key: cord-0880665-picekqlo authors: Nikita, Saxena; Raman, Ruchir; Rathore, Anurag S. title: A chemical engineer's take of COVID‐19 epidemiology date: 2021-07-05 journal: AIChE J DOI: 10.1002/aic.17359 sha: 061b8b29274def87368dd190e746e38c96c3e326 doc_id: 880665 cord_uid: picekqlo SARS‐CoV‐2, a novel coronavirus spreading worldwide, was declared a pandemic by the World Health Organization 3 months after the outbreak. Termed as COVID‐19, airborne or surface transmission occurs as droplets/aerosols and seems to be reduced by social distancing and wearing mask. Demographic and geo‐temporal factors like population density, temperature, healthcare system efficiency index and lockdown stringency index also influence the COVID‐19 epidemiological curve. In the present study, an attempt is made to relate these factors with curve characteristics (mean and variance) using the classical residence time distribution analysis. An analogy is drawn between the continuous stirred tank reactor and infection in a given country. The 435 days dataset for 15 countries, where the first wave of epidemic is almost ending, have been considered in this study. Using method of moments technique, dispersion coefficient has been calculated. Regression analysis has been conducted to relate parameters with the curve characteristics. The highly contagious coronavirus disease , originated from China and is caused by the virus SARS-CoV-2, which is part of a family of coronaviruses that have in the past caused severe acute respiratory syndrome (SARS) and Middle East respiratory syndrome (MERS). 1 Since December 2019, COVID-19 has rapidly spread to over 200 countries worldwide, causing more than 40 million infections and 1 million deaths till November, 2020. 2 Compared to SARS and MERS, the fatality rate of COVID-19 is lower, however as the disease is more infectious, the total number of fatalities is much higher. 3 On March 11, COVID-19 was officially declared a pandemic by the WHO. Several models have been developed based on different approaches, with the initial attempts resulting more in confusion than clarity. 4, 5 Underreporting and inaccurate reporting of cases and deaths has made it difficult to fully understand the impact of the disease including ambiguity regarding spread, severity and duration of pandemic. Validity of models based on artificial intelligence has been questioned due to limitation of the training dataset. 6 Forecasting day level data based on prior patterns has been attempted, although prediction of changes is not in its scope 7 Agent based models, depending on population movement, distancing and virus infectivity characteristics, have been difficult to simulate. 8 Conventionally, differential equation models considering susceptible (S), infective (I), and recovered (R) fractions have been used for predicting pandemic dynamics. However, the efficiency of most of the SIR models developed to predict the impact was higher for short-term intervals in comparison to the long term. 9 Modified versions of SIR models are the SEIR models, which also incorporate the exposed (E) population but demand more data for development. 10 As a result, COVID-19 poses a distinctive difficulty in attempting to control the disease and limit the number of infections. Due to the lack of a vaccine and public health infrastructure designed to handle an outbreak of this magnitude, preventative measures have become necessary. All over the world, governments, healthcare systems, and economic systems have implemented measures to slow the spread of the disease and minimize its impact. 11, 12 This includes, but is not limited to, enforcing lockdowns, closing borders, school and work closures, social distancing, increasing sanitation and hygiene, and using facemasks. 13 As the stringency of these measures has varied by country, the size of the outbreak has as well. A case study relating outbreak in China with government and individual action demonstrate different effect of these actions on daily cases. 14 In countries such as the USA, Brazil, and India, governments have struggled with a coordinated, effective, and timely response to COVID-19, which have dis- proportionately affected vulnerable and economically challenged populations in these countries. 15 Comparatively, the majority of Western European countries have managed to "flatten the curve," reaching a plateau with the cumulative number of cases during the timeline considered for the analysis in this study. 16 With so many components influencing the spread of COVID-19, looking at the effect of various factors on the trajectory of the outbreak can provide an insight into how the spread of the disease can be slowed down. This article attempts to examine patterns in COVID-19 data, demographic factors, lockdown stringency, and country characteristics using residence time distribution (RTD) analysis. RTD is a theoretical modeling technique used to predict the distribution of residence times, typically in continuous flow systems. With applications in many biomedical sciences, RTD is most often used to analyze industrial units such as chemical reactors, fluidized beds, flotation cells, and mixers. 17, 18 One key application of RTD is in chemical engineering, where the technique is used to analyze the residence times of particles in chemical reactors. However, we demonstrate that the RTD concept can be applied toward examining the epidemiological data related to COVID-19 and new insights can be acquired. The residence time theory deals with the particles that enter, flow and leave the system. There are situations when the reactor fluid is F I G U R E 1 Total number of infected cases and deaths for the countries considered in the study. . Similar profiles are seen in COVID-19 daily cases trends for different countries. Also, tracer response for tank in series system follows same behavior neither perfectly mixed nor perfectly in plug flow. In such cases, RTD analysis helps in estimating the time the fluid has spent inside the reactor. Two model approaches, viz., one parameter approach and two parameter approach, are used commonly for simulating non ideal reactors. In this article, one parameter approach has been considered to deal with tank in series and axial dispersion model. RTD has been determined using the tracer injected in the reactor at time t = 0 in the form of pulse. It is assumed that the age of the particles while entering the system is zero and while leaving the system is equal to the residence time. [19] [20] [21] If the path of a particle is traced using a tracer with concentration, c(t), then the tracer amount, ΔN, leaving the reactor between time t and t + Δ is c (t)νΔt; ν is effluent volumetric flow rate. 22 For pulse injecting, the RTD function, E(t), is defined as On integrating the outlet concentration, N 0 can be obtained For constant ν, the RTD then becomes, The base properties of the distribution function are defined by its moments. It is common to compare RTD using moments instead of full distribution. For order r, the general moment is defined as Equation (4). The zeroth moment, r = 0, depicts the area under the distribution function. The first moment, r = 1, tells the centroid position indicating the mean or the expectation of residence time The physical meaning of the mean is related the volume/mass of the system per volumetric/mass out flow rate. Higher order moments are used to find out the experimental errors and for parameter estimation of the distribution function. Second moment (r = 2) gives the variance of the distribution (σ 2 t ) and is usually calculated around the mean value that is, central moment. In order to compare residence time distributions for the different system, the dimensionless form, σ 2 , is used which is given as: . Method of moments technique is applied to determine the dispersion coefficients. For any closed system, the relation between the tracer concentration and the model parameter can be obtained by solving unsteady state mass balance Equation (5). where D is dispersion coefficient, c is tracer concentration, and U is superficial velocity. For pulse input, Equation (5) where, φ ¼ c c0 ;λ ¼ z L ; θ ¼ tU L . Applying Danckwerts boundary conditions at λ = 0 and λ = 1, then solving numerically for mean residence time, t m and σ 2 can be estimated as shown in Equation (7) Pe T A B L E 1 COVID-19 epidemiologic curve characteristics Pe D = u*L/Pe where, u ¼ L tm , u is mean velocity of particle (m/s), L is length of fluidized bed (m), and t m is mean time. In the present study, the countries (Figure 1 The trend for the countries as shown in Figure 4 seems to be Gaussian that is, bell-shaped curves with different means (first moment) and variances (second moment). For normalization, the exit age distribution curves, E curve, is obtained for all the countries ( Figure 5 ). Then the mean, variance, Peclet number, and dispersion coefficient were calculated as listed in Table 1 Table S1 and for data refer to Tables S2-S4 . For performing regression, partial least square method The proposed model is applied to Belgium for validation. The demographic factors were obtained and the mean and variance were calculated using Equations (7) and (10) . The Gaussian curve was plotted for the predicted mean and variance based on the average data for 435 days. The X (predictors) and Y variables are mention in Table 2 . The main objective of PLS is to explain X space, Y space and the greatest relation between the two. For the present study, PLS is performed using the JMP software. The prediction formula obtained for mean and variance are given by Equations (7) and (10). y 1 ¼ 264:73163 þ 59:26697 7:58169e À11 x 1 À 4:8084e À7 x 2 À þ1:9651e À8 x 3 À 1:03225e À2 x 4 þ 1:032254e À2 x 5 À 5:6080078e À3 x 6 À 4:7682615e À3 x 7 þ 1:603318e À9 x 8 À 2:25189382e À7 x 9 þ 7:1562e À3 x 10 À 2:19540e À2 x 11 þ 2:4598e À2 x 12 þ 2:401391e À4 x 13 À 0:14032033x 14 À 3:01809e À3 x 15 þ 9:1029e À3 x 16 Þ ð9Þ y 2 ¼ 18903:738728414 þ 3554:7815 À5:75194e À9 x 1 À þ7:498057e À6 x 2 þ 2:19167e À7 x 3 À 0:055827x 4 þ 0:0558271x 5 þ 0:220175x 6 À 0:172117x 7 þ 3:703774761e À10 x 8 þ 3:036815e À6 x 9 À 0:2695x 10 À 0:022033x 11 À 0:0016909x 12 þ 0:000338473x 13 þ 0:085650x 14 þ 0:00142608x 15 À 0:00295370x 16 Þ ð10Þ Figure 6 shows the prediction efficiency for mean and variance. In particular, two notable strategies that largely helped mitigate the pandemic spread were the proactive approach taken by Denmark 26 and the high testing and contact tracing approach implemented by South Korea. 27 It is also of note that Djibouti managed to control the spread despite being the only lower-middle income country in our analysis, largely due to its response plan being aligned with WHOs four pillars (testing, isolating, early case management, and contact tracing). 28 To summarize, the key common factors leading to mitigation in the pandemic spread were proactiveness in implementing model-based data-driven decisions in policymaking and effective communication and trust between the governments and the public. We believe that the presented regression analysis-based approach can be used to predict the curve characteristics for different country. This will help us to estimate the level of the pandemic and plan for the suitable strategies to avoid the spread. The data that supports the findings of this study are available in the supplementary material of this article. 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