While effective prediction methods of the future dynamics of the COVID-19 pandemic can significantly improve the quality of the outbreak`s containment, the number of such models specifically for Ukraine is rather low. We applied a compartment epidemiological model with heuristics along with machine learning techniques in order to create an effective method of modeling and prediction of the COVID-19 epidemic in Ukraine. The stages of the proposed method are building a SEIRD compartment model with vital dynamics, estimating its parameters, calculating and predicting the difference between the SEIRD model solution and the observed data using the ARIMA model, and adjusting model prediction using this newly obtained data on the residuals. The proposed method was tested on the data on the epidemic`s dynamic in Ukraine obtained from a Ukrainian finance analytics website. The validation results indicate the method`s aptitude to real-world usage.
As the coronavirus pandemic continues to rattle the world, humanity craves for means to alleviate the situation if not overcome the crisis entirely. Quality estimations and predictions of future dynamics of the disease spread will ensure better prevention and thorough preparation for exacerbations of the problem (such as expected rises in infection cases after the holidays or lockdown lifts). Rational use of resources may help avoid future boiling points for the healthcare and other systems critical to the delivery of the COVID-19 response.
While the patterns of the epidemic`s dynamics may be similar across countries, each country has specifics in demographics, economics, epidemic containment methods, amount of available resources, and cultural particularities, and therefore should be considered separately by researchers and scientists aiming for creating models with potential for practical usage. As shown in Figure 1, the World Health Organization reports that Ukraine has one of the highest numbers of daily increase in the number of infected individuals. Multiple models have been proposed as methods for modeling and prediction of the epidemic around the world. In contrast, the papers count for Ukraine remains relatively low. Perfecting the techniques of epidemic modeling specifically for Ukrainian statistics by independent researchers will accelerate the process of finding optimal tools and algorithms for the best possible results in models` performance. Networking and spreading awareness on novice helpful solutions and findings are crucial to this process.
The SEIR model replicates the “time-history” of any epidemic or pandemic outbreak, and it
presents the model of dynamic interaction between people with four different health conditions or
phases of the pandemic, namely the susceptible (S), exposed (E), infective (I), and recovered (R).
SEIRD model, as a generalization of the SEIR model, has an additional variable – Deceased
individuals. A “Formal Characterization and Model Comparison Validation” based on the SEIRD
model, which uses the data from Korea and Spain, is proposed by Casas et al. [
A data-driven model to predict the spread of Covid-19 for an upcoming week using the SEIRD
model is studied and tested for datasets obtained from Italy, India, and Russia [
The proposed method was tested on the data on the epidemic`s dynamic in Ukraine obtained from
a Ukrainian finance analytics website [
May 7
September 28 1 73 1319 13691 24012 60995 110085 201305 1. Cumulative infected people as of each date (the total number of diagnosed people until each date); 2. Cumulative recovered people from the start of the outbreak (the total number of no longer ill people who gained immunity until each date); 3. Deceased people from the start of the outbreak. The key dates in the dynamics of the outbreak are stay-at-home advisories enactments and other government-enforced restrictions (March 12, March 23, April 6, July 22, August 26) and their lifts (May 7, June 1). The last day of observation used while building this method is September 28. The observed data as of those dates is reported in Table 1. The data as of later dates (up until October 19) is used for validation of the proposed method. Since lockdown and other introduced measures didn`t significantly drop the outbreak`s spread rate, they aren`t considered in the proposed model, and basic model parameters are proposed to take as fixed. Due to the small number of cases of re-infection, all recovered individuals are assumed to have absolute immunity against COVID-19.
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Mortality Rate Expon. Trend Line
As reflected in Figure 2, the COVID-19 mortality rate has decreased and stabilized over time, which was reflected in the dynamic model. This can be explained by continuous scientific efforts to cure the disease more efficiently as well as the proportion of asymptomatic and undiagnosed cases that aren’t reflected in statistics. The relatively stable mortality rate observed in later months proves the disease to be lethal to a small portion of the population and is expected to stay at this level or slightly decrease. The data instances used while working with the model are represented in percents of the country`s population. 2.2.
Upon investigation, we introduce a novice model based on an enhanced SEIRD model and ARIMA model. As shown in Figure 2, the stages of the proposed method are building a SEIRD compartment model with vital dynamics, estimating its parameters, calculating and predicting the difference between the SEIRD model solution and the observed data using the ARIMA model, and finally adjusting model prediction using this newly obtained data on the residuals.
This model consists of such stages: 1. At the first one, we estimate SEIRD model parameters using historical data, trying to lessen the difference between the model`s output and observed data. This model is responsible for longterm prediction (i.e., 60 days or 100 days). 2. Calculate residuals between observed infected, recovered, and deceased percentage of the population and corresponding solutions of the SEIRD model. 3. Build three ARIMA models on the time-series of each of these residuals. Prediction of these ARIMA models will compensate residuals between the SEIRD model and historical data in order to make predictions mode accurate. 4. Validate the prediction of the obtained model using the data on the number of infected, recovered, and deceased individuals as of the most recent days, data on which was not included while working with the model on previous stages. 2.3.
A basic compartment model in epidemiology is the SIR model [
A SEIRD model simulates the flow of the population between Susceptible, Exposed, Infected, Recovered, and Deceased groups (or compartments). While traditionally compartment models are built for closed systems, in this method, the total population size is not fixed due to the introduction of birth and mortality rates. This allows us to model the pandemic more accurately. The COVID-19 mortality rate is represented by an inverse exponential function with two parameters rather than a constant. Based on the analysis shown in Figure 1, it was proved to be useful to model mortality rate as an inverse exponential function, which is another heuristic to the proposed method for the same reason.
The compartments of the model are as follows:
( ): Susceptible individuals - stock of healthy people who may be infected; population inflow due to births is taken into account.
( ): Exposed individuals - virus carriers in the latent stage, during which they are not virus spreaders. Usually corresponds to an asymptomatic phase of the disease.
( ): Infectious individuals - virus carriers able to spread the disease to individuals in contact with them.
( ): Recovered individuals - stock of healthy people who are immune to COVID-19. ( ): Deceased individuals - population loss due to the disease, natural deaths included. The model itself is comprised of a system of differential equations: = = = − µ − − (µ + ) − ( ç + µ) = (1 − µ
( )) − µ = µ ( ) (1) with constraints at time t=0 S= 0, E= 0,I= 0,R = 0, D= 0 and parameters µ of infection; formula µ – population`s birth rate; µ – population`s mortality rate; susceptible and an infectious individual; ( ) =
− .
– rate of virus transmission, which is the probability of transmitting disease between a – rate of latent individuals becoming infectious (average duration of incubation is 1/ ); ç – recovery rate, which can be initially estimated as = 1/ , where is the average duration ( ) – death rate due to COVID-19, which is estimated by an inverse exponential The population size ( ) = ( ) + ( ) + ( ) + ( ) is not fixed due to its global birth and mortality rates taken into account at any given time t. 2.4.
To use the model proposed in the previous section, firstly, we need to specify its parameters so it will fit the historical data. Moreover, we estimate not only the model parameters but also initial conditions for susceptible and exposed compartments of the model. The reason of it that we still don’t know the percentage of the population that is insusceptible to the virus (they will suffer from the disease in a mild form and don’t infect others). Regarding the exposed population, we also don’t have the exact number of exposed passengers that came to Ukraine at the time of the COVID outbreak.
As soon as the dataset consists of cumulative data, we calculated the number of currently infected individuals as a difference between cumulative infected and recovered ones. After this step, data was rescaled from the absolute numbers to the percent of the population.
To fit model parameters and initial conditions, we use the Basin-hopping algorithm [
One of the key reasons for choosing this instrument is the algorithm`s ability to reach global
optima even after finding several local ones, as it is not restricted to the best candidates at each step.
As a measure of quality between differential equation solution and historical data, we use MAE/mean
metrics that were described and investigated in [
and co-dependencies of observations. subtracted from the value at time t-1.
In this step, the difference between data by SEIRD algorithm and observed data is estimated and corrected using the ARIMA model (stands for Auto-Regressive Integrated Moving Average).
The structure of this model includes autoregression and moving average as the main components. The autoregression algorithm uses a certain number of past data instances (also called the number of lagged observations) to make a prediction about variable value at each new point, exploring trends
Differentiation of raw data is performed to ensure stationarity of variable: each value at time t is stationarity and seasonality is used. in order to increase its performance. 2.6.
The third part, moving average, also makes use of dependencies in the data, but this time between an observation and a residual error from applying the moving average algorithm to a number of lagged observations.
To each of these parts corresponds a parameter [
In our case, an algorithm that finds the best set of parameters and runs statistical tests of The obtained prediction of residuals is subtracted from data predicted by the compartment model During the validation stage, we gather new data that was not used in SEIRD model parameter estimation and ARIMA models fitting. We will use such measures of quality:
Mean average error, given by equation (2) (3) =1 =1 Mean squared logarithmic error, given by equation ( , ̂) =
( , ̂) ( )
( , ̂) ( ) ⋅ ( ̂)
finance analytics website [
Optimized
value (7) In this section, we will provide results of hybrid model approbation on data from the Ukrainian In this subsection, we estimate some parameters and initial conditions of the SEIRD model using the Basin-hopping algorithm and build rough long-term predictions of pandemic development. We optimize only initial values of susceptible and exposed fraction of the population, whilst infected, recovered, and deceased initial conditions are set to zero. Global birth and death rate are also not optimized and are set according to actual values for the annual 2020 birth and death rate in Ukraine.
ç 0 0
estimated as = 1/ , where is the average duration of infection
enhancements in treatment
0 0 0 0 0 0.4 0.1 1 0.1 0.3 0.1 1
In Table 2, boundaries and optimized values for the SEIRD model parameters and initial values disease in a mild form or even asymptomatically. Interestingly recovery rate is very low, which means that if a person suffers from the disease in a severe form, it takes a lot of time to recover. The rate of becoming infectious is also shallow, which proves that it takes a lot of time for the disease to be able to spread itself since acquiring a new host - the incubation period of COVID-19 is quite large. While all of the parameters have a real-life context to them and represent rates of transitions between compartments and initial conditions of the SEIRD model, they were estimated using mathematical algorithms, and that worked with available data that doesn’t entirely reflect the reality. Therefore, the estimated values of some parameters such as the incubation period and recovery rate may differ from the data collected at hospitals and estimates of other researchers.
The pure SEIRD model can be used for the long-term rough predictions of the pandemic dynamic.
In Figure 4, long-term predictions for infected, recovered, and deceased fractions of the population are displayed. Based on the figures, we can conclude that number of infected people will continue to rise till summer 2021with a relatively stable rate.
Based on Figure 4 and Table 3, we can conclude that the SEIRD model fits historical data quite well. The best fit is observed for recovered and infected compartments of the model. Unnormalized measures are the lowest for the infected fraction population, which is the most informative data timeseries among the studied ones. 3.2.
At this step, we calculate residuals between the fitted SEIRD model and historical data and train
ARIMA models on the residuals for each category (infected, recovered, deceased). While having its
limitations [
Table 4
autoregressive model (P) 0 0 0
The degree of differencing (D)
As shown in Figure 6, until early August, the losses from COVID-19 are quite accurately modeled. It is safe to assume that some people who passed away due to the disease were undiagnosed or misdiagnosed.
Therefore the data on those cases was not taken into account in COVID statistics, which explains the observed number of deceased people being slightly lower. In later months we observe a gradual rise – the medical system isn`t well prepared for the pressure of the pandemic and struggles to cope with the growing inflow of patients.
Hopefully, there will be a decline in the COVID death rate due to the development and spreading of treatment protocols and medical research that allow selecting the most effective medicine. In the meanwhile, despite all the measures of previous months, the predicted number of deceased individuals rises quite sharply.
The proposed method describes the observed number of recovered individuals very accurately (Figure 7) with some minor deviations, while in the future stages of the outbreak, the number of people recovered is expected to be lower than the SEIRD model suggests. It can be explained by a lack of techniques and materials to treat the patients and the already beginning congestion of the medical system of the country. 3.3.
Validation of any method is an essential step that helps understand how the final model will perform in the future with new previously unseen data. The method was validated on the most recent data - the last three weeks (from 29.09.2020 to 19.10.2020) of the pandemic. The validation dataset was taken from the same source and therefore has the same structure.
As shown in Table 5, all measures of the prediction quality for the infected, recovered, and deceased fractions of the population are very low. Normalized MAE values show that: 1. Average difference between the actual number of infected individuals and predicted one is only 3.6%; 2. Average difference between the actual number of recovered individuals and predicted one is only 11%; 3. Average difference between the actual number of deceased individuals and predicted one is only 8.4%; 4. Based on the maximum deviation column, we can conclude that for the next 60 days starting from the last day of model training: 5. Maximum deviation between the predicted and actual number of infected individuals will not exceed 8.6% with the probability of 95%. 6. Maximum deviation between the predicted and actual number of recovered individuals will not exceed 15.4% with the probability of 95%. 7. Maximum deviation between the predicted and actual number of deceased individuals will not exceed 15.5% with the probability of 95%.
The proposed hybrid model consists of a dynamic SEIRD model with vital dynamics and decaying COVID mortality rate and three ARIMA models that cancel out dynamic model residuals and enhance prediction quality. The model was tested on Ukrainian COVID statistic data. Obtained validation results allow us to draw conclusions that the proposed hybrid model has good prediction ability and decent performance. Obtained long-term predictions reflect the general dynamic of the outbreak and are especially useful for the healthcare system workers and government officials. Obtained short-term predictions allow us not only to forecast the future number of infected, recovered, and deceased patients but only estimate forecast error under adverse or optimistic circumstances.
Key method`s standouts include: 1. Using a Basin-hopping algorithm to fit parameters and initial conditions of the model for this specific disease. 2. Including into the SEIRD model exponentially decaying mortality rate, which reflects historic dynamics over the year of 2020. 3. Correction of model residuals using the ARIMA model with automatically selected parameters.
Here are some perspective ways of further development of the proposed method: 1. Parameter estimation with different algorithms and boundaries; 2. Testing the method on COVID statistics other countries; 3. Develop alternative methods for residue prediction.
Enhancing the proposed hybrid model depends on profound research results about COVID-19. That’s why monitoring recent research in the field and quickly adjusting the model according to the new data is crucial.
In conclusion, the proposed method has proved its predictive capability and can be used as an effective tool for prediction and analysis of the dynamics regarding the number of infected, recovered and deceased individuals due to the COVID-19 pandemic in Ukraine. The predicted optimistic and pessimistic scenarios of the infection spread for the nearest future are very similar, so we can conclude with sufficient confidence. Unfortunately, these conclusions give reasons to believe that the most difficult times are still ahead of us. Such results are extremely important in terms of planning disease containment measures on all levels - from governmental to personal. The analysis of obtained data indicates the forthcoming of a crisis - most importantly, in medical and economical spheres, and naturally suggests that all possible rational preemptive actions should be taken immediately.