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Background: The emergence and spread of antibiotic-resistant pathogens have led to the exploration of antibiotic combinations to enhance clinical effectiveness and counter resistance development. Synergistic and antagonistic interactions between antibiotics can intensify or diminish the combined therapy's impact. Moreover, these interactions can evolve as bacteria transition from wildtype to mutant (resistant) strains. Experimental studies have shown that the antagonistically interacting antibiotics against wildtype bacteria slow down the evolution of resistance. Interestingly, other studies have shown that antibiotics that interact antagonistically against mutants accelerate resistance. However, it is unclear if the beneficial effect of antagonism in the wildtype bacteria is more critical than the detrimental effect of antagonism in the mutants. This study aims to illuminate the importance of antibiotic interactions against wildtype bacteria and mutants on the deacceleration of antimicrobial resistance. Methods: To address this, we developed and analyzed a mathematical model that explores the population dynamics of wildtype and mutant bacteria under the influence of interacting antibiotics. The model investigates the relationship between synergistic and antagonistic antibiotic interactions with respect to the growth rate of mutant bacteria acquiring resistance. Stability analysis was conducted for equilibrium points representing bacteria-free conditions, all-mutant scenarios, and coexistence of both types. Numerical simulations corroborated the analytical findings, illustrating the temporal dynamics of wildtype and mutant bacteria under different combination therapies. Results: Our analysis provides analytical clarification and numerical validation that antibiotic interactions against wildtype bacteria exert a more significant effect on reducing the rate of resistance development than interactions against mutants. Specifically, our findings highlight the crucial role of antagonistic antibiotic interactions against wildtype bacteria in slowing the growth rate of resistant mutants. In contrast, antagonistic interactions against mutants only marginally affect resistance evolution and may even accelerate it. Conclusion: Our results emphasize the importance of considering the nature of antibiotic interactions against wildtype bacteria rather than mutants when aiming to slow down the acquisition of antibiotic resistance. [ABSTRACT FROM AUTHOR]

Background The emergence and spread of antibiotic-resistant pathogens have led to the exploration of antibiotic combinations to enhance clinical effectiveness and counter resistance development. Synergistic and antagonistic interactions between antibiotics can intensify or diminish the combined therapy’s impact. Moreover, these interactions can evolve as bacteria transition from wildtype to mutant (resistant) strains. Experimental studies have shown that the antagonistically interacting antibiotics against wildtype bacteria slow down the evolution of resistance. Interestingly, other studies have shown that antibiotics that interact antagonistically against mutants accelerate resistance. However, it is unclear if the beneficial effect of antagonism in the wildtype bacteria is more critical than the detrimental effect of antagonism in the mutants. This study aims to illuminate the importance of antibiotic interactions against wildtype bacteria and mutants on the deacceleration of antimicrobial resistance. Methods To address this, we developed and analyzed a mathematical model that explores the population dynamics of wildtype and mutant bacteria under the influence of interacting antibiotics. The model investigates the relationship between synergistic and antagonistic antibiotic interactions with respect to the growth rate of mutant bacteria acquiring resistance. Stability analysis was conducted for equilibrium points representing bacteria-free conditions, all-mutant scenarios, and coexistence of both types. Numerical simulations corroborated the analytical findings, illustrating the temporal dynamics of wildtype and mutant bacteria under different combination therapies. Results Our analysis provides analytical clarification and numerical validation that antibiotic interactions against wildtype bacteria exert a more significant effect on reducing the rate of resistance development than interactions against mutants. Specifically, our findings highlight the crucial role of antagonistic antibiotic interactions against wildtype bacteria in slowing the growth rate of resistant mutants. In contrast, antagonistic interactions against mutants only marginally affect resistance evolution and may even accelerate it. Conclusion Our results emphasize the importance of considering the nature of antibiotic interactions against wildtype bacteria rather than mutants when aiming to slow down the acquisition of antibiotic resistance.

A tethered towing system provides an effective method for capturing pieces of space debris and dragging them out of orbit. This paper focuses on the in-plane stability analysis and libration control of a two-segment tethered towing system. The first segment is the same as the traditional single-tether towing system. The second segment is similar to a simplified space tether net. The dynamic equations are established in the orbit frame. Considering the elasticity of the tethers, the equilibrium solutions are obtained and the stability of equilibrium solutions is proved. An in-plane libration controller based on the sliding mode control scheme is designed to ensure the safety of the towing mission and save fuel. The controller suppressed the librations of the in-plane angles in the desired state by applying two external torques. Finally, simulation results are provided to validate the effectiveness of the proposed controller. [ABSTRACT FROM AUTHOR]

It is a known fact that there are a particular set of people who are at higher risk of getting COVID-19 infection. Typically, these high-risk individuals are recommended to take more preventive measures. The use of non-pharmaceutical interventions (NPIs) and the vaccine are playing a major role in the dynamics of the transmission of COVID-19. We propose a COVID-19 model with high-risk and low-risk susceptible individuals and their respective intervention strategies. We find two equilibrium solutions and we investigate the basic reproduction number. We also carry out the stability analysis of the equilibria. Further, this model is extended by considering the vaccination of some non-vaccinated individuals in the high-risk population. Sensitivity analyses and numerical simulations are carried out. From the results, we are able to obtain disease-free and endemic equilibrium solutions by solving the system of equations in the model and show their global stabilities using the Lyapunov function technique. The results obtained from the sensitivity analysis shows that reducing the hospitals' imperfect efficacy can have a positive impact on the control of COVID-19. Finally, simulations of the extended model demonstrate that vaccination could adequately control or eliminate COVID-19. [ABSTRACT FROM AUTHOR]

The paper studies a system of first order Hamilton–Jacobi equations with discontinuous coefficients, arising from a model of deterministic optimal debt management in infinite time horizon, with exponential discount and currency devaluation. The existence of an equilibrium solution is obtained by a suitable concatenation of backward solutions to the system of Hamilton–Jacobi equations. A detailed analysis of the behavior of the solution as the debt-ratio-income tends to infinity is provided. [ABSTRACT FROM AUTHOR]

The goal of the lockdown is to mitigate and if possible prevent the spread of an epidemic. It consists in reducing social interactions. This is taken into account by the introduction of a factor of reduction of social interactions q, and by decreasing the transmission coefficient of the disease accordingly. Evaluating q is a difficult question and one can ask if it makes sense to compute an average coefficient q for a given population, in order to make predictions on the basic reproduction rate ℛ0, the dynamics of the epidemic or the fraction of the population that will have been infected by the end of the epidemic. On a very simple example, we show that the computation of ℛ0 in a heterogeneous population is not reduced to the computation of an average q but rather to the direct computation of an average coefficient ℛ0. Even more interesting is the fact that, in a range of data compatible with the Covid-19 outbreak, the size of the epidemic is deeply modified by social heterogeneity, as is the height of the epidemic peak, while the date at which it is reached mainly depends on the average ℛ0 coefficient. This paper illustrates more technical results that can be found in [4], with new numerical computations. It is intended to draw the attention on the role of heterogeneities in a population in a very simple case, which might be difficult to apprehend in more realistic but also more complex models.

A tethered towing system provides an effective method for capturing pieces of space debris and dragging them out of orbit. This paper focuses on the in-plane stability analysis and libration control of a two-segment tethered towing system. The first segment is the same as the traditional single-tether towing system. The second segment is similar to a simplified space tether net. The dynamic equations are established in the orbit frame. Considering the elasticity of the tethers, the equilibrium solutions are obtained and the stability of equilibrium solutions is proved. An in-plane libration controller based on the sliding mode control scheme is designed to ensure the safety of the towing mission and save fuel. The controller suppressed the librations of the in-plane angles in the desired state by applying two external torques. Finally, simulation results are provided to validate the effectiveness of the proposed controller.

In this paper, we construct an evolutionary (time-dependent) split variational inequality problem and show how to reformulate equilibria of the dynamic traffic network models of two cities as such problem. We also establish existence result for the proposed model. Primary numerical results of equilibria illustrate the validity and applicability of our results. [ABSTRACT FROM AUTHOR]

In this work, we study a mathematical model for the interaction of sensitive-resistant bacteria to antibiotics and analyse the effects of introducing random perturbations to this model. We compare the results of existence and stability of equilibrium solutions between the deterministic and stochastic formulations, and show that the conditions for the bacteria to die out are weaker in the stochastic model. Moreover, a corresponding optimal control problem is formulated for the unperturbed and the perturbed system, where the control variable is prophylaxis. The results of the optimal control problem reveal that, depending on the antibiotics, the costs of the prophylaxis, such as implementation, ordering and distribution, have to be much lower than the social costs, to achieve a bacterial resistance effective control.

It is a known fact that there are a particular set of people who are at higher risk of getting COVID-19 infection. Typically, these high-risk individuals are recommended to take more preventive measures. The use of non-pharmaceutical interventions (NPIs) and the vaccine are playing a major role in the dynamics of the transmission of COVID-19. We propose a COVID-19 model with high-risk and low-risk susceptible individuals and their respective intervention strategies. We find two equilibrium solutions and we investigate the basic reproduction number. We also carry out the stability analysis of the equilibria. Further, this model is extended by considering the vaccination of some non-vaccinated individuals in the high-risk population. Sensitivity analyses and numerical simulations are carried out. From the results, we are able to obtain disease-free and endemic equilibrium solutions by solving the system of equations in the model and show their global stabilities using the Lyapunov function technique. The results obtained from the sensitivity analysis shows that reducing the hospitals’ imperfect efficacy can have a positive impact on the control of COVID-19. Finally, simulations of the extended model demonstrate that vaccination could adequately control or eliminate COVID-19.

We correct the proof of Theorem 2 in the mentioned paper concerning finite-difference equilibrium solutions.

In this manuscript two BMO estimates are obtained, one for Linear Elasticity and one for Nonlinear Elasticity. It is first shown that the BMO-seminorm of the gradient of a vector-valued mapping is bounded above by a constant times the BMO-seminorm of the symmetric part of its gradient, that is, a Korn inequality in BMO. The uniqueness of equilibrium for a finite deformation whose principal stresses are everywhere nonnegative is then considered. It is shown that when the second variation of the energy, when considered as a function of the strain, is uniformly positive definite at such an equilibrium solution, then there is a BMO-neighborhood in strain space where there are no other equilibrium solutions. [ABSTRACT FROM AUTHOR]

We study variants of the SEIR model for interpreting some qualitative features of the statistics of the Covid-19 epidemic in France. Standard SEIR models distinguish essentially two regimes: either the disease is controlled and the number of infected people rapidly decreases, or the disease spreads and contaminates a significant fraction of the population until herd immunity is achieved. After lockdown, at first sight it seems that social distancing is not enough to control the outbreak. We discuss here a possible explanation, namely that the lockdown is creating social heterogeneity: even if a large majority of the population complies with the lockdown rules, a small fraction of the population still has to maintain a normal or high level of social interactions, such as health workers, providers of essential services, etc. This results in an apparent high level of epidemic propagation as measured through re-estimations of the basic reproduction ratio. However, these measures are limited to averages, while variance inside the population plays an essential role on the peak and the size of the epidemic outbreak and tends to lower these two indicators. We provide theoretical and numerical results to sustain such a view. [ABSTRACT FROM AUTHOR]

In the present work, it is analyzed existence and stability of equilibrium solutions of the following nonlinear reaction-diusion equation: ut = auxx + wu + k ln(u²)u. Explicit formulas for a family of equilibrium solutions to the former equation which decay to zero at innity are provided. The instability of those solutions is obtained by detailed spectral analysis of the linear operator which approximates the solutions of the equation around the equilibrium solutions. A result about the instability of any non-trivial equilibrium solution of the equation is also established. [ABSTRACT FROM AUTHOR]

In this paper, two different mechanisms are used to study a homogeneous Cournot duopoly in a market characterized by the downward sloping and concave price function. Two firms, which have constant marginal costs, use adaptive, low-rationality mechanisms to adjust their production levels toward equilibrium. In particular, the stability of the equilibrium for two different mechanisms is studied. However, complex dynamics arise, especially when the reaction coefficient increases. Finally, we compare the obtained results of the two models.

We consider a coupled model surface-deep ocean effect, where an Energy Balance Model (EBM) is used for modelling the surface temperature and a two-dimensional heat equation represents the evolution of the temperature of the deep ocean. Although the model under study is based on that proposed by Watts & Morantine (1990), here we consider a modified model that incorporates other processes, such as the nonlinear diffusion and the action of coalbedo, depending on the temperature. The stationary states of the model under study, taking the solar constant as the parameter, are numerically attained. The results of the simulation are depicted in a {(Q,u)} plot where u is the temperature in the surface and Q is the solar constant. The numerical solution is achieved by means of a finite volume scheme with Weighted Essentially Non-Oscillatory (WENO) reconstruction in space and third order Runge-Kutta scheme, which verifies the Total Variation Diminishing (TVD) property, for time integration. The equilibrium states are accomplished by evolving in time the numerical solution until the stationary solutions are reached. The main novel results of this work concern the numerical obtention of the stationary solutions of both the EBM and the coupled model EBM-deep ocean and the agreement of these results with the theoretically obtained in previous works, where an interval of values of the solar constant Q was obtained with the existence of at least three stationary solutions. In this work, we have numerically obtained more than three stationary solutions for such interval of Q.

The uniqueness of absolute minimizers of the energy of a compressible, hyperelastic body subject to a variety of dead-load boundary conditions in two and three dimensions is herein considered. Hypotheses under which a given solution of the corresponding equilibrium equations is the unique absolute minimizer of the energy are obtained. The hypotheses involve uniform polyconvexity and pointwise bounds on derivatives of the stored-energy density when evaluated on the given equilibrium solution. In particular, an elementary proof of the uniqueness result of Fritz John (Commun. Pure Appl. Math. 25:617-634, 1972) is obtained for uniformly polyconvex stored-energy densities. [ABSTRACT FROM AUTHOR]

En este artículo se formula un modelo matemático simple que describe la interacción entre bacterias sensibles y resistentes a múltiples antibióticos de acción bactericida y bacteriostática de forma simultánea, en el supuesto de que la adquisición de resistencia bacteriana se da a través de mutaciones espontáneas y adquiridas por la exposición a diferentes antibióticos. El análisis cualitativo revela la existencia de un equilibrio libre de bacterias, un equilibrio solo con bacterias resistentes y un equilibrio endémico donde coexisten ambas poblaciones de bacterias. Abstract. In this work we formulate a simple mathematical model that describes the population dynamics of bacteria exposed simultaneously to multiple bactericidal and bacteriostatic antibiotics, assuming that resistance is acquired through mutations due to antibiotic exposure. Qualitative analysis reveals the existence of a free-bacteria equilibrium, resistant-bacteria equilibrium and an endemic equilibrium where both bacteria coexist.

In this article, we introduce a model of bi-matrix games based on crisp parametric payoffs via utilizing the method of interval value function. Then, we get that equilibrium solutions of bi-matrix games on the basis of fuzzy payoffs and equilibrium solutions of the game model are of equal value. Furthermore, it is concluded that equilibrium solutions of the game can be converted to optimal solutions of discrete nonlinear optimization problems with parameters. Lastly, the proposed methodology is illustrated by an example.

In this paper, two different mechanisms are used to study a homogeneous Cournot duopoly in a market characterized by the downward sloping and concave price function. Two firms, which have constant marginal costs, use adaptive, low-rationality mechanisms to adjust their production levels toward equilibrium. In particular, the stability of the equilibrium for two different mechanisms is studied. However, complex dynamics arise, especially when the reaction coefficient increases. Finally, we compare the obtained results of the two models. [ABSTRACT FROM AUTHOR]

In this paper, a model of two teams of predators interacting with two teams of preys is proposed. The predator teams help each other and so do the prey teams. Therefore, some stability analysis are carried out on the model to detect the stability and instability of the system's equilibrium solutions. Also, the persistence of the solutions is investigated. Finally, the optimal control is applied to the unstable equilibrium solutions. Some numerical simulations are carried out to verify the analytic results. [ABSTRACT FROM AUTHOR]

In this paper, we give necessary conditions for Lie-stability of the equilibrium solutions of autonomous and periodic Hamiltonian systems with n degrees of freedom which possess multiple resonances of arbitrary order. Necessary conditions for instability in the sense of Lyapunov in some cases of multiple resonances are provided. These conditions depend on the order of the resonance and the coefficients of the Hamiltonian function. [ABSTRACT FROM AUTHOR]

It is a known fact that there are a particular set of people who are at higher risk of getting COVID-19 infection. Typically, these high-risk individuals are recommended to take more preventive measures. The use of non-pharmaceutical interventions (NPIs) and the vaccine are playing a major role in the dynamics of the transmission of COVID-19. We propose a COVID-19 model with high-risk and low-risk susceptible individuals and their respective intervention strategies. We find two equilibrium solutions and we investigate the basic reproduction number. We also carry out the stability analysis of the equilibria. Further, this model is extended by considering the vaccination of some non-vaccinated individuals in the high-risk population. Sensitivity analyses and numerical simulations are carried out. From the results, we are able to obtain disease-free and endemic equilibrium solutions by solving the system of equations in the model and show their global stabilities using the Lyapunov function technique. The results obtained from the sensitivity analysis shows that reducing the hospitals' imperfect efficacy can have a positive impact on the control of COVID-19. Finally, simulations of the extended model demonstrate that vaccination could adequately control or eliminate COVID-19.

Abstract: The main purpose of this paper is to investigate the existence and stability of periodic and non-periodic equilibrium solutions related to the nonlinear heat equation: The existence of periodic equilibriums with a fixed period L is deduced from the Theory of Jacobian Elliptical Functions and the Implicit Function Theorem. We show that these periodic equilibriums tend to the non-periodic positive equilibrium solution in the real line. Our stability/instability results are obtained trough the spectral study of the linear operator associated to the linearized stability problem as well as the study of a certain scalar quantity. [Copyright &y& Elsevier]

Highlights: [•] We model the interaction between sensitive and resistant bacteria to multiple antibiotics. [•] We consider the most basic mechanisms of acquisition of bacterial resistance to antibiotics. [•] Conditions that determine outcome of bacterial elimination or progression are established. [ABSTRACT FROM AUTHOR]

We consider a class of dynamic tournaments in which two contestants are faced with a choice between two courses of action. The first is a riskless option ('hold') of maintaining the resources the contestant already has accumulated in her turn and ceding the initiative to her rival. The second is the bolder option ('roll') of taking the initiative of accumulating additional resources, and thereby moving ahead of her rival, while at the same time sustaining a risk of temporary setback. We study this tournament in the context of a jeopardy race game (JRG), extend the JRG to $$N > 2$$ contestants, and construct its equilibrium solution. Compared to the equilibrium solution, the results of three experiments reveal a dysfunctional bias in favor of the riskless option. This bias is substantially mitigated when the contestants are required to commit in advance how long to pursue the risky course of action. [ABSTRACT FROM AUTHOR]

We correct the proof of Theorem 2 in the mentioned paper concerning finite-difference equilibrium solutions. [ABSTRACT FROM AUTHOR]

Abstract: Numerical schemes to approximate the Cahn–Hilliard equation have been widely studied in recent times due to its connection with many physically motivated problems. In this work we propose two type of linear schemes based on different ways to approximate the double-well potential term. The first idea developed in the paper allows us to design a linear numerical scheme which is optimal from the numerical dissipation point of view meanwhile the second one allows us to design unconditionally energy-stable linear schemes (for a modified energy). We present first and second order in time linear schemes to approximate the CH problem, detailing their advantages over other linear schemes that have been previously introduced in the literature. Furthermore, we compare all the schemes through several computational experiments. [Copyright &y& Elsevier]

In the present survey paper, basic convergence results for gradient-like systems relying on the Łojasiewicz gradient inequality are recalled in a self-contained way. A uniform version of the gradient inequality is used to get directly convergence and the rate of convergence in one step and a new technical trick, consisting in the evaluation of the integral of the velocity norm from t to 2t is introduced. A short idea of the state of the art without technical details is also given. [ABSTRACT FROM AUTHOR]

Abstract: We are interested in the investigation of the dynamics of the canonical system of differential equations deriving from the galactic type Hamiltonian given by the quartic function from an analytical point of view according to the real parameters and . Similar Hamiltonians have been considered by several investigators in order to study the motion in the central parts of elliptical galaxies. [ABSTRACT FROM AUTHOR]

Based on a numerical large-scale geostrophic ocean thermohaline circulation model, the influence of time step size in modeling large-scale temperature and salinity fields with the use of an implicit time integration method is investigated. It is shown that for a more realistic description of the processes of deep vertical convection and ocean thermohaline circulation, it is necessary to use time steps less than 10 days. At such time steps, the influence of numerical viscosity (diffusion) is insignificant. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear size-structured population model with functional response, which describes the dynamics of a predator-prey system living in a common habitat. We present a kind of functional response for the prey being a plant or algae, and explain its biological meanings. When the vital rates depend both on the individual’s size and on the total population or only depend on the former, we obtain the existence of equilibrium solutions. [ABSTRACT FROM AUTHOR]

Abstract: A characteristic feature of particulate systems that evolve due to competition between aggregation and breakage is that they sometimes produce non-trivial steady-state particle size distributions. If such solutions satisfy detailed balance conditions, then they are equilibrium solutions. The conditions that must be satisfied by aggregation and fragmentation rate kernels in order for equilibrium solutions to be produced are elaborated, and it is shown that the rate kernels are uniquely determined by the aggregation and breakage rate constants for the reactions involving monomers. Consequently, for equilibrium systems there is a significant reduction in the amount of information needed in order to infer the general form for aggregation or breakage kernels, and we explore implications for constructing rate kernels by using atomistic simulations such as molecular dynamics. [Copyright &y& Elsevier]