Before revision:  There have been numerous attempts to define and mathematically formulate this concept in various  scientific fields in literature . The current indirect effects formulations for an economic system represent the sum of the subsequent gross outputs of goods and services at each step of a production chain to supply the final demands, generally after the first gross outputs from each sector in the system.
After revision:  There have been numerous attempts to define and mathematically formulate this concept in various scientific fields in literature as well. The existing indirect effects formulations for an economic system represent the sum of the subsequent gross outputs of goods and services at each step of a production chain to supply the final demands, generally after the first gross outputs from each sector in the system.