This paper is concerned with the mathematical analysis of the stochastic infectious model under subclinical infections and vaccination and waning immunity. In the modern society with the advanced medical technology, we have still various kinds of the infectious disease threat including Coronavirus disease (Covid-19). Hence, the control and the analysis of infection diseases is one of major problems in epidemiology. In the control of the infectious diseases, vaccination plays an important role. In the realistic spread of the infectious disease, environmental change and individual difference cause some kinds of random fluctuations in the infection and the recovery rates. Moreover, noting that one of characteristics of Covid-19 is the existence of subclinical infections, we propose the stochastic infectious model under subclinical infections and vaccination with waning immunity. Since the stability analysis of the infectious model is effective in the control of the spread of the infectious disease, we analyze the stability of the stochastic infectious model. We derive the stability conditions for the proposed stochastic infectious model to be stable. By numerical simulations, we show the efficacy of the stability theorems derived in this paper and we study the influence of the random noise and vaccination on the stability.

The topic we address in this paper concerns the minimization of a Hamiltonian function for an Ising model through the application of simulated annealing algorithms based on (single-site) Glauber dynamics and stochastic cellular automata (SCA). Some rigorous results are presented in order to justify the application of simulated annealing for a particular kind of SCA. After that, we compare the SCA algorithm and its variation, namely the ε-SCA algorithm, studied in this paper with the Glauber dynamics by analyzing their accuracy in obtaining optimal solutions for the max-cut problem on Erdös-Rényi random graphs, the traveling salesman problem (TSP), and the minimization of Gaussian and Bernoulli spin glass Hamiltonians. We observed that the SCA performed better than the Glauber dynamics in some special cases, while the ε-SCA showed the highest performance in all scenarios.

This paper addresses the stability analysis problem of recurrent neural networks (RNNs) with activation functions being rectified linear units (ReLUs). In particular, we focus on the linear nonegative properties of the input-output signals of ReLUs. To capture these linear properties within the framework of integral quadratic constraint (IQC), we introduce a copositive multiplier constructed from a copositive matrix. This enables us to capture the linear input-output properties of ReLUs in quadratic form. By using the copositive multipliers in additon to existing multipliers, we can reduce the conservatism of the IQC-based stability analysis for RNNs. We also show that the proposed IQC-based stability condition with copositive multipliers can be viewed as an extension of a recently proposed scaled small-gain stability condition based on the L₂₊ induced norm.