The coronavirus disease 2019 (COVID-19) pandemic placed epidemic modeling at the forefront of worldwide public policy making. Nonetheless, modeling and forecasting the spread of COVID-19 remain a challenge. This talk begins with a review of the historical use of epidemic models and addresses the challenges of choosing a model in the early stages of a worldwide pandemic. The spread of COVID-19 has illustrated the heterogeneity of disease spread at different population levels. With finite size populations, chance variations may lead to significant departures from the mean. In real-life applications, finite size effects arise from the variance of individual realizations of an epidemic course about its fluid limit. I will illustrate how to model this variance with a martingale formulation consisting of a deterministic and a stochastic component, along with estimates for the size of the variance compared to real world data and simulations. Another cause of heterogeneities is the differing attitudes at the population level for control measures such as mask-wearing and physical distancing. Often, individuals form opinions about their behaviors from social network opinions. I will show some results from a two-layer multiplex network for the coupled spread of a disease and conflicting opinions. We model each process as a contagion. We develop approximations of mean-field type by generalizing monolayer pair approximations to multilayer networks; these approximations agree well with Monte Carlo simulations for a broad range of parameters and several network structures. We find that lengthening the duration that individuals hold an opinion may help suppress disease transmission, and we demonstrate that increasing the cross-layer correlations or intra-layer correlations of node degrees may lead to fewer individuals becoming infected with the disease.

In this talk, we will introduce first introduce Toeplitz structured random matrices and review some asymptotic results of these matrices. We will briefly show the limiting Toeplitz law and recent result of central limit theorem for linear statistics of a specific strucetured Toeplitz matrix based on moment methods. Some challenges, future directions and applications of random Toeplitz matrices will also be mentioned in this talk. Secondly, we will introduce the nonlinear random matrices in random neural networks. In the linear proportional regime, the limiting eigenvalue distributions of conjugate matrices and empirical neural tangent kernels at initial have been studied via Stieltjes transform, which will help us better understand the deep neural networks. We will finally present a recent result of nonlinear random matrix theory beyond linear regime, where a deformed a Wigner law will appear.