A new mathematical model of Coronavirus (2019-nCov) using piecewise hybrid fractional order derivatives is given in this paper. Moreover, in order to be consistent with the physical model problem, a new parameter μ is presented. The boundedness, existence, and positivity of the solutions for the proposed model are discussed. Two improved numerical methods are presented in this paper. The Caputo proportional constant nonstandard modified Euler–Maruyama method is introduced to study the fractional stochastic model, and the Grünwald–Letnikov nonstandard finite difference method is presented to study the hybrid fractional order deterministic model. Comparative studies with real data from Spain and Wuhan are presented. [ABSTRACT FROM AUTHOR]
The COVID-19 pandemic is having a strong impact on the economies of all countries, negatively affecting almost all sectors. This paper compares Spanish electricity and natural gas prices in the first half-year of 2020 with the prices expected for that period at the end of 2019. The half-year of 2020 selected coincides with the period of greatest impact of COVID-19 on Spanish society. Expected prices and their future probability distributions are calculated using a stochastic model with deterministic and stochastic parts; the stochastic part includes mean-reverting and jumps behaviour. The model is calibrated with 2016–2019 daily spot prices for electricity and with day-ahead prices for natural gas. The results show large monthly differences between the prices expected at the end of the year 2019 and the actual prices for the half-year; in May 2020, wholesale electricity prices are found to be EUR 31.60/MWh lower than expected, i.e., 60% lower. In the case of natural gas, the prices in the same month are EUR 8.96/MWh lower than expected, i.e., 62% lower. The spark spread (SS) is positive but lower than expected and also lower than in the same months of the previous year. [ABSTRACT FROM AUTHOR]