Dynamic properties of an SARS-CoV-2 epidemic model via stochastic PINNs.

Stochastic modeling predicts various outcomes from stochasticity in the data, parameters and dynamical system. Stochastic models are deemed more appropriate than deterministic models accounting in terms of essential and practical information about a system. The objective of the current investigation is to address the issue above through the development of a novel deep neural network referred to as a stochastic epidemiology-informed neural network. This network learns knowledge about the parameters and dynamics of a stochastic epidemic vaccine model. Our analysis centers on examining the nonlinear incidence rate of the model from the perspective of the combined effects of vaccination and stochasticity. Based on empirical evidence, stochastic models offer a more comprehensive understanding than deterministic models, mainly when we use error metrics. The findings of our study indicate that a decrease in randomness and an increase in vaccination rates are associated with a better prediction of nonlinear incidence rates. Adopting a nonlinear incidence rate enables a more comprehensive representation of the complexities of transmitting diseases. The computational analysis of the proposed method, focusing on sensitivity analysis and overfitting analysis, shows that the proposed method is efficient. Our research aims to guide policymakers on the effects of stochasticity in epidemic models, thereby aiding the development of effective vaccination and mitigation policies. Several case studies have been conducted on nonlinear incidence rates using data from Tennessee, USA.

By using the semidiscrete method of differential equations, a new version of discrete analogue of stochastic fuzzy BAM neural networks was formulated, which gives a more accurate characterization for continuous‐time stochastic neural networks than that by the Euler scheme. Firstly, the existence of the 2p‐th mean almost periodic sequence solution of the discrete‐time stochastic fuzzy BAM neural networks is investigated with the help of Minkowski inequality, Hölder inequality, and Krasnoselskii’s fixed point theorem. Secondly, the 2p‐th moment global exponential stability of the discrete‐time stochastic fuzzy BAM neural networks is also studied by using some analytical skills in stochastic theory. Finally, two examples with computer simulations are given to demonstrate that our results are feasible. The main results obtained in this paper are completely new, and the methods used in this paper provide a possible technique to study 2p‐th mean almost periodic sequence solution and 2p‐th moment global exponential stability of semidiscrete stochastic fuzzy models.

In this paper, we investigate the problem of robust stability for uncertain stochastic neural network systems with interval time-varying delay. By means of singular model transformation technique, special Lyapunov-Krasovskii functional approach, similar Leibniz-Newton formula and linear matrix inequality (LMI) concept, some new stability conditions are derived for above systems. There are three main parts concerning our research results. The first result is to propose both delay-independent and delay-dependent criteria for guaranteeing the asymptotic stability of stochastic switched Hopfield neural network systems with constant parameters and time delay. The second result is to present several new delay-dependent criteria for testing the mean-square exponential stability of stochastic cellular neural network systems with interval time-varying delay. The third result is to provide sufficient conditions for ensuring asymptotic stability of stochastic neural network systems with uncertain parameters and interval time-varying delay. In the results, we do not assume that the network's activation functions are with the property of sigmoid function. The purpose of introducing the singular model transformation is to improve the results on bounds of the inner product of two vectors. Our results do not need the solution of Lyapunov equation or Riccati equation. Compared with existing results in the literature, our method is shown to be superior to other ones. Numerical examples are given to demonstrate the effectiveness of the proposed approach. Besides, our approach can be also applied to the stability testing problem for large-scale stochastic neural network systems with uncertain parameters and time-varying delays.

The learning capability of neural networks is equivalent to modeling physical events that occur in the real environment. Several early works have demonstrated that neural networks belonging to some classes are universal approximators of input-output deterministic functions. Recent works extend the ability of neural networks in approximating random functions using a class of networks named stochastic neural networks (SNN). In the language of system theory, the approximation of both deterministic and stochastic functions falls within the identification of nonlinear no-memory systems. However, all the results presented so far are restricted to the case of Gaussian stochastic processes (SPs) only, or to linear transformations that guarantee this property. This paper aims at investigating the ability of stochastic neural networks to approximate nonlinear input-output random transformations, thus widening the range of applicability of these networks to nonlinear systems with memory. In particular, this study shows that networks belonging to a class named non-Gaussian stochastic approximate identity neural networks (SAINNs) are capable of approximating the solutions of large classes of nonlinear random ordinary differential transformations. The effectiveness of this approach is demonstrated and discussed by some application examples.

Stochastic neural network is a hierarchical network of stochastic neurons which emit 0 or 1 with the probability determined by the values of inputs. We have developed an efficient training algorithm so as to maximize the likelihood of such a neural network. This algorithm enables us to apply the stochastic neural network to a practical problem like prediction of fall or rise of Tokyo Stock Price Index (TOPIX). We trained it with the data from 1994 to 1996 and predicted the fall or rise of 1 day ahead of TOPIX for the period from 1997 to 2000. The result is quite promising. The accuracy of the prediction of the stochastic network is the 60.28%, although those of non-stochastic neural network, autoregressive model and GARCH model are 50.02, 51.38 and 57.21%, respectively. However, the stochastic neural network is not so advantageous over other networks or models for prediction of the TOPIX used for training. This means that the stochastic neural network is less over fitting to the training data than others, and results in the best prediction. We will demonstrate how the stochastic neural network learns well non-linear structure behind of the data in comparison to other models or networks, including Generalized Linear model (GLM).

This paper focuses on solving the problem of checking the exponential stability of a class of stochastic fuzzy cellular neural networks with Markovian jumping parameters, time-varying delays and distributed delays. By constructing suitable Lyapunov functional and applying stochastic analysis, we firstly developed some sufficient conditions to guarantee the almost surely exponential stability and the exponential stability in the mean square of this kind of neural networks. We then showed, by developing several corollaries, that our results could be specialized to various cases including some published studies. Finally, a numerical example proves the effectiveness of our method. Keywords: Distributed time delays, exponential stability, fuzzy systems, Markovian jumping parameters, stochastic neural networks, time-varying delays.

Preface. List of Notations. 1: Random Media. 1.1. Markov Chains. 1.2. Ergodicity and Reducibility of Markov Chains. 1.3. Markov Renewal Processes. 1.4. Semi-Markov Processes. 1.5. Jump Markov Processes. 1.6. Wiener Processes and Diffusion Processes. 1.7. Martingales. 1.8. Semigroups of Operators and their Generators. 1.9. Martingale Characterization of Markov and Semi-Markov Processes. 1.10. General Representation and Measurability of Biological Systems in Random Media. 2: Limit Theorems for Difference Equations in Random Media. 2.1. Limit Theorems for Random Evolutions. 2.2. Averaging of Difference Equations in Random Media. 2.3. Diffusion Approximation of Difference Equations in Random Media. 2.4. Normal Deviations of Difference Equations in Random Media. 2.5. Merging of Difference Equations in Random Media. 2.6. Stability of Difference Equations in Random Media. 2.7. Limit Theorems for Vector Difference Equations in Random Media. 3: Epidemic Models. 3.1. Deterministic Epidemic Models. 3.2. Stochastic Epidemic Model (Epidemic Model in Random Media). 3.3. Averaging of Epidemic Model in Random Media. 3.4. Merging of Epidemic Models in Random Media. 3.5. Diffusion Approximation of Epidemic Models in Random Media. 3.6. Normal Deviations of Epidemic Model in Random Media. 3.7. Stochastic Stability of Epidemic Model. 4: Genetic Selection Models. 4.1. Deterministic Genetic Selection Models. 4.2. Stochastic Genetic Selection Model (Genetic Selection Model in Random Media). 4.3. Averaging of Slow Genetic Selection Model in Random Media. 4.4. Merging of Slow Genetic Selection Model in Random Media. 4.5. Diffusion Approximation of Slow Genetic Selection Model in Random Media. 4.6. Normal Deviations of Slow Genetic Selection Model in Random Media. 4.7. Stochastic Stability of Slow Genetic Selection Model. 5: Branching Models. 5.1. Branching Models with Deterministic Generating Function. 5.2. Branching Models in Random Media. 5.3. Averaging of Branching Models in Random Media. 5.4. Merging of Branching Model in Random Media. 5.5. Diffusion Approximation of Branching Process in Random Media. 5.6. Normal Deviations of Branching Process in Random Media. 5.7. Stochastic Stability of Branching Model in Averaging and Diffusion Approximation Schemes. 6: Demographic Models. 6.1. Deterministic Demographic Model. 6.2. Stochastic Demographic Models (Demographic Models in Random Media). 6.3. Averaging of Demographic Models in Random Media. 6.4. Merging of Demographic Model. 6.5. Diffusion Approximation of Demographic Model. 6.6. Normal Deviations of Demographic Models in Random Media. 6.7. Stochastic Stability of Demographic Model in Averaging and Diffusion Approximation Schemes. 7: Logistic Growth Models. 7.1. Deterministic Logistic Growth Model. 7.2. Stochastic Logistic Growth Model (Logistic Growth Model in Random Media). 7.3. Averaging of Logistic Growth Model in Random Media. 7.4. Merging of Logistic Growth Model in Random Media. 7.5. Diffusion Approximation of Logistic Growth Model in Random Media. 7.6. Normal De

In this paper, the stability of stochastic Hopfield neural network with distributed parameters is studied. To discuss the stability of systems, the main idea is to integrate the solution to systems in the space variable. Then, the integration is considered as the solution process of corresponding neural networks described by stochastic ordinary differential equations. A Lyapunov function is constructed and Ito formula is employed to compute the derivative of the mean Lyapunov function along the systems, with respect to the space variable. It is difficult to treat stochastic systems with distributed parameters since there is no corresponding Ito formula for this kind of system. Our method can overcome this difficulty. Till now, the research of stability and stabilization of stochastic neural networks with distributed parameters has not been considered.

For Part I, see ibid., p.275-82. The authors introduce a neural computation architecture based on a stochastic Hopfield neural network model for solving job-shop scheduling. A computation circuit computes the total completion times (costs) of all jobs, and the cost difference is added to the energy function of the stochastic neural network. Using a simulated annealing algorithm, the temperature of the system is slowly decreased according to an annealing schedule until the energy of the system is at a local or global minimum. By choosing an appropriate annealing schedule, near-optimal and optimal solutions to job-shop problems can be found. The architecture of the system is presented at both the functional and circuit levels. Simulation results are presented. >

This paper is concerned with the problem of delay-dependent exponential stability analysis for a class of stochastic Hopfield type neural networks with constant time delays. By employing an augmented Lyapunov-Krasovskii functional, together with the linear matrix inequality approach, a delay-dependent condition guaranteeing the global exponential stability (in the mean square sense) of the considered stochastic neural network is presented. A numerical example is provided to demonstrate the effectiveness of the proposed stability condition.

We prove the global asymptotic stability of the disease-free and the endemic equilibrium for general SIR and SIRS models with nonlinear incidence. Instead of the popular Volterra-type Lyapunov functions, we use the method of Dulac functions, which allows us to extend the previous global stability results to a wider class of SIR and SIRS systems, including nonlinear (density-dependent) removal terms as well. We show that this method is useful in cases that cannot be covered by Lyapunov functions, such as bistable situations. We completely describe the global attractor even in the scenario of a backward bifurcation, when multiple endemic equilibria coexist.

In this paper, a stochastic SVEIR epidemic model with saturated incidence and partial immunity is proposed. First, we define the basic reproduction number [Formula: see text] and study the global asymptotic stability of disease-free equilibrium and endemic equilibrium of the deterministic epidemic model. Then we establish the existence and uniqueness of global positive solution, along with delineating sufficient conditions for disease extinction. By constructing appropriate Lyapunov functions, we analyze the asymptotic behavior of solutions to the stochastic epidemic model around the disease-free equilibrium and endemic equilibrium points of the deterministic epidemic model. Under certain conditions, the solution of the stochastic model fluctuates around the disease-free equilibrium point and the endemic equilibrium point, with the intensity of fluctuation proportional to the intensity of white noise. The existence of ergodic stationary distribution is proved by the Khasminskii method. Finally, numerical simulations are presented to illustrate our analysis results.

Mean-square exponential stability of stochastic Hopfield neural networks with time-varying discrete and distributed delays

Synchronization for an array of coupled stochastic discrete-time neural networks with mixed delays

Context. To reduce the computational resource time in the problems of diagnosing and recognizing distorted images based on a fully connected stochastic pseudospin neural network, it becomes necessary to thin out synaptic connections between neurons, which is solved using the method of diagonalizing the matrix of synaptic connections without losing interaction between all neurons in the network. Objective. To create an architecture of a stochastic pseudo-spin neural network with diagonal synaptic connections without loosing the interaction between all the neurons in the layer to reduce its learning time. Method. The paper uses the Hausholder method, the method of compressing input images based on the diagonalization of the matrix of synaptic connections and the computer mathematics system MATLAB for converting a fully connected neural network into a tridiagonal form with hidden synaptic connections between all neurons. Results. We developed a model of a stochastic neural network architecture with sparse renormalized synaptic connections that take into account deleted synaptic connections. Based on the transformation of the synaptic connection matrix of a fully connected neural network into a Hessenberg matrix with tridiagonal synaptic connections, we proposed a renormalized local Hebb rule. Using the computer mathematics system “WolframMathematica 11.3”, we calculated, as a function of the number of neurons N, the relative tuning time of synaptic connections (per iteration) in a stochastic pseudospin neural network with a tridiagonal connection Matrix, relative to the tuning time of synaptic connections (per iteration) in a fully connected synaptic neural network. Conclusions. We found that with an increase in the number of neurons, the tuning time of synaptic connections (per iteration) in a stochastic pseudospin neural network with a tridiagonal connection Matrix, relative to the tuning time of synaptic connections (per iteration) in a fully connected synaptic neural network, decreases according to a hyperbolic law. Depending on the direction of pseudospin neurons, we proposed a classification of a renormalized neural network with a ferromagnetic structure, an antiferromagnetic structure, and a dipole glass.