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Because of mutations and selection, pathogens can manage to resist to drugs. However, the evolution of an asexual population (e.g., viruses, bacteria and cancer cells) depends on some external factors (e.g., antibiotic concentrations), and so understanding the impact of the environmental changes is an important issue. This paper is devoted to model this problem with a nonlocal diffusion PDE, describing the dynamics of such a phenotypically structured population, in a changing environment. The large-time behaviour of this model, with particular forms of environmental changes (linear or periodically fluctuations), has been previously developed. A new mathematical approach (limited to isotropic mutations) has been developed recently for this problem, considering a very general form of environmental variations, and giving an analytic description of the full trajectories of adaptation. However, recent studies have shown that an anisotropic mutation kernel can change the evolutionary dynamics of the population: some evolutive plateaus can appear. Thus the aim of this paper is to mix the two previous studies, with an anisotropic mutation kernel, and a changing environment. The main idea is to study a multivariate distribution of (2n) "fitness components". Its generating function solves a transport equation, and describes the distribution of fitness at any time. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a control strategy for applying the Sterile Insect Technique (SIT) to eliminate the population of Aedes mosquitoes which are vectors of various deadly diseases like dengue, zika, chikungunya... in a wide area. We use a system of reaction-diffusion equations to model the mosquito population and study the effect of releasing sterile males. Without any human intervention, and due to the so-called hair-trigger effect, the introduction of only a few individuals (eggs or fertilized females) can lead to the invasion of mosquitoes in the whole region after some time. To avoid this phenomenon, our strategy is to keep releasing a small number of sterile males in the treated zone and move this release forward with a negative forcing speed c to push back the invasive front of wild mosquitoes. By using traveling wave analysis, we show in the present paper that the strategy succeeds in repulsing the population while consuming a finite amount of mosquitoes in any finite time interval even though we treat a moving half-space {x > ct}. Moreover, we succeed in constructing a 'forced' traveling wave for our system moving at the same speed as the releases. We also provide some numerical illustrations for our results. [ABSTRACT FROM AUTHOR]

Many researchers began doing studies about pandemic COVID-19 which began to spread from Wuhan, China in 2019 to all around the world and so far, numerous researches have been done around the world to control this contagious disease. In this paper, we proposed a MSIlIhR-V mathematical model to study the spreading of pandemic COVID-19. This paper is aimed to study the vaccination effect in the control of the disease propagation rate. Another goal of this paper is to find the maximum number of susceptible people, minimum number of infected people, and the best value for number of vaccination people. The Jacobian matrix was obtained in the virus absenteeism equilibrium point for the proposed dynamical system. The spectral radius method was applied to find the analytical formula for the reproductive number. Reproductive number is one of the most benefit and important tools to study of epidemic model's stability and instability. In the following, by adding a controller to the model and also using the optimal control strategy, model performance was improved. To validate of the proposed models with controller and without controller we use the real data of COVID-19 from 4 January, 2021 up to 14 June, 2021 in Iran. Maple and MATLAB software's will be used for programming. We will use Maple software for analytical parts and MATLAB software for numerical and simulation parts. [ABSTRACT FROM AUTHOR]

This paper is concerned with a stochastic population model with Allee effect and jumps. First, we show the global existence of almost surely positive solution to the model. Next, exponential extinction and persistence in mean are discussed. Then, we investigated the global attractivity and stability in distribution. At last, some numerical results are given. The results show that if attack rate a is in the intermediate range or very large, the population will go extinct. Under the premise that attack rate a is less than growth rate r, if the noise intensity or jump is relatively large, the population will become extinct; on the contrary, the population will be persistent in mean. The results in this paper generalize and improve the previous related results. [ABSTRACT FROM AUTHOR]

This paper presents a mathematical model for tumour-immune response interactions in the perspective of immunotherapy by immune checkpoint inhibitors (ICIs). The model is of the nonlocal integro-differential Lotka-Volterra type, in which heterogeneity of the cell populations is taken into account by structuring variables that are continuous internal traits (aka phenotypes) present in each individual cell. These represent a lumped "aggressiveness", i.e., for tumour cells, malignancy understood as the ability to thrive in a viable state under attack by immune cells or drugs – which we propose to identify as a potential of de-differentiation–, and for immune cells, ability to kill tumour cells, in other words anti-tumour efficacy. We analyse the asymptotic behaviour of the model in the absence of treatment. By means of two theorems, we characterise the limits of the integro-differential system under an a priori convergence hypothesis. We illustrate our results with a few numerical simulations, which show that our model reproduces the three Es of immunoediting: elimination, equilibrium, and escape. Finally, we exemplify the possible impact of ICIs on these three Es. [ABSTRACT FROM AUTHOR]

New technologies play a central role in damage prevention of artistic and cultural heritage. The literature is ourishing of mathematical models that describe the process of corrosion due to weathering and exposition to pollutants. These models consist of differential equations or partial differential equations that need to be solved approximately by numerical methods. This paper aims to describe the mathematical models in the literature and the numerical methods used for their solution. We focus in particular on the studies of corrosion of pieces of art made of stone, lime mortar concrete and metal. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in modeling the production of mucus by diatoms under the constraint of a nutrient limitation and carbon overabundance. The initial questioning comes from the observation of the so-called "liga" on the Acquitaine coast. The biological origin of the phenomenon is presented and discussed based on the existing literature. An original mathematical model incorporating mucus secretion is proposed and its theoretical properties are analized: well-posedness and differentiability with respect to the model parameters. Finally, numerical experiments are provided, investigating the possibility of parameter identification for the model using chemostat-type experiments. [ABSTRACT FROM AUTHOR]

Rotating an axially symmetric rigid body on a horizontal plane is rather a common and simple experience, but this experience has attracted a great deal of interests due to it exhibiting novel features and containing fairly complicated mechanics. This paper is concerned with the threedimensional rotational motion of a rigid tube on a plane.We present the governing dynamical equations of this motion and give a numerical treatment, based on which we discuss the nutation of tube and simulate the trajectory of tube end. We also discuss how fast the angular velocity should be in order to initiate an uninterrupted, steady rotational motion. Then the air lift related to such a three-dimensional rotation of tube is modeled by using Kutta-Joukowski law. By employing this model, we show that the air lift indeed "lift" the tube head during rotating. [ABSTRACT FROM AUTHOR]

Cocoa mirid, Sahlbergella singulars, is the major pest of cocoa (Theobroma cacao) responsible of several damage in plots in West Africa and particularly in Cameroon. Occasional damage accounts for 30 40% of pod losses. However, when miridae affect the foliage, gradual wilting occurs and eventually, tree death. A few studies have focused on describing the time evolution of Miridae in the plot in Cameroon, yet numerous questions remain. The aim of this paper is to estimate and control the losses of production caused by the bites of miridea. To do this, we will formulate and study a mathematical model for the dynamics of pods that takes into account the feeding and egg-laying of adults miridae on pods. We present the theoretical analysis of the model. More precisely, we compute equilibria and derive a threshold parameter that determines the presence or not of miridae in the plot. Throughout numerical simulations, we found that miridae can cause approximately 39.21% of production losses (which represents approximatively USD 1276.8 revenue losses) when initially, one has 1200 plants in the plot. After, we aim to increase cocoa production through optimal control. Optimal control consists in reducing density the number of nymphs and adults miridae in the plot. We studied the controlled model and we found that losses with control shrink to 20.58% which corresponds to USD670.32 income revenue. [ABSTRACT FROM AUTHOR]

Developing new approaches that help control the spread of infectious diseases is a critical issue for public health. Such approaches must consider the available resources and capacity of the healthcare system. In this paper, we present a new mathematical approach to controlling an epidemic model by investigating the optimal control that aims to bring the output of the epidemic to target a desired disease output yd = (ydi)i∈{0,...,N}. First, we use the state-space technique to transfer the trajectory controllability to optimal control with constraints on the final state. Then, we use the fixed point theorems to determine the set of admissible controls and solve the output trajectory controllability problem. Finally, we apply our method to the model of a measles epidemic, and we give a numerical simulation to illustrate the findings of our approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider a steady state multi-anticipative traffic model and we provide necessarily and sufficient conditions for the existence of traveling solutions. In our work, the word "traveling" means that the distance between two consecutive vehicles travels continuously between two different states. As application to our result, we show that taking a strictly concave optimal velocity, we can construct a traveling solution such that the distance between two vehicles decreases. The existence, uniqueness and the study of the asymptotic behavior of such solutions is done at the level of the Hamilton-Jacobi equation. [ABSTRACT FROM AUTHOR]

This paper investigates a three-strategy (cooperators, toxin producers, and cheaters) N-player division of labor game in bacterial populations. We construct the replicator equation to discuss the evolution of the frequency of the three strategies. Firstly, we prove that the interior equilibrium is always unstable, the three strategies cannot coexist. Secondly, according to Sotomayor's theorem, the system undergoes transcritical bifurcation. Furthermore, the sensitivity of the two-dimensional evolutionary state diagrams to the third parameter (toxin rate, absorption rate, toxin quantity, etc) is analyzed. In summary, high toxicity rates, high levels of toxins, and low levels of competition tend to promote cooperation. All players choose to perform the task, and the cheater disappears. When the absorption rate of cooperators is high enough, only cooperators exist in the population over time. When the absorption rate of the cooperator is low, and the absorption rate of the toxin producer is greater than the threshold, the cooperator and the toxin producer coexist. All players perform the task. Finally, the triangle diagrams and three-dimensional diagrams are presented, which show the initial conditions of the three strategies also affect the dynamic results. As the amount of toxin increases, the range of players who choose to perform tasks widens. [ABSTRACT FROM AUTHOR]

In this paper, we present and analyze an age-structured tumor immune model. Based on the fact that tumor cells of different ages tend to exhibit different physiological behaviors, we consider the age structure of tumor cells, age-based proliferation function and age-dependent death function in the model. The threshold R0 for the existence of tumor-free steady state is derived. It is found that if R0 > 1, the tumor-free steady state is not only locally stable but also globally stable. Moreover, numerical simulation shows that the threshold R0 may be regarded as an index to reflect the ability of "tumor immune surveillance", that is, the smaller the R0, the better the ability of tumor immune surveillance. If R0 > 1, it is proved that the tumor steady state is existent and uniformly persistent. The local stability of the tumor steady state is investigated under some further conditions besides R0 > 1. In the end, we estimate the system parameters, verify the theoretical results and analyze some system parameters' sensitivities. [ABSTRACT FROM AUTHOR]

The Spatio-temporal pattern induced by self-diffusion of a predator-prey model with Boiling type III functional response that incorporates the hunting cooperation between predators has been investigated in this paper. For the local model without structure, the stability of non-negative equilibria with or without collaborative hunting in predators is studied. For the Spatio-temporal model, we analyze the effect of hunting cooperation term on diffusion-driven Turing instability of the homogeneous positive equilibria. To get an idea about patterns formation near the Turing bifurcation, we derive and give a detailed study of the amplitude equation using the multiple-scale analysis. Our result shows that hunting cooperation plays a crucial role in determining the stability and the Turing bifurcation of the model, which is in sharp contrast to the case without cooperation in hunting. Furthermore, some numerical simulations are illustrated to visualize the complex dynamic behavior of the model. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SEIR epidemic model is studied with alertness and vaccination. The goal is to stabilize the infectious disease system quickly. The dynamic behavior of the model is analyzed and an integral sliding mode controller with distributed compensation is designed. By using Lyapunov function method, the sufficient conditions for the existence and uniqueness of global positive solutions and the existence of ergodic stationary distributions are obtained. The stochastic center manifold and stochastic average method are used to simplify the system into a one-dimensional Markov diffusion process. The stochastic stability and Hopf bifurcation are analyzed using singular boundary theory. An integral sliding mode controller with non-parallel distributed compensation is designed by linear matrix inequality (LMI) method, which realizes the stability of system and prevents the outbreak of epidemic disease. The correction of theoretical analysis and the effectiveness of controller are validated using numerical simulation performed in MATLAB/Simulink. [ABSTRACT FROM AUTHOR]

A large necrotic core increases the risk of atherosclerotic plaque instability. Statins can delay the growth of necrotic core in plaques, but the kinetic mechanism of statins in slowing down the necrotic core has not yet been addressed in detail. In this paper, a mathematical model is governed by a system of advection-diffusion-reaction equations coupling of the porous nature of vessel wall is established and applied to illustrate the plaque growth with lipid-rich necrotic core (LRNC) with and without statins using finite element method. We study the influence of LRNC plaque growth for different drug concentrations at different time intervals. The results showed that the drug use at different time points has a significant impact on the treatment efficacy. Compared with short-term, low-dose treatment, early statin treatment with high dose showed more pronounced effects on reducing the low-density lipoprotein (LDL) cholesterol, decreasing the volume of necrotic core, changing the characteristics of plaques, and improving the plaque stability. The model is validated by comparing with the clinical data, and may be used to predict the progression of LRNC plaque and the effects of statin therapy. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the convergence of the Faedo-Galerkin approximations, in a strong sense, to a strong T-periodic solution of the torso-coupled bidomain model where T is the period of activation of the inner wall of the heart. First, we define the torso-coupled bidomain operator and prove some of its more important properties for our work. After, we define the abstract evolution system of the equations that are associated with torso-coupled bidomain model and give the definition of a strong solution. We prove that the Faedo-Galerkin's approximations have the regularity of a strong solution, and we find that some restrictions can be imposed over the initial conditions, so that this sequence of Faedo-Galerkin fully converges to a strong solution of the Cauchy problem. Finally, these results are used for showing the existence a strong T-periodic solution. [ABSTRACT FROM AUTHOR]

This research paper proposes and discusses a conceptual modelling of both growth of tumours in presence of immortal multipotent cancer stem cells (CSCs) and of several lineages of differentiated tumour cells (CCs). The replication of CSCs is assumed symmetric or asymmetric with a prescribed mean ratio and mitosis and apoptosis are taken into account for the CCs aging. Replication can be hindered by the local crowding of the cells in the vicinity of the mother cell. The model is implemented in the framework of 3D cellular automata (CA) whose dynamics is governed by stochastic rules. Some simulations are displayed showing the growth of a tumour and the fractions of different lineages and age classes of CCs. Then, an approach that considers the same dynamics of aging, replication, and apoptosis, but with the aim to study the time evolution of the fractions of the different lineages and age classes of cells averaged over the total volume is presented. The dynamics is governed by a system of ordinary differential equations (ODEs), hence by deterministic rules. Numerical simulations of the solution of this system show qualitative similarity with the CA results, although the crowding effect is no longer a local effect, but also averaged over the total volume. The Appendix provides the proof of the mathematical well-poscdness of this model in a general framework. [ABSTRACT FROM AUTHOR]

In this work we have studied a stochastic predator-prey model where the prey grows logistically in the absence of predator. All parameters but carrying capacity have been perturbed with telephone noise. The prey's growth rate and the predator's death rate have also been perturbed with white noises. Both of these noises have been proved extremely useful to model rapidly fluctuating phenomena Dimentberg (1988). The conditions under which extinction of predator and prey populations occur have been established. We also give sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution of the positive solution, red which in stochastic predator-prey systems means that the predator and prey populations can be persistent, that is to say, the predator and prey populations can be sustain a quantity that is neither too much nor too little. In our analysis, it is found that the environmental noise plays an important role in extinction as well as coexistence of prey and predator populations. It is shown in numerical simulation that larger white noise intensity will lead to the extinction of the population, while telephone noise may delay or reduce the risk of species extinction. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic SIRS epidemic model with nonlinear incidence and Markovian switching. By using the stochastic calculus background, we establish that the stochastic threshold ℜswt can be used to determine the compartment dynamics of the stochastic system. Some examples and numerical simulations are presented to confirm the theoretical results established in this paper. [ABSTRACT FROM AUTHOR]

In this paper, by using the new concepts of monotonicity, pseudomonotonicity and hemicontinuity associated with the considered curvilinear integral functional, we investigate the well-posedness and well-posedness in generalized sense for a class of controlled variational inequality problems. More precisely, by introducing the approximating solution set of the considered class of controlled variational inequality problems, we formulate and prove some characterization results on well-posedness and well-posedness in generalized sense. Also, the theoretical developments presented in the paper are accompanied by illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we construct a model to describe the transmission of HIV in a homogeneous host population. By considering the specific mechanism of HIV, we derive a model structured in three successive stages: (i) primary infection, (ii) long phase of latency without symptoms, and (iii) AIDS. Each HIV stage is stratified by the duration for which individuals have been in the stage, leading to a continuous age-structure model. In the first part of the paper, we provide a global analysis of the model depending upon the basic reproduction number ℜ0. When ℜ0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable and the infection is cleared in the host population. On the contrary, if ℜ0 > 1, we prove the epidemic's persistence with the asymptotic stability of the endemic equilibrium. By performing the sensitivity analysis, we then determine the impact of control-related parameters on the outbreak severity. For the second part, the initial model is extended with intervention methods. By taking into account antiretroviral therapy (ART) interventions and the probability of treatment drop out, we discuss optimal intervention methods which minimize the number of AIDS cases. [ABSTRACT FROM AUTHOR]

In this paper, optimal control of excessive water waves in a canal system, modeled by a nonlinear improved Boussinesq equation, is considered. For this aim, well-posedness and controllability properties of the system is investigated. Suppressing of the waves in the canal system is successfully obtained by means of optimally determining of canal depth control function via maximum principle, which transforms to optimal control problem to solving an nonlinear initial-boundary-terminal value problem. The beauty of the present paper than other studies existing in the literature is that optimal canal depth control function is analytically obtained without linearization of nonlinear term. In order to show effectiveness and robustness of the control actuation, several numerical examples are given by MATLAB in tables and graphical forms. [ABSTRACT FROM AUTHOR]

A nonlocal theory for thermoelastic materials with voids based on Mindlin's strain gradient theory was derived in this paper with some qualitative properties. We have also established the size effect of nonlocal heat conduction with the aids of extended irreversible thermodynamics and generalized free energy. The obtained system of equations is a coupling of three equations with higher gradients terms due to the length scale parameters ϖ and l. This poses some new mathematical difficulties due to the lack of regularity. Based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions to the one dimensional problem. By an approach based on the Gearhart-Herbst-Prüss-Huang theorem, we prove that the associated semigroup is exponentially stable; but not analytic. [ABSTRACT FROM AUTHOR]

Analysis of longitudinal vibration in a nanorod is an important subject in science and engineering due to its vast application in nanotechnology. This paper introduces a port-Hamiltonian formulation for the longitudinal vibrations in a nanorod, which shows that this model is essentially hyperbolic. Furthermore, it investigates the spectral properties of the associated system operator. Standard distributed control and feedback are shown not to be controllable nor stabilizing. [ABSTRACT FROM AUTHOR]

As cyber attacks have become more frequent, cyber insurance premiums have increased, resulting in the need for better modeling of cyber risk. Toward this direction, Jevtić and Lanchier [Insur. Math. Econ.91 (2020) 209–223] proposed a dynamic structural model of aggregate loss distribution for cyber risk of small and medium-sized enterprises under the assumption of a tree-based local-area-network topology that consists of the combination of a Poisson process, homogeneous random trees, bond percolation processes, and cost topology. Their model assumes that the contagion spreads through the edges of the network with the same fixed probability in both directions, thus overlooking a dynamic cyber security environment implemented in most networks, and their results give an exact expression for the mean of the aggregate loss but only a rough upper bound for the variance. In this paper, we consider a bidirectional version of their percolation model in which the contagion spreads through the edges of the network with a certain probability of moving toward the lower level assets of the network but with another probability of moving toward the higher level assets of the network, which results in a more realistic cyber security environment. In addition, our mathematical approach is quite different and leads to exact expressions for both the mean and the variance of the aggregate loss, and therefore an exact expression for the insurance premiums. [ABSTRACT FROM AUTHOR]

This paper presents a numerical study of spatial transformations of a radially symmetric flow in a shallow fluid-filled cylindrical cavity partially covered by a solid non-deformable film. The upper central part of the liquid has a free surface where an intense light beam is focused to produce a hot spot on the symmetry axis. The heating generates a divergent thermocapillary motion on the free surface, which causes the fluid to flow under the immovable solid film. The edge of this film induces thermal perturbations, which, at specific heat generation values, begin to increase and give rise to a gradual radial flow symmetry breakdown that visually demonstrates the onset of vorticity in the azimuthal plane. Three-dimensional calculations have been carried out based on the interfacial hydrodynamics equations using Comsol Multiphysics software. The results of numerical calculations confirm that the instability occurs in the area corresponding to the boundary between the free surface and the solid film. The motion in the azimuthal direction becomes more evident with the growth of heating intensity. The vorticity in the azimuthal plane makes the flow structure significantly more complex compared to the axisymmetric radial flow. Thus, there is a predominantly radial flow in the area of free surface and, at the same time, the vortices with the azimuthal velocity component are observed under the film. As in the experiment, the number of vortices is determined by the ratio of the area of free surface to the covered area. It is shown that the appearance of motion in the azimuthal direction depends on the joint action of several physical mechanisms, which have been initialized in the theoretical model by means of various piecewise thermal and mechanical conditions at the upper boundary. [ABSTRACT FROM AUTHOR]

A hyperbolic model for diffusion, nonlfnear transport (or advection) and production of a scalar quantity, is considered. The model is based on a constitutive law of Cattaneo-Maxwell type expressing non-Fickian diffusion by means of a relaxation time relation. The production or source term is assumed to be of logistic type. This paper studies the existence and spectral stability properties of spatially periodic traveling wave solutions to this system. It is shown that a family of subcharacteristic periodic waves emerges from a local Hopf bifurcation around a critical value of the wave speed. These waves have bounded fundamental period and small-amplitude. In addition, it is shown that these waves are spectrally unstable as solutions to the hyperbolic system. For that purpose, it is proved that the Floquet spectrum of the linearized operator around a wave can be approximated by a linear operator whose point spectrum intersects the unstable half plane of complex numbers with positive real part. [ABSTRACT FROM AUTHOR]

In this paper, we establish a rigorous connection between a microscopic and a macroscopic pedestrians model on a convergent junction. At the microscopic level, we consider a "follow the leader" model far from the junction point and we assume that a rule to enter the junction point is imposed. At the macroscopic level, we obtain the Hamilton-Jacobi equation with a flux limiter condition at x = 0 introduced in Imbert and Monneau [Ann. Sci. l'École Normale Supér. 50 (2017) 357-414], To obtain our result, we inject using the "cumulative distribution functions" the microscopic model into a non-local PDE. Then, we show that the viscosity solution of the non-local PDE converges locally uniformly towards the solution of the macroscopic one. [ABSTRACT FROM AUTHOR]

The Sterile Insect Technique (SIT) is a classic vector control method that has been successfully applied to fight against diverse insect plagues since the 1950s. In recent years, this strategy has been used to control mosquito populations, in order to limit the spread of the diseases they transmit. In this paper, we consider a system of reaction-diffusion equations to model the mosquito population and study the effect of the release of sterile mosquito males. Then, we propose to analyze the release in a limited area inside a wider area containing a natural mosquito population. We are interested in protecting a mosquito free region from invasion by mosquitoes from an exterior domain by controlling the population in a release band at the border between the two regions: we construct a barrier blocking the invasion of mosquitoes from the exterior. We adapt the geometric method of Lewis and Keener (see Lewis and Keener [SIAM J. Appl. Math.61 (2000) 293-316]) in this framework and extend their main result to find relations on the size of the release region and the density of the released sterile males that allow us to block the invasion. Numerical simulations are also performed to illustrate our results. [ABSTRACT FROM AUTHOR]

Atherosclerosis is a chronic disease which involves the build up of cholesterol and fatty deposits within the inner lining of the artery. It is associated with a progressive thickening and hardening of the arterial wall that result in narrowing of the vessel lumen and restriction of blood flow to vital organs. These events may cause heart attack or stroke, the commonest causes of death worldwide. In this paper we study the early stages of atherosclerosis via a mathematical model of partial differential equations of reaction-diffusion type. The model includes several key species and identifies endothelial hyperpermeability, believed to be a precursor on the onset of atherosclerosis. For simplicity, we reduce the system to a monotone system and provide a biological interpretation for the stability analysis according to endothelial functionality. We investigate as well the existence of solutions of traveling waves type along with numerical simulations. The obtained results are in good agreement with current biological knowledge. Likewise, they confirm and generalize results of mathematical models previously performed in literature. Then, we study the non monotone reduced model and prove the existence of perturbed solutions and perturbed waves, particularly in the bistable case. Finally, we extend the study by considering the complete model proposed initially, perform numerical simulations and provide more specific results. We study the consistency between the reduced and complete model analysis for a certain range of parameters, we elaborate bifurcation diagrams showing the evolution of inflammation upon endothelial permeability and LDL accumulation and we consider the effect of anti-inflammatory process on the system behavior. In this model, the regulation of atherosclerosis progression is mediated by anti-inflammatory responses that, up to certain extent, lead to plaque regression. [ABSTRACT FROM AUTHOR]

Ion channel proteins have many conformational (metastable) states and, for this reason, they exhibit hysteresis. This fact is responsible for the non-Markovian stochastic nature of single ion channel recordings. It is suggested in the paper that the stochastic single channel recordings can be modeled as the random outputs of rectangular hysteresis loops driven by stochastic processes. The latter problem can be mathematically treated as an exit problem for stochastic processes or by using the theory of stochastic processes on graphs. It is also demonstrated in the paper that the collective action of sodium and potassium channels responsible for the generation and propagation of action potentials exhibit hysteresis. This demonstration is accomplished by using the inverse problem approach to the nonlinear Hodgkin-Huxley diffusion equation. [ABSTRACT FROM AUTHOR]

Electroporation ablation is a promising non surgical and minimally invasive technique of tumor ablation, for which no monitoring is currently available. In this paper, we present the recent advances and challenges on the numerical modeling of clinical electroporation ablation of liver tumors. In particular, we show that a nonlinear static electrical model of tissue combined with clinical imaging can give crucial information of the electric field distribution in the clinical configuration. We conclude the paper by presenting some important questions that have to be addressed for an effective impact of computational modeling in clinical practice of electroporation ablation. [ABSTRACT FROM AUTHOR]

This paper investigates the maximum principle for a nonlinear size structured model that describes the optimal management of the fish resources taking into account harvesting the fish and putting the fry. First, we show the existence of a unique non-negative solution of the system, and give a comparison principle. Next, we prove the existence of optimal policies by using maximizing sequence and Mazur's theorem in convex analysis. Then, we obtain necessary optimality conditions by using tangent-normal cones and adjoint system techniques. Finally, some examples and numerical results demonstrate the effectiveness of the theoretical results in our paper. [ABSTRACT FROM AUTHOR]

In this paper, an age structured epidemic Susceptible-Infected-Quarantined-Recovered-Infected (SIQRI) model is proposed, where we will focus on the role of individuals that leave the R-class before being completely recovered and thus will participate again to the disease transmission. We investigate the asymptotic behavior of solutions by studying the stability of both trivial and positive equilibria. In order to see the impact of the different model parameters like the relapse rate on the qualitative behavior of our system, we firstly, give an explicit expression of the basic reproduction number R0, which is a combination of the classical basic reproduction number for the SIQR model and some other model parameters, corresponding to the individuals infected by the relapsed ones. It will be shown that, if R0 ≤ 1, the disease free equilibrium is globally asymptotically stable and becomes unstable for R0 > 1. Secondly, while R0 > 1, a suitable Lyapunov functional is constructed to prove that the unique endemic equilibrium is globally asymptotically stable on some subset Ω0. [ABSTRACT FROM AUTHOR]

The current paper is devoted to stochastic damped higher order KdV equation driven by Poisson process. We establish the well-posedness of stochastic damped higher-order KdV equation, and prove that there exists an unique invariant measure for non-random initial conditions. Some discussion on the general pure jump noise case are also provided. Some numerical simulations of the invariant measure are provided to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper the stability of the zero equilibrium of a system with time delay is studied. The critical case of a multiple zero root of the characteristic equation of the linearized system is treated by applying a Malkin type theorem and using a complete Lyapunov-Krasovskii functional. An application to a model for malaria under treatment considering the action of the immune system is presented. [ABSTRACT FROM AUTHOR]

In this paper, we implemented the generalized (G′/G) $(\frac{G^{'}}{G})$ ( G ′ G) and extended (G′/G) $(\frac{G^{'}}{G})$ ( G ′ G) methods to solve fractional-order biological population models. The fractional-order derivatives are represented by the Caputo operator. The solutions of some illustrative examples are presented to show the validity of the proposed method. First, the transformation is used to reduce the given problem into ordinary differential equations. The ordinary differential equation is than solve by using modified (G′/G) $(\frac{G^{'}}{G})$ ( G ′ G) method. Different families of traveling waves solutions are constructed to explain the different physical behavior of the targeted problems. Three important solutions, hyperbolic, rational and periodic, are investigated by using the proposed techniques. The obtained solutions within different classes have provided effective information about the targeted physical procedures. In conclusion, the present techniques are considered the best tools to analyze different families of solutions for any fractional-order problem. [ABSTRACT FROM AUTHOR]

In this paper, we aim to determine an optimal insurance premium rate for health-care in deterministic and stochastic SEIR models. The studied models consider two standard SEIR centres characterised by migration fluxes and vaccination of population. The premium is calculated using the basic equivalence principle. Even in this simple set-up, there are non-intuitive results that illustrate how the premium depends on migration rates, the severity of a disease and the initial distribution of healthy and infected individuals through the centres. We investigate how the vaccination program affects the insurance costs by comparing the savings in benefits with the expenses for vaccination. We compare the results of deterministic and stochastic models. [ABSTRACT FROM AUTHOR]

In this paper, we examine a Bidirectional Associative Memory neural network model with distributed delays. Using a result due to Cid [J. Math. Anal. Appl.281 (2003) 264–275], we were able to prove an exponential stability result in the case when the standard Lipschitz continuity condition is violated. Indeed, we deal with activation functions which may not be Lipschitz continuous. Therefore, the standard Halanay inequality is not applicable. We will use a nonlinear version of this inequality. At the end, the obtained differential inequality which should imply the exponential stability appears 'state dependent'. That is the usual constant depends in this case on the state itself. This adds some difficulties which we overcome by a suitable argument. [ABSTRACT FROM AUTHOR]

In this paper we the study of dynamics of time since infection structured vector born model with the direct transmission. We use standard incidence term to model the new infections. We analyze the corresponding system of partial differential equation and obtain an explicit formula for the basic reproduction number ℜ0. The diseases-free equilibrium is locally and globally asymptotically stable whenever the basic reproduction number is less than one, ℜ0 < 1. Endemic equilibrium exists and is locally asymptotically stable when ℜ0 > 1. The disease will persist at the endemic equilibrium whenever the basic reproduction number is greater than one. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the non-constant stationary solutions of a general Gause-type predator-prey system with self- and cross-diffusions subject to the homogeneous Neumann boundary condition. In the system, the cross-diffusions are introduced in such a way that the prey runs away from the predator, while the predator moves away from a large group of preys. Firstly, we establish a priori estimate for the positive solutions. Secondly, we study the non-existence results of non-constant positive solutions. Finally, we consider the existence of non-constant positive solutions and discuss the Turing instability of the positive constant solution. [ABSTRACT FROM AUTHOR]

The behavior of neutral disturbance in arterial vessels has attracted more and more attention in recent decades because it carries some important information which can be applied to predict and diagnose related heart disease, such as arteriosclerosis and hypertension, etc. Because of the complexity of blood flow in arteries, it is very necessary to construct accurate mathematical model and analyze the mechanical behavior of neutral disturbance in arterial vessels. In this paper, start from the basic equations of blood flow and the two-dimensional Navier–Stokes equation, the vorticity equation describing the disturbance flow is presented. Then, by use of multi-scale analysis and perturbation expansion method, the ZK equation is put forward which can reflect the behavior of the neutral perturbation flow in arterial vessels. Compared with the traditional KdV model, the model established in the paper can show the propagation of the disturbance flow in the radius direction. Furthermore, the time-fractional ZK equation is derived by semi-inverse method and Agrawal's method, which is more convenient and accurate for discussing the feature of neutral disturbance in arterial vessels and can provide more information for analyzing some related heart disease. Meanwhile, with the help of the modified extended tanh method, the above mentioned equation is solved. The results show that neutral disturbance exists in arterial vessels and propagates in the form of solitary waves. By calculating, we find the relation of the stroke volume with vascular radius, blood flow velocity as well as the fractional order parameter α, which is very meaningful for preventing and treating related heart disease because the stroke volume is closely linked with heart disease. [ABSTRACT FROM AUTHOR]

This paper describes an SIR model with logistic growth rate of susceptible population, non-monotonic incidence rate and saturated treatment rate. The existence and stability analysis of equilibria have been investigated. It has been shown that the disease free equilibrium point (DFE) is globally asymptotically stable if the basic reproduction number is less than unity and the transmission rate of infection less than some threshold. The system exhibits the transcritical bifurcation at DFE with respect to the cure rate. We have also found the condition for occurring the backward bifurcation, which implies the value of basic reproduction number less than unity is not enough to eradicate the disease. Stability or instability of different endemic equilibria has been shown analytically. The system also experiences the saddle-node and Hopf bifurcation. The existence of Bogdanov-Takens bifurcation (BT) of co-dimension 2 has been investigated which has also been shown through numerical simulations. Here we have used two control functions, one is vaccination control and other is treatment control. We have solved the optimal control problem both analytically and numerically. Finally, the efficiency analysis has been used to determine the best control strategy among vaccination and treatment. [ABSTRACT FROM AUTHOR]

This paper studies a fractional Bloch equation pertaining to Hilfer fractional operator. Bloch equation is broadly applied in physics, chemistry, nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI) and many more. The sumudu transform technique is applied to obtain the analytic solutions for nuclear magnetization M = (Mx, My, Mz). The general solution of nuclear magnetization M is shown in the terms of Mittag-Leffler (ML) type function. The influence of order and type of Hilfer fractional operator on nuclear magnetization M is demonstrated in graphical form. The study of Bloch equation with composite fractional derivative reveals the new features of Bloch equation. The discussed fractional Bloch model provides crucial and applicable results to introduce novel information in scientific and technological fields. [ABSTRACT FROM AUTHOR]

In this paper, we propose and investigate a fractional diffusive model for hepatitis B virus (HBV) infection with capsids and immune response presented by cytotoxic T lymphocyte (CTL) cells. We derive the conditions for global asymptotic stability of the steady states of the model in terms of the basic reproduction number R0 and the immune response reproduction number R1. By constructing appropriate Lyapunov functionals, it is shown that the infection-free equilibrium is globally asymptotically stable when R0 ≤ 1, the immune-free infection equilibrium is globally asymptotically stable when R1 ≤ 1 < R0 and the infection equilibrium with CTL immune response is globally asymptotically stable when R1 > 1. Numerical simulations are performed to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

This paper focuses on the dynamics of a three species ratio-dependent food chain model with diffusion and double free boundaries in one dimensional space, in which the free boundaries represent expanding fronts of top predator species. The existence, uniqueness and estimates of the global solution are discussed firstly. Then we prove a spreading–vanishing dichotomy, specifically, the top predator species either successfully spreads to the entire space as time t goes to infinity and survives in the new environment, or fails to establish and dies out in the long run. The long time behavior of the three species and criteria for spreading and vanishing are also obtained. Besides, our simulations illustrate the impacts of initial occupying area and expanding capability on the dynamics of top predator for free boundaries. [ABSTRACT FROM AUTHOR]

In this paper, we study the essential spectrum of the operator obtained by linearizing at traveling waves that occur in the one-dimensional version of the M5-model for mesenchymal cell movement inside a directed tissue made up of highly aligned fibers. We show that traveling waves are spectrally unstable in L2(ℝ; ℂ3) as the essential spectrum includes the imaginary axis. Tools in the proof include exponential dichotomies and Fredholm properties. We prove that a weighted space Lw2(ℝ; ℂ3) with the same function for the tree variables of the linearized operator is no suitable to shift the essential spectrum to the left of the imaginary axis. We find a pair of appropriate weight functions whereby on the weighted space Lwα2(ℝ; ℂ2) × Lwε2(ℝ; ℂ) the essential spectrum lies on {Reλ<0}, outside the imaginary axis. [ABSTRACT FROM AUTHOR]

A generalized form of (2+1)-dimensional Hirota bilinear (2D-HB) equation is considered herein in order to study nonlinear waves in fluids and oceans. The present goal is carried out through adopting the simplified Hirota's method as well as ansatz approaches to retrieve a bunch of rational wave structures from multiple soliton solutions to breather, rational, and complexiton solutions. Some figures corresponding to a series of rational wave structures are provided, illustrating the dynamics of the obtained solutions. The results of the present paper help to reveal the existence of rational wave structures of different types for the 2D-HB equation. [ABSTRACT FROM AUTHOR]

Finding a mathematical model that incorporates various stochastic aspects of neural dynamics has proven to be a continuous challenge. Among the different approaches, the noisy leaky integrate-and-fire and the escape rate models are probably the most popular. These two models are generally thought to express different noise action over the neural cell. In this paper we investigate the link between the two formalisms in the case of a neuron subject to a time dependent input. To this aim, we introduce a new general stochastic framework. As we shall prove, our general framework entails the two already existing ones. Our results have theoretical implications since they offer a general view upon the two stochastic processes mostly used in neuroscience, upon the way they can be linked, and explain their observed statistical similarity. [ABSTRACT FROM AUTHOR]