Reaction-diffusion Type Research Articles

Application of a Kinetic Model for Studying the Spatial Spread of COVID-19

A one-dimensional model based on a kinetic-type equation is proposed for studying the dynamic distribution density of virus carriers in time and space while taking into account their distribution from a dedicated center. This model is new and fundamentally different from known models of the diffusion–reaction type. The analytical solution is built; for obtaining a series of calculations, numerical methods are also used. The model and real data from Italy, Russia, and Chile are compared. In addition to the rate of infection, the “rate of recovery” is considered. When the wave of recovery passes through a territory with the greater part of the commonwealth, a conclusion is made about the onset of global recovery, which corresponds to real data. The predictions are proved to have been accurate also for the second wave of the pandemic in Russia. The model is expected to be able also to describe adequately subsequent epidemics instead of only the development of COVID-19.

Numerical method for a boundary value problem for a linear singularly perturbed parabolic delay differential equation of reaction-diffusion type with discontinuous source term

A singularly perturbed boundary value problem for a linear parabolic second order delay differential equation of reaction-diffusion type with discontinuous source term is considered in the rectangle domain Ω = {(x, t) : 0 < x < 2,0 < t ⩽ T}. Discontinuity occurs in the source term at (d, t) < ∈ Ω. As the highest order space derivative is multiplied by a singular perturbation parameter, the components of the solution exhibit boundary layers at (x,t) = (0,t) and (x,t) = (2,t) and interior layers at (x, t) = (1, t) and/or at (x,t) = (d,t) and (x,t) = (1 + d, t) with respect to the position of (d, t) in Ω. A numerical method which uses classical finite difference scheme on a Shishkin piecewise uniform mesh is suggested to approximate the solution. The method is proved to be first order convergent uniformly for all the values of singular perturbation parameters. Numerical illustrations are presented so that the theoretical results are supported.

Wave Propagation for Reaction-Diffusion Equations on Infinite Random Trees

The asymptotic wave speed for FKPP type reaction-diffusion equations on a class of infinite random metric trees are considered. We show that a travelling wavefront emerges, provided that the reaction rate is large enough. The wavefront travels at a speed that can be quantified via a variational formula involving the random branching degrees $\vec{d}$ and the random branch lengths $\vec{\ell}$ of the tree. This speed is slower than that of the same equation on the real line $\mathbb{R}$, and we estimate this slow down in terms of $\vec{d}$ and $\vec{\ell}$. Our key idea is to project the Brownian motion on the tree onto a one-dimensional axis along the direction of the wave propagation. The projected process is a multi-skewed Brownian motion, introduced by Ramirez [Multi-skewed Brownian motion and diffusion in layered media, Proc. Am. Math. Soc., Vol. 139, No. 10, pp.3739-3752, 2011], with skewness and interface sets that encode the metric structure $(\vec{d}, \vec{\ell})$ of the tree. Combined with analytic arguments based on the Feynman-Kac formula, this idea connects our analysis of the wavefront propagation to the large deviations principle (LDP) of the multi-skewed Brownian motion with random skewness and random interface set. Our LDP analysis involves delicate estimates for an infinite product of $2\times 2$ random matrices parametrized by $\vec{d}$ and $\vec{\ell}$ and for hitting times of a random walk in random environment. By exhausting all possible shapes of the LDP rate function (action functional), the analytic arguments that bridge the LDP and the wave propagation overcome the random drift effect due to multi-skewness.

Spiral wave chimeras in reaction-diffusion systems: Phenomenon, mechanism and transitions

Spiral wave chimeras (SWCs), which combine the features of spiral waves and chimera states, are a new type of dynamical patterns emerged in spatiotemporal systems due to the spontaneous symmetry breaking of the system dynamics. In generating SWC, the conventional wisdom is that the dynamical elements should be coupled in a nonlocal fashion. For this reason, it is commonly believed that SWC is excluded from the general reaction-diffusion (RD) systems possessing only local couplings. Here, by an experimentally feasible model of a three-component FitzHugh–Nagumo-type RD system, we demonstrate that, even though the system elements are locally coupled, stable SWCs can still be observed in a wide region in the parameter space. The properties of SWCs are explored, and the underlying mechanisms are analyzed from the point view of coupled oscillators. Transitions from SWC to incoherent states are also investigated, and it is found that SWCs are typically destabilized in two scenarios, namely core breakup and core expansion. The former is characterized by a continuous breakup of the single asynchronous core into a number of small asynchronous cores, whereas the latter is featured by the continuous expansion of the single asynchronous core to the whole space. Remarkably, in the scenario of core expansion, the system may develop into an intriguing state in which regular spiral waves are embedded in a completely disordered background. This state, which is named shadowed spirals, manifests from a new perspective the coexistence of incoherent and coherent states in spatiotemporal systems, generalizing therefore the traditional concept of chimera states. Our studies provide an affirmative answer to the observation of SWCs in typical RD systems, and pave a way to the realization of SWCs in experiments.

Computational method for singularly perturbed delay differential equations of the reaction-diffusion type with negative shift

A numerical method for solving singularly perturbed delay differential equations with a layer or oscillatory behavior for which a small shift is introduced in the reaction term is presented. The stability and convergence of the method have been investigated. To demonstrate the efficiency of the method, two model examples have been presented. Numerical results obtained by the present scheme described that the finding of the present method is accurate than the findings of the previous studies.

Uniformly convergent finite difference method for reaction-diffusion type third order singularly perturbed delay differential equation

A class of third order reaction-diffusion type singularly perturbed ordinary delay differential equations is considered in this article. A fitted finite difference method on Shishkin mesh is suggested to solve the problem. Moreover, we present a class of nonlinear problems. An error estimation is obtained based on the maximum norm and it is of almost first order convergence. Numerical results are given to support theoretical claims.

ON APPROXIMATE OPTIMAL CONTROL FOR THE REACTION-DIFFUSION PROCESS IN MICROINHOMOGENEOUS MEDIUM

The problem of constructing an approximate optimal control for controlled processes of chemical kinetics in microinhomogeneous medium is considered. Such processes are described by semilinear parabolic equations of the reaction-diffusion type with coefficients of the form . The preference of an approximate control as the optimal control in the problem with averaged coefficients is justified. An example of the construction of such a control is discussed and its efficiency is demonstrated.

Robust stability of the global attractor of the reaction-diffusion system

In this paper we consider the problem of robust stability for a nonlinear system of equations in partial derivatives of the reaction-diffusion type. An undisturbed system is considered to have a global attractor. The main task is to estimate the deviation of the trajectory of the perturbed system from the global attractor of the perturbed system depending on the magnitude of the perturbations. Such an estimate can be obtained in the framework of the theory of input-to-state stability (ISS). The paper does not impose any conditions on the derivative of the nonlinear interaction function, so the unity of the solution of the initial problem is not ensured. The paper proposes a new approach to obtaining estimates of robust stability of the attractor in the case of a multivalued evolutionary decoupling operator. In particular, it is proved that the multivalued decoupling operator generated by weak solutions of a nonlinear reaction-diffusion system has the property of asymptotic gain (AG) with respect to the attractor of the undisturbed system.

An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations

A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction–diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. This is a new, fractional version of the Alternating Direction Implicit (ADI) method, where the source term is balanced, in that its effect is split in the three space directions, and it may be relevant, especially in the case of anisotropy. The method is unconditionally stable, second-order in space, and third-order in time. A strategy is devised in order to improve its speed of convergence by means of an extrapolation method that is coupled to the PageRank algorithm. Some numerical examples are given.

Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations

Abstract One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction–diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen–Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes.

Physical Modeling of a Sliding Clamp Mechanism for the Spreading of ParB at Short Genomic Distance from Bacterial Centromere Sites.

SummaryBacterial ParB partitioning proteins involved in chromosomes and low-copy-number plasmid segregation are cytosine triphosphate (CTP)-dependent molecular switches. CTP-binding converts ParB dimers to DNA clamps, allowing unidimensional diffusion along the DNA. This sliding property has been proposed to explain the ParB spreading over large distances from parS centromere sites where ParB is specifically loaded. We modeled such a “clamping and sliding” mechanism as a typical reaction-diffusion system, compared it to the F plasmid ParB DNA binding pattern, and found that it can account neither for the long range of ParB binding to DNA nor for the rapid assembly kinetics observed in vivo after parS duplication. Also, it predicts a strong effect on the F plasmid ParB binding pattern from the presence of a roadblock that is not observed in ChIP-sequencing (ChIP-seq). We conclude that although “clamping and sliding” can occur at short distances from parS, another mechanism must apply for ParB recruitment at larger genomic distances.

Protein-protein interactions in neurodegenerative diseases: A conspiracy theory.

Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are associated with the prion-like propagation and aggregation of toxic proteins. A long standing hypothesis that amyloid-beta drives Alzheimer’s disease has proven the subject of contemporary controversy; leading to new research in both the role of tau protein and its interaction with amyloid-beta. Conversely, recent work in mathematical modeling has demonstrated the relevance of nonlinear reaction-diffusion type equations to capture essential features of the disease. Such approaches have been further simplified, to network-based models, and offer researchers a powerful set of computationally tractable tools with which to investigate neurodegenerative disease dynamics. Here, we propose a novel, coupled network-based model for a two-protein system that includes an enzymatic interaction term alongside a simple model of aggregate transneuronal damage. We apply this theoretical model to test the possible interactions between tau proteins and amyloid-beta and study the resulting coupled behavior between toxic protein clearance and proteopathic phenomenology. Our analysis reveals ways in which amyloid-beta and tau proteins may conspire with each other to enhance the nucleation and propagation of different diseases, thus shedding new light on the importance of protein clearance and protein interaction mechanisms in prion-like models of neurodegenerative disease.

Spreading speed for a KPP type reaction-diffusion system with heat losses and fast decaying initial data

In this work we consider a two-components reaction-diffusion system of KPP type with heat losses posed in a straight cylinder and equipped with fast decaying initial data. We derive a criteria condition – expressed in term of the sign of a suitable elliptic eigenvalue – for extinction and for propagation of the solutions. In the case of propagation we derive a spreading speed property and we obtain an asymptotic expansion in the large times for the location of the front, that is strongly related with the minimal wave speed of the travelling waves. We also obtain decay estimates for the solutions ahead the front.

Radial symmetric stationary solutions for a MEMS type reaction–diffusion equation with spatially dependent nonlinearity

We consider the radial symmetric stationary solutions of $$u_{t}=\varDelta u-|x|^{q}u^{-p}$$ . We first give a result on the existence of the negative value functions that satisfy the radial symmetric stationary problem on a finite interval for $$p \in 2{\mathbb{N}}$$ , $$q\in{\mathbb{R}}$$ . Moreover, we give the asymptotic behavior of u(r) and $$u'(r)$$ at both ends of the finite interval. Second, we obtain the existence of the positive radial symmetric stationary solutions with the singularity at $$r=0$$ for $$p\in{\mathbb{N}}$$ and $$q\ge -2$$ . In fact, the behavior of solutions for $$q>-2$$ and $$q=-2$$ are different. In each case of $$q=-2$$ and $$q>-2$$ , we derive the asymptotic behavior for $$r \rightarrow 0$$ and $$r \rightarrow \infty $$ . These facts are studied by applying the Poincare compactification and basic theory of dynamical systems.

A simple reaction-diffusion model for initial stages of biofouling in reverse osmosis membranes

Biofouling is a critical issue in membrane water and wastewater treatment as it greatly compromises the efficiency of the treatment processes and consequently increases operational and maintenance costs. It is difficult to control this operational challenge, so the development of effective biofouling monitoring and control methods and strategies is a critical issue for membrane technology and applications. In this work, we develop a simulation approach for evaluating the operational time of reverse osmosis (RO) membranes based on a reaction-diffusion (RD) type of model. This approach would help to understand different factors involved in the formation of biofilms including microbial population dynamics (replication and death rates of microbial cells) and nutrient consumption. The model is focused on the initial stages of the membrane biofouling that is initiated by attachment of microbial species to the membrane leading to pore blocking followed by the formation of thick cake layer. We applied this approach to study the RO membrane biofouling by Picochlorum algae, the most common biofouling agent in the seawater of the Arabian Gulf, at known contents of total organic carbon and essential nutrients. We found that the biofilm growth dynamics on an RO membrane is mainly defined by the ratio of the replication and death rates of microbial cells. The proposed approach should be useful for fast evaluation of the RO membrane performance in different environmental conditions without using significant computational resources. This methodology allows generalization for multi-microbial and multi-nutrient systems. The establishment of effective fouling control strategies should decrease operational and maintenance costs of RO membrane systems.

Temporal Forcing Induced Pattern Transitions Near the Turing–Hopf Bifurcation in a Plankton System

We explore the impact of time-periodic forcing on pattern transitions in a plankton system of reaction–diffusion type. Here, we mainly focus on the forced states near the Turing–Hopf bifurcation. A normal form analysis leads to the finding that weak forcing exhibits a destabilizing effect on the dynamics by exciting the transitions from a spatially homogeneous stationary state to a periodic oscillation in time. The results are obtained by studying the amplitude equations derived using weakly nonlinear analysis in the presence of forcing, which enables us to calculate the changes of states. Examples are given to confirm the theoretical results.

Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of effectors (the activator and the inhibitor respectively), which participate in the reaction. Diffusion waves propagating over zero background with finite velocity form an essential class of solutions of these systems. The existence of such solutions is possible because the parabolic type of equations degenerates if unknown functions are equal to zero. We study the analytic solvability of a boundary value problem with the degeneration for the reaction-diffusion system. The diffusion wave front is known. We prove the theorem of existence of the analytic solution in the general case. We construct a solution in the form of power series and suggest recurrent formulas for coefficients. Since, generally speaking, the solution is not unique, we consider some cases not covered by the proved theorem and present the example similar to the classic example of S.V. Kovalevskaya.

Isogeometric analysis for singularly perturbed high-order, two-point boundary value problems of reaction–diffusion type

We consider two-point, reaction–diffusion type, singularly perturbed boundary value problems of order 2ν∈Z+, and the approximation of their solution using isogeometric analysis. In particular, we use a Galerkin formulation with B-splines as basis functions, defined using appropriately chosen knot vectors. We prove robust exponential convergence in the energy norm, independently of the singular perturbation parameter. Numerical examples are also presented, which illustrate (and extend) the theory.

Numerical simulation and applications of the convection–diffusion–reaction equation with the radial basis function in a finite-difference mode

This paper develops two local mesh-free methods for designing stencil weights and spatial discretization, respectively, for parabolic partial differential equations (PDEs) of convection–diffusion–reaction type. These are known as the radial-basis-function generated finite-difference method and the Hermite finite-difference method. The convergence and stability of these schemes are investigated numerically using some examples in two and three dimensions with regularly and irregularly shaped domains. Then we consider the numerical pricing of European and American options under the Heston stochastic volatility model. The European option leads to the solution of a two-dimensional parabolic PDE, and the price of the American option is given by a linear complementarity problem with a two-dimensional parabolic PDE of convection–diffusion–reaction type. Then we use the operator splitting method to perform time-stepping after space discretization. The resulting linear systems of equations are well conditioned and sparse, and by numerical experiments we show that our numerical technique is fast and stable with respect to the change in the shape parameter of the radial basis function. Finally, numerical results are provided to illustrate the quality of approximation and to show how well our approach converges with the results presented in the literature.

Multiscale modeling of vertebrate limb development.

We review the current state of mathematical modeling of cartilage pattern formation in vertebrate limbs. We place emphasis on several reaction-diffusion type models that have been proposed in the last few years. These models are grounded in more detailed knowledge of the relevant regulatory processes than previous ones but generally refer to different molecular aspects of these processes. Considering these models in light of comparative phylogenomics permits framing of hypotheses on the evolutionary order of appearance of the respective mechanisms and their roles in the fin-to-limb transition. This article is categorized under: Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Mechanistic Models Developmental Biology > Developmental Processes in Health and Disease Analytical and Computational Methods > Analytical Methods.