Mathematical models have been developed to study the emergence of instability in the operation of complex socio-economic systems under pandemic conditions and systemic risks. These models allow for the assessment of mechanisms behind the onset of instabilities and their consequences for various sectors of the economy. Risk assessment is performed based on the proximity of system parameters to their bifurcation values using methods from the theory of smooth functions. The socio-economic impacts of these instabilities are calculated using a six-sector Lorenz model with variable coefficients, which integrates sectors of the economy into a unified structure described uniformly. Each sector is analyzed in terms of productivity levels, employment rates, and structural disruptions. The models enable forecasting of crisis phenomena, selecting strategies to ensure a specified level of security, investigating the emergence of regimes with rapid, abrupt changes in system behavior, ranking various types of threats, and identifying weak links that significantly influence the formation of instability and distortions in the security space.