key: cord-0547761-7d61zh05
authors: Stochictoiu, Radu D.; Rebedea, Traian; Popescu, Ionel; Leordeanu, Marius
title: A self-supervised neural-analytic method to predict the evolution of COVID-19 in Romania
date: 2020-06-23
journal: nan
DOI: nan
sha: b07703dc4963e38d5575961a87289ebbea8e0f3d
doc_id: 547761
cord_uid: 7d61zh05

Analysing and understanding the transmission and evolution of the COVID-19 pandemic is mandatory to be able to design the best social and medical policies, foresee their outcomes and deal with all the subsequent socio-economic effects. We address this important problem from a computational and machine learning perspective. More specifically, we want to statistically estimate all the relevant parameters for the new coronavirus COVID-19, such as the reproduction number, fatality rate or length of infectiousness period, based on Romanian patients, as well as be able to predict future outcomes. This endeavor is important, since it is well known that these factors vary across the globe, and might be dependent on many causes, including social, medical, age and genetic factors. We use a recently published improved version of SEIR, which is the classic, established model for infectious diseases. We want to infer all the parameters of the model, which govern the evolution of the pandemic in Romania, based on the only reliable, true measurement, which is the number of deaths. Once the model parameters are estimated, we are able to predict all the other relevant measures, such as the number of exposed and infectious people. To this end, we propose a self-supervised approach to train a deep convolutional network to guess the correct set of Modified-SEIR model parameters, given the observed number of daily fatalities. Then, we refine the solution with a stochastic coordinate descent approach. We compare our deep learning optimization scheme with the classic grid search approach and show great improvement in both computational time and prediction accuracy. We find an optimistic result in the case fatality rate for Romania which may be around 0.3% and we also demonstrate that our model is able to correctly predict the number of daily fatalities for up to three weeks in the future.

While cases of the new coronavirus, COVID-19, appeared in China as early as December 2019 and January 2020 (Organization [2020] ), the first documented cases of COVID-19 in Romania emerged in the middle of March 2020. Early understanding of the dynamics of an infectious disease is fundamental to being able to act on time and take the best safety measures for the population. Existing powerful mathematical models based on differential equations are able to predict reasonably well the evolution of the different curves (e.g. number of infected and hospitalized people, fatalities), given the correct set of model parameters. However, the inverse learning problem of finding the best parameters given the observed data is not an easy task, especially when the problem is not convex and several distinctive sets of parameters constitute relatively good local optima. The existence of multiple solutions is an important aspect, as we want to be able to predict future outcomes as well as learn fundamental parameters, such as the reproduction number and the fatality rate.

We take a dual neural-analytic approach, which effectively combines the power of the analytical solutions to model and predict data with a relatively small set of meaningful parameters with the power of deep neural networks to learn the inverse problem, that of estimating the correct parameters from the observed data. We propose an effective self-supervised training and prediction scheme, in which the two pathways, one classic, analytical (using differential equations) and the other based on machine learning (using a deep convolutional networks) can feed each other, in tandem. During the self-supervised training phase, we start from random parameters of an improved, state-of-the-art modified SEIR model (Chowdhury et al. [2020] , Goh [2020] ) (which we will refer to as Modified-SEIR), within medically acceptable intervals to predict fatality curves for a given period. Then we train the neural net to predict the known generator set of parameters, given the generated curves, by the analytical model. At test time, the network is used to rapidly estimate the correct set of parameters from the real, observed, curve of daily fatalities. Then, the set of parameters is further optimized, by stochastic coordinate descent to minimize the L2-norm between the predicted curve (by the analytical Modified-SEIR model Chowdhury et al. [2020] ) and the real curve of daily fatalities. Note that we estimate the correct set of model parameters solely from the number of fatalities, since, as mentioned above, that number is the only one that could be measured correctly. The number of true infected people is impossible to know, given mainly the limitations in testing and relatively large portion of the population that is asymptomatic.

Among the first measures taken by the authorities in Romania was to impose a very strict social distancing plan. This restriction has a major impact on the basic reproduction number of the SEIR model, usually reducing it by a percentage that lies between 40% and 80% (Kelso et al. [2009] , Read et al. [2020] ). As the social distancing norms were alleviated on May 15, 2020, we consider two simulated scenarios, so we could analyse and compare the impact of the heavy vs. moderate isolation restrictions. We now know, based on Qian et al. [2020] , that there are Gaussian simulations that could help us make a more educated guess for a date when to lift the lockdown completely. Considering the fact that Romania is among the most religious out of the 34 countries in Europe, ranked 1st in Europe by the combined index (Center [2018] and Jonathan et al. [2018] ), we are also aware of the possibility of Easter (April 19. 2020), when people tend to gather in larger groups, to influence the dynamics of COVID-19.

The Romanian authorities have not presented yet any official analysis or predictions for the evolution of the pandemic. However, Romania urgently needs a rigorous data-driven approach, based on actual cases and statistics, with thorough computational models, for dealing with such pandemics. Note that using and learning from data collected from a specific country is important, as it is well-known that a model that best applies to one part of the world might not be optimal for another. From this perspective, our work brings a genuine value, being the first to propose a successful computational approach for estimating the evolution of COVID-19 in Romania.

The two main contributions that we make in this paper are:

1. We present the first computational approach to predict the evolution of COVID-19 in Romania, based on publicly available data. For prediction we implement a recently proposed mathematical model based on SEIR (Chowdhury et al. [2020] , Goh [2020] ) (refered to as Modified-SEIR) for estimating the evolution of infectious diseases. We optimize the model to fit the data provided by the Romanian health authorities for the COVID-19 pandemic, using a novel deep learning approach (see the second contribution), trained in a self-supervised fashion. We estimate that on August 1, 2020, there will be 1640 total deaths by keeping the heavy social containment and up to 1861 deaths by slightly alleviating it. Our predictions are right around the bounds predicted by IHME (1614 deaths by August 1, as of June 12, 2020 -IHME COVID-19 health service utilization forecasting team [2020] ) and the ML-based approach presented in Section 2 (1776 deaths by August 1, as of June 12, 2020 - Gu [2020] ).

2. We introduce a novel self-supervised deep learning approach for fast optimization and learning of the parameters of the anlytical Modified-SEIR model. More specifically, the convolutional network is trained on many outputs of the Modified-SEIR model, generated by random sets of parameters, within medically acceptable ranges, to predict precisely the same generator parameters set, in each case. After being trained on hundreds of thousands of such synthetic cases, the neural network becomes able to take us directly in the neighborhood of the best fitting parameters, when presented with the real data curve. Then, a refinement coordinate descent procedure is applied at the end, to obtain the final solution.

Our experiments clearly show that the proposed deep learning approach to optimization greatly improves speed and accuracy over the baseline (grid search with coordinate gradient) optimization approach.

Shortly after the first cases of COVID-19 appeared in the world, an important research movement has began, which aims at finding bounds for the characteristics of the infectiousness of this new coronavirus. In Kucharski et al. [2020] authors show that the basic reproduction number is significantly influenced by travel restrictions, ranging from 2.35 to 1.05. In their procedures they used an estimate of the incubation period equal to 5.2 days, but which can be as low as 2 days, according to a study conducted in Wuhan (Li et al. [2020] ). This study also found that the value of 2.2 is a good approximation for the basic reproduction number, which is also similar to the findings in our research reported here.

In authors use a SEIR (Susceptible-Exposed-Infectious-Recovered) model to estimate the basic reproduction number at 2.68 on January 25 in Wuhan. At the same time, it reveals a worldwide incubation time that ranges from 4.6 to 6.9. We see in this article the positive correlation between the basic reproduction number and the probability of creating an exponential epidemic starting with a single infected person.

The official report of the World Health Organization (WHO [2020]) introduces several observed parameters, including a basic reproduction number in the range of 2 − 2.5, an incubation period with an average in the range of 5 − 6, a minimum hospitalization rate around 6.1%, represented by critically ill patients, and a maximum of 20%, to which are added those 13.8% in severe condition, a period from incubation to death in a wide range of 2 − 8 weeks and a recovery time for mild cases topped by 12 days and for severe cases up to 14 days of hospitalization.

In Another important studied factor is how strict the rules of social distancing must be in order to stop the number of cases from increasing. In Read et al. [2020] it is estimated that a reduction in the basic reproduction number by 58 − 76% is needed to stop the increase in the number of infected people, considering the base reproduction number equal to 3.11.

At the core of our prediction model, it is the Modified-SEIR (Chowdhury et al. [2020] ), with a freely available toolbox, the Epidemic Calculator (Goh [2020] ), which we use as a stepping stone in our own implementation and validation of our model and experiments.

There is thorough research focused on the hospital needs (IHME COVID-19 health service utilization forecasting team [2020] ) that predicts a total number of deaths equal to 1625 caused by a first wave of the infection. An ML-based project (Gu [2020] ) assuming a mid-May reopening predicted 1776 deaths in Romania on Aug 1, 2020, as of June 12, 2020. While Artificial Intelligence (AI) solutions are sometimes biased and considered inadequate, we know that their true power arrives from grasping the inner complexity of a problem by learning from experience (real data), which a simpler heuristic method cannot.

Figure 1: Modified SEIR model: the "Removed" case is split into: 1) Recovered from mild symptoms; 2) Recovered from severe symptoms; 3) Deceased. The diagram follows the differential equations of the Modified-SEIR model presented in Table 1 .

For solving a very difficult prediction problem, such as the ones revolving around COVID-19, Mihaela van der Schaar [2020] proposes AI-powered ways to manage limited healthcare resources, to develop personalized patient management and treatment plans, to inform policies, to enable effective collaboration and to understand and account for uncertainty better.

A limited number of published medical studies Popescu et al. [2020] , Gherghel and Bulai [2020] present and discuss the epidemiology, clinical preparedness and medical challenges of the COVID-19 pandemic in Romania. However, ours, as mentioned previously, is the first to consider COVID-19 and its evolution from a computational perspective and offer an efficient model for learning and prediction. As shown next, our model accurately fits the real data, predicts future events (not seen during training) and estimates important core characteristics of the virus specific to Romania, such as the reproduction number, the length of infectiousness, time to recovery and fatality rate. Note that the related work presented above provides specific ranges, established in the medical literature, within which we optimize the parameters of the Modified-SEIR.

In order to predict the evolution and understand key factors of COVID-19 we use the recent mathematical model (Chowdhury et al. [2020] ) based on the classic SEIR (Susceptible, Exposed, Infectious, Removed; Hethcote [2000]), which is a widely accepted standard for modeling the evolution of infectious diseases. The Modifed-SEIR model follows the usual steps in which an infectious disease evolves. The evolution along with key elements and measures are fully described by a set of differential equations ( Table 1 ) that we present in this section. We also offer a visual representation of the model in Figure 1 .

We further divide the Removed section into three categories: recovered from mild symptoms (R M ), recovered from severe symptoms (R V ) and deceased (R F ). There is also an extra layer of differential equations in the middle which helps us better shape and understand the dynamics of the disease. Following the Modified-SEIR model (Chowdhury et al. [2020] ) and the Epidemic Calculator (Goh [2020] ), we assume that all fatalities come from hospitals, and that all severe cases are admitted to hospitals immediately after the infectious period ends.

The advantage of having an explicit analytic model versus a pure deep learning approach is that each parameter has a clear meaning that is easy to interpret. In this case, the meaning of each parameter used is described and summarized in Table 2 . The ranges in which we search for the optimal parameters are fixed according to recently published medical research and their confidence intervals, as presented in Section 2. Note that we use constant values for some parameters, such as the total size of the population or the time from severe severe symptoms to hospitalization (in accordance with Goh [2020] ). 

In order to solve the time-based differential equations and produce the different evolution curves we use a fourth order Runge-Kutta integrator (Tan Delin [2012] ).

As the number of new fatalities per day it is easily known and it is not influenced by the number of actual tests run in any location, we consider it as ground truth in our experiments. Even though this number mixes patients with comorbidites (whose health is also influenced by other conditions) with those without comorbidities, the actual measured number of infected people who are dying is ultimately the only estimation that can be considered certain, in the current research. We use the data uploaded daily by Johns Hopkins CSSE.

Our purpose is to find a set of parameters that best approximate with a curve modelled by the modified SEIR, the real curve of daily number of fatalities. We denote D real as the vector (curve) of reported daily number of deaths in a specific time interval and D θ as the vector (curve) generated by the model with parameters θ, for the same time interval. Thus, the cost function that we minimize to find the best fitting parameters is the square root of sum of squared errors between the real and the predicted curve, as shown in Equation (1). We denote the optimal set of parameters by θ * , as defined in (2). The valid search (optimization) ranges for each parameter are presented in Table 2 , as mentioned previously, and constitute relatively large search regions, as unions over the ranges published in recent medical literature.

We test and compare two main ways of finding the optimal parameters. One is a baseline, which starts by a classic grid search procedure followed by a stochastic coordinate descent refinement. The second optimization method, which is our main technical novelty, is to make an initial guess of the parameters using the self-supervised trained convolutional network followed by the same refinement step using stochastic coordinate descent. Each optimization module (grid search, neural network and coordinate descent) is described in the next sections. 

There are 11 parameters that we optimize over:

Therefore a full and very fine grid search is computationally infeasible. However, we can divide each range in (2-4) smaller ranges and look for a decent approximation to start with. The grid search module is followed by the coordinate descent refinement procedure (Section 4.4).

Because of the long computational time required by grid search optimization, we propose a deep learning approach, using a convolutional neural network trained in a self-supervised manner, as discussed previously, that is able to bring the solution in the neighborhood of the optimum very fast. The neural network optimization module is also followd by the same final coordinate descent procedure. Interestingly enough, it turns out that the results, when using the neural net optimization, are vastly superior in both speed and accuracy to the grid search approach.

Below we present the exact steps taken for the self-supervised scheme in which the neural network learns to guess the right set of parameters, given a curve of daily deaths for a given period of time.

1. Create a dataset • Take 100000 random samples from a uniform distribution of the 11 parameters in the ranges indicated in Table 2 ; • Generate a curve of daily fatalities using the Modified-SEIR model (Table 1) for each set of model parameters picked at the previous step. Even though the estimated total number of deaths is cumulative (the R F variable), we take the daily deaths number as daily increments in the total number of fatalities. • For each curve we randomly select a fixed number L of consecutive days of daily deaths (L is defined by the number of days in certain time ranges in our experiments, such that for [March 22, May 3], L = 43). This vector of L consecutive numbers, representing the fatalities for the corresponding L days, modeled by Modified-SEIR, along with the corresponding set of parameters will constitute the 100K training pairs used in training the neural net optimizer presented below. Note that the start of the L-day sequence is chosen randomly (so it could be at the beginning or towards the end of the pandemic). We try to mirror the real case, when we really do not know which day should be considered the first of the pandemic.

2. Multi-head neural network optimizer, trained in a self-supervised way, to predict the Modified-SEIR model parameters, given the L-element curve of daily fatalities (generated by precisely the same set of parameters that should be predicted by the network).

• Hidden layers We compare the grid search approach with the neural network predictions. The advantage of the latter approach is that it offers almost instantaneous predictions. In Figure 2 we show the percentage of tasks (problems that are randomly generated by the model using random initialization) where the samples (sets of parameters) found by grid search produce superior curves (closer to the truth) than the ones predicted by the neural network. The percentage is a function of time, since for an infinite amount of time we expect grid search, with a sufficiently fine grid, to beat the neural network. The plot shows how vastly superior in terms of speed the neural network is. Almost 7 hours of running for grid search, on an Intel c Core TM i9-9980HK CPU @ 2.40GHz x 8, are not enough to surpass the neural net predictions.

We can expect that neither of the two approaches (grid search and neural net optimization) will directly produce an optimal solution, even though we do expect them to output a set of parameters that are close to a local optimum. In order to refine our results we further apply an iterative stochastic coordinate descent approach similar to Wright [2015] . Starting from the best predictions of a given first-stage module (neural net or grid search), we take random subsets of two parameters at a time, divide their search ranges into 20-40 parts around the current best solution and replace the next values of the chosen parameters with the ones that minimize the cost function. We iterate the procedure until we reach a convergence of 10 −3 absolute error.

We make use of the stochastic property iteratively because we assume that some parameters influence the cost more than others. Thus, by choosing random subsets of parameters to optimize over, we avoid the risk of spending valuable time optimizing over subsets of parameters that do not bring much value. Figure 2 : Comparison between the grid search and the neural network solutions. We present how often grid search produces better solutions than the neural net. We clearly see that 7 hours of computation are far from sufficient for grid search to beat the neural net. Note that in the plot, the final refinement procedure is not used by either of the two approaches.

We consider three particular data sets on which we optimize and search for the best parameters of our model: We know that on May 15 2020 the Romanian authorities changed the policy from state of emergency to state of alert. Because the time from incubation to death is greater than one week, we assume the reported daily fatalities from May 15 2020 to May 21 2020 are not influenced by the changed policy. This enables us to search for optimal parameters for the interval March 22 2020 to May 21 2020, as it can be modelled by Modified-SEIR.

For every data set, we present in Table 3 the optimal model parameters found with neural net optimization followed by the refinement step, as defined in Section 4. As stated previously, they minimize the L2 distance between the real and the predicted curves of fatalities from March 22 to May 3, May 14, and May 21, respectively (all in year 2020, of course). For each set of optimal parameters we compute the error of prediction for the following dates: May 3, May 15, May 21, June 3, June 8, June 9, June 10, June 11 (all in 2020).

As the prediction errors are inversely proportional to the data set size, we prove that our approach is data-driven and provides better solutions for bigger data sets. In other words, every new day of observations is important in finding the real parameters that shape the evolution of the COVID-19 infectious disease.

It is worth mentioning that in general the reversed problem of finding the correct model parameters from the observation of partial data (in our case we observe only the values of one output variable, the number of daily fatalities) is ill posed, since the problem is not necessarily convex and many different sets of parameters can produce similar output. However, this is a common case in machine learning, in which several valid solutions exist. The AI system usually learns to predict the most probable output (in this case, the set of parameters) given the observed data, based on its training experience. One good example is the case of vision, in which many different 3D worlds with different semantic interpretations could produce the same 2D image. Nevertheless, the visual system learns to pick the most likely interpretation (given its prior experience) out of infinitely many. In our specific case, we expect that the synthetic generation of many curves from different sets of parameters, will help the neural network implicitly learn the priors in the data model, such that the network will learn to output, from the many different solutions, one that is most likely to have produced the given curve. We can easily imagine that there are certain distinct neighborhoods of parameters that generate similar curves and that a set of parameters coming from a larger neighborhood is more likely than one coming from a smaller one. In other words, sets of model parameters for which the output curve is more stable (low curve gradient w.r.t parameters) are probably more likely than sets of parameters for which the curve changes rapidly in their immediate neighborhood. Such subtle priors in the space of parameters should be implicitly learned during the self-supervised training, if sufficiently many pairs of model generated curves (input) -parameters (output) are presented to the network.

It is clear by now that searching for the best parameters is a complex task, especially if the cost function is not convex, thus admitting multiple local minima. Besides the intuitive discussion, we also experimentally analyze this issue by comparing different 4D plots where the 3D space is defined by three different parameters and the fourth dimension, the final cost function, is defined by color. In Figure 3 one can see that there are regions where there is a linear dependency between two parameters preserving the minimum cost and that there are multiple intervals with local minima for sets of three parameters. The latter finding is especially interesting as it tells us that our prediction might not be the correct one even if it has the lowest cost. Or, in other words, different distinctive sets of parameters may generate the same final cost. Thus, learning from larger sets of reported data, over longer periods of time, may shift the balance towards a different set of optimal parameters which can immediately modify the foreseen dynamics of the coronavirus infectiousness and fatality. This situation, which is commonly encountered in many AI tasks, reveal the inner ambiguity and difficulty of the problem tackled. What is however, important and relveant here, is that we are able to learn sets of parameters which are plausible, often matching many independent findings in the literature, which predict with surprising accuracy the evolution of COVID-19 in Romania.

Starting from the knowledge that a particular parameter set might not be necessarily the correct one even if it has the lowest cost on the limited training data, we compare the predictions of two different sets of parameters that have similar L2 costs (very close to the lowest cost found for the interval from March 22, 2020, to May 3, 2020). In Figure 4 we present the evolution curves corresponding to these two sets of parameters. Both of them seem to fit the given data equally well. Surprisingly, when we extrapolate the curves into the future (by running the models according to the corresponding parameters into the future), we notice substantially different evolution curves. In Figure 5 we notice how one approximation goes down quickly, while the other continues to increase for another three months.

This subtle change in the way the two fitted curves diverge at the end of the data set has drastic outcomes regarding, among other characteristics such as the number of active infections, the total number of fatalities. Figure 6 shows us that while the fitted curves for the observed interval (March 22, 2020 , to May 3, 2020 are similar, one set of parameters predicts a total of 2290 deaths, while the other predicts a total of 10225. We compare their predictions in Table 4 .

As discussed in the previous section, the approach based on a neural network optimization followed by a stochastic alternative coordinate descent seems to overfit the data set. In order to help our model learn parameters that generalize better in the future, we add a relative future cost based on the real numbers to the cost presented in Section 4, so its new formula is introduced in Equation 3. 

R date ≡ Reported number of f atalities on date (5) P date ≡ P redicted number of f atalities on date (6) We present in Table 5 the best model parameters found with neural net optimization followed by the refinement step. As stated previously, they minimize the L2 distance between the real and predicted curves of fatalities, from March 22, 2020, to May 21, 2020, while trying not to overfit the data set by looking ahead until the last available reported date. Several interesting observations are worth making: the estimated basic reproduction number was found to be 2.21, which is very close to the the value estimated in the literature. Since it is above 1, it defines an exponential growth in the number of infected people unless reduced by the social distancing measures. The imposed measures of containment indeed reduced the reproduction number by 60% (from 2.2 to 0.884), so the curve starts decreasing towards zero. The continuous decrease definitely helps our medical staff to manage patients better and safer, which may also explain why the fatality rate found is so small, of only 0.245%. Note that this value is significantly lower than other numbers reported in the literature so far, which is very good news but it is highly dependent on the number of tested people and on the social dynamics of the observed population.

Based on this set of parameters, we analyse two cases that influence the daily deaths curve:

1. Heavy social distancing, meaning that the enforced norms will not be diminished until the end of the pandemic period; 2. Moderate social distancing, meaning that on May 15, 2020, the social interaction increased by 10%.

Here, we assume that people will adopt a careful behavior and the heavy social distancing rules will apply long after they were proposed (R 0 becomes 40% of its initial value). In Figure 7 we plot our best fitting for the daily deaths approximation using the parameters from Table 5 . You can notice a change in the convexity of the prediction (blue) curve when the heavy social distancing norms have been adopted (on intervention day).

Using the RK4 integrator, as presented in Section 3, we extrapolate around 200 days from the first day of reported data. In this way, we predict when there are going to be less than 5, 3 or 1 deaths per day, as you can see in Figure 8 . A key insight is that the maximum number of daily deaths, the peak of our extrapolation, has already passed on April 18, 2020 (which is right before the Orthodox Easter on April 19-21), meaning that the curve is going to stay under 24 deaths per day.

A subject that is of interest is the total number of fatalities the virus is going to cause. Using the daily numbers extrapolation, we create a cumulative curve which tells us the total number of fatalities by a certain date. Thus, we predict a total of around 1730 using data from March 22, 2020, to May 21, 2020, as presented in Figure 9 .

There is the possibility of optimizing the parameters for fitting the cumulative number of fatalities instead of their daily number. The reason we opted for the latter is three-fold. First, the amount of information that each specific day brings to the cumulative function becomes smaller and smaller each day, converging to 0 at infinity. The cumulative number thus increases with each day and the meaningful information for a given day becomes much smaller than the total cumulated number. It is thus expected that learning could suffer from numerical issues. Second, the number of daily fatalities contains a certain amount of noise that can help us generalize better for our predictions. And third, the curve of daily numbers shows better and clearer when the pandemic peaks and when it is expected to diminish to a non-threatening state.

We know that the real number of actively infectious people is hard to obtain, as the number of tested people at a time is just a fraction of the whole population. However, once the right parameters are learned, the model can estimate a total number of infectious people and predict that we need to wait until August 25, 2020, to have less than 1000 such individuals. The results are shown in Figure 10 .

While we do not know for sure how the social dynamics influence the evolution of the coronavirus, we conduct an experiment assuming that on May 15, 2020, when the social distancing norms became less constraining, the already reduced basic reproduction number (by the heavy social distancing norms) increased by 10%, while the rest of the parameters remained the same. Figure 10: Total number of infectious people extrapolation. We predict that we will have less than 10000 infections on June 2, 2020, less than 5000 infections on June 8, 2020, and less than 1000 infections on August 25, 2020.

In Figure 11 we show in our prediction that the period until there will be less than 5 daily deaths is prolonged until the end of August, 2020. We do not see a second peak because the basic reproduction number does not get to be over 1 again, but the pandemic is predicted not to end until the end of 2020.

The total number of deaths jumps to 2361, shown in Figure 12 . Compared to Gu [2020] we seem a bit pessimistic as the number of deaths predicted by us by Aug 1, 2020, is 1861 while their machine learning approach predicts 1776, as of June 12, 2020. Thus, we are still inside their confidence bounds: [1586 -2187] . Please note that their approach is also data dependent and can offer significantly different predictions by using more data.

In Figure 13 one could see the effect of a slight change in the prevention norms, as the total number of actively infected people takes way more time to go down.

Ultimately, such modeling is important not only to fit the available observed data and estimate various model parameters, such as the fatality rate, but to predict future events. The ability to predict future outcomes is its real value, in order to prepare the best courses of action in advance. Acting on time, especially in the face of a pandemic, is vital. Thus, in order to test the validity of our model (as it is usually done, in fact, in machine learning) we compare the future predictions made on May 21, 2020 (based on observed data until that date), with the latest information available in the meantime, Figure 11 : Daily fatalities extrapolation with increased mobility (moderate social distancing) from May 15, 2020. The red plot represents the officially reported daily number of deaths in Romania. Our extrapolation is the blue line and the green lines represent the days when we estimate less that 5, 3 or 1 daily fatalities. Figure 12 : Cumulative deaths extrapolation with increased mobility from May 15, 2020. The red plot represents the officially reported cumulative number of deaths in Romania. Our extrapolation of the cumulative number of deaths is the blue line.

until June 11, 2020. Our model is surprisingly accurate with the heavy social distancing assumptions. The predicted total number of fatalities is close to the reported values: 1296 true reported deaths on June 3, 2020, versus 1294 (predicted by the heavy social distancing model) and 1312 (predicted by the moderate social distancing model) and 1369 true reported fatalities on Jun 11, 2020, versus 1375 (predicted by the heavy social distancing model) and 1411 (predicted by the moderate social distancing model).

Regarding the number of infected people, we already know that the number of reported active cases is lower than the real one, so, as mentioned previously, we can use our model to guess the actual number of active infections. Testing our predictions against future data is shown in Table 6 .

Now that we have two models, one following heavy social distancing norms and the other following moderate social distancing norms, that have prediction errors < 4% so far, we can use them as bounds for our future predictions. We care about how the coronavirus will evolve for the next several months, so we summarize our findings in Table 7 . Figure 13 : Total number of infectious people extrapolation with less social distancing from May 15, 2020. We predict that we will have less than 1000 only in 2021. our novel approach can be useful in a fast paced pandemic, maybe not only for the case of Romania. A notable finding is that the case fatality rate in all the local optima sets seems to be significantly less than 1%, mostly around 0.245% and 0.3% -and this is a optimistic surprise, when compared to the estimates in the literature.

We know that access to more data could change the optimal parameters, but, based on the results presented in this paper, we are confident that in the case of Romania, the case fatality rate is significantly smaller (≈ 0.3%) than the worldwide average.

Considering that our model trained on data collected until May 21, 2020, accurately describes the future evolution (future unseen data) of the number of fatalities until June 11, 2020, we conclude that both the model and its inner parameters found, provide answers that is very close to the true ones.

The results strongly indicate that we should seriously consider data-driven computational approaches, in combination with machine learning, in the analysis and decision making process, with respect to fundamental aspects of our lives (such as it is the case of COVID-19 pandemics), for the future and greater good of the society.

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Coordinate descent algorithms. Mathematical Programming

Nowcasting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study

In this paper we propose the first computational model to predict the evolution of COVID-19 in Romania and estimate key factors of the pandemics such as the fatality rate, incubation period, infectiousness period and reproduction number, based on the state of the art Modified-SEIR model Chowdhury et al. [2020] . Our technical novelty consists in the way we optimize the parameters of the model, through a self-supervised deep learning approach, in which a convolutional neural network learns from synthetic data, produced by the analytical Modified-SEIR model for random sets of parameters, to predict the correct parameter set -which is known, since it is the one used to generate the synthetic data. Our results show beyond any doubt that our novel self-supervised approach is effective and learning a set of parameters which are not only able to fit the observed data but also to accurately predict in the future, for the three weeks period tested (which is a relatively large period in the case of a rapidly evolving pandemics).At the conclusion of our study, we highlight some important findings comprising the total number of fatalities by following the heavy social distancing norms (1730), the total number of deaths following a small decrease in the prevention norms on May 15, 2020, (2361) and the fact that we already passed the peak of the daily number of deaths on April 18, 2020 (one day before the Orthodox Easter).Our predictions are right inside and around the bounds predicted by IHME (1614 deaths by August 1, as of June 12, 2020; IHME COVID-19 health service utilization forecasting team [2020] ) and the ML-based approach presented in Section 2 (1776 deaths by August 1, as of June 12, 2020; Gu [2020] ). This and the fact that our set of found parameters are close to the ones presented in the latest literature (for example, an optimal basic reproduction number of 2.21) only empowers the idea that