Mathematical Modeling and Estimation for Next Wave of COVID19 in Poland

We investigate the problem of mathematical modeling of new corona virus (Covid19) in Poland and tries to predict the upcoming wave. A Gaussian mixture model is proposed to characterize the COVID-19 disease and to predict a new / future wave of COVID-19. This prediction is very much needed to prepare for medical setup and continue with the upcoming program. Specifically, data related to the new confirmed cases of COVID-19 per day are considered, and then we attempt to predict the data and statistical activity. A close match between actual data and analytical data by using the Gaussian mixture model shows that it is a suitable model to present new cases of COVID-19. In addition, it is thought that there are N waves of COVID-19 and that information for each future wave is also present in current and previous waves as well. Using this concept, predictions of a future wave can be made.


INTRODUCTION
Currently, the world is threatened by new upcoming of waves of Covid19, a new disease transmitted by the corona family virus. Many countries around the world have noticed a large number of cases of Covid 19 from December 2019 onwards Let us recall some basic facts about the pandemic and recommended behaviors. People with low immunity, aging, and lung-related medical problems are more prone to Covid 19 infections [1,2,3,4]. Symptoms of Covid19 are coughs, colds, respiratory problems that are very similar to the flu. It is clear from doctors that a person infected with Covid19 recovers within 14 -16 days because the incubation period of the novel corona virus is 14 days. Covid19 protective measures should be protected by frequent hand washing, to avoid touching the mouth, nose, and face, and to keep the public (2 m or 6 feet) away from other people. Covid19 is now a pandemic as announced by the World Health Organization (WHO) [1]. Therefore, preparation for health care should be adequate worldwide. As the disease is contagious, the number of young infected people is increasing rapidly. If hospitals are not well prepared, health workers will not be able to function properly. In this case, it is inevitable that we will have an accurate measure of new Covid19 cases, which can help medical managers and administrators. Now three waves of Covid 19 have appeared in Poland and in several other countries as well. An important question about the current state of Covid 19, that is, when will the next wave of Covid 19 emerge? The conviction of the Polish authorities is rather optimistic, although very cautious. The daily announcements of the ministry of health (July 2021) predict the fourth wave of the pandemic, most often including fall dates. However, it results rather from premonitions and informal forecasts of an increase in prohibitions, e.g. in connection with the beginning of the school year in the stationary mode planned by the authorities. The anticipated appearance of another wave of infections is also reinforced by the growing aversion to vaccinations of the so far unvaccinated part of the population still below the threshold of general social immunity (in July 2021, about 41% of the population was vaccinated with the second dose [34]). Therefore, virus spread models and prediction methods are still an extremely important challenge for the world of science, including the part that uses statistical inference and artificial intelligence. The authors of this work considered confirmed cases as the most important time series, ignoring others, which does not mean disregarding them, but only an attempt to model reality in this way -with one variable accumulating the influence of many predictors. In previous publications, the problem of predicting the spread of the virus in Poland has been discussed many times [19,24,26,32], often along with the analysis of the situation in other countries [14,22,23,25]. In the above-mentioned studies, statistical data on Poland are directly used to exemplify changes in the spread of COVID-19 [19,24,26,31,33]. Moreover, there are many studies presenting general models that can be applied to quite any country and the selected data are only a pretext for confirming the methodological correctness of the applied model and the prognostic method [7, 12, 13, 16, 18, 27, 28, 29 30, 31]. Some of them are devoted to the analysis of the situation in a selected country, eg in the USA [10,13] or India [8,9]. Interesting and inspiring may be the analysis in terms of the applied models and predictive methods, mainly emerging from artificial intelligence and machine learning algorithms. Data for work were collected from sources such as [1, 5, 6, 34].
In this paper, we propose a Gaussian composite model to present new confirmed cases of Covid 19. Limited estimates can be made using this model. We have read and discussed details of Poland. The proposed model can also be used in other countries as well. Approval intensity is demonstrated by comparing available data with analytical results. In addition, an estimate of the next wave of COVID-19 wave can be made using this model.

PROPOSED MODEL
We consider that there are total N Gaussian waves. Let us write a mixture Gaussian function with the help of [21] as where 1 and 2 are constants, is the exponential function; , , and are the weight, mean, and standard deviation of k-th Gaussian wave. It can be noticed from (1)  Actual data of daily new cases of Covid 19 in Poland and analytical data is plotted in Fig. 1. Analytical data is calculated by using (2). The actual data of daily confirmed cases in Poland of Covid 19 is considered from 04/03/2020 to 09/07/2021 [5]. It can be noticed from actual data [5] that first peak of Covid 19 was observed on 07/11/2020 in Poland with 27875 fresh cases, whereas the analytical value is approx. 26570, which is quite close to the true value. Further, a spike can be seen in actual data of value 37596 on 23/11/2020, it is not modelled properly with the analytical relation given by (2). Then a very small second wave can be observed from Fig. (1), which started in mid of December 2020 and vanished in first week of February 2021. It's peak value was around 14,216, as seen from the figure. Third wave of Covid 19 started in the month of February 2021, just after second wave. It was more severe as compared to first and second wave. It's peak value was around 35,246 [5] and it was observed in last week of March 2021. If we calculate the analytical peak of third wave, it is approximately 33,000, which is very close to the actual value. Therefore, it can be concluded that (2) characterizes the new cases of Covid 19 in Poland efficiently.

RESULTS OD THE PRDICTION OF THE NEXT WAVE OF COVID-19
In this section, it is proposed to estimate the forthcoming Covid 19 wave, in particular, proposing a predictor of the next covid wave along with its maximum value. To explain this approach, let us first consider the measurement of mean, standard deviation, and the peak value of a Gaussian wave by utilizing its sample values.

Estimation of mean, standard deviation, and the peak value of a Gaussian wave:
We can write a Gaussian function as where p, m, s are the height, mean, and standard deviation of the Gaussian wave provided by (3). The plot of (3) is also shown in Fig. 2. Let us assume that some samples of ( ) are available as shown in Fig. 3 and we need to estimate p, m, s by using these samples. From (3), we obtain and By using (4) and (5), we get log ( 1 )  (2) Similarly, by using (5) and (6), we can write log ( 2 ) With the help of (7) and (8), after some algebra, we obtain 2(( 2 − 1 ) log and = √ .
By using (4), (5), and (6), we get 6 The value of p can be obtained by substituting the values of m and s from (9) and (10), respectively, into (11). Further, it can be observed that only three samples are sufficient to get a Gaussian wave.

3.2.Estimation of mean, standard deviation, and the peak value of a Gaussian wave with N samples:
In this subsection, we assume that N samples of a Gaussian function is available, as in the case of Covid 19 new wave, in that case following procedure can be used: 1) Divide all samples in the group of three, if it is not possible leave some old samples and take all recent samples, which can make a group of three. 2) Estimate p, m, s by using the method given in subsection A for all groups.
3) Estimate p, m, s by performing averaging of p, m, s obtained by different groups.

Prediction of next wave of Covid 19:
In the current situation of Covid 19, the most important question is that when will next wave of Covid 19 start for the proper arrangements performed by hospitals and other authorities as well. It can be noticed from (1) that any in any sample of X(t) , i.e. from the information of fresh cases of Covid 19, one can have many more details of Covid 19. It is because of the fact that the X(t) contains the data of preceding waves, present Covid 19 wave and upcoming Covid 19 waves also. Another important fact is that Gaussian function decreases as it moves away from its mean value, therefore, it can be assumed that the governing waves are only nearby Gaussian waves, all other will contribute negligibly. Let us rewrite (1) in more detailed form as Where N can take any positive integer value, large or small. It can be seen from (12) In (13), 1 and 2 can be obtained by curve fitting. It is assumed that first wave has not been fully arrived, so by following the method given in III-A, we can find the values of 1 , 1 and 1 . As more samples of first Covid 19 wave received, estimation of 1 will be better. Moreover, these values will also supply some details of second Covid 19 wave also. If there is no upcoming Covid 19 wave, then X(t) must be zero after the first Covid 19 wave. It shows that after sufficient time, i.e., after dying out the first wave, there must be zero or very small daily new cases of Covid 19 patients. But if it is not so then ( ) − 1 − ( − 1 ) 2 2 1 2 are the values which may generate second wave of Covid 19. By using these sample values, one can estimate the upcoming covid 19 wave and so on.

3.4.Prediction of next wave of Covid 19 in Poland:
We want to estimate about the next or fourth Covid 19 wave in Poland. From (2), we have the characteristics of the first, second, and third waves of Covid 19. The calculation of the parameters of the second wave and third wave of Covid 19 is performed using the method given in section III-A. The relationship of new Covid 19 cases daily is given in (2). If there is no fourth wave then third wave will die out completely and new covid cases count will be zero. But if it is not so then we need to calculate   Fig. 4 and 5, that the prediction of upcoming next wave is largely dependent upon the data samples used. Further, it can also be observed that almost one month delay or advancement can be noticed, if we change set of sample values.

DISCUSSION AND CONCLUSIONS
A mathematical model has been proposed to characterize the Covid 19 daily new cases. A close match between actual and analytical data is evident from numerical results provided in the paper. By choosing different parameter values in the proposed function, the plots can be obtained for other countries also. Specifically, the characterization has been performed of Covid 19 daily new cases in Poland. Moreover, an estimation method has been developed for Gaussian function by utilizing its samples. It can be concluded that the estimated mean, variance and peak of Gaussian function largely depends upon the set of samples used.
The results obtained for Poland do not seem to be in line with the so-called common expectation. However, the authors present and justify the method, obtaining results that are consistent with historical data so far. The model does not take into account many potentially significant predictors that would e.g. occur in regression models (e.g., time series of tests, deaths, recovered, vaccinations, restrictive actions, health service status, etc.). In this matter, it is worth returning to valuable already published and partially cited works [7,9,13,16,18,27,29].
Most of the works cited here, those aimed at forecasting methods, limit their aspirations to rather short prediction horizons, often several days, weeks, and less often months, usually stipulating a wide range of the possible forecasted result. In the model with the use of Gaussian waves, we are far ahead, perhaps risking our reputation. However, we show that such far predictive horizons can be achieved, as usual, with the risk of error. We also assume that there are countries for which this method will be more effective than for Poland, as well as those for which it will be less successful.
Here is an example of the philosophy of computational intelligence.
With large data set, estimation will also be improved. In addition, the process of estimating the upcoming Gaussian wave has been also proposed. It is shown by estimation that with more samples obtained, a better estimate can be made.

DECLARATIONS:
Fundingnot applicable; Conflict of interest: The Authors declare no conflict interests.
Availability of data and material: The data used in the work is available on the following pages: Author's contributions: