**Bayesian Model for COVID-19 Spread Prediction**

At present time, there are different methods, approaches, data sets for for modeling COVID-19 spread [1, 2, 3, 4, 5, 6]. For the predictive analytics of COVID-19 spread, we used a logistic curve model. Such model is very popular nowadays. To estimate model parameters, we used Bayesian regression [7, 8, 9]. This approach allows us to receive a posterior distribution of model parameters using conditional likelihood and prior distribution. In the Bayesian inference, we can use informative prior distributions which can be set up by an expert. So, the result can be considered as a compromise between historical data and expert opinion. It is important in the cases when we have a small number of historical data. Probabilistic approach makes it possible to receive the probability density function for the target variable. Logistic curve model with Bayesian regression approach can be written as follows:

where Date0 is a start day for observations in the historical data set, it is measured in weeks. The data for the analysis were taken from here [2]. α parameter denotes maximum cases of coronavirus, β parameter is an empirical coefficient which denotes the rate of coronavirus spread. For solving Bayesian models, numerical Monte-Carlo methods are used. Gibbs and Hamiltonian sampling are the popular methods of finding posterior distributions for the parameters of probabilistic mode [7, 8, 9]. Bayesian inference makes it possible to obtain probability density functions for model parameters and estimate the uncertainty that is important in risk assessment analytics. In Bayesian regression approach, we can take into account expert opinions via information prior distribution. For Bayesian inference calculations, we used pystan package for Stan platform for statistical modeling [9]. The next figure shows the box plots for β parameters of coronavirus spread model for different countries:

Next figures show the predictions for coronavirus spread cases using current historical data:

In practical analytics, it is important to find the maximum of coronavirus cases per day, this point means the estimated half time of coronavirus spread in the region under investigation. New historical data will correct the distributions for model parameters and forecasting results. The ’Bayesian Model for COVID-19 spread Prediction’ package can be loaded here [10] for free use.

**COVID-19 Impact on the Stock Market**

Coronavirus outbreak has a huge impact on the stock market. It is very important , e.g. for forming stable portfolios, to understand how different crises impact stock prices. We consider the impact of coronavirus crisis on stocks and compare it to the crisis of 2008 and market downturn of 2018. For this, we can use the regression approach using OLS regression and Bayesian regression. Bayesian inference makes it possible to obtain probability density functions for coefficients of the factors under investigation and estimate the uncertainty that is important in the risk assessment analytics. In Bayesian regression approach, we can analyze extreme target variable values using non gaussian distributions with fat tails. We took the following time periods for each of crises – crisis 2008: [2008-01-01,2009-01-31], down turn 2018: [2018-10- 01,2019-01-03], coronavirus: [2020-02-18,2020-03-25]. For each of above mentioned crises, we created a regression variable which is equal to 1 in the crisis time period and 0 in other cases. Next figure shows the time series for S&P500 composite index:

As a target variable, we consider the daily price return. Knowing the daily price return, changes in crises periods, one can estimate the ability of investors to understand trends and recalculate portfolios. These results were received using Bayesian inference. For Bayesian inference calculations, we used Python pystan package for Stan platform for statistical modeling [9]. The next figure shows the box plots of impact weights of each crisis on S&P500 composite index:

The wider box for coronavirus weight can be caused by shorter time period of investigation comparing with other crises and consequently larger uncertainty. For our investigations, we took a random set of tickers from S&P list. The next figure shows top negative price returns in coronavirus crises:

The next figure shows the tickers with positive price return in coronavirus crisis:

The next figure shows the weights for different crises for arbitrarily chosen stocks:

We calculated the distributions for crises weights using Bayesian inference. The next figure shows the box plots for crisis weights for different stocks:

The logistic curve model can be used with Bayesian regression for predictive analytics of coron- avirus spread. Such a model can be effective when the exponential growth of number of coron- avirus confirmed cases takes place. In practical analytics, it is important to find the maximum of coronavirus cases per day, this point means the estimated half time of coronavirus spread in the region under investigation. New historical data will correct the distributions for model parameters and forecasting results. ’Bayesian Model for COVID-19 spread Prediction’ package can be loaded here for free use.

The obtained results show that different crises with different reasons have different impact on the same stocks. It is important to analyze their impact separately. Bayesian inference makes it possible to analyze the uncertainty of crisis impacts. The uncertainty of crisis impact weights can be measured as a standard deviation for weight probability density functions. The uncertainty of coronavirus crisis is larger comparing to other crises that can be caused by shorter analysis time. Knowing the uncertainty allows making risk assessment for portfolios and other financial and business processes. In Bayesian regression, we can receive a quantitative measure for the uncertainty that can be a very useful information for experts in model selection and stacking. An expert can also set up informative prior distributions for stacking regression coefficients of models, taking into account the domain knowledge information.

**References**

[1] IHME COVID, Christopher JL Murray, et al. Forecasting covid-19 impact on hospital bed-days, icu-days, ventilator-days and deaths by us state in the next 4 months. https://www.medrxiv.org/content/10.1101/2020.03.27.20043752v1

[2] COVID19 Global Forecasting (Week 2). Kaggle.Com. https://www.kaggle.com/c/covid19-global-forecasting-week-2

[3] CSSE COVID-19 Dataset. GitHub.Com. https://github.com/CSSEGISandData/COVID-19/tree/master/csse_covid_1…

[4] Coronavirus (Covid-19) Data in the United States. GitHub.Com. https://github.com/nytimes/covid-19-data

[5] COVID-19 reports. https://www.imperial.ac.uk/mrc-global-infectious-disease-analysis/c…

[6] COVID-19 Kaggle community contributions. Kaggle.Com. https://www.kaggle.com/covid-19-contributions

[7] John Kruschke. Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan. Aca- demic Press, 2014.

[8] Andrew Gelman, John B Carlin, Hal S Stern, David B Dunson, Aki Vehtari, and Donald B Rubin. Bayesian data analysis. Chapman and Hall/CRC, 2013.

[9] Bob Carpenter, Andrew Gelman, Matthew D Hoffman, Daniel Lee, Ben Goodrich, Michael Betancourt, Marcus Brubaker, Jiqiang Guo, Peter Li, and Allen Riddell. Stan: A probabilistic programming language. Journal of statistical software, 76(1), 2017.

[10] Bayesian Model for COVID-19 Spread Prediction. GitHub.Com. https://github.com/pavlyshenko/covid19