyzing the relevant eigenvalue problem with the aid of decoupling method. Hence, the occurrence conditions of Turing bifurcation and Hopf bifurcation for the model at this steady state are obtained. Meanwhile, the comparative simulations on the stability regions of the steady state between the spatially discretized diffusive Brusselator model and its counterpart in continuous space are given. Furthermore, the approximate expressions of the bifurcating periodic solutions are derived according to Hopf bifurcation theorem. The bifurcating spatially nonhomogeneous periodic solutions show the formation of a special kind of periodic structures for this model. Finally, numerical simulations are given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

An autonomous system of ordinary differential equations describing nonlinear oscillations on the plane is considered. The influence of non-autonomous perturbations decaying at infinity in time is investigated. Such systems are usually called asymptotically autonomous and arise, in particular, as a result of the reduction of multidimensional autonomous and non-autonomous systems. In this paper, we consider a special class of oscillatory perturbations that satisfy the non-resonance condition and do not vanish at the equilibrium of the unperturbed system. We construct a near-identity transformation that averages the system in the first asymptotic terms at infinity in time, and study the structure of the simplified equations. Under some natural assumptions, we describe possible long-term asymptotic regimes for solutions and analyse their stability by constructing Lyapunov functions. In particular, we show how oscillatory terms of perturbations can break the stability of the system and discuss conditions under which the perturbed system behaves like the corresponding unperturbed autonomous system or has new stable regimes. The results obtained are applied to some examples of oscillatory systems with time-decaying oscillatory perturbations. [ABSTRACT FROM AUTHOR]

Glioblastoma (GBM) is an aggressive primary brain cancer that currently has minimally effective treatments. Like other cancers, immunosuppression by the PD-L1-PD-1 immune checkpoint complex is a prominent axis by which glioma cells evade the immune system. Myeloid-derived suppressor cells (MDSCs), which are recruited to the glioma microenviroment, also contribute to the immunosuppressed GBM microenvironment by suppressing T cell functions. In this paper, we propose a GBM-specific tumor-immune ordinary differential equations model of glioma cells, T cells, and MDSCs to provide theoretical insights into the interactions between these cells. Equilibrium and stability analysis indicates that there are unique tumorous and tumor-free equilibria which are locally stable under certain conditions. Further, the tumor-free equilibrium is globally stable when T cell activation and the tumor kill rate by T cells overcome tumor growth, T cell inhibition by PD-L1-PD-1 and MDSCs, and the T cell death rate. Bifurcation analysis suggests that a treatment plan that includes surgical resection and therapeutics targeting immune suppression caused by the PD-L1-PD1 complex and MDSCs results in the system tending to the tumor-free equilibrium. Using a set of preclinical experimental data, we implement the approximate Bayesian computation (ABC) rejection method to construct probability density distributions that estimate model parameters. These distributions inform an appropriate search curve for global sensitivity analysis using the extended fourier amplitude sensitivity test. Sensitivity results combined with the ABC method suggest that parameter interaction is occurring between the drivers of tumor burden, which are the tumor growth rate and carrying capacity as well as the tumor kill rate by T cells, and the two modeled forms of immunosuppression, PD-L1-PD-1 immune checkpoint and MDSC suppression of T cells. Thus, treatment with an immune checkpoint inhibitor in combination with a therapeutic targeting the inhibitory mechanisms of MDSCs should be explored. [ABSTRACT FROM AUTHOR]