A Model for Optimal Economic Lockdown to Contain Epidemic

Abstract

Marco Tognoli

This paper is concerned with the application of operations research in defining the optimal lockdown of economic activities to contain epidemic. The problem of optimal lockdown consists in deciding as best economic sectors can be lockdown with respect to fundamental sectors (essential goods and services) while disruptive impacts are minimized on the economy as a whole. Many countries around the world are currently implementing the lockdown of most economic activities to contain the spread of the Covid-19 pandemic. The lockdown brings health benefits for the society as it contains the spread of the virus, reducing the number of infections and allowing the health system to treat those infected better. This paper describes a Boolean linear programming model to deal with the problem of selecting several economic sectors to be shutdown. The objective function is linear and the constraints are linear inequalities related to the Leontief’s input-output table. The model permits to analyze the feasibility of national economic system in which some elements of the input-output table are set equal to zero. The mathematical approach to the shutdown problem permits to identify the greatest number of economic sectors that can be closed without destroying the fundamental sectors. Since solution of the shutdown problem is the greatest number of lockdown economic sectors, author believes the model allows to oppose effectively the spread of virus. Once spread of the virus decreases, another feature of the model is to support decision makers in assigning priorities to the economic sectors to be gradually unlocked.

Standard input-output models are able to reveal how different sectors of an economy are interconnected and how changes in one sector affect all other sectors. Besides the use of the input-output data for descriptive analyses of economic interrelations, this table also provides the empirical fundament for a wide scale of impact analysis. input- output models greatly differ in size and possible applications. Simple static input-output models are used for comparative-static impact (scenario) analysis. With the help of input-output quantity models, statements can be made about direct and indirect effects based on exogenous changes in demand. More complex dynamic input-output models largely resolve the limitations and inherent assumptions of static input-output models. Time is considered explicitly; quantity and price reactions are modelled endogenously in a holistic approach and feedback effects are captured. Last models are not only suitable for scenario analysis and for forecasting as well. In “A Linear Programming Solution to Dynamic Leontief type Models” Harvey M. Wagner presents a general dynamic model of an economy and investigates a number of questions related to the feasibility of certain time profiles of demand and the rate of substitution between economic activities. The Inoperability Input-Output Model [1-4]. was developed to understand better the infrastructure interdependencies. Based on the Leontief Input- Output Model, the model is demand-driven, wherein perturbations to the final demand levels are considered the initiating event, and the impact to sectors’ production outputs are the direct and indirect effects resulting from sector interdependencies. Case study shows that the different perturbation classes (demand or supply, quantity or price) yield different rank orders of sectors impacted by the initiating event, thus providing various perspectives of impacts.

The present study differs from most previous studies in two aspects. First, linear programming techniques have been widely used in Input-Output model: from the primary and dual formulation of the problem to the analysis of Lagrange multipliers in order to assess the mutual relative importance of sectors. Second, the impact of lockdown is measured in terms of annihilated interdependencies rather than in terms of canceled economic values such as revenues and monetary exchanges. For this reason, the proposed model takes into account the adjacency matrix got from input-output Table rather than the Table itself. This analysis is static in character (it is a static I-O model) because it does not take care explicitly of time, quantity and price reactions, consumption and production lags, growth of final demand (these ones are known as dynamic I-O models). The study simply considers the interruption of several activity as a consequence of a government measure adopted to reduce the risk of contagion. The proposed model answers to the basic question if the annihilation of N sectors is feasible, secondary the model identifies in the Lagrange multipliers the means to sequencing the sectors to be reopened.

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