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In this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme’s application is demonstrated through two test problems. [ABSTRACT FROM AUTHOR]

We consider a non-local tumour growth model of phase-field type, describing the evolution of tumour cells through proliferation in presence of a nutrient. The model consists of a coupled system, incorporating a non-local Cahn-Hilliard equation for the tumour phase variable and a reaction-diffusion equation for the nutrient. First, we establish novel regularity results for such a model, by applying maximal regularity theory in weighted L p spaces. This technique enables us to prove the local existence and uniqueness of a regular solution, including also chemotaxis effects. By leveraging time-regularisation properties and global boundedness estimates, we further extend the solution to a global one. These results provide the foundation for addressing an optimal control problem, aimed at identifying a suitable therapy, guiding the tumour towards a predefined target. Specifically, we prove the existence of an optimal therapy and, by studying the Fréchet-differentiability of the control-to-state operator and introducing the adjoint system, we derive first-order necessary optimality conditions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of time-varying impulsive recursive motions. We investigate the connections among minimal sets, non-wandering points, the Birkhoff center, and attractors. As an illustration, the obtained results are applied to a semilinear reaction-diffusion equation with impulses and to an integrate-and-fire neuron model. [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation phenomena for the nonlocal reaction-diffusion monostable equation with delay of the form ∂ u (t , x) ∂ t = d Δ u (t , x) + f (u (t , x) , ∫ R J (x − y) u (t − τ , y) d y) , t > 0 , x ∈ R. It is well-known that if we take J (x) = δ (x) and τ = 0 , there exists a minimal wave speed c ⁎ > 0 , such that this equation has no traveling wave front for 0 < c < c ⁎ and a traveling wave front for each c ≥ c ⁎ , which is unique up to translation and is globally asymptotically stable. Furthermore, when J is symmetry and exponentially bounded and τ > 0 , Wang et al. (2008) [27] considered the effects of delay and nonlocality on the spreading speed and proved that (i) if ∂ 2 f (0 , 0) > 0 , then the delay can slow the spreading speed of the wave fronts and the nonlocality can increase the spreading speed; and (ii) if ∂ 2 f (0 , 0) = 0 , then the delay and nonlocality do not affect the spreading speed. However, when J is asymmetry or exponentially unbounded , the question was left open. In this paper we obtain a rather complete answer to this question. More precisely, we show that for exponentially bounded kernels the minimal speed of traveling waves exists and coincides with the spreading speed. We also investigate the case of exponentially unbounded kernels where we prove the non-existence of traveling wave solutions and obtain upper and lower bounds for the position of any level set of the solutions. These bounds allow us to estimate how the solutions spread, depending on the kernels. [ABSTRACT FROM AUTHOR]

A space distributed model based on reaction–diffusion equations, which was previously developed, is generalized and applied to COVID-19 pandemic modeling in Ukraine. Theoretical analysis and a wide range of numerical simulations demonstrate that the model adequately describes the second wave of the COVID-19 pandemic in Ukraine. In particular, comparison of the numerical results obtained with the official data shows that the model produces very plausible total numbers of the COVID-19 cases and deaths. An extensive analysis of the impact of the parameters arising from the model is presented as well. It is shown that a well-founded choice of parameters plays a crucial role in the applicability of the model. [ABSTRACT FROM AUTHOR]

Four formulations based on the Kirchhoff transformation and time linearization for the numerical study of one-dimensional reaction–diffusion equations, whose heat capacity, thermal inertia and reaction rate are only functions of the temperature, are presented. The formulations result in linear, two-point boundary-value problems for the temperature, energy or heat potential, and may be solved by either discretizing the second-order spatial derivative or piecewise analytical integration. In both cases, linear systems of algebraic equations are obtained. The formulation for the temperature is extended to two-dimensional, nonlinear reaction–diffusion equations where the resulting linear two-dimensional operator is factorized into a sequence of one-dimensional ones that may be solved by means of any of the four formulations developed for one-dimensional problems. The multidimensional formulation is applied to a two-dimensional, two-equation system of nonlinearly coupled advection–reaction–diffusion equations, and the effects of the velocity and the parameters that characterize the nonlinear heat capacities and thermal conductivity are studied. It is shown that clockwise-rotating velocity fields result in wave stretching for small vortex radii, and wave deceleration and thickening for counter-clockwise-rotating velocity fields. It is also shown that large-core, clockwise-rotating velocity fields may result in large transient periods, followed by time intervals of apparent little activity which, in turn, are followed by the propagation of long-period waves. [ABSTRACT FROM AUTHOR]

The reaction-diffusion equation is widely used in mathematical models of cancer. The calibration of model parameters based on limited clinical data is critical to using reaction-diffusion equation simulations for reliable predictions on a per-patient basis. Here, we focus on cell-level data as routinely available from tissue biopsies used for clinical cancer diagnosis. We analyze the spatial architecture in biopsy tissues stained with multiplex immunofluorescence. We derive a two-point correlation function and the corresponding spatial power spectral distribution. We show that this data-deduced power spectral distribution can fit the power spectrum of the solution of reaction-diffusion equations that can then identify patient-specific tumor growth and invasion rates. This approach allows the measurement of patient-specific critical tumor dynamical properties from routinely available biopsy material at a single snapshot in time. [ABSTRACT FROM AUTHOR]

A necessary and sufficient condition for the existence or nonexistence of global solutions to the following fractional reaction-diffusion equations u t = Δ α u + ψ (t) f (u) , in R N × (0 , ∞) , u (· , 0) = u 0 ≥ 0 , in R N , has not been known and remained as an open problem for a few decades, where N ≥ 2 , Δ α = - - Δ α / 2 denotes the fractional Laplace operator with 0 < α ≤ 2 , ψ is a nonnegative and continuous function, and f is a convex function. The purpose of this paper is to resolve this problem completely as follows: There is a global solution to the equation if and only if ∫ 1 ∞ ψ (t) t N α f ϵ t - N α d t < ∞ , for some ϵ > 0. [ABSTRACT FROM AUTHOR]

The functioning of several cellular processes in neuron cells relies on the interplay between multiple systems, such as calcium ([Ca2+]), inositol 1, 4, 5-trisphosphate (IP3), and dopamine. But, their individual dynamics provide very little insight into the various regulatory and dysregulatory cellular processes. The interaction of two systems dynamics offers some useful information about cell functioning in neurons. But, no attempt has been noted in the literature about the cooperation of three systems dynamics of [Ca2+], IP3, and dopamine in neurons. A mathematical model was utilized to examine the dynamic interactions of [Ca2+], IP3, and dopamine in neurons, considering their spatiotemporal aspects. Numerical findings were obtained using the finite element technique in conjunction with the Crank–Nicholson scheme. The effects of different component events like IP3-receptor (IP3R), sodium–calcium exchanger (NCX), calbindin-D28K buffer, etc. on the synergetic calcium, IP3, and dopamine dynamics have been studied in neuronal cells. The present model offers novel insights into the effects of regulation and dysregulation in different mechanisms like IP3R, NCX, calbindin-D28K, etc. on the synergetic systems of [Ca2+], IP3 and dopamine in neurons and their association with multiple neurological disorders, including Alzheimer's disease and Parkinson's disease. [ABSTRACT FROM AUTHOR]

In this paper, the Fuzzy Sumudu Transform Iterative method (FSTIM) was applied to find the exact fuzzy solution of the fuzzy space-time fractional telegraph equations using the Fuzzy Caputo Fractional Derivative operator. The Telegraph partial differential equation is a hyperbolic equation representing the reaction-diffusion process in various fields. It has applications in engineering, biology, and physics. The FSTIM provides a reliable and efficient approach for obtaining approximate solutions to these complex equations improving accuracy and allowing for fine-tuning and optimization for better approximation results. The work introduces a fuzzy logic-based approach to Sumudu transform iterative methods, offering flexibility and adaptability in handling complex equations. This innovative methodology considers uncertainty and imprecision, providing comprehensive and accurate solutions, and advancing numerical methods. Solving the fuzzy space-time fractional telegraph equation used a fusion of the Fuzzy Sumudu transform and iterative approach. Solution of fuzzy fractional telegraph equation finding analytically and interpreting its results graphically. Throughout the article, whenever we draw graphs, we use Mathematica Software. We successfully employed FSTIM, which is elegant and fast to convergence. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of a time-fractional nonlinear diffusion equation. According to Kaplan's first eigenvalue method, we first prove the blow-up of the solutions in finite time under some sufficient conditions. We next provide sufficient conditions for the existence of global solutions by using the results of Zhang and Sun. In conclusion, we find the second critical exponent for the existence of global and non-global solutions via the decay rates of the initial data at spatial infinity. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of a class of reaction-advection-diffusion equations of the form u t − Δ u + β (y) u x = f (t , u) in cylinders, where f is a tristable or multistable nonlinearity satisfying f (⋅ , 0) = f (⋅ , 1) = 0. We establish the alternative of wave solutions, that is, either there is a unique, asymptotically stable periodic traveling wave connecting 0 to 1 directly, or there is a unique propagating terrace connecting 0 to 1. Under the assumption that all wave speeds in propagating terrace are not equal to each other, we further obtain that the propagating terrace is asymptotically stable if it exists. Moreover, the sufficient and necessary conditions for the existence of periodic traveling waves connecting 0 to 1 are given respectively. Our arguments are based on the parabolic maximum principle, sliding method and the super- and sub-solution method. [ABSTRACT FROM AUTHOR]

We consider a general impulsive reaction-diffusion equation with shifting environment and birth pulse, both induced by climate changes, to capture essential features of the dynamics of species exhibiting distinct stages of reproduction and dispersal. We convert this model into a discrete-time semiflow, and hence to a discrete-time recursion system. We establish the existence of forced waves and asymptotic spreading properties of solutions, and obtain sufficient conditions for the global asymptotic stability of the forced waves. [ABSTRACT FROM AUTHOR]

This paper aims to demonstrate a numerical strategy via finite difference formulations for time fractional reaction–diffusion models which are ubiquitous in chemical and biological phenomena. The time-fractional derivative is considered in the Caputo sense for both linear and nonlinear problems. First, the Caputo derivative is replaced with a quadrature formula, then an implicit method is used for the remaining part. In the linear case, the proposed strategy reduces the time fractional models into linear simultaneous equations. In nonlinear cases, Quasilinearization is utilized to tackle the nonlinear parts. With this strategy, solutions of the fractional system transform into linear algebraic systems which are easy to solve. Next, the Von Neumann method is implemented to examine the stability of the scheme which discloses that the scheme is unconditionally stable. Further, the applicability of the presented scheme is tested with different linear and nonlinear models which include the one dimensional Schnakenberg and Gray–Scott models, and one and two dimensional Brusselator models. To analyze the accuracy of the present technique two norms namely, L ∞ and L 2 , and relative error are addressed. Moreover, the obtained outcomes are shown tabulated and graphically which identifies that the scheme properly works for the time fractional reaction–diffusion systems. [ABSTRACT FROM AUTHOR]

We propose a malaria model involving the sensitive and resistant strains, which is described by reaction-diffusion equations. The model reflects the scenario that the vector and host populations disperse with distinct diffusion rates, susceptible individuals or vectors cannot be infected by both strains simultaneously, and the vector population satisfies the logistic growth. Our main purpose is to get a threshold type result on the model, especially the interaction effect of the two strains in the presence of spatial structure. To solve this issue, the basic reproduction number (BRN) Ri0 and invasion reproduction number (IRN) Ri0 of each strain (i = 1 and 2 are for the sensitive and resistant strains, respectively) are defined. Furthermore, we investigate the influence of the diffusion rates of populations and vectors on BRNs and IRNs. [ABSTRACT FROM AUTHOR]

A statistical linearization method is proposed for determining the response of the chain-like multi-degree-of-freedom (MDOF) hysteretic system endowed with fractional derivative elements and subjected to combined periodic and stochastic excitation. The method is developed based on representing the system response into a combination as a deterministic and a zero-mean stochastic component. Specifically, first, the equation of motion can be transformed into two sets of coupled nonlinear differential equations for the deterministic and the stochastic response components, respectively. Next, the deterministic and the stochastic differential equations are treated by the harmonic balance method and the statistical linearization method, respectively, yielding two sets of coupled nonlinear algebraic equations, which can be solved simultaneously via standard numerical methods for the deterministic response time history and the stochastic response second moment. Pertinent numerical examples demonstrate the applicability of the proposed method. [ABSTRACT FROM AUTHOR]

Graph neural networks (GNNs) have gained significant attention in diverse domains, ranging from urban planning to pandemic management. Ensuring both accuracy and robustness in GNNs remains a challenge due to insufficient quality data that contains sufficient features. With sufficient training data where all spatiotemporal patterns are well-represented, existing GNN models can make reasonably accurate predictions. However, existing methods fail when the training data are drawn from different circumstances (e.g., traffic patterns on regular days) than test data (e.g., traffic patterns after a natural disaster). Such challenges are usually classified under domain generalization. In this work, we show that one way to address this challenge in the context of spatiotemporal prediction is by incorporating domain differential equations into graph convolutional networks (GCNs). We theoretically derive conditions where GCNs incorporating such domain differential equations are robust to mismatched training and testing data compared to baseline domain agnostic models. To support our theory, we propose two domain-differential-equationinformed networks: Reaction-Diffusion Graph Convolutional Network (RDGCN), which incorporates differential equations for traffic speed evolution, and the Susceptible-Infectious-Recovered Graph Convolutional Network (SIRGCN), which incorporates a disease propagation model. Both RDGCN and SIRGCN are based on reliable and interpretable domain differential equations that allow the models to generalize to unseen patterns. We experimentally show that RDGCN and SIRGCN are more robust with mismatched testing data than state-of-the-art deep learning methods. [ABSTRACT FROM AUTHOR]

This paper is concerned with pullback measure attractors of the non-autonomous stochastic reaction-diffusion equations defined in thin domains. We first prove the existence and uniqueness of pullback measure attractors for the inhomogeneous Markov process associated with the stochastic equations. We also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback measure attractors. Then we establish the upper semi-continuity of these attractors as a family of thin domains collapses into a lower-dimensional domain. [ABSTRACT FROM AUTHOR]

It is usually not possible to solve partial differential equations, especially the delay type, with analytical methods. Therefore, in this article, we present an efficient method for solving differential equations of the difference delayed reaction-diffusion type, which can be generalized to other delayed partial differential equations. In the proposed approach, we first convert the delayed equation into an equivalent non-delayed equation by inserting the corresponding delay function with an effective technique. Then, using a pseudo-spectral method, we discretize the obtained equation in the Legendre-Gauss-Lobatto collocation points and present an algebraic system with an equal number of equations and unknowns which can be solved by quasi-Newton methods such as Levenderg-Marquardt algorithm. The approximate solutions can be obtained with exponential accuracy. The convergence analysis of the method is fully discussed and four examples are presented to evaluate the results and compare with one of the conventional methods used to solve partial differential equations, that is, the compact finite difference method. [ABSTRACT FROM AUTHOR]

In this paper, we study the two-parameter spectral element method based on weighted shifted orthogonal polynomials for solving singularly perturbed diffusion equation on an interval [0, 1] which are modeled with singular parameters. We continue our study to estimate the lower bound of the weighted orthogonal polynomial coefficient and the upper bound of a posteriori error estimates of the method through different weighted norms to minimize the computational cost. Numerical examples are implemented to study the applicability and efficiency of the technique. The obtained error bounds for the coefficient of orthogonal polynomials and the posteriori estimates fall within the bounds derived in the theoretical section. It is also observed that the two weighted norms decreases when the values of N 1 and N 2 increases for the three choices of and for different values of x and y. The quality and accuracy of the solution can be realized through figures and tables. [ABSTRACT FROM AUTHOR]

Biological models for cardiac regeneration and remodeling, along with the effects of cytokines or chemokines during the therapy with mesenchymal stem cells after a myocardial infarction, are of crucial importance for understanding the complex underlying mechanisms. This paper presents a mathematical model composed of three coupled partial differential equations that describes the dynamics of stem cells, nutrients and chemokines, highlighting the fundamental role of the chemokines during the myocardial tissue regeneration process. The system is solved numerically using mimetic difference operators and the MOLE library for MATLAB. The results show the tissue regeneration process in the necrotic part closest to the cell implantation area. [ABSTRACT FROM AUTHOR]

Inspired by a recent study associating shifting temperature conditions with changes in the efficiency with which predators convert prey to offspring, we propose a predator-prey model of reaction-diffusion type to analyze the consequence of such effects on the population dynamics and spread of the predator species. In the model, the predator conversion efficiency is represented by a spatially heterogeneous function depending on the variable ξ = x - c1t for some given c1 > 0. Using the Hamilton-Jacobi approach, we provide explicit formulas for the spreading speed of the predator species. When the conversion function is monotone increasing, the spreading speed is determined in all cases and non-local pulling is possible. When the function is monotone decreasing, we provide formulas for the spreading speed when the rate of shift of the conversion function is sufficiently fast or slow. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the stability analysis for a contaminant convection-reaction-diffusion model of the recovered fracturing fluid (RFFM, for short), which couples a nonlinear and non-smooth stationary incompressible Navier-Stokes equation with a multivalued frictional boundary condition, and a nonlinear reaction-diffusion equation with mixed Neumann boundary conditions. First, we introduce a family of perturbation problems corresponding to (RFFM), and present the variational formulation of perturbation problem which is a perturbation elliptic hemivariational inequality driven by a perturbation nonlinear variational equation. Then, the existence of solutions and the uniform bound of the solution set to the perturbation problem are obtained. Finally, it is established that, as the perturbation parameter tends to zero, the solution set of the perturbation problems converge to the solution set of (RFFM) in the sense of the Kuratowski upper limit. This shows that (RFFM) is stable with respect to the perturbation data. [ABSTRACT FROM AUTHOR]

Nonlinear differential equations often arise in many real-world problems. The complexity of solving nonlinear systems arises from the strong interdependence between the variables of the system and the boundary conditions. An immobilized glucose isomerase-based mathematical model for the enzymatic isomerization process that converts glucose to fructose is presented. The model's kinetic mechanism is stated using the nonlinear reaction-diffusion equation for MichalisMenten kinetics. The general approximate analytical formulas for the glucose molar concentration and flux inside packed-bed reactors are determined by solving the nonlinear equation using Akbari-Ganji's method. The effects of the kinetic parameters and pore-level Thiele modulus on concentration and flux were discussed. Estimating the kinetic parameters from current density is suggested. It has been shown that this method reduces processing time without affecting the quality of the solution. This method works well for many different types of nonlinear systems, making it useful in engineering and other fields of science. [ABSTRACT FROM AUTHOR]

We revisit the problem of optimising (either maximising or minimising) the total biomass or wider classes of criteria in logistic-diffusive models. Since [24] , a lot of effort has been devoted to the qualitative understanding of the following question: how should one spread resources in order to maximise or minimise the total biomass? This question was studied in detail, mostly in situations where the carrying capacity of the environment and the growth rate of the population are equal. Following recent contributions [7,11,40] , we propose a mathematical treatment of this question in contexts where the growth rate and the carrying capacity are de-correlated. We investigate the optimisation of generic criteria with respect to the carrying capacity. We settle the problem of existence by providing the proper relaxation of the optimisation problems, using homogenisation-type arguments. At a qualitative level, we obtain theoretical results backing up the observations and other results of [7,11,40] regarding the optimality of homogeneous environments. We similarly investigate in details the question of minimising the total biomass. We also provide several stationarity results (in the context of large diffusivities) for the minimisation of the biomass and related criteria. [ABSTRACT FROM AUTHOR]

The investigation presented in this study delves into the analysis of Lie symmetries for the bistable Allen-Cahn (BAC) equation with a quartic potential, specifically applied to the biofilm model. By employing the Lie symmetry method, we have acquired the Lie infinitesimal generators for the considered model. Using a transformation method, the nonlinear partial differential equations (NPDEs) are converted into various nonlinear ordinary differential equations (NLODEs), providing the numerous closed-form solitary wave solutions. The obtained solutions manifest in various forms including dark, bright, kink, anti-kink, and periodic types using diverse strategies. To enhance the physical interpretation, the study presents 3D, 2D, and contour plots of the acquired solutions. Every graph's wave-like structure contains information about the structural behaviour of the bacteria that build biofilms on surfaces where rectangles have different densities. This analysis enhances comprehension of the complex dynamics present in areas like fluid dynamics, fiber optics, biology, ocean physics, coastal engineering, and nonlinear complex physical systems. [ABSTRACT FROM AUTHOR]

In this article, we formulate and solve the representation problem for diffusion equations: giving a discretization of the Laplace transform of a diffusion equation under a space discretization over a space scale determined by an increment h > 0 , can we construct a continuous in h family of Cauer ladder networks whose constitutive equations match for all h > 0 the discretization. It is proved that for a finite differences discretization over a uniform geometric space scale, the representation problem over fractal Cauer networks is possible if and only if the coefficients of the diffusion are exponential functions in the space variable. Such diffusion equations admit a (Laplace) transfer function with a fractional behavior whose exponent is explicit. This allows us to justify previous works made by Sabatier and co-workers in [15-16] and Oustaloup and co-workers [14]. [ABSTRACT FROM AUTHOR]

Studying the calcium dynamics within a fibroblast cell individually has provided only a restricted understanding of its functions. However, research efforts focusing on systems biology approaches for such investigations have been largely neglected by researchers until now. Fibroblast cells rely on signaling from calcium (C a 2 +) and nitric oxide (NO) to maintain their physiological functions and structural stability. Various studies have demonstrated the correlation between NO and the control of C a 2 + dynamics in cells. However, there is currently no existing model to assess the disruptions caused by various factors in regulatory dynamics, potentially resulting in diverse fibrotic disorders. A mathematical model has been developed to investigate the effects of changes in parameters such as buffer, receptor, sarcoplasmic endoplasmic reticulum C a 2 + -ATPase (SERCA) pump, and source influx on the regulation and dysregulation of spatiotemporal calcium and NO dynamics in fibroblast cells. This model is based on a system of reaction-diffusion equations, and numerical simulations are conducted using the finite element method. Disturbances in key processes related to calcium and nitric oxide, including source influx, buffer mechanism, SERCA pump, and inositol trisphosphate (I P 3) receptor, may contribute to deregulation in the calcium and NO dynamics within fibroblasts. The findings also provide new insights into the extent and severity of disorders resulting from alterations in various parameters, potentially leading to deregulation and the development of fibrotic disease. [ABSTRACT FROM AUTHOR]

In order to study the effects of reverse transcriptase inhibitors, protease inhibitors and flavonoids on two target cells infected by HIV in a heterogeneous environment, an HIV mathematical model at the cellular level was established. Research shows that infected cells can be categorized into immature infected cells, latent infected cells, and mature infected cells based on the infection process. The basic reproduction number R 0 was established, and it is proved that R 0 serves as a threshold parameter: When R 0 < 1 , the disease-free steady state is globally asymptotically stable, and the disease is extinct; when R 0 > 1 , the solution of the system is uniformly persistent, and the virus exists. Considering the huge advantages of drug intervention in controlling HIV infection, the optimal control problem was proposed under the condition that the constant diffusion coefficient is positive, so as to minimize the total number of HIV-infected cells and the cost of drug treatment. To illustrate our theoretical results, we performed numerical simulations in which the model parameters were obtained with reference to some medical studies. The results showed that: (1) as R 0 increases, the risk of HIV transmission increases; (2) pharmacological interventions are important in early treatment of HIV spread and control of viral load in the body; (3) the treatment process must consider the heterogeneity of medication, otherwise it will not be conducive to suppressing the spread of the virus and will increase costs. In order to study the effects of reverse transcriptase inhibitors, protease inhibitors and flavonoids on two target cells infected by HIV in a heterogeneous environment, an HIV mathematical model at the cellular level was established. Research shows that infected cells can be categorized into immature infected cells, latent infected cells, and mature infected cells based on the infection process. The basic reproduction number was established, and it is proved that serves as a threshold parameter: When , the disease-free steady state is globally asymptotically stable, and the disease is extinct; when , the solution of the system is uniformly persistent, and the virus exists. Considering the huge advantages of drug intervention in controlling HIV infection, the optimal control problem was proposed under the condition that the constant diffusion coefficient is positive, so as to minimize the total number of HIV-infected cells and the cost of drug treatment. To illustrate our theoretical results, we performed numerical simulations in which the model parameters were obtained with reference to some medical studies. The results showed that: (1) as increases, the risk of HIV transmission increases; (2) pharmacological interventions are important in early treatment of HIV spread and control of viral load in the body; (3) the treatment process must consider the heterogeneity of medication, otherwise it will not be conducive to suppressing the spread of the virus and will increase costs. [ABSTRACT FROM AUTHOR]

The main objective of this manuscript is to define the concepts of F-(⋏,h)-contraction and (α , η) -Reich type interpolative contraction in the framework of orthogonal F -metric space and prove some fixed point results. Our primary result serves as a cornerstone, from which established findings in the literature emerge as natural consequences. To enhance the clarity of our novel contributions, we furnish a significant example that not only strengthens the innovative findings but also facilitates a deeper understanding of the established theory. The concluding section of our work is dedicated to the application of these results in establishing the existence and uniqueness of a solution for a fractional differential equation of anomalous diffusion. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an initial-boundary value problem of a reaction–diffusion equation in a bounded domain with a Robin boundary condition and introduce some particular parameters to consider the non-zero flux on the boundary. This problem arises in the study of mosquito populations under the intervention of the population replacement method, where the boundary condition takes into account the inflow and outflow of individuals through the boundary. Using phase plane analysis, the present paper studies the existence and properties of non-constant steady-state solutions depending on several parameters. Then, we prove some sufficient conditions for their stability. We show that the long-time efficiency of this control method depends strongly on the size of the treated zone and the migration rate. To illustrate these theoretical results, we provide some numerical simulations in the framework of mosquito population control. [ABSTRACT FROM AUTHOR]

Animal skin patterns are increasingly explained using the Turing pattern model proposed by Alan Turing. The Turing model, a self-organizing model, can produce spotted or striped patterns. However, several animal patterns exist that do not correspond to these patterns. For example, the body patterns of the ornamental carp Nishiki goi produced in Japan vary randomly among individuals. Therefore, predicting the pattern of offspring is difficult based on the parent fish. Such a randomly formed pattern could be explained using a majority voting model. This model is a type of cellular automaton model that counts the surrounding states and transitions to high-number states. Nevertheless, the utility of these two models in explaining fish patterns remains unclear. Interestingly, the patterns generated by these two models can be detected among very closely related species. It is difficult to think that completely different epidermal formation mechanisms are used among species of the same family. Therefore, there may be a basic model that can produce both patterns. Herein, the Turing pattern and majority voting method are represented using cellular automata, and the possibility of integrating these two methods is examined. This integrated model is equivalent to both models when the parameters are adjusted. Although this integrated model is extremely simple, it can produce more varied patterns than either one of the individual models. However, further research is warranted to determine whether this model is consistent with the mechanisms involved in the formation of animal fish patterns from a biological perspective. [ABSTRACT FROM AUTHOR]

Despite the rapid advance of unsupervised object intrusion detection of metro rail, existing methods require to train separate models for different classes. In this work, a unified detection model based on the fast diffusion model is proposed to address the problem. The diffusion model empowers the unified model to detect intrusions of different classes by training only on normal samples. The diffusion model restores the anomaly part in the image to normal and compares the reconstructed image and input to locate intrusion objects. It is found that by properly designing the sampling steps schedule, the diffusion model can directly restore the abnormal part of the input to normal without the diffusion process compared to previous work. To accelerate the sampling process of the diffusion model, we first analytically divide the ordinary differential equations of the diffusion model into the linear part and non-linear part. Then we estimate the non-linear part and precisely calculate the linear part. When it comes to comparing the restored image and input, the encoder part of the diffusion model is applied to extract feature maps of both images and calculate the hierarchic distance between them to output an anomaly heatmap. The results of experiments show that the proposed algorithm can achieve 99.05% and 97.62% area under the curve of the receiver operating characteristic in image level and pixel level while the number of function evaluations is only 4. The research proposes a unified intrusion detection algorithm for metro rail that is trained on normal samples, which can effectively solve the problem that the single model cannot apply to multiple views. [ABSTRACT FROM AUTHOR]

Schistosomiasis, a neglected tropical disease caused by parasitic trematodes of the genus Schistosoma, affects millions of people in tropical and subtropical regions lacking access to clean water and proper hygiene. With its impact on health and well-being, the World Health Organization aspires to eliminate schistosomiasis by 2030. This work addresses the challenge of effective control in endemic areas by integrating diffusion in each sub-population using reaction-diffusion equations. The proposed model includes treated individuals who have undergone massive drug administration and a time-dependent function that models the change in human behavior. We present a Partial Differential Equation (PDE) model of schistosomiasis spread that incorporates population movement and human behavior change. Mathematical analysis explores the system’s dynamics according to the infection threshold R0, shedding light on the disease’s behavior. Sensitivity analysis is used to identify the key parameters affecting disease spread. Numerical simulations under different scenarios elucidate the impact of human behavior on disease dynamics. This research contributes to a deeper understanding of schistosomiasis transmission and provides insights into control strategies. [ABSTRACT FROM AUTHOR]

We study positive stationary solutions for the following Neumann problem in one-dimension space arising from population genetics: { u t = D u x x + g (x) u 2 (1 − u) in (0 , 1) × (0 , ∞) , u x (0 , t) = u x (1 , t) = 0 in (0 , ∞) , where g changes sign once in (0 , 1) and D is a positive parameter. This equation has a stationary positive solution u , where u − 1 has n zeros in (0 , 1). We denote this solution by an M (n) -solution (n = 1 , 2 , ⋯). We show that the M (n) -solution branch bifurcates from the trivial solution u = 1 and the M (n) -solution branch does not meet other bifurcating points and is extended globally to D → 0. When D is sufficiently small, the M (n) -solution has a very characteristic shape. Especially when n = 1 , we will show all possible shapes of M (1) -solutions. [ABSTRACT FROM AUTHOR]

Microphysiological models (MPS) are increasingly getting recognized as in vitro preclinical systems of pathophysiology and drug discovery. However, there is also a growing need to adapt and advance MPS to include the physiological contributions of the capillary vascular dynamics, because they undergo angiogenesis or vasculogenesis to deliver soluble oxygen and nutrients to its organs. Currently, the process of formation of microvessels in MPS is measured arbitrarily, and vascularized MPS do not include oxygen measurements in their analysis. Sensing and measuring tissue oxygen delivery is extremely difficult because it requires access to opaque and deep tissue, and/or requires extensive integration of biosensors that makes such systems impractical to use in the real world. Here, a finite element method-based oxygen transport program, called AngioMT, is built in MATLAB. AngioMT processes the routinely acquired 2D confocal images of microvascular networks in vitro and solves physical equations of diffusion-reaction dominated oxygen transport phenomena. This user-friendly image-to-physics transition in AngioMT is an enabling tool of MPS analysis because unlike the averaged morphological measures of vessels, it provides information of the spatial transport of oxygen both within the microvessels and the surrounding tissue regions. Further, it solves the more complex higher order reaction mechanisms which also improve the physiological relevance of this tool when compared directly against in vivo measurements. Finally, the program is applied in a multicellular vascularized MPS by including the ability to define additional organ/tissue subtypes in complex co-cultured systems. Therefore, AngioMT serves as an analytical tool to enhance the predictive power and performance of MPS that incorporate microcirculation. [ABSTRACT FROM AUTHOR]

The paper's primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles within this system. We employ qualitative and quantitative analysis methods rooted in bifurcation theory, dynamical theory, and numerical simulation. We also delve into studying the boundedness of solutions and investigating the stability and existence of equilibrium points within the system. Leveraging Sotomayor's theorem, we establish the presence of both the saddle-node and transcritical bifurcations. The analysis of the Hopf bifurcation is carried out using the normal form theorem. The model under consideration is extended to the fractional reaction–diffusion model which captures non-local and long-range effects more accurately than integer-order derivatives. This makes fractional reaction–diffusion systems suitable for modeling phenomena with anomalous diffusion or memory effects, improving the fidelity of simulations in turn. An adaptable numerical technique for solving this class of differential equations is also suggested. Through simulation results, we observe that one of the Lyapunov exponents has a negative value, indicating the potential for the emergence of a stable-limit cycle via bifurcation as well as chaotic and complex spatiotemporal distributions. We supplement our analytical investigations with numerical simulations to provide a comprehensive understanding of the system's behavior. It was discovered that both the prey and predator populations will continue to coexist and be permanent, regardless of the choice of fractional parameter. [ABSTRACT FROM AUTHOR]

Two (consecutive) reductions of the complete Gatenby–Gawlinski model for cancer invasion are proposed in order to investigate the mathematical framework, mainly from a computational perspective. After a brief overview of the full model, we proceed by examining the case of a two-equations-based and one-equation-based reduction, both obtained by means of a quasi-steady-state assumption. We focus on invasion fronts, exploiting a numerical strategy based on a finite volume approximation, and perform corresponding computational simulations to study the sharpness/smoothness of the traveling waves. Then, we employ a space-averaged wave speed estimate—referred to as the LeVeque–Yee formula—to quantitatively approach the propagation phenomenon. Concerning the one-equation-based model, we propose a scalar degenerate reaction-diffusion equation, which proves to be effective in order to qualitatively recover the typical trends arising from the Gatenby–Gawlinski model. Finally, we carry out some numerical tests in a specific case where the analytical solution is available. [ABSTRACT FROM AUTHOR]

Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points. [ABSTRACT FROM AUTHOR]

This paper uses the attached flow method for solving nonlinear second-order differential equations of the reaction–diffusion type. The key steps of the method consist of the following: (i) reducing the differentiability order by defining the first derivative of the variable as a new variable called the flow and (ii) a forced decomposition of the derivative-free term so that the flow appears explicitly in it. The resulting reduced equation is solved using specific balancing rules. Only step (i) would lead to an Abel-type equation with complicated integral solutions. Completed with (ii) and with the graduation procedure, the attached flow method used in the paper, without requiring such a great effort, allows for the obtaining of accurate analytical solutions. The method is applied here to a subclass of reaction–diffusion equations, the generalized Dodd–Bulough–Mikhailov equation, which includes a translation of the variable and nonlinearities up to order five. The equation is solved for each order of nonlinearity, and the solutions are discussed following the values of the parameters involved in the equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider 3D steady convection-diffusion equations in cylindrical domains. Instead of applying the finite difference methods (FDM) or the finite element methods (FEM), we propose a difference finite element method (DFEM) that can maximize good applicability and efficiency of both FDM and FEM. The essence of this method lies in employing the centered difference discretization in the z-direction and the finite element discretization based on the P1 conforming elements in the (x; y) plane. This allows us to solve partial differential equations on complex cylindrical domains at lower computational costs compared to applying the 3D finite element method. We derive stability estimates for the difference finite element solution and establish the explicit dependence of H1 error bounds on the diffusivity, convection field modulus, and mesh size. Finally, we provide numerical examples to verify the theoretical predictions and showcase the accuracy of the considered method. [ABSTRACT FROM AUTHOR]

This article focuses on the analysis of dynamics emerging in a network of Hodgkin–Huxley reaction–diffusion equations. The network has three levels. The three neurons in level 1 receive a periodic input but do not receive inputs from other neurons. The three neurons in level 2 receive inputs from one specific neuron in level 1 and all neurons in level 3. The neurons in level 3 (all other neurons) receive inputs from all other neurons in levels 2 and 3. Furthermore, the right-hand side of pre-synaptic neurons is connected to the left-hand side of the post-synaptic neurons. The synchronization phenomenon is observed for neurons in level 3, even though the system is initiated with different functions. As far as we know, it is the first time that evidence of the synchronization phenomenon is provided for spatially extended Hodgkin–Huxley equations, which are periodically forced at three different sites and embedded in such a hierarchical network with space-dependent coupling interactions. [ABSTRACT FROM AUTHOR]

In this paper, we concentrate on the propagation dynamics of stochastic reaction–diffusion equations, including the existence of travelling wave solution and asymptotic wave speed. Based on the stochastic Feynman–Kac formula and comparison principle, the boundedness of the solution of stochastic reaction–diffusion equations can be obtained so that we can construct a sup-solution and a sub-solution to estimate the upper bound and the lower bound of wave speed. [ABSTRACT FROM AUTHOR]

In this study, we introduce a novel framework for exploring the dynamics of tumor growth and an evolution model for two-stage carcinogenic mutations in two-dimensions based on a system of reaction-diffusion equations. It is shown theoretically that the system is globally stable in the absence of both delay and diffusion. The inclusion of diffusion does not destabilize the system, while including delay does capture the key elements of how normal cells convert into cancer cells. To further validate these results, several numerical experiments are performed for different parameter values involved in the model equation. These parameter values are chosen in the sense that they have some biological meanings using the steady states of the equilibrium points. For the purpose of simulation, a stable Euler scheme is used for temporal discretization, while a Fourier spectral method is used for space variables, which is a natural choice due to the periodic boundary conditions in the model equation. The numerical simulation results further confirm our theoretical justification. [ABSTRACT FROM AUTHOR]

Synchronization of stochastic reaction-diffusion Hopfield neural networks with s-delays via sliding mode control is investigated in this article. To begin with, we choose suitable functional space for state variables, then the system is transformed into a functional differential equation in an infinite-dimensional Hilbert space by using appropriate functional analysis technique. Based on above preliminary preparation, sliding mode control (SMC) is constructed to drive the error trajectory into the designed switching surface. Specifically, the switching surface is constructed as linear combination of state variables, which is related to control gains. Then novel SMC law is designed which involving delay, reaction diffusion term, and reaching law. Furthermore, the criterion of mean-square exponential synchronization for stochastic delayed reaction-diffusion Hopfield neural networks with s-delays is given in the form of matrix form. This criterion is less restrictive and easy to check in computer. Meanwhile, a different novel Lyapunov-Krasovskii functional (LKF) mixed with Itô's formula, Young inequality, Hanalay inequality is employed in this proof procedure. At last, a numerical example is presented to validate the availability of theoretical result. The simulation is based on the finite difference method, and numerical result coincides with the theoretical result proposed. [ABSTRACT FROM AUTHOR]

Modulation instability of one-dimensional plane wave is demonstrated in nonlinear Kerr media with sine-oscillatory nonlocal response function and pure quartic diffraction. The growth rate of modulation instability, which depends on the degree of nonlocality, coefficient of quartic diffraction, type of the nonlinearity and the power of plane wave, is analytically obtained with linear-stability analysis. Different from other nonlocal response functions, the maximum of the growth rate in media with sine-oscillatory nonlocal response function occurs always at a particular wave number. Theoretical results of modulation instability are confirmed numerically with split-step Fourier transform. Modulation instability can be controlled flexibly by adjusting the degree of nonlocality and quartic diffraction. [ABSTRACT FROM AUTHOR]

The Schwarz domain decomposition (SDD) method is known for its high efficacy in solving large-scale systems of partial differential equations, primarily due to its parallelizability. However, the method's reliance on iteration introduces substantial computational expenses. In this study, we propose a reduced-order Schwarz domain decomposition (ROSDD) method tailored specifically for the convection-diffusion equation. Utilizing the proper orthogonal decomposition (POD) technique, the ROSDD method significantly reduces computational costs by considering only a small number of unknowns. Since the equation typically exhibits convection-dominated behavior, we employ the characteristic finite element discretization scheme. Consequently, we derive optimal a priori error estimates to assess the accuracy of our approach. Finally, we conduct some numerical experiments to validate the superiority of the ROSDD method. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new results on the existence of positive solutions for a singular tempered sub-diffusion fractional equation involving a changing-sign perturbation and a lower-order sub-diffusion term of the unknown function. By employing multiple transformations, we transform the changing-sign singular perturbation problem to a positive problem, then establish some sufficient conditions for the existence of positive solutions of the problem. The asymptotic properties of solutions are also derived. In deriving the results, we only require that the singular perturbation term satisfies the Carathéodory condition, which means that the disturbance influence is significant and may even achieve negative infinity near some time singular points. [ABSTRACT FROM AUTHOR]

In this work, the investigation dedicated to the study of the control of the dynamics of the propagation of cancer using the math model suggested in article [18] is provided. Here we show if the class of control functions chosen at the beginning of the characteristics of the diseases then there exists such control operator by which influence can be cured or, at least, stop the growth of cancer [ABSTRACT FROM AUTHOR]