A regime switching on Covid19 analysis and prediction in Romania
Marian Petrica, Radu D. Stochitoiu, Marius Leordeanu, Ionel Popescu
AA regime switching on Covid-19 analysis and prediction inRomania
Marian Petrica
Faculty of Mathematics and Computer ScienceUniversity of [email protected]
Radu D. Stochitoiu
Faculty of Automatic Control and ComputersUniversity Politehnica of [email protected]
Marius Leordeanu
Institute of Mathematics of the Romanian AcademyUniversity Politehnica of [email protected]
Ionel Popescu
Faculty of Mathematics and Computer ScienceUniversity of BucharestInstitute of Mathematics of the Romanian [email protected]
Abstract
In this paper we propose a regime separation for the analysis of Covid19 on Romania combinedwith mathematical models of SIR and SIRD.The main regimes we study are the free spread of the virus, the quarantine and partial relaxationand the last one is the relaxation regime.The main model we use is SIR which is a classical model, but because we can not fully trust thenumbers of infected or recovered people we base our analysis on the number of deceased people whichis more reliable. To actually deal with this we introduce a simple modification of the SIR model toaccount for the deceased separately. This in turn will be our base for fitting the parameters.The estimation of the parameters is done in two steps. The first one consists in training a neuralnetwork based on SIR models to detect the regime changes. Once this is done we fit the main parametersof the SIRD model using a grid search.At the end, we make some predictions on what the evolution will be in a timeframe of a monthwith the fitted parameters.
Infectious disease pandemics have had a major impact on the evolution of mankind and have playeda critical role in the course of history. Over the ages, pandemics made countless victims, decimatingentire nations and civilizations. As medicine and technology have made remarkable progress in the last1 a r X i v : . [ q - b i o . P E ] A ug entury, the means of fighting pandemics have become significantly more efficient. Another problem isthat globalization, the development of the commerce and the ease to travel all over the would facilitatesthe transmission mechanism of a new disease much more than it did in the past.In 2019 a new pneumonia was associated to a new virus from the corona virus family known now asCOVID-19. This spread out very quickly around the wolds as the next subsection shows. On 31 st of December 2019, the first cases of infection with an unknown virus causing symptoms similarto those of pneumonia were reported to the World Health Organization in China. Shortly after that thenew virus was identified as a new coronavirus, and the public health organizations feared that the situa-tion may degenerate in one similar to the SARS epidemic in 2003. The SARS outbreak was also caused bya newly identified coronavirus and it caused 8000 infection cases, 774 deaths and significant financial losses.Within less than 3 months COVID-19 outbreak has become a global pandemic, spreading across almostall countries all over the world. As of 29 th of March 2020, there were 725980 COVID-19 confirmed caseswhich caused 35186 deaths. Although the outbreak started in China, the virus has spread rapidly allover the globe, the most affected countries being now Italy, Spain, China, Iran, France, United States ofAmerica and United Kingdom.The fast-evolving spread of the new coronavirus, which has been officially declared a pandemic, isrepresented below. The following charts show the countries where there have been reported at least 1000cases of COVID-19 infection: (a) 9 th of February 2020. (b) 8 th of March 2020.(c) 29 th of March 2020. While at the beginning of February 2020 the virus was still affecting mainly China, it has startedto spread rapidly to other countries, causing infections especially in western Europe countries at the2eginning of March 2020. In less than a month, by the end of March 2020, the outbreak was present onall continents, affecting most of the countries in the world, which led the World Health Organization toofficially name it a pandemic.
In the fight with the COVID-19, quarantine was one of the main measures, at least when the hospitalswere overwhelmed with patients and the virus propagation and its inside body working was not wellunderstood.A basic tool in analyzing the spread of the virus is the mathematical modeling. There is a growingbody of mathematical models used at the moment as for a small sample by no means exhaustive see[WDM19, CGR07, GF08, KRD +
20, SNC20, SSSK20, FLNG +
20, TLSB20]. This turned out to be avaluable tool that can be used in the assessment, prediction and control of infectious diseases, as it isthe COVID-19 pandemic, which significantly impacted almost all countries, with important social andeconomical implications worldwide. The main purpose of this work is to develop a predictive model thatcan accurately assess the transmission dynamic of COVID-19.In this paper we use as a starting point the standard SIR model initiated in [Ros] and later investigatedin depth in [KM91a, KM91b, KM33]. The basic SIR model uses the assumption that the parameters areconstant over time.By starting with given initial conditions and observing the solution at a single day, we can determinein a unique way the parameters of the model which we prove in Proposition 1. This suggests that by fixinginitial conditions and observing the solutions on a given day we can determine the parameters uniquely.As a new tool here we use a neural network having as input the solutions on a certain number ofdays for the SIR model with the output the parameters of the model. Now using the data of infectedand recovered in Romania, we use the neural network to guess in the first round what the parametersare from day to day. This is not very accurate, because there is a lot of uncertainty in the data. Forinstance we can not have an accurate estimate of the infected number of individuals, nor can we accuratelyestimate the number of recovered, particularly when there are so many asymptotic cases and not muchtesting as it happened at the beginning of the pandemic. By analyzing the data, we draw the conclusionthat assuming the parameters constant in time is not a valid assumption. The parameters of the modelvary over time due to the measures implemented in the effort to mitigate the spread of the disease. Byexamining the data, we noticed that we can split the time frame into separate regimes, with transitionsperiods between them, and we can consider the parameters constant on each such regime with a transitionallowed between regimes modeled by a logistic function.The actual fit of the parameters is based on a grid search around the averages suggested by the neuralnetwork output we described above. This is worked out by fitting the number of deaths reported and thesolution to the differential equation which drives the dynamic of the number of deaths.The organization of the paper is as follows. In Section 2 we introduce briefly the SIR model and providethe main result that knowledge of solution to the system at any given time determines the parameters in aunique way. In section 3 we provide details on the construction of the neural network and the rough guessof the parameters based on the public data. In Section 4 we give a description of the SIRD model in whichwe separate the deceased numbers from the recovered and discuss how one can arrive at a differential3quation for the deaths alone. Next, in Section 5 we introduce the regimes idea and how we model it.Here we also describe how we fit the paramters and in Section 6 we discuss the predictions based on thismodel. Finally Section 7 includes the main conclusions.
The first attempts of developing a mathematical model of the infectious diseases spreading were madeat the beginning of the twentieth century. One of the most important models that can describe infectiousdiseases is the SIR model. The first ones that developed SIR epidemic models were Bernoulli, Ross,Kermack-McKendrick and Macdonald.The SIR model is a mathematical model that can be used in epidemiology in order to analyze, at agiven time for a specific population, the interactions and dependencies between the number of individualswho are susceptible to get an infectious disease, the number of people who are currently infected andthose who have already been recovered or have died as cause of the infection. This model can be usedto describe diseases that can be contracted just one time, meaning that a susceptible individual gets adisease by contracting an infectious agent, which is afterwards removed (death or recovery).It is assumed that an individual can be in either one of the following three states: susceptible (S),infected (I) and removed/ recovered (R). This can be represented in the following mathematical schema:Susceptible Infected Recovered β γ where: • β = infection rate • γ = recovery rate.We consider N as the total population in the affected area. We assume N to be fixed, with no birthsor deaths by other causes, for a given period of n days. Therefore, N is the sum of the three categoriespreviously defined: the number of susceptible people, the ones infected, and the ones removed: N = ¯ S + ¯ I + ¯ R. Therefore, we analyze the following SIR model: at time t , we consider ¯ S ( t ) as the number of susceptibleindividuals, ¯ I ( t ) as the number of infected individuals, and ¯ R ( t ) as the number of removed/recoveredindividuals. The equations of the SIR model are the following: d ¯ Sdt = − ¯ β ¯ S ¯ INd ¯ Idt = ¯ β ¯ S ¯ IN − γ ¯ I d ¯ Rdt = γ ¯ I (1)where: • d ¯ Sdt is the rate of change of the number of individuals susceptible to the infection over time;4 d ¯ Idt is the rate of change of the number of individuals infected over time; • d ¯ Rdt is the rate of change of the number of individuals recovered over time.Because there is no canonical choice of N , we will transform the system (1) by dividing it by N andconsidering S ( t ) = ¯ S ( t ) /N , I ( t ) = ¯ I/N and R ( t ) = ¯ R ( t ) /N . It is customary to choose N = 10 forconvenience but this is just an arbitrary choice. For instance, analysis on smaller communities, or citiesinvolves less than 10 , however 10 is a common choice because countries number their populations inmultiples of 10 . With these notations we translate (1) into dSdt = − βSI dIdt = − βSI − γI dRdt = γI (2)where β = ¯ β/N and γ is the same as in (1).Notice that now we actually have that S ( t ) + I ( t ) + R ( t ) = S + I + R = 1 for all t ≥
0. Since weare interested in the reverse problem, namely determining the parameters β, γ from the observations, weput this as a formal mathematical result as follows.
Proposition 1.
Referring to the system (2) , if we know I , S and the values I ( t ) , S ( t ) for some t > ,these determine uniquely the parameters β and γ of the system. Notice there the main assumption, that the parameters β, γ do not change in time.
Proof.
The first step is to notice that by assumption, β, γ constants in time yields in the first place that I (cid:48) ( t ) S (cid:48) ( t ) = − βS ( t ) I ( t ) − γI ( t ) βS ( t ) I ( t ) = − γβS ( t ) . which in turn gives that I (cid:48) ( t ) = − S (cid:48) ( t ) + γβ S (cid:48) ( t ) S ( t )and finally integrating this shows that (here we denote ρ = γ/β ) I ( t ) + S ( t ) − ρ log( S ( t )) constant in t. In particular, this means that I ( t ) + S ( t ) − ρ log( S ( t )) = I + S − ρ log( S ) . (3)Typically the initial value of S is close to 1 and I is relatively small. In particular, if we assume thatthe epidemic ends somewhere then we definitely have I ( t ) = 0 and thus S ( t ) solves the equation S − ρ log( S ) = I + S − ρ log( S ) . (4)In particular if we assume that I ( t ∞ ) = 0 and S ( t ) converges as t → t ∞ , then we get in the limit that S ( t ∞ ) solves (4). One consequence of this argument is that for all time 0 ≤ t ≤ t ∞ , we have that S ( t ) − ρ log( S ( t )) ≤ α := I + S − ρ log( S ). 5nother important consequence of this model is that if we assume S and I fixed (obviously R willalso be determined) but, for a given time t = t >
0, knowing S ( t ) and I ( t ) (therefore R ( t ) as well),we can determine uniquely the parameters β and γ . Indeed this is clearly seen from (3) which gives ρ = I + S − I ( t ) − S ( t )log( S ) − log( S ( t )) . On the other hand, from (3) in the first line of (2), then we obtain that dSdt = − βS ( I + S − ρ log( S ) − S + ρ log( S )) (5)The problem is that we can not integrate explicitly this to obtain an analytic expression for S ( t ). However,what we can still show is that by knowing I , S , I ( t ) , S ( t ) we can determine the parameter β . As wealready pointed out, we know how to determine ρ = γ/β , thus we can rewrite (5) in the form S (cid:48) S ( I + S − ρ log( S ) − S + ρ log( S )) = − β. (6)Now, for α := I + S − ρ log( S ) > ρ > x such that α > x − ρ log( x ),Φ( x ) = (cid:90) x dss ( α − s + ρ log( s ))and notice that using this function, integrating (6), we arrive atΦ( S ( t )) − Φ( S (0)) = βt from which it is clear that β is completely determined by S ( t ) , S , I . Knowing β and ρ , we can imme-diately solve for γ = ρβ , thus all parameters are determined. Our next goal is to get estimates on the parameters β, γ of the SIR model. There are two basic ideashere. The first one is to train a neural network using a typical inverse problem. The second one is to usethis neural network combined with the data to estimate the regimes of the parameters. In a real world theparameters do not stay constant, they change slightly and we would like to catch part of this behavior.We combine the neural network with this day by day estimate to get an indication of the regimes for β and γ . In fact, what we do is we try to detect the regimes where the parameters stay more or lessconstant. As we will see the regime change is confirmed by the quarantine imposed as a fighting measureagainst the virus. To deal with the parameter estimates, we do the following. First we discretize β by considering 200points equally spaced in the interval [0 .
1; 1 .
5] and for γ we consider 100 points equally spaced in theinterval [0 .
05; 0 . β i , γ j ), for 50 days, for a population of10 individuals, and we store the results in a dataset.We train a neural network on the resulting dataset so that the input is of the form: XT rain = (
Day,
Susceptible,
Inf ected,
Recovered )or in the terminology of the previous paragraph, we have
XT rain = ( t, S ( t ) , I ( t ) , R ( t )) where t = 0 , , . . . , . and the output is exactly the pair Y T rain = ( β, γ ) , which generated the solution above. We fixed the initial conditions S = 1 − I , I = 2 /N and R = 0.We started with 2 infected people because there were two initial individuals who traveled in outside thecountry in exposed regions and were first spotted as the original spreaders.The neural network we used is of the following form1. Dense 64, activation ReLU, with input dimension=4, dropout =.22. Dense 128, activation ReLU, dropout=.23. Dense 256, activation ReLU, dropout=.24. Dense 512, activation ReLU, dropout=.25. output ( β, γ ) with optimazer Adam and loss MSE. Before we move on with the results of the day by day estimates, we point out that the result ofProposition 1 guarantees that the parameters estimated should be well-determined by the network aseach triple ( t, S ( t ) , I ( t )) determines in a unique way the parameters ( β, γ ).Once the model had been trained, we use it to predict the day by day β and γ for Romania Covid-19reported numbers. What this means is that we try to predict a set of parameters such that for a givenday t , what we observe is exactly the number of suspected, infected and recovered reported on that dayby the officials. Therefore, we assume and try to predict a single set of parameters for the time period[0 , t ], t here being the corresponding day. The results are represented in the chart below.7igure 2: The prediction of day by day neural network trained on SIR models.What this suggests is that we can identify three regimes. The first one is characterized by uncertainty,with a high infection spread and big variation of the two parameters values from day to day. This isapproximately for the first 15 days or so. This may be due to the fact that, even if there were not manycases reported yet and the restrictions have not already been imposed, people were starting to be awareof the severity of the situation. On the other hand, the last 25 days have a lower volatility for bothparameters, which can be a consequence of the measures taken by the authorities. The intermediateregime can be considered a transition between the first one and the last one. This is roughly centered onthe 30th day with a period of ±
10 days of regime switching.We should also commment on the fact that the data that is available shows the number of individualsthat have been tested positive, but it is very likely that the real number of people infected is in factmuch higher, as there are also asymptomatic individuals, people that are not being tested although theypresent the specific symptoms, so they are not part of the official reports. Another aspect that should betaken into consideration is that the long incubation period characteristic to this virus determines a delaybetween the moment when a person has been infected and the moment when that person has been testedpositive.In order to reduce the effects of the above deficiencies, we consider another model below which accountsthe number of deceased as separate rubric. In this process we keep in mind all the points discussed untilnow, an important one being that the parameters are not constants for all period, they can be at mostconstant on pieces, reflecting in fact the regime imposed on the population.
In the sequel we propose a model in which we modify the SIR model in two different directions. Thefirst one is to consider an interaction between the recovered and the susceptible on one hand and theother direction is to have an account of the deaths in this analysis. If we want to take into account the8nteraction between the recovered and the susceptible, we really need to separate the deceased ones fromthe recovered ones. At the end of the day the number of deaths is probably the most reliable number wecan account for, as the number of infected people is wildly unknown and the number of recovered is alsolargely unknown.Thus we have four variables changing with time now. These are S ( t ), I ( t ), R ( t ) and D ( t ) where R ( t )is the proportion of recovered and alive people while the D ( t ) is the proportion of deceased people. Weset the interaction as follows dSdt = − βSI dIdt = βSI − ( γ + γ ) I dRdt = γ I dDdt = γ I. (7)Notice that in this setup the recovered population bifurcates into recovered ones, accounted by R andthe dead ones accounted by D . The point of this model is to see how many people die in the long runfrom this disease. Of course this is not complete as there are other factors which should be taken intoaccount, but we are going to use this simple model.Notice that for R taken as the sum of the two factors R + D above we fall into the classical SIR model.There are two points here for the model. One is that we separate the dead people from the recoveredones which are mixed up in the classical SIR model.We are going to manipulate these equations and reduce the computations to a single equation involvingonly one of these quantities, the most reliable one, namely D ( t ). To do this we will write all the otherquantities as functions of D as follows: S = u ( D ) , I = v ( D ) , R = w ( D ) . The easiest to deal with is R because from the last two equations we get dRdt = γ γ dDdt from which we deduce that R ( t ) = γ γ ( D ( t ) − D ) + R .Now, we deal with the function u from S ( t ) = u ( D ( t )). Dividing the first and the last we get u (cid:48) ( D ) = − βγ u ( D )which can be integrated and gives S in terms of D as S = S exp (cid:18) − βγ ( D − D ) (cid:19) . On the other hand this allows us to solve for I = v ( D ). First we notice that dSdt + dIdt = − ( γ + γ ) I = − γ + γ γ dDdt from which we deduce that S + I + γ + γ γ D = S + I + γ + γ γ D . D ) dDdt = γ I − ( γ + γ )( D − D ) + γ S (cid:20) − exp (cid:18) − βγ ( D − D ) (cid:19)(cid:21) (8)This last deduction works in the case the parameters β, γ , γ are all assumed constant in time.However, if they vary with time, then, the equation is a little bit different, the main equation becomesnow D (cid:48) ( t ) = γ ( t ) I − γ ( t ) (cid:90) t (cid:18) γ ( s ) γ ( s ) + 1 (cid:19) D (cid:48) ( s ) ds + γ ( t ) S (cid:20) − exp (cid:18) − (cid:90) t β ( s ) γ ( s ) D (cid:48) ( s ) ds (cid:19)(cid:21) (9)The main idea from here is to use the data on the death cases to estimate the coefficients involved.As we already pointed out, the number of perished people is the most reliable data, since all the otherdata is very rough. For instance, the proportion of people which are infected is grossly underestimatedsince there are probably more infected people than the reported cases tested usually in the hospitals. It isalso true that even the dead numbers are probably overestimated as many people perish due to existingconditions which lead to complications which in the end makes the task of deciding the cause of deathmuch more difficult. However this is the most reliable data we can trust.From the technical standpoint, equation (9) is not easy to handle and we will use equation (8) insteadtogether with a regime switch and a piecewise parameter fit. In other words, we fit the number of deceasedon pieces where we assume that the parameters do not change. We talked about the existence of different regimes in the spreading of the disease because of themeasures that have been taken which had a significant impact on the evolution of the infection rate. Wechoose to apply an approach in which we have separate scenarios for the free spread period and the caseof quarantine, as well as a transition period between these two scenarios.First we define the sigmoid function: σ ( x, c, s ) = e s ( x − c ) where c models the turning point and s models how swift the transition between these two regimes is taking place.Considering 2 different regimes and a transition period between them the SIRD model becomes dSdt = − ( β · σ ( t, c, s ) + β · (1 − σ ( t, c, s ))) · S · I dIdt = ( β · σ ( t, c, s ) + β · (1 − σ ( t, c, s ))) · S · I − ( γ · σ ( t, c, s ) + γ · (1 − σ ( t, c, s ))) · I − ( γ σ ( t, c, s ) + γ (1 − σ ( t, c, s ))) I dRdt = ( γ · σ ( t, c, s ) + γ · (1 − σ ( t, c, s ))) · I dDdt = ( γ · σ ( t, c, s ) + γ · (1 − σ ( t, c, s ))) · I. (10)where • β , β represent the infection rates in the first regime respectively in the second one; • γ , γ represent the recovery rates in the first regime respectively in the second one; • γ , γ represent the fatality rates in the first regime respectively in the second one10ith the same notations and same logic we will consider β, γ , γ in (8) as follows: • β = β · σ ( t, c, s ) + β · (1 − σ ( t, c, s )) • γ = γ · σ ( t, c, s ) + γ · (1 − σ ( t, c, s )) • γ = γ · σ ( t, c, s ) + γ · (1 − σ ( t, c, s ))We have previously shown that we can derive all the parameters using the number of dead people,as it is the most reliable one, using equation (8). In other words we will use the above substitutions for β, γ , γ in equation (8) and we aim at finding the solution of the following problem:minimize β , β , γ , γ , γ , γ t = n (cid:88) t =1 (cid:18) D ( t ) − Data ( t ) (cid:19) (11)It is important to mention that we fix some of the parameters. We take as follows • c = 30 based on the moment when the recommendations were made or restrictions were imposed,when people started to be aware of the severity of this virus • s = 0 . • n = 60 we solve the problem using the data from the first 60 days. • The optimization problem is solved by using a grid search around the average values of the β, γ obtained from the neural network on each of the regimes outlined in the section above and plottedin Figure 2. In particular we use here the fact that γ + γ = γ , for the first regime and similarlywe have γ + γ = γ for the second regime. We interpreted γ = pγ and γ = (1 − p ) γ andthus we in fact search for p on a certain grid for the best fit in (11).We point out that the last two parameters depend on the country, as different countries have acteddifferently.Solving the optimum problem in these conditions we obtain:11igure 3: The parameters are as follows: β = 0 . , γ = 0 . , γ =0 . , β = 0 . , γ = 0 . , γ = 0 . , c = 30 , s = 0 . ± ±
15% and ±
20% estimates in the shaded areas.We plot here for 2 different timeframes: 60 days and 90 days. An important aspect that we haveto keep in mind is that the prediction of the disease evolution can not be done accurately for a longtimeframe, as the new restrictions or other measures, treatment and medication developed over the timedefinitely impact the model. Considering this idea, the prediction of the infectious disease evolutionshould always incorporate in the model the new information that arises as the time passes and we shouldsplit the timeframe in separate regimes.
Starting 15th of May, the Romanian Government started to gradually relax the COVID-19 restrictionsand on the 15th of June the secound round of relaxations were issued. During this time, various restrictionshave been lifted and a new package of relaxation measures has been released every two weeks. In order togenerate reliable predictions, we now have to take into consideration the effect of lifting the restrictionsand to adapt the model to this new regime. Therefore, we analyze how the measures of relaxation haveimpacted the parameters of the model and we generate a new prediction. We expect the number of deathsto grow.With the same logic that we used before we consider these new parameters: • β represents the infection rate in the third regime • γ represents the recovery rate in the third regime • γ represents the fatality rate in the third regime;Now, in order to make a new prediction, we will consider β, γ , γ in (8) as follows: • β = [ β · σ ( t , c , s ) + β · (1 − σ ( t , c , s ))] · σ ( t , c , s ) + β · (1 − σ ( t , c , s ))12 γ = [ γ · σ ( t , c , s ) + γ · (1 − σ ( t , c , s ))] · σ ( t , c , s ) + γ · (1 − σ ( t , c , s )) • γ = [ γ · σ ( t , c , s ) + γ · (1 − σ ( t , c , s ))] · σ ( t , c , s ) + γ · (1 − σ ( t , c , s )) • c = 115 corresponding to 15th of June when the last round of relaxations were put in place, inparticular the opening of restaurants, bars, hotels and pools were regulated. With this we take s = .
22 which corresponds to a switch of approximately 5 days from one regime to the other.Solving the equation with the third regime included we obtain the following prediction:Figure 4: The parameters for each region is as follows: β = 0 . , γ = 0 . , γ = 0 . , β =0 . , γ = 0 . , γ = 0 . , β = 0 . , γ = 0 . , γ = 0 . c = 115, s = 0 . (cid:15) = 10% , ,
20% adjustment done as follows. With the parameters from above we constructed a gridaround the mean values of β and γ and took the range of (1 ± (cid:15) ) β , (1 ± (cid:15) ) γ and split this into a mesh with10 values in each range. Furthermore, we evaluate the number of deaths for each such combination andarrange them according to difference from the number of deaths predicted with β , γ . Then we take thevalues for which the number of deaths is the most extreme at the end of the period, both underestimatedand overestimated. This will give our ( β , max / min , γ , max / min ). These will give the boundary curves foreach (cid:15) = 10% , , γ /γ as constant.The left figure is predicting the evolution until 1st of August based on the training till 15th of July, whilethe second figure gives a prediction until 1st of September.On the new regime, the model predicts that in 30 days (with the fitting based on 141 days, whichis up until July 15th), the number of deaths will reach deaths on 1 st of August (the 160th day), deaths by Aug 15th and deaths by Sep 1 st and the corresponding confidence intervals outlinedabove.It is worth comparing the previous scenario, when we had only two regimes, with the new situationwhen we have three such regimes: 13igure 5: The figures of the regime without the adjusting of the parameters (left) and with the adjustingof the parameters (right). One can see that the prediction does not fit the real data any longer.The left figure gives the estimated prediction curve together with the ± ±
15% and ±
20% estimatesin the shaded areas. The right figure is the same as the left picture in Figure 4.The two graphs above show that the partial lifting of the restrictions will have a negative impact onthe number of deaths caused by COVID-19. In the initial approach the number of predicted deaths on 1 st of August would have been while in the second approach which includes the relaxation period, thepredicted number of deaths increases significantly, to . We can easily notice that the first approachhaving one regime when the pandemic starts and one regime when restrictions are imposed, seems tounderstand the evolution of the epidemic better. Even so, considering the social and economical aspectsand the behavior of the population and governments in the affected countries, we can deduce that theapproach which adds the relaxation regime fits better the real situation.We use now the same parameters for SIRD model and in Figure 6 is the plot of the evolution of allthe main categories. 14igure 6: The predicted evolution using the SIRD model with three different regimes. The estimates ofthe main categories for the period Feb 26 - Sep 1 st , which amounts for 190 days. Probably the mostinteresting part is that by then 80% of the population has been exposed to the virus already. We shouldnote that part of assumptions here is the fact that some form of immunity is built up and the reinfectionis not taken into account. Now we summarize what we did here. The main idea is that within the models we used, be it SIRor more realistic SIRD, we split the problem according to various regimes. In this paper we take threeregimes. One is the regime before any measures were taken. The second regime is the one in which thequarantine was imposed on the population. We also model the transition from one regime to another.The third regime we consider is the one following the relaxation. The transition is also modeled with thehelp of logistic function.The fit is done using the number of deaths and using the SIRD model. The search of the parameters isdone around the values of β provided by the neural network constructed based on the simpler SIR model.We believe that this methodology is a general one and can be extended to any country provided thatwe have data, in particular some information about the regime switch for each of the regimes.As a disclaimer, there are several assumptions made here. One of them is that people build upimmunity to this virus and the reinfections are negligible. References [CGR07] Marc Choisy, Jean-Fran¸cois Gu´egan, and P Rohani,
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