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This paper handles with a time-space SIR model in two regions which have a common interface. The model is defined as coupled reaction–diffusion system with a robin boundary in interface to predict the immigration between the two regions. As known, the immigration has an influence on the spread of disease, this is caused by immigrants who could carry the disease from their areas to other places. For this reason, this paper deals with the existence of solutions to SIR model with disease spread between different regions by the Fadeo–Galerkin method and validate this result by the numerical simulations.

This paper presents a brucellosis disease model with reaction–diffusion and time delay. The model takes into account both the direct and indirect transmission of infected animals and pathogens in the environment. By analyzing the associated characteristic equation, the local stability of the unique positive equilibrium point is established. The existence of Hopf bifurcations at the positive equilibrium point is also examined by considering the discrete time delay as a bifurcation parameter. Additionally, an optimal control analysis is conducted to minimize disease outbreaks and control costs. This includes reducing the exposure of susceptible animals to infected animals, removing infected animals from herds, and reducing emissions of brucella into the environment. By constructing Hamiltonian function and applying Pontryagin’s maximum principle, the necessary conditions for the existence of optimal control are given. Finally, the existence of bifurcation periodic solutions and the effectiveness of control strategies are illustrated through numerical simulations.

During the outbreak of major emerging infectious diseases, virus droplets can survive in the environment as aerosols for hours to days, and their impact on human infectious diseases is often overlooked. In addition, the speed of transmission of infectious diseases is often closely related to transportation. Therefore, studying the impact of environmental viruses and transportation on disease development is significant for effective infectious disease prevention and control. We proposed a degenerate reaction–diffusion infectious disease model ($SEAIR$) considering environmental virus interference and established a well-posedness and threshold system for this model. We have obtained the system solution approaches the disease-free equilibrium (${𝔼}_0$) when the basic reproduction number ${{ℛ}}_0\le 1$. The system has at least one positive steady state solution (PSS) when ${{ℛ}}_0>1$. This paper used a non-standard finite difference method discretization model while data fitting and parameter estimation were performed based on data provided by the Health Commission. Further sensitivity analysis was conducted on ${{ℛ}}_0$. At the same time, we also discussed the impact of various parameters in the early stages of the outbreak of major emerging infectious diseases on the development of the disease. Research found that even if the contact rate between people is controlled at a shallow level, the disease may persist. In the early stages of major emerging outbreak of infectious diseases, immediately reducing the use of transportation can effectively reduce the speed of disease spread.

We study the Gierer–Meinhardt model of reaction-diffusion on a site-disordered square lattice. Let p be the site occupation probability of the square lattice. For p greater than a critical value p_c, the steady state consists of stripe-like patterns with long-range connectivity. For p < p_c, the connectivity is lost. The value of p_c is found to be much greater than that of the site percolation threshold for the square lattice. In the vicinity of p_c, the cluster-related quantities exhibit power-law scaling behavior. The method of finite-size scaling is used to determine the values of the fractal dimension d_f, the ratio, γ/ν, of the average cluster size exponent γ and the correlation length exponent ν. The values appear to indicate that the disordered GM model belongs to the universality class of ordinary percolation.

We study the discrete Gierer–Meinhardt model of reaction–diffusion on three different types of networks: regular, random and scale-free. The model dynamics lead to the formation of stationary Turing patterns in the steady state in certain parameter regions. Some general features of the patterns are studied through numerical simulation. The results for the random and scale-free networks show a marked difference from those in the case of the regular network. The difference may be ascribed to the small world character of the first two types of networks.

We present a cellular-automaton model of a reaction-diffusion excitable system with concentration dependent inhibition of the activator, and study the dynamics of mobile localizations (gliders) and their generators. We analyze a three-state totalistic cellular automaton on a two-dimensional lattice with hexagonal tiling, where each cell connects with 6 others. We show that a set of specific rules support spiral glider-guns (rotating activator-inhibitor spirals emitting mobile localizations) and stationary localizations which destroy or modify gliders, along with a rich diversity of emergent structures with computational properties. We describe how structures are created and annihilated by glider collisions, and begin to explore the necessary processes that generate this kind of complex dynamics.

We consider a Reactive-Diffusive Lattice Boltzmann (RDLB) model for environmental applications, as for instance the computation of the flux of a metal ion M towards a consuming surface in presence of several ligands. In order to simulate such a environmental system, in which chemical rate constants may vary in a wide range of values, we need a multi-scale approach using grid refinement methods. In this paper we consider three different grid refinement techniques based on the conservation of lattice velocities, relaxation parameters and time steps respectively. We discuss the advantages and disadvantages of each way of refining the grid, in terms of computer speed and computational accuracy. Moreover, an application to a natural mixture of ligands of environmental interest is shown.

We study the operator-split method Gillespie-MultiParticle method (GMP) for a diffusion-limited reaction in a finite medium by looking at the detailed distribution of reaction times for a broad range of time scales. The diffusion-limited reaction lies at the lower limits of the applicability of the GMP method. We propose an empirical distribution for the inter-reaction times (duration of the dissociated state) that accounts for the power-law and exponential regimes. The knowledge of this distribution can be used to change the reactive method of GMP and enable it to use coarser discretization, with consequent improvement in performance.

Multi-species reaction-diffusion systems, with more-than-two-site interaction on a one-dimensional lattice are considered. Necessary and sufficient constraints on the interaction rates are obtained, that guarantee the closure of the time evolution equation for , the expectation value of the product of certain linear combination of the number operators on n consecutive sites at time t.

We study the evolution of unstable front in reaction–diffusion systems analytically and derive the evolution equation for the fronts of a bistable system in the presence of advection. As an issue, we investigate the stability of planar fronts and explicitly show that the coupling to advection on the basis of the model of active fluids’ or, alternatively, on the model of an interfacial reaction can lead to an instability of the front, eventually leading to a fractalization of the chemical fronts.

This paper investigates the problem of exponential passive filter design for switched neural networks with time-delay and reaction-diffusion terms. With the aid of a suitable Lyapunov–Krasovskii functional and some inequalities, a linear matrix inequality-based design method is developed that not only makes the filtering error system exponentially stable but also forces it to be passive from external interference to output error. Then, the filter design is extended to the complex-valued case via separating the system into real-valued and complex-valued parts. Finally, a numerical example is utilized to illustrate the effectiveness of the filter design methods for the real-valued and complex-valued cases, respectively.

The work discusses the formation of Voronoi diagrams in spatially extended nonlinear systems taking experimental and theoretical results into account. Concerning experimental systems a number of chemical systems used previously as prototype chemical processors and a barrier gas-discharge system are investigated. Although the underlying microscopic processes are very different, both types of systems show self-organized Voronoi diagrams for suitable parameters. Indeed certain chemical systems exhibit Voronoi diagrams as an output state for two distinct sets of parameters one that corresponds to the interaction of stable forced trigger waves and the other that corresponds to the spontaneous initiation and interaction of waves due to point instabilities in the system. In the case of the chemical systems front initiation, propagation and interaction (annihilation) are the primary mechanisms for Voronoi diagram formation, in the case of the barrier gas-discharge system regions of vanishing electric field define the medial axes of the Voronoi diagram. On the basis of cellular automata models the general concept of the formation of Voronoi diagrams has been explained, and related mechanisms have been simulated. Another intuitive approach towards the understanding of self-organized Voronoi diagrams has been given on the basis of reaction–diffusion models explaining the formation of Voronoi diagrams as a result of the mutual interactions of trigger fronts. The variety of systems exhibiting Voronoi diagrams as stationary states indicates that Voronoi diagrams are a generic and natural pattern formation phenomenon.

Stochastic effects on convergence dynamics of reaction–diffusion recurrent neural networks (RNNs) with constant transmission delays are studied. Without assuming the boundedness, monotonicity and differentiability of the activation functions, nor symmetry of synaptic interconnection weights, by skillfully constructing suitable Lyapunov functionals and employing the method of variational parameters, M-matrix properties, inequality technique, stochastic analysis and non-negative semimartingale convergence theorem, delay independent and easily verifiable sufficient conditions to guarantee the almost sure exponential stability, mean value exponential stability and mean square exponential stability of an equilibrium solution associated with temporally uniform external inputs to the networks are obtained, respectively. The results are compared with the previous results derived in the literature for discrete delayed RNNs without diffusion or stochastic perturbation. Two examples are also given to demonstrate our results.

We consider a diffusion equation on a domain Ω with a cubic reaction at the boundary. It is known that there are no patterns when the domain Ω is a ball, but the existence of such patterns is still unkown in the more general case in which Ω is convex. The goal of this paper is to present numerical evidence of the existence of nonconstant stable equilibria when Ω is the unit square. These patterns are found by continuation of families of unstable equilibria that bifurcate from constant solutions.

Among reaction–diffusion systems showing Turing patterns, the diffusive Gray–Scott model [Pearson, 1993], stands out by showing self-replicating patterns (spots), which makes it the ideal simple model for developmental research. A first study of the influence of noise in the Gray–Scott model was performed by Lesmes et al. [2003] concluding that there exists an optimal noise intensity for which spot multiplication is maximal. Here we show in detail the transition from nonspotlike to spotlike pattern, with the identification of a wide range of noise intensities instead of an optimal value for which this transition occurs. Additional studies also reveal that noise produces a shift and a shrinkage of the regions of spatial patterns in the parameters space, without introducing qualitative changes to the diagram.

For a large class of reaction–diffusion bidirectional associative memory (RDBAM) neural networks with periodic coefficients and general delays, several new delay-dependent or delay-independent sufficient conditions ensuring the existence and global exponential stability of a unique periodic solution are given, by constructing suitable Lyapunov functionals and employing some analytic techniques such as Poincaré mapping. The presented conditions are easily verifiable and useful in the design and applications of RDBAM neural networks. Moreover, the employed analytic techniques do not require the symmetry of the bidirectional connection weight matrix, the boundedness, monotonicity and differentiability of activation functions of the network. In several ways, the results generalize and improve those established in the current literature.

Exponential stability of reaction–diffusion fuzzy recurrent neural networks (RDFRNNs) with time-varying delays are considered. By using the method of variational parameters, M-matrix properties and inequality technique, some delay-independent or delay-dependent sufficient conditions for guaranteeing the exponential stability of an equilibrium solution are obtained. One example is given to demonstrate the theoretical results.

Stochastic effects on the stability property of reaction–diffusion generalized Cohen–Grossberg neural networks (GDCGNNs) with time-varying delay are considered. By skillfully constructing suitable Lyapunov functionals and employing the method of variational parameters, inequality technique and stochastic analysis, the delay independent and easily verifiable sufficient conditions to guarantee the mean-value exponential stability of an equilibrium solution associated with temporally uniform external inputs to the networks are obtained. One example is given to illustrate the theoretical results.

An analytical solution characterizing initial conditions leading to action potential firing in smooth nerve fibers is determined, using the bistable equation. In the first place, we present a nontrivial stationary solution wave, then, using the perturbative method, we analyze the stability of this stationary wave. We show that it corresponds to a frontier between the initiation of the travelling waves and a decay to the resting state. Eventually, this analytical approach is extended to FitzHugh–Nagumo model.

In this paper, a class of impulsive fuzzy cellular neural networks (FCNNs) with mixed delays and diffusion is formulated and investigated. By establishing an intergro-differential inequality, applying M-matrix theory and inequality technique, several sufficient conditions are obtained to ensure the existence, uniqueness and global exponential stability of an equilibrium point for impulsive FCNNs with mixed delays and diffusion. In particular, the estimate of the exponential convergence rate is also provided, which depends on the system parameters and impulses. These results generalize and improve the earlier publications. Some examples are given to show the effectiveness of the obtained results. It is believed that these results are significant and useful for the design and applications of FCNNs.