key: cord-0689493-7o5zfzim authors: Islam, M. S.; Hoque, M. E.; Amin, M. R. title: Integration of Kalman filter in the epidemiological model: a robust approach to predict COVID-19 outbreak in Bangladesh date: 2020-10-20 journal: nan DOI: 10.1101/2020.10.14.20212878 sha: 3349225738d16a5b305872952e6265d5027adfd3 doc_id: 689493 cord_uid: 7o5zfzim As one of the most densely populated countries in the world, Bangladesh have been trying to contain the impact of a pandemic like COVID-19 since March, 2020. Although government announced an array of restricted measures to slow down the diffusion in the beginning of the pandemic, the lockdown has been lifted gradually by reopening all the industries, markets and offices with a notable exception of educational institutes. As the physical geography of Bangladesh is highly variable across the largest delta, the population of different region and their life style also differ. Thus, to get the real scenario of the current pandemic across Bangladesh, it is essential to analyze the transmission dynamics over the individual districts. In this article, we propose to integrate the Unscented Kalman Filter (UKF) with classic SIRD model to explain the epidemic evolution of individual districts in the country. We show that UKF-SIRD model results in a robust prediction of the transmission dynamics for 1-4 months. Then we apply the robust UKF-SIRD model over different regions in Bangladesh to estimates the course of the epidemic. Our analysis demonstrate that in addition to the densely populated areas, industrial areas and popular tourist spots are in the risk of higher COVID-19 transmission. In the light of these outcomes, we provide a set of suggestions to contain the pandemic in Bangladesh. All the data and relevant codebase is available at https://mjonyh.github.io. 1 Introduction 1 respiratory droplets or small airborne droplets [3] . Usually the virus infected person 14 shows symptoms like fever, shortness in breath, cough, diarrhea, pneumonia and loss of 15 smell, whereas the corona positive patient with premedical conditions such as diabetics, 16 pneumonia and high blood pressure might experience the fatal death if proper medical 17 treatment cannot be sought for in time [4, 5] . Since there are no recommended 18 medicines or remedies available for treating COVID-19 disease as of now, many 19 countries are setting up an array of measures including wearing face mask, early 20 detection of infected people, contact tracing, isolation of infected people, increasing the 21 number of testing per day, and imposing lockdown to curb the COVID-19 22 outbreak [6, 7] . Recent studies also support the effectiveness of wearing face mask, social 23 distancing, and imposing lockdown on the declining of both infection and death rate [8] . 24 Bangladesh is one of the most densely populated (ranked 8 th worldwide) and under 25 developing countries in the world. The recent economic development of this country 26 makes it very competent in the South Asian region, especially in terms of foreign 27 exchange reserves, reaching as high as second position in South Asia [9] . The first 28 COVID-19 positive case was identified in Bangladesh on March 8, 2020 . The 29 Government of Bangladesh imposed a set of strict measures and closed all educational 30 institutions, government, non-government offices including garments and factories [10] 31 on March 26th, 2020. As a result, the number of infection and fatality rates are less 32 compared to India and Pakistan. In Bangladesh, the contamination numbers spilled 33 over 275,000 confirmed cases with death toll of more than 3,500 and it is still growing 34 each day [11] . 35 To understand the dynamics of a pandemic, mathematical model can play a major 36 role. The classic mean-field Susceptible-Infected-Recovery-Death (SIRD) model by 37 Kermack and McKendrick [10, 12] , is frequently used to illustrate a quantitative picture 38 of COVID-19 outbreak for many countries [6] . However, spread of an epidemic is a 39 sophisticated process that depends largely on the mutated strains of the virus and its 40 principle vectors [13, 14] . Moreover, the recorded data of infected, recovered and death 41 cases might miss significant amount of information due to many kinds of biases [15] . For 42 instance, time series of recovered numbers are heavily unreliable as it is solely based on 43 how a country traces its asymptomatic or mildly symptomatic patients. Therefore, 44 usual mean-field, compartment-based models and stochastic spatial epidemic models 45 might estimate the dynamics of an epidemic with higher error margin [16] [17] [18] [19] . The SIRD model is associated with a set of differential equations which compute 47 values for an instantaneous event, whereas the time series of COVID-19 cases that we 48 observe everyday is a discrete event. Therefore, all the daily events of COVID-19 cases 49 lack the instantaneous effect in the differential equation based SIRD model. Here, we 50 use the prediction based Unscented Kalman Filter (UKF) model to derive the dynamics. 51 UKF is a classic non-linear estimation algorithm that accurately and timely forecasts 52 the dynamic state in a nonlinear system. UKF has been used in various areas such as, 53 navigation, target tracking, structural dynamics and vehicle positioning due to its high 54 accuracy and rapid convergence merits [20, 21] . In this study, we integrate the UKF with classic SIRD model to examine the 56 underlying process of COVID-19 transmission in a more precise way. As Bangladesh has 57 64 districts which are uneven in terms of population density, economical significance 58 and geographical locations, we estimate the transmission trend of 64 districts to 59 evaluate the complete picture of the outbreak in Bangladesh. We used the standard SIRD model to estimate the number of COVID-19 active cases, 62 recovered cases and death cases in Bangladesh. A homogeneous immunity has been 63 October 14, 2020 2/13 considered for the whole country with no transmission from animals and no significant 64 difference between natural death and birth. We segmented the total population size, N , 65 into four stages of disease: S, susceptible; I, infected; R, recovered; D, death and can 66 be written by N = S + I + R + D [10, 12, 22, 23] . The interaction between these four 67 stages is illustrated in the Figure 1 . The differential equations of SIRD model can be written as: Where t denotes the time duration, S(t) represents the number of susceptible people 70 at time t, I(t) shows the number of people infected at time t, R(t) stands for the 71 number of people who have recovered from the infection and D(t) indicates the number 72 of deaths. The population, N = S(t) + I(t) + R(t) + D(t), is a conserved quantity for 73 every time step [24, 25] . The constant β, γ and µ illustrate the rate of infection, 74 recovery and deaths, respectively. The Eq. 1 -4 have been discretized by forward Euler method as , An augmented state vector X is presented for simplicity as, October 14, 2020 3/13 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted October 20, 2020. . and the discrete-time augmented SIRD model, Eq. 5 -8, can be written as, where F is the nonlinear term and W is the zero mean Gaussian uncertainty with 79 covariance Q f . All types of reported cases, the cumulative confirmed, recovered, deaths 80 and active cases, are incorporated with the model using the vector as, where ν is the uncertainties due to the SERS-CoV-2 test results. The considered 82 zero mean Gaussian uncertainty has the known covariance R f . Let us consider an estimated vector stateX t for the Unscented Kalman Filter where J F (X t ) is the Jacobian matrix of F (X t ), given by, Hence the algorithm of UKF is given by, Predict: Update: Here, P (t|t) and C represent the posterior estimate covariance matrix and the data 94 augmented matrix respectively. We integrated this updated formula of UKF in SIRD 95 model to estimate the dynamics of the SERS-CoV-2 in Bangladesh. The dynamical behaviour of the class of infected people is described by the basic 97 (R 0 ), effective (R e ) and time-varying (R t ) reproduction number. In the case of SIRD 98 model, this are defined as, Using Eq. 2, one can find that S(t) = N/R 0 and R e = 1, at I = I max . Besides, one 100 can estimate the onset confirmed (C ) cases as, C = Ce γ (Rt−1) = Ce θ , where 101 θ = γ (R t − 1) observes random walk and γ vary independently [26] . Thus, October 14, 2020 4/13 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 20, 2020. . If this reproduction number becomes larger than the ratio between the total 103 population and initial susceptible people, there will be a proper epidemic outbreak [27] . 104 Otherwise, the disease will never lead to a proper outbreak. cases at 100 days (third week of June, 2020) relative to 90,000 real cases, whereas SIRD 114 model estimates 120,000 active cases at the same time in Bangladesh. Therefore, these 115 results illustrate the fact that integrated UKF in SIRD model is a robust method which 116 is capable of exclusive prediction of COVID-19 transmission in Bangladesh. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 20, 2020. . cases. In addition, the daily growth rate also follows a high trajectory trend for Dhaka 135 compared to Narsingdi, Shariatpur and Bagerhat, as shown in the Fig. 3 approaching to < 1.0 for all twelve districts in Fig. 3 .1, which could lead to the end of 144 this pandemic if the current trend continues for the next 3-4 months. October 14, 2020 6/13 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 20, 2020. . https://doi.org/10.1101/2020.10.14.20212878 doi: medRxiv preprint . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 20, 2020. . https://doi.org/10.1101/2020.10.14.20212878 doi: medRxiv preprint . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 20, 2020. . https://doi.org/10.1101/2020.10.14.20212878 doi: medRxiv preprint These sectors are playing significant role in national economy by contributing 18% to 172 country's GDP as mentioned in the Bangladesh Economic Review. Thus, next we 173 estimate the COVID-19 cases in several districts of Bangladesh that are highly 174 associated with agricultural and fisheries sectors. The UKF-SIRD model estimates 175 ∼ 1, 500 confirmed cases at the end of October for Noakhali district compared to 176 ∼ 1, 000 cases for Patuakhali and Dinajpur region. In addition, haors (large shallow 177 water body or backswamp) and wetlands area, such as Sunamganj has approximately 178 ∼ 1, 200 cases, whereas Satkhira has ∼ 700 Corona virus positive cases in October, 2020. 179 Notably, the model predicts lower number of confirmed cases (< 1, 000) and a smaller 180 growth rate for high lands areas namely, Rangamati and Khagrachhari districts. Since 181 these districts also have low R t values (< 0.9) at the moment, there is a high chance 182 that Rangamati and Khagrachhari will wipe out the COVID-19 outbreak if the current 183 transmission trend continues for next 3-4 months. Taken together, this study represents 184 that the COVID-19 pandemic is less serious in agricultural and high land areas 185 compared to industrial areas in Bangladesh. October 14, 2020 9/13 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 20, 2020. . https://doi.org/10.1101/2020.10.14.20212878 doi: medRxiv preprint Epidemiological models are the powerful tools that are designed for the mathematical 188 representations of the disease transmission and their associated dynamics [28, 29] . In 189 addition, models can be used to estimate the trend of an epidemic outbreak under 190 certain conditions so that necessary actions can be taken at various scales to restrain 191 the disease diffusion in a community [30] . However, to make a series of policy decisions, 192 it is a critical prerequisite to have a robust epidemiological model that is suitable for the 193 intended purpose and shows accurate and precise prediction of a pandemic [31] . In this 194 study, we integrated UKF with classic SIRD model and estimate COVID-19 diffusion in 195 different parts of Bangladesh. Dhaka is the 6 t h largest densely populated city in the World, and the key financial 197 hub which holds the major portions of the Bangladesh's economy [32, 33] . Unfortunately, 198 we observe that Dhaka is the epicenter of the COVID-19 pandemic with the highest 199 number of infected cases due to its high population density in Bangladesh. Although 200 the government has implemented zone-coded lockdown in small areas within the Dhaka 201 city [34] , we observe that the mobility between zones could not be maintained strictly 202 due to economic activities. If the government had the capacity to test and isolate each 203 positive cases, only then the zone-coded lockdown system in Bangladesh could have 204 been successful. Several reports show the association of population density with higher 205 rates of transmission, infection, and mortality from COVID-19 [35, 36] . Strikingly, our 206 nationwide analysis of COVID-19 dynamics also demonstrate the relationship between 207 population density and the COVID-19 diffusion rates for Bangladesh. with the current trend. Therefore, industrialists from Chattogram must ensure proper 214 guideline for social distancing and the hygiene rules at the industrial areas to save their 215 workers. Moreover, to reduce the public mobility, we suggest that public transportation 216 between high risk areas should also be suspended for a time until the situation gets 217 under control. To support this claim we would like to mention that the high rate of 218 COVID-19 positive cases in the New York city of USA could have been controlled by 219 implementing the proper lockdown and public mobility suspension [37] . Since Bangladesh is in emergency situation, we deal with incomplete and raw data of 231 the current outbreak. For instance, if a person from a family becomes COVID-19 232 positive, the whole family might turn out to be positive. But, in many families only one 233 of the members were tested for COVID-19 and if positive then counted as one despite 234 the necessity to test for the entire family. Moreover, there were many cases who died 235 with COVID-19 symptoms, but could not be tested, and conversely all the death cases 236 counted in the analysis due to COVID-19 might have died by other diseases. In 237 addition, the recovered cases count only considered those who went through two 238 negative COVID-19 test results. Therefore, our UKF-SIRD model might not exemplify 239 the pandemic situation accurately. 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