Impact studies of nationwide measures COVID-19 anti-pandemic: compartmental model and machine learning

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 forecasting is done for Senegal. We show a comparison between deterministic and two machine learning technics for forecasting.


INTRODUCTION
The COVID-19 pandemic is testing the entire world so that measures are being taken in most nations to stem its development. These measures can generally be of different kinds such as social distancing, partial or total confinement, etc. It would therefore be interesting to be able to effectively analyze the effects of the measures taken on the spread of the pandemic. Here we offer an analysis of the impact of the measures taken that we apply to the case of Senegal. In previous papers [4], [12] and [13] we proposed a start of study using the results of [10]. In [10], a useful method for the study of the evolution of COVID-19 pandemic has been resented by using a compartmental model with Susceptible, Infected, Infected reported and unreported (SIRU). In [4], the author use 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 continuation of the work done in [4]. We do a two-step analysis. The first uses the SIRU epidemic model introduced in [10], and the second uses two machine learning tools: Predict of Wolfram Mathematica using Neural Networks method and Prophet. We conduct the work in the following way. In the first section we will study the effect of the nationwide measures, 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 the forth section, making a conclusion and advancing perspectives.

ANALYSIS a. Analysis of the measures
In [4] a classic SIR model was studied using results from the paper [10]. The aim was to analyze the effect of the nationwide measures using the data after nationwide measures. In fact, throughout t 0 to T , we fitted an exponential function to the data of the total cases of infection of this period. T represents the date of the nationwide measures, and t 0 is the starting time of the epidemic. We consider that the effects of the measures are such that they lead to a reduction in the contact rate. To describe this reduction, we chose a slowly decreasing function over time. We consider that the measures taken are not strong enough to systematically drop the contact rate to 0. This new function corresponds to the first one and the data in the period before measures from t 0 to T , then takes a slower trajectory than that of the first function. In other words, from the date T , the new curve goes under the old one. We consider that if on the dates t > T the data goes under the new curve obtained with a contact rate after measures then, we can say that they affect the 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 square method, is T NI(t) = b exp(ct) − a with a = 13.9324, b = 9.61779 and c = 0.100095 (figure 1a). In this paper we fit a new exponential function to the data from 2020 March 02 to April 25 (figure 1b). For this new function, we have a = 99.9214, b = 81.325, c = 0.0371767. The cumulative numbers of confirmed, deaths and recovered cases are illustrated in Figure 2.  To continue in our analysis, we will use an epidemiological model introduced in [10]. This model is as follows: The initial conditions are S(t 0 ) = S 0 ≥ 0, I(t 0 ) = I 0 ≥ 0 I R (t 0 ) = I R0 ≥ 0, and I U (t 0 ) = I U0 ≥ 0. We can also consider a version with a removed compartment: Let's present the parameters. β is the contact rate. 1/ν is the average time during which asymptomatic infectious are asymptomatic. γ is the fraction of asymptomatic infectious individuals that become reported symptomatic infectious. 1/η is the average time symptomatic infectious have symptoms. γν is the rate at which asymptomatic infectious become reported symptomatic. (1 − γ)ν is the rate at which asymptomatic infectious become unreported symptomatic. α is the proportion of recovered and 1 − α is the proportion of death due to the infection.
We make the assumption that the function we fit the total number of reported infected cases is given by γν t t0 I(s)ds.
We consider that after the measures taken at the time T , the contact rate depends on time following a formula we 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 where δ and p are parameters to choose. The second one is: where ϕ is a parameter to choose. Then the new model to solve is: By solving (4), we getĨ(t) from which we can calculate the total number of infected with the formulã T NI(t) = γν t t0 I(s)ds. We choose values of parameters ofβ such thatT NI(t) follows the same path that T NI(t). Then, by way of parameters inβ , we can evaluate the measures.
Now we study the maximal value of infection peak. For that, we consider at time T 2 , the nation applies stronger measures are taken. We simply interpret as the contact rateβ is reduced by a factor φ ∈ [0, 1].
When the value of φ is equal to 0, it means that there are no measures, while when φ is equal to 1, it means that the strongest measures are taken. Hence the measures are quantified with the values of proportion φ . Then the contact rate become:β We solve the new model withβ replaced byβ 2 . We do a parametric solve concerning the parameter φ , and we plot the result for different values of φ . Then, we show different values of the peak, depending on the values of φ .

b. Artificial Neural Networks
Artificial neural networks are part of artificial intelligence. Biological neural networks inspire them. Biological neural networks are part of the animal brain. One of the main functions of the brain is to process information, and the primary information processing element is the neuron. This specialized brain cell combines (usually) several inputs to generate a single output. Depending on the animal, an entire brain can contain anywhere from a handful of neurons to more than a hundred billion, wired together. The output of one cell feeding the input of another, to create a neural network capable of remarkable feats of calculation and decision making (See [14]). If we could qualify the brain as a computer, then we would say that it is the best of computers. For this reason, the engineer seeks to improve mechanical computers to be closer to the biological computer, i.e., the brain. The more neural connections there are, the more the network can solve complex 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, the neural network needs to be trained. Then, neural networks can predict the outcome of entirely new datasets based on training from old data sets. Most neural network structures use some type of neuron, node, or unit. An algorithm called a neural network will 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 entered into 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 program also describes the binary input as using a bipolar system with true as 1 and false as −1. An artificial neuron multiplies each of these inputs by a weight. It then adds these multiplications and transmits this sum to an activation function given by: with the variables x and w represent the input and the weights of the neuron, n is the number of input and weight. Some neural networks do not use an activation function. To read more about Neural Networks one can see [1], [6], [5] and [14].
c Forecasting using Prophet number of infected cases and the contact rate. In the second case, the aim is to do forecasting by considering the 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 dataβ given by (3).  c. Forecasting using Prophet In this section, we develop another machine learning tool for forecasting to compare with the previous SIRU models and Neural Networks method. We use Prophet [15], a procedure for forecasting time series data based on an additive model where non-linear trends are fit with yearly, weekly, and daily seasonality, plus holiday effects.
For the average method, the forecasts of all future values are equal to the average (or âȂIJmeanâȂİ) of the historical data. If we let the historical data be denoted by y 1 , ..., y T , then we can write the forecasts aŝ The notationŷ T +h|T is a short-hand for the estimate of y T +h based on the data y 1 , ..., y T . A prediction interval gives an interval within which we expect y t to lie with a specified probability. For example, assuming that the forecast errors are normally distributed, a 95% prediction interval for the h-step forecast iŝ 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 become negative. To counter this, we round negative values to zero. Also, no tweaking of seasonality-related parameters and additional regressors are performed.

NUMERICAL SIMULATIONS a. Numerical analysis
The data for Senegal, we use, is obtained from daily press releases on the COVID-19 from the Ministry of Health and Social Action (http://www.sante.gouv.sn/).  (c) Parameric plot, with as maximum value on the ordinate axis 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 ordinate axis 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 ordinate axis fixed at 10000, of the total number of infected casẽ T NI(t).
(f) Parameric plot, with as maximum value on the ordinate axis fixed at 5000, of the total number of infected casẽ T NI(t), the reported I R (t) and unreported I U (t) infected case.  (c) Parameric plot, with as maximum value on the ordinate axis 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 ordinate axis 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 ordinate axis fixed at 10000, of the total number of infected casẽ T NI(t).
(f) Parameric plot, with as maximum value on the ordinate axis fixed at 5000, of the total number of infected casẽ T NI(t), the reported I R (t) and unreported I U (t) infected case.

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, to May 12, 2020 we carry out simulations for a longer time and forecast the potential trends of the COVID-19 pandemic in Senegal. The predicted cumulative number of confirmed cases are first plotted for periods until May 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 the value of the confidence interval).
The figures 10 shows forecasting using Neural Networks method of Predict. Particularly the figure 10 shows two forecasting, one based only on data and an other obtained by training the Neural Networks method with data and contact rate. The prediction are done until 2020, May 26, June 10 and September 18.        Forecasting using cumulative data only in yellow curve with confidence interval in orange, using both cumulative and contact rate data in green curve with confidence interval in blue, and data in red dotted. In the left plot using data set 1 and in the right plot using data set 2.

DISCUSSION
Analysis of the new trend in the data from March 2 to April 24, 2020 shows a change in the trajectory of the total number of cases. That causes a reduction of the maximum value of the peak compared to what it would have 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 of May 10, 2020, we note that the maximum value of the peak decreases according to the force of the measures taken. 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 closing of markets, shops, stores and other public places, with an opening calendar. We have therefore chosen May 10, 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 exponential nature or become slower. The prediction with neural networks and Prophet show an optimistic situation. The forecasting based only on data and those on contact rates show a slow evolution as shown in figures 10 and 11. We see that the curve obtained using the contact rate function in training of the neural networks, goes below that obtained only using the data on the total number of cases.

CONCLUSION AND PERSPECTIVES
In this paper, we have analyzed the impact of anti-pandemic measures in Senegal. First, we used techniques of fitting function to the data of the total number of cases. The choice of the data fit function is crucial for the results since it allows the estimation of the parameters of the compartmental model used. In a second work, we used neural networks to also predict the future evolution of the pandemic in Senegal. Also, we were 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 exponential nature. It would be interesting, in the following, to use other functions of a slow nature like the logistical function to fit data and thus obtain different results. A stochastic study using Brownian motions applied to the SIRU compartmental model would also be interesting.