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Given a set of securities or assets it is of interest to find an optimal way of investing in these assets. What is optimal has to specified. The objective is to optimize the return consistent with the specified objective. When there are several assets it is unlikely all the assets will increase if they are correlated. It is necessary to diversify one’s assets for a secure return. To deal with the different assets a combination of the assets should be considered with constraints as needed. One approach is the Markowitz mean-variance model where the mean variance is minimized including constraints. In this paper neural networks and machine learning are used to extend the ways of dealing with portfolio asset allocation. Portfolio selection problem in an efficient way. The use of heuristic algorithms in this case is imperative. In the past some heuristic methods based mainly on evolutionary algorithms, tabu search and simulated annealing have been developed. The purpose of this paper is to consider a particular neural network model, the Hopfield network, which has been used to solve some other optimisation problems and apply it here to the portfolio selection problem, comparing the new results to those obtained with previous heuristic algorithms. Although great success has been achieved for portfolio analysis with the birth of Markowitz model, the demand for timely decision making has significantly increased especially in recent years with the advancement of high frequency trading (HFT), which combines powerful computing servers and the fastest Internet connection to trade at extremely high speeds. This demand poses new challenges to portfolio solvers for real-time processing in the face of time-varying parameters. Neural networks, as one of the most powerful machine learning tools has seen great progress in recent years for financial data analysis and signal processing ([1], [14]). Using computational methods, e.g., machine learning and data analytics, to empower conventional finance is becoming a trend widely adopted in leading investment companies ([3]). 

An epidemic disease caused by coronavirus has spread all over the world with a strong contagion rate. We present simulations of epidemic models constructed using real data to give a clear perspective and confirmation on the effect of quarantine on the evolution of the infection and the number of infected, recovered, and dead because of this epidemic in South Carolina in a time window (December 1, 2020, to June 1, 2021) when the epidemic was relatively strong. We use CDC data for infected and dead populations covering the period December 1, 2020, to June 1, 2021 in South Carolina to develop models and do simulations. There were no data available for recovered populations in this period. Part of our goal is to estimate the number of recovered for the entire period. The models and results are consistent with the data. The infection and recovery increasing in South Carolina do not show improvement in this period. The number of dead people in this period tended to increase although by small amount. Optimal control methodologies are considered where transmission, recovery, relapse of immunity and death rates are considered as decision variables in minimizing the difference between the real and computed COVID-19 infection and dead data. Effect of quarantine as intervention strategy is also considered as it is critical issue. What we want to show is what could have been the outcome if quarantine had been implemented from the very beginning. The progress of an infection in general is related not only to the present states, but also to its historical states. To account for the effect of past evolution we add fractional differential equations models.