Impact studies of nationwide measures COVID-19 anti-pandemic: compartmental model and machine learning
IImpact studies of nationwide measures COVID-19anti-pandemic: compartmental model and machine learning ∗ Mouhamadou A.M.T. Baldé a , Coura Baldé b and Babacar M. Ndiaye a University of Cheikh Anta Diop.BP 45087, 10700. Dakar, Senegal. a Laboratory of Mathematics of Decision and Numerical Analysis (LMDAN).Department of Mathematics of Decision (DMD)[email protected]@ucad.edu.sn b Laboratory of Applied Mathematics (LMA)[email protected]
Abstract
In this paper, we deal with the study of the impact of worldwide nation measures for COVID-19 anti-pandemic. We drive two processes to analyze COVID-19 data considering measures, and then forecastingis done for Senegal. We show a comparison between deterministic and two machine learning technics forforecasting.
Keywords—
Nationwide measures analysis, COVID-19, SIR model, fitting, forecasting, machine learning
1. I
NTRODUCTION
The COVID-19 pandemic is testing the entire world so that measures are being taken in most nations tostem its development. These measures can generally be of different kinds such as social distancing, partialor total confinement, etc. It would therefore be interesting to be able to effectively analyze the effects ofthe measures taken on the spread of the pandemic. Here we offer an analysis of the impact of the measurestaken that we apply to the case of Senegal. In previous papers [4], [12] and [13] we proposed a start of studyusing the results of [10].In [10], a useful method for the study of the evolution of COVID-19 pandemic has been resented by using acompartmental model with Susceptible, Infected, Infected reported and unreported (SIRU). In [4], the authoruse that method to study the COVID-19 spread in Senegal with a classical SIR model.In this work we aim to analyze the impact of the anti pandemic measures taken in Senegal. It is a continuationof the work done in [4]. We do a two-step analysis. The first uses the SIRU epidemic model introducedin [10], and the second uses two machine learning tools: Predict of Wolfram Mathematica using NeuralNetworks method and Prophet.We conduct the work in the following way. In the first section we will study the effect of the nationwidemeasures, using a model presented in [10] and machine learning tools. We will show, in the second section,the numerical results. Then in the third section, we will discuss the results. Finally we will end with theforth section, making a conclusion and advancing perspectives. ∗ Support of the Non Linear Analysis, Geometry and Applications (NLAGA) Project a r X i v : . [ q - b i o . P E ] M a y ANALYSIS
2. A
NALYSIS a. Analysis of the measures
In [4] a classic SIR model was studied using results from the paper [10]. The aim was to analyze theeffect of the nationwide measures using the data after nationwide measures. In fact, throughout t to T ,we fitted an exponential function to the data of the total cases of infection of this period. T represents thedate of the nationwide measures, and t is the starting time of the epidemic. We consider that the effectsof the measures are such that they lead to a reduction in the contact rate. To describe this reduction, wechose a slowly decreasing function over time. We consider that the measures taken are not strong enough tosystematically drop the contact rate to 0. This new function corresponds to the first one and the data in theperiod before measures from t to T , then takes a slower trajectory than that of the first function. In otherwords, from the date T , the new curve goes under the old one. We consider that if on the dates t > T thedata goes under the new curve obtained with a contact rate after measures then, we can say that they affectthe evolution of the pandemic.In the previous paper [4], the function we fitted to the data from 2020 March 02 to March 31 by least squaremethod, is T NI ( t ) = b exp ( ct ) − a with a = . , b = . c = . a = . , b = . , c = . (a) Fit with data of the total cases: T NI ( t ) is the blueline and the data are the red dotted. From 2020 March02 to March 31. (b) Fit with data of the total cases: T NI ( t ) is the blueline and the data are the red dotted. From 2020 March02 to April 25. Figure 1: Plot of the exponential curves fitting the total number of case Senegal’s data.To continue in our analysis, we will use an epidemiological model introduced in [10]. This model is asfollows: dSdt = − β S ( t )( I ( t ) + I U ( t )) dIdt = β S ( t )( I ( t ) + I U ( t )) − ν I ( t ) dI R dt = γν I ( t ) − η I R ( t ) dI U dt = ( − γ ) ν I ( t ) − η I U ( t ) (1)The initial conditions are S ( t ) = S ≥ , I ( t ) = I ≥ I R ( t ) = I R ≥
0, and I U ( t ) = I U ≥ Analysis of the measures 2 ANALYSIS
Figure 2: Plot of data of total confirmed, death and recovered from 2020 March 02 to May 12.We can also consider a version with a removed compartment: dSdt = − β S ( t )( I ( t ) + I U ( t )) dIdt = β S ( t )( I ( t ) + I U ( t )) − ν I ( t ) dI R dt = γν I ( t ) − η I R ( t ) dI U dt = ( − γ ) ν I ( t ) − η I U ( t ) dRdt = αη I ( t ) dDdt = ( − α ) η I ( t ) (2)Let’s present the parameters. β is the contact rate. 1 / ν is the average time during which asymptomaticinfectious are asymptomatic. γ is the fraction of asymptomatic infectious individuals that become reportedsymptomatic infectious. 1 / η is the average time symptomatic infectious have symptoms. γν is the rate atwhich asymptomatic infectious become reported symptomatic. ( − γ ) ν is the rate at which asymptomaticinfectious become unreported symptomatic. α is the proportion of recovered and 1 − α is the proportion ofdeath due to the infection.We make the assumption that the function we fit the total number of reported infected cases is given by γν (cid:90) tt I ( s ) ds .Following the calculation in [10] and [4], we have: 3 Analysis of the measures 2 ANALYSIS • t = ln ( a ) − ln ( b ) c . • I ( t ) = I ( t ) exp ( c ( t − t )) , with I ( t ) = I = ac γν . • I U ( t ) = I U ( t ) exp ( c ( t − t )) , with I U ( t ) = I U = ( − γ ) ac γ ( η + c ) = ( − γ ) νη + c I . • I R ( t ) = I R ( t ) exp ( c ( t − t )) , with I R ( t ) = I − I U = ac η + c = γνη + c I . • R ( t ) = αη ( I ( t ) − I ) c . • D ( t ) = ( − α ) η ( I ( t ) − I ) c . • β = c + ν S η + c ( − γ ) ν + η + c . • R = c + νν η + c ( − γ ) ν + η + c ( + ( − γ ) νη ) .We consider that after the measures taken at the time T , the contact rate depends on time following a formulawe choose. We use three formulas. One of them was introduced in [4], and the second one was proposed in[11]. The first one is : ˜ β ( t ) = (cid:40) β if t ∈ [ t , T ] β ( Tt ) δ / p if t > T , (3)where δ and p are parameters to choose. The second one is:˜ β ( t ) = (cid:26) β if t ∈ [ t , T ] β exp ( − ϕ ( t − T )) if t > T , (4)where ϕ is a parameter to choose.Then the new model to solve is: dSdt = − ˜ β S ( t )( R ( t ) + U ( t )) dIdt = ˜ β S ( t )( R ( t ) + U ( t )) − ν I ( t ) dI R dt = γν I ( t ) − η I R ( t ) dI U dt = ( − γ ) ν I ( t ) − η I U ( t ) (5)By solving (4), we get ˜ I ( t ) from which we can calculate the total number of infected with the formula˜ T NI ( t ) = γν (cid:90) tt I ( s ) ds . We choose values of parameters of ˜ β such that ˜ T NI ( t ) follows the same path that T NI ( t ) . Then, by way of parameters in ˜ β , we can evaluate the measures.Now we study the maximal value of infection peak. For that, we consider at time T , the nation appliesstronger measures are taken. We simply interpret as the contact rate ˜ β is reduced by a factor φ ∈ [ , ] .4 Artificial Neural Networks 2 ANALYSIS
When the value of φ is equal to 0, it means that there are no measures, while when φ is equal to 1, it meansthat the strongest measures are taken. Hence the measures are quantified with the values of proportion φ .Then the contact rate become: ˜ β ( t ) = (cid:26) ˜ β ( t ) if t ∈ [ t , T ]( − φ ) ˜ β ( T ) if t > T , (6)We solve the new model with ˜ β replaced by ˜ β . We do a parametric solve concerning the parameter φ , andwe plot the result for different values of φ . Then, we show different values of the peak, depending on thevalues of φ . b. Artificial Neural Networks Artificial neural networks are part of artificial intelligence. Biological neural networks inspire them. Bio-logical neural networks are part of the animal brain. One of the main functions of the brain is to processinformation, and the primary information processing element is the neuron. This specialized brain cell com-bines (usually) several inputs to generate a single output. Depending on the animal, an entire brain cancontain anywhere from a handful of neurons to more than a hundred billion, wired together. The output ofone cell feeding the input of another, to create a neural network capable of remarkable feats of calculationand decision making (See [14]). If we could qualify the brain as a computer, then we would say that it is thebest of computers. For this reason, the engineer seeks to improve mechanical computers to be closer to thebiological computer, i.e., the brain. The more neural connections there are, the more the network can solvecomplex problems. Pattern recognition is a task that neural networks can easily accomplish. For this task,introducing as input a pattern to a neural network, yields as output a pattern back (See [6]).In general, neural network problems involve a dataset used to predict values for later datasets. For that, theneural network needs to be trained. Then, neural networks can predict the outcome of entirely new datasetsbased on training from old data sets.Most neural network structures use some type of neuron, node, or unit. An algorithm called a neural networkwill generally be made up of individual interconnected neurons, see figure 3.The artificial neuron receives input from one or more sources, which may be other neurons or data enteredinto the network from a computer program see figure 4. This entry is usually a floating-point or binary.Often the binary input is coded floating point representing true or false like 1 or 0. Sometimes the programalso describes the binary input as using a bipolar system with true as 1 and false as −
1. An artificial neuronmultiplies each of these inputs by a weight. It then adds these multiplications and transmits this sum to anactivation function given by: f ( x i , w i ) = φ ( n ∑ i = ( w i · x i )) , (7)with the variables x and w represent the input and the weights of the neuron, n is the number of input andweight.Some neural networks do not use an activation function. To read more about Neural Networks one can see[1], [6], [5] and [14].Neural networks are implemented in machine learning tools of many software like Wolfram Mathematica[17].We use the machine learning tool “Predict” to forecast the evolution of the COVID-19 pandemic in Senegal.We can choose different method of regression algorithm: “RandomForest”, “LinearRegression”, “Neural-Network”, “GaussianProcess”, “NearestNeighbors”, etc.We use the “NeuralNetwork” regression algorithm, which predicts using artificial neural networks.We consider two cases in the forecasting. The first case, only use the total number of infected case datain the training of the neural networks. While in the other forecasting, we use two types of data: the total5 Forecasting using Prophet 2 ANALYSIS number of infected cases and the contact rate. In the second case, the aim is to do forecasting by consideringthe effect of the contact rate. It is a way to show the effect of the nationwide measures taken at the time T ,as specified in section 2. For the contact rate we use as data ˜ β given by (3).Figure 3: Example of artificial neuron from [6].Figure 4: Example of artificial neural networks from [6]. I1, I2 and I3 mean input 1,2 and 3. N1, N2 and N3mean neuron 1,2 and 3. O means output. c. Forecasting using Prophet In this section, we develop another machine learning tool for forecasting to compare with the previous SIRUmodels and Neural Networks method. We use Prophet [15], a procedure for forecasting time series databased on an additive model where non-linear trends are fit with yearly, weekly, and daily seasonality, plusholiday effects. 6
NUMERICAL SIMULATIONS
For the average method, the forecasts of all future values are equal to the average (or â ˘AIJmeanâ ˘A˙I) of thehistorical data. If we let the historical data be denoted by y , ..., y T , then we can write the forecasts asˆ y T + h | T = ¯ y = ( y + y + ... + y T ) / T The notation ˆ y T + h | T is a short-hand for the estimate of y T + h based on the data y , ..., y T . A prediction intervalgives an interval within which we expect y t to lie with a specified probability. For example, assuming thatthe forecast errors are normally distributed, a 95% prediction interval for the h -step forecast isˆ y T + h | T ± .
96 ˆ σ h where σ h is an estimate of the standard deviation of the h -step forecast distribution.For the data preparation, when we are forecasting at the country level, for small values, forecasts can be-come negative. To counter this, we round negative values to zero. Also, no tweaking of seasonality-relatedparameters and additional regressors are performed.
3. N
UMERICAL S IMULATIONS a. Numerical analysis
The data for Senegal, we use, is obtained from daily press releases on the COVID-19 from the Ministry ofHealth and Social Action ( ).The figures 5, 6, 7 and 8 show results related to a. The figures 5, 6 correspond to ˜ β β γ = . , ν = / , η = / , the total population of Senegal is N = t = . , I = . , I U = . I R = . S = N − I = . × , R = . β = . × − .The time of the nationwide measures in Senegal taken at 2020, March 23 is considered. Then T =
23. For˜ β given by (3), results are shown by figures 5a and 5b. For ˜ β given by (4), results are shown by figures 7aand 7b. We see that the maximal number of reported case goes up to 340000 with the time peak at t = φ =
0. Hence, we see that the time of the peak becomes T , the date where stronger measures are taken.We choose as sample T =
70 which correspond to 2020, May 10.The figures 5d, 7d, 5e, 7e, 5f, 7f show parametric plot of the infected I ( t ) , the reported I R ( t ) and unreported I U ( t ) and total number of infected ˜ T NI ( t ) , with varying values of the parameter φ .The figures 6 and 8 show again, in different range of the ordinate axis, parametric plot of the infected I ( t ) ,the reported I R ( t ) and unreported I U ( t ) and total number of infected ˜ T NI ( t ) , with varying values of the pa-rameter φ . 7 Numerical analysis 3 NUMERICAL SIMULATIONS (a) Plot of the SIRU model(2) using (3). The fit function tototal number of infected data
T NI ( t ) in red line, The totalnumber of infected case ˜ T NI ( t ) in yellow, the reported case I R ( t ) in blue line, the unreported case I U ( t ) in black line, thefirst fit function to total number of infected data in green line. (b) Plot, with maximal ordinate value fixed to 1000, of theSIRU model(2) using (3). The fit function to total numberof infected data T NI ( t ) in red line and the total number ofinfected case ˜ T NI ( t ) in yellow have the same path.(c) Plot of the SIRU model(2) using (3), with stronger mea-sures taken at a time T and φ =
1. The fit function to totalnumber of infected data
T NI ( t ) in red line and the total num-ber of infected case ˜ T NI ( t ) in yellow have the same path. (d) Parameric plot of the infected case I ( t ) of the SIRUmodel(2) using (3), with stronger measures taken at a time T . From top to down coresponding to increasing values ofthe parameter φ from 0 to 1 by step of 0 . I R ( t ) and unreported I U ( t ) infected case of the SIRU model(2) using (3), with strongermeasures taken at a time T . From top to down corespondingto increasing values of the parameter φ from 0 to 1 by stepof 0 .
1. (f) Parameric plot of the total number infected case ˜
T NI ( t ) ofthe SIRU model(2) using (3), with stronger measures takenat a time T . From top to down coresponding to increasingvalues of the parameter φ from 0 to 1 by step of 0 . Figure 5: Plot and parametric plot of the SIRU model(2), using (3) with δ = p = φ as parameter, fitted to data of Senegal. On the abscissa axis, the graduation 55 represents 2020, April 25.8 Numerical analysis 3 NUMERICAL SIMULATIONS (a) Parameric plot, with as maximum value on the ordinateaxis fixed at 400000, of the reported I R ( t ) and unreported I U ( t ) infected case. (b) Parameric plot, with as maximum value on the ordinateaxis fixed at 100000, of the reported I R ( t ) and unreported I U ( t ) infected case.(c) Parameric plot, with as maximum value on the ordinateaxis fixed at 40000, of the reported I R ( t ) and unreported I U ( t ) infected case. (d) Parameric plot, with as maximum value on the ordinateaxis fixed at 40000, of the reported I R ( t ) and unreported I U ( t ) infected case.(e) Parameric plot, with as maximum value on the ordinateaxis fixed at 10000, of the total number of infected case˜ T NI ( t ) . (f) Parameric plot, with as maximum value on the ordi-nate axis fixed at 5000, of the total number of infected case˜ T NI ( t ) , the reported I R ( t ) and unreported I U ( t ) infected case. Figure 6: View of different sizes parametric plot of the SIRU model(2) using (4), with stronger measurestaken at a time T . From top to down corresponding to increasing values of the parameter φ from 0 to 1 bystep of 0 .
1. On the abscissa axis, the graduation 55 represents 2020, April 25.9
Numerical analysis 3 NUMERICAL SIMULATIONS (a) Plot of the SIRU model(2) using (4). The fit function tototal number of infected data
T NI ( t ) in red line, The totalnumber of infected case ˜ T NI ( t ) in yellow, the reported case I R ( t ) in blue line, the unreported case I U ( t ) in black line, thefirst fit function to total number of infected data in green line. (b) Plot, with maximal value fixed to 1000, of the SIRUmodel(2) using (4). The fit function to total number of in-fected data T NI ( t ) in red line and the total number of in-fected case ˜ T NI ( t ) in yellow have the same path.(c) Plot of the SIRU model(2) using (4), with stronger mea-sures taken at a time T and φ =
1. The fit function to totalnumber of infected data
T NI ( t ) in red line and the total num-ber of infected case ˜ T NI ( t ) in yellow have the same path. (d) Parameric plot of the infected case I ( t ) of the SIRUmodel(2) using (4), with stronger measures taken at a time T . From top to down coresponding to increasing values ofthe parameter φ from 0 to 1 by step of 0 . I R ( t ) and unreported I U ( t ) infected case of the SIRU model(2) using (4), with strongermeasures taken at a time T . From top to down corespondingto increasing values of the parameter φ from 0 to 1 by stepof 0 .
1. (f) Parameric plot of the total number infected case ˜
T NI ( t ) ofthe SIRU model(2) using (4), with stronger measures takenat a time T . From top to down coresponding to increasingvalues of the parameter φ from 0 to 1 by step of 0 . Figure 7: Plot and parametric plot of the SIRU model(2), using (4) with ϕ = − and (6) with φ as param-eter, fitted to data of Senegal. On the abscissa axis, the graduation 55 represents 2020, April 25.10 Comparative forecasting with machine learning 3 NUMERICAL SIMULATIONS (a) Parameric plot, with as maximum value on the ordinateaxis fixed at 400000, of the reported I R ( t ) and unreported I U ( t ) infected case. (b) Parameric plot, with as maximum value on the ordinateaxis fixed at 100000, of the reported I R ( t ) and unreported I U ( t ) infected case.(c) Parameric plot, with as maximum value on the ordinateaxis fixed at 40000, of the reported I R ( t ) and unreported I U ( t ) infected case. (d) Parameric plot, with as maximum value on the ordinateaxis fixed at 40000, of the reported I R ( t ) and unreported I U ( t ) infected case.(e) Parameric plot, with as maximum value on the ordinateaxis fixed at 10000, of the total number of infected case˜ T NI ( t ) . (f) Parameric plot, with as maximum value on the ordi-nate axis fixed at 5000, of the total number of infected case˜ T NI ( t ) , the reported I R ( t ) and unreported I U ( t ) infected case. Figure 8: View of different sizes parametric plot of the SIRU model(2) using (6), with stronger measurestaken at a time T . From top to down corresponding to increasing values of the parameter φ from 0 to 1 bystep of 0 .
1. On the abscissa axis, the graduation 55 represents 2020, April 25. b. Comparative forecasting with machine learning
The forecasting is done with two data set. For both data from March 02, to April 25, 2020 and March 02, toMay 12, 2020 we carry out simulations for a longer time and forecast the potential trends of the COVID-1911
Comparative forecasting with machine learning 3 NUMERICAL SIMULATIONS pandemic in Senegal. The predicted cumulative number of confirmed cases are first plotted for periods untilMay 26, June 10 and Sept. 18, 2020 ahead forecast with Prophet, with 95% prediction intervals.The confirmed predictions for Senegal, using Prophet are given in Figure 11 (see Tables 2, 3 and 4 for thevalue of the confidence interval).The figures 10 shows forecasting using Neural Networks method of Predict. Particularly the figure 10 showstwo forecasting, one based only on data and an other obtained by training the Neural Networks method withdata and contact rate. The prediction are done until 2020, May 26, June 10 and September 18.Table 1: Predicted cumulative confirmed cases using SIRU model. Forecasting until the dates May 26, June10 and September 18, 2020.
Until date
J J − J − J − J − JJ = J = J = (a) Plot until May 26,2020 corresponding to the graduations85. (b) Plot Plot until June 10,2020 corresponding to the gradu-ations 100.(c) Plot until September 18, 2020 corresponding to the grad-uations 200. Figure 9: Plot the function
T NI ( t ) in blue line with data set 1 in red dotted, until the dates 2020, May 26,June 10 and September 18 corresponding to the graduations 85, 100 and 200 on the abscissa axis.12 Comparative forecasting with machine learning 3 NUMERICAL SIMULATIONS
Table 2:
Data set 2. Predicted cumulative confirmed cases ∼ May 26, 2020, from top to down using Prophet and NeuralNetworks method of Predict. On the down, from left to right with data set & contact rate and data set only.
Date ˆ y ˆ y lower ˆ y upper P β P β lower P β upper P P lower P upper Table 3:
Data set 2. Predicted cumulative confirmed cases ∼ June 10, 2020, from top to down using Prophet and NeuralNetworks method of Predict. On the down, from left to right with data set & contact rate and data set only.
Date ˆ y ˆ y lower ˆ y upper P β P β lower P β upper P P lower P upper Table 4:
Data set 2. Predicted cumulative confirmed cases ∼ Sept 18, 2020, from top to down using Prophet and NeuralNetworks method of Predict. On the down, from left to right with data set & contact rate and data set only.
Date ˆ y ˆ y lower ˆ y upper P β P β lower P β upper P P lower P upper Comparative forecasting with machine learning 3 NUMERICAL SIMULATIONS (a) The forecasting is until 2020, May 26 which correspondto the graduation 85 on the abscissa axis. (b) The forecasting is until 2020, May 26 which correspondto the graduation 85 on the abscissa axis.(c) The forecasting is until 2020, June 10 which correspondto the graduation 100 on the abscissa axis. (d) The forecasting is until 2020, June 10 which correspondto the graduation 100 on the abscissa axis.(e) The forecasting is until 2020, September 18 which corre-spond to the graduation 200 on the abscissa axis. (f) The forecasting is until 2020, September 18 which corre-spond to the graduation 200 on the abscissa axis.
Figure 10: Comparative forecasting, using Neural Networks, of the total number of infected cases. On theabscissa axis the graduations 85, 100 and 200 correspond to the dates 2020, May 26, June 10 and September18. Forecasting using cumulative data only in yellow curve with confidence interval in orange, using bothcumulative and contact rate data in green curve with confidence interval in blue, and data in red dotted. Inthe left plot using data set 1 and in the right plot using data set 2.14
DISCUSSION (a) The forecasting is until the date 2020, May 26. (b) The forecasting is until the date 2020, May 26.(c) The forecasting is until the date 2020, June 10. (d) The forecasting is until the date 2020, June 10.(e) The forecasting is until the date 2020, September 18. (f) The forecasting is until the date 2020, September 18.
Figure 11: Forecasting, using Prophet, of the total number of infected cases until the dates 2020, May 26,June 10 and September 18, from top to down. In left plot using data set 1 and in the right plot using data set2.
4. D
ISCUSSION
Analysis of the new trend in the data from March 2 to April 24, 2020 shows a change in the trajectory of thetotal number of cases. That causes a reduction of the maximum value of the peak compared to what it wouldhave been without the measures taken on March 23, 2020. See figures 5a, 5b, 7b, 7a.By considering new nationwide measures of Senegal, which we have chosen in this study to fix on the date ofMay 10, 2020, we note that the maximum value of the peak decreases according to the force of the measurestaken. Likewise, the time of the peak is postponed as shown by the parametric plots in figures 5, 6, 7 and 8.Since the first measures on March 23, 2020, Senegal has laughed at additional measures such as the closing15
EFERENCES of markets, shops, stores and other public places, with an opening calendar. We have therefore chosen May10, 2020 as a date from which the additional measures can begin to take effect in reducing contamination.We see that depending on the strength of these measures, the evolution of the disease can lose its exponentialnature or become slower.The prediction with neural networks and Prophet show an optimistic situation. The forecasting based onlyon data and those on contact rates show a slow evolution as shown in figures 10 and 11. We see that thecurve obtained using the contact rate function in training of the neural networks, goes below that obtainedonly using the data on the total number of cases.
5. C
ONCLUSION AND PERSPECTIVES
In this paper, we have analyzed the impact of anti-pandemic measures in Senegal. First, we used techniquesof fitting function to the data of the total number of cases. The choice of the data fit function is crucial forthe results since it allows the estimation of the parameters of the compartmental model used. In a secondwork, we used neural networks to also predict the future evolution of the pandemic in Senegal. Also, wewere able to integrate the effects of the measures into the prediction.Depending on the measures taken, the pandemic’s trajectory may become slower or lose its exponentialnature. It would be interesting, in the following, to use other functions of a slow nature like the logisticalfunction to fit data and thus obtain different results. A stochastic study using Brownian motions applied tothe SIRU compartmental model would also be interesting. A CKNOWLEDGEMENT
The authors thanks the Non Linear Analysis, Geometry and Applications (NLAGA) Project for supportingthis work. R EFERENCES [1] E. Alpaydin, Introduction to Machine Learning 2nd ed, 584. Adaptive Computation and MachineLearning, (2010).[2] R. M. Anderson and R. M. May, Infectious Diseases of Humans, 768. Oxford University Press, Oxford,(1992).[3] A. Antonov, Coronavirus propagation modeling considerations, https://github.com/antononcube/SystemModeling , (2020)[4] M.A.M.T. BaldÃl’, Fitting SIR model to COVID-19 pandemic data and comparative forecasting withmachine learning, medRxiv preprint doi: https://doi.org/10.1101/2020.04.26.20081042 .(2010).[5] I. Goodfellow, Y. Bengio and A. Courville, Deep Learning, MIT Press. (2016).[6] J. Heaton, AIFH Volume 3: Deep Learning and Neural Networks, Heaton Research, Inc, 268. TracyHeaton (2015).[7] H. W. Hethcote, The Mathematics of Infectious Diseases, Society for Industrial and Applied Mathe-matics, Vol. 42, No. 4, pp. 599â ˘A ¸S653 (2000).[8] H. P. Langtangen and G. K. Pedersen, Scaling of Differential Equations, 152. Springer, (2016).16
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