key: cord-0198517-dhzb3jfb authors: Stosic, Borko D. title: Dynamics of COVID-19 mitigation inefficiency in Brazil date: 2021-04-20 journal: nan DOI: nan sha: 7442542faeba2ecbf37e3df8e8ffe83ec70e2f95 doc_id: 198517 cord_uid: dhzb3jfb In this work Data Envelopment Analysis (DEA) is employed in thirty-day windows to quantify temporal evolution of relative pandemic mitigation inefficiency of Brazilian municipalities. For each thirty-day window the results of inefficiency scores of over five thousand Brazilian municipalities are displayed on maps, to address the spatial distribution of the corresponding values. This phenomenological spatiotemporal approach reveals location of the hotspots, in terms of relative pandemic inefficiency of the municipalities, at different points in time along the pandemic. It is expected that the current approach may subsidize decision making through comparative analysis of previous practice, and thus contribute to the future pandemic mitigation efforts. A year into the COVID-19 pandemic, Brazil, as many other countries, is currently experiencing a severe second wave of the disease, currently being described as the world's epicenter of the pandemic. Coronavirus disease 2019 (COVID-19), a highly contagious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), was declared officially a pandemic by World Health Organization on March 11, 2020 [1] . Emerging in China, and initially being mostly contained in that country, the virus has been spreading worldwide ever since. This dispersion has invoked stringent protective measures against the spread in most countries in the world, ranging from self-isolation and mandatory quarantine, to curfews and travel restrictions. The current situation remains precarious: different countries have been implementing different measures to mitigate the propagation of the virus ever since the pandemic start, but the outcome of the effectiveness of these diverse measures has yet to be quantified. In Brazil, a continental size country with over two hundred million inhabitants and over five thousand municipalities, a wide range of mitigation measures has been implemented by the local governments, at different time frames. The overall result of these measures is currently often described in the news as disastrous, with an overloaded health system and a large number of deaths of people waiting for a bed in intensive care units. On the other hand, Brazil has a very high organizational vaccination capacity through the highly organized public health system (Sistema Unico de Saude -SUS), but the world's capacity of vaccine production has turned insufficient to provide the number of vaccine doses necessarily to reach collective heard immunity. This is the current problem in most countries, indicating that a prolonged difficult period still lies ahead. Consequently, it appears paramount to expand the analysis of the available data in different directions, in the attempt of providing complementary insight into the phenomenon, in order to aid the policymakers in their decisions. To this end, Data Envelopment Analysis (DEA) is employed in this work to quantify mitigation inefficiency of 5442 Brazilian municipalities along the year of the pandemic, in a spatially explicit manner. It is expected that the results of this work may aid a governor of a state, or a major of a city, to compare their own measures, taken at different times, to those of neighboring states or cities, and draw conclusions as to the efficiency of their own disease mitigation measures. The database used in this study [2] , available at https://github.com/wcota/covid19br, contains data on 5570 Brazilian municipalities, including population size, location (latitude and longitude), and daily numbers of confirmed cases and deaths, since February 25, 2020 (the data used in this work are up to April 12, 2021) . It was found that 128 (very small) municipalities have no reported deaths along the period under study, and as the subsequent analysis is sensitive to outliers they were removed from this study, which was performed with the remaining 5442 municipalities. In what follows Data Envelopment Analysis (DEA) method is used to address mitigation inefficiency of the Brazilian municipalities. DEA was introduced in econometrics, it represents a powerful, non-parametric method for measurement of relative production efficiency of a given set of comparable profit organizations. The concept of DEA was originally introduced by Farrell [3] , and subsequently formulated as a Linear Programming based technique by Charnes, Cooper and Rhodes [4] (the reader is referred to a recent textbook by Charnes et al. [5] for a comprehensive coverage of the subject). The method is achieved by considering empirical efficiency frontiers, spanned by such organizations (so called Decision Making Units -DMUs), which use minimal input to produce maximal output within the observed sample. While DEA was originally developed for comparative efficiency measurement of economic entities within a strict profit maximization concept (banks, industries, firms, etc.), it has been subsequently applied to various multiple input and output situations, where an input is understood as a quantity that should be minimized, and an output as a quantity that should be maximized, in order to reach the so called , ≥ 0 = 1, … , Effectively, this represents scaling of all of the input and output variables which maximizes the output versus input ratio of the considered DMU , while making sure (by imposing restriction (2)) that similar ratios (comparative efficiencies) for all of the DMUs remain bounded by unity. The conditions (3) guarantee the required positivity of the input and output variables after scaling. It is evident that any solution to relations (1-3), represented by sets of u's and v's, is invariant to uniform scaling, achieved by multiplying all the u's and the v's by the same (arbitrary, positive) constant. As this generates an infinite set of solutions, the usual approach to overcome this problem and remove the solution degeneracy is to impose the additional constraints, either ∑ =1 = 1 (so called input orientation), or ∑ =1 = 1 (output orientation). In the case of input orientation, the procedure is now reduced to determining , ≥ 0 = 1, … , ; = 1, … , which is called input oriented DEA "multipler formulation". Finally, it turns out that every linear programming problem has two equivalent formulations, and that the dual formulation (so called "envelopment form") is numerically more advantageous (as there are more DMUs than inputs and outputs), so the common input oriented DEA formulation found in the literature is given by Here are positive coefficients to be adjusted (for each DMU ) in order to minimize (instead of scale factors and in the multiplier formulation (4-7)). For each DMU with efficiency below unity, this input oriented DEA formulation indicates as a goal the reduction of inputs necessary for reaching the efficiency frontier. In the output oriented DEA version, by normalizing the linear combination of outputs, , ≥ 0 = 1, … , Finally, the linear programming problem dual to relations (12-15) yields the envelopment form subject to conditions and the efficiency of the -th DMU is given by = 1/ . To examine the "efficiency frontier", in the current context of COVID 19 the numbers of confirmed cases and deaths can be taken as an input, and municipality population as output, in which case the so called input oriented DEA is adequate, which considers the reduction of inputs for which the observed DMU reaches the efficiency frontier. On the other hand, if the main interest is to identify the most inefficient municipalities (the "inefficiency frontier"), population can be taken as input, and the numbers of confirmed cases and deaths taken as outputs: if one of two particular municipalities, with the same population size, has more confirmed cases, and/or more deaths, it is considered to exhibit a higher mitigation inefficiency than the other. In this case the so called output oriented DEA is implemented, which considers the increase of the quantity of outputs (here confirmed cases and deaths) at which the observed DMU reaches the "inefficiency frontier". The spatial distribution of the values is obtained in this work by interpolation using standard isotropic Inverse Distance Weighting (IDW) technique, introduced by Shepard [7] . Interpolated value of the quantity ( ) at point is found from the known values ( ) at the neighboring data points , = 1, … , , using and ( , ) ≠ 0 is the distance between the points and . The actual value of the parameter is taken to be = 2 throughout this work, as originally proposed by Shepard [7] . The It is seen from Tab. 1, and Fig. 1 that population, confirmed cases and number of deaths span several orders of magnitude, suggestive of power law behavior. In fact, the validity of Zipf's law [8, 9] (also called rank-size rule) has already been shown for Brazilian cities [10] , and for COVID-19 cases for a number of countries [11] . Zipf's law holds for a given dataset if for the ordered values 1 > 2 > ⋯ > > ⋯ > the plot of ( ) versus ( ) is linear. In the current case, all three quantities follow the Zipf's law for a wide range of values, with a rather similar exponent (slope on the log-log plot), as can be seen in Fig. 2 . The spatial distribution of population, confirmed cases and deaths over the study period is shown in Fig. 3 (again on logarithmic scale), together with the death rate (number of deaths divided by the number of confirmed cases, on linear scale). It is seen that the spatial distribution of confirmed cases and deaths closely matches the municipality population size distribution, while the death rate spatial distribution reveals a rather different pattern, that should be more scrutinized by local governments of municipalities shown in red on Fig. 3d . in the former approach lies on highlighting small cities with few confirmed cases and deaths, and tends to underrepresent the cities with low efficiency. By comparing the CRS with the VRS versions ( Fig. 7a with Fig. 7b , and Fig. 7c with Fig.7d ) it can be concluded that taking into account scaling effect does not bring about a significant change of the spatial distribution of results, probably because the choice of input variables already accounts for scale in terms of population size, for the full study period. As already mentioned, the efficiency frontier approach is rather sensitive to presence of outliers (small cities with zero confirmed cases and/or deaths), and it is found that it yields reasonable results only for the whole period of study, while the time evolution in 30-day windows enhances the outlier effect. The time evolution study is therefore implemented here only for the inefficiency frontier, with the results displayed in Figs 8 and 9. 5 and 6 it is seen that both the number of deaths and the death rate have increased from January to February 2021 in the ongoing pandemic peak, while the confirmed cases number has not grown considerably, as can be seen in Fig. 4 . This means that the inefficiency frontier has shifted forward, to municipalities with an increased number of deaths, but this is observed only in a subset of the municipalities that were the most inefficient in February 2021. The DEA results presented in Figs. 8 and 9 highlight the inefficiency scores relative to the most inefficient ones in each particular period of time, those that comprise the inefficiency frontier. Therefore, the authorities of the municipalities that show as red in February 2021 should be even more concerned than those of the municipalities shown as red for January, while in March 2021the situation becomes rather uniformly alerting among most municipalities in Brazil, perhaps less so for the northeast region. The current study aims to provide a contribution to the assessment of the current observational data on the COVID-19 virus outbreak in Brazil, from a phenomenological point of view. The current analysis of the data reveals the spatial distribution of the hotspots in this continental size country over the entire period of the pandemic, as well as the evolution on monthly scale. [12] . Additional calculation results (for these, or for different timeframes, in the form of data tables, images and/or videos) are available from the author upon request, in the hope that such information may subsidize decision making through comparative analysis of previous practice, and thus contribute to pandemic mitigation efforts. Monitoring the number of COVID-19 cases and deaths in brazil at municipal and federative units level The Measurement of Productive Efficiency Measuring the efficiency of decision making units Data Envelopment Analysis: Theory, Methodology and Applications Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis A two-dimensional interpolation function for irregularly-spaced data The psycho-biology of language: An introduction to dynamic philology Human behavior and the principle of least effort: An introduction to human ecology Zipf law for Brazilian cities On the authenticity of COVID-19 case figures Dynamics of COVID-19 mitigation inefficiency in Brazil supplementary material, Figshare Funding: The author acknowledges support of Brazilian agency CNPq through the research grant 307445/2018-6. The author declares that he has no competing interests. The data is available at (1). All the code was developed by the author in C programming language (with inline assembler code, for DEA calculations) and CUDA, C and Windows APIs (for interpolation and visualization). The source code, Visual Studio projects and executables are available from the author upon request.