A NEW COMPUTATIONAL MODELLING FOR PREDICTION OF COVID-19 POPULATION AND TO APPROXIMATE EPIDEMIC EVOLUTION OF THE DISEASE

. Infectious diseases are growing at a fast rate. The early prediction and monitoring of the progress of such diseases, such as the recent outbreak of COVID-19, are fundamental to and infection control and rapid recovery. This paper presents (i) a new computational program based on a make blobs generator designed to predict the number of daily COVID-19 cases over a period of 30 days and (ii) a deterministic compartmental SEIQHRV model based on a system of ordinary differential equations. The objective of both the computational and mathematical models is to describe the dynamics of COVID-19 over time to understand the inﬂuence of speciﬁc parameters on its spread, while also predicting the approximate epidemic evolution of the disease. Furthermore, we evaluate and validate the new computational model by comparing our simulation results with the mathematical model, treating the latter as a reference model. Numerical simulations of the proposed models are applied to a randomly selected group of individuals. The results show that the curves obtained by the SEIQHRV mathematical model and those obtained by each scenario of the proposed computational model are approximately similar.


INTRODUCTION
The outbreak of coronavirus disease 2019  has led to one of the greatest global challenges in contemporary history, with more than 642 million infections and more than 6 million deaths worldwide [1]. While such health statistics are staggering, it would, however, be a serious mistake to describe this challenge only as a health crisis. It is a large-scale humanitarian crisis that has caused untold misery and human suffering, driving the socio-economic wellbeing of people to the brink of collapse. The COVID-19 pandemic has also had a significant negative influence on the global health system, leading many governments to adopt stringent measures to prevent the further spread of the epidemic and to ensure the proper functioning of their health systems [2]. Although the enforcement of such regulations and vaccines mitigated the effects of the pandemic initially, several vaccinated countries have recently seen a recurrence of the disease with new variations. Thus, the virus's extremely contagiousness and the emergence of harmful mutations are still impacting public health.
Understanding the nature of transmission and correctly forecasting the course of the epidemic are therefore two of the most crucial components in combating COVID-19, especially in countries with large populations. Furthermore, precise forecasting can provide feedback on whether an implemented policy is successful in reducing the strain on a particular nation's healthcare system. Governments can also use forecasts to evaluate mitigation strategies and regulate policies. For instance, by using mathematical models like the SIR and SEIR models, researchers have been able to accurately anticipate the COVID-19 reproduction parameter for the early prevention of the pandemic, thus highlighting the necessity for trustworthy forecasting models [3].
In this sense, mathematical models can be used to track the evolution of the epidemic by studying the behaviour of the virus. Using this approach, numerous researchers have examined COVID-19 from various perspectives [4,5,6,7,8]. Particulary, the study in [9] produced the traditional SEIR model, also known as fractional models for the examination of the global epidemic of COVID-19. The establishment of new quarantine regulations and the hospitalisation of infected patients (SEIQR model), which are regarded as epidemic parameters for COVID -19, are the foundations of this development [10]. Xu et al. [11] used the generalised fractional order SEIR model to forecast the epidemic trend of COVID-19 in the USA. Two improved models, SEIQRP and SEIQRPD, then successfully captured the development process of COVID- 19, providing an important reference for comprehending the trend of the outbreak. Meanwhile, the authors of [12] propose an optimal control problem based on a SEUIRD model adapted to the COVID-19 pandemic, which takes into consideration the epidemiological situation in Morocco.
Furthermore, Ould Beinane et al. [13] studied the dynamics of the COVID-19 model SEIQHR with a standard incidence rate, considering fractional Caputo derivatives in order to gain a better insight into the disease.
Based on the aforementioned findings and previous literature, with the lack of effective treatment for COVID-19, vaccination is still one of the possible solutions for managing this infectious disease. In order to assess the impact of a vaccine programme on COVID-19 transmission, we propose an an extension of the classical models by using them to review the impact that vaccination has on infection rate hospitalised individuals (H), quarantined individuals (Q), and recovered individuals (R). The SEIQHRV goal of the epidemic model is to describe the dynamic behaviour of the disease and predict its propensity to spread.
The literature on this topic has until now indicated that mathematical modelling can play a crucial role in forecasting, detecting, and preventing COVID-19, while certain artificial intelligence approaches [14,15,16] can be used to estimate future COVID-19 viral transmission. See, for example, machine learning models (a hybrid approach based on a regression tree [17], long short-term memory (LSTM) networks [18], polynomial neural network (PNN) [19], FPASSA-ANFIS [20], SVR [21], etc.), and other types of models [22]. Another study [23] used deep learning to anticipate the pandemic trend for Brazil, India, and Russia -three nations that rank among the top ten in terms of global impact according to a thorough analysis by healthcare professionals. To forecast the number of cases and the spread of COVID-19 in these countries, three distinct deep learning models were used. In [24], the authors proposed a multi-source deep transfer learning-based novel approach to robustly forecast the COVID-19 transmission in a province within a country.
Additionally, significant progress has been made in detecting, preventing, and predicting COVID-19's harmful effects via computational, dynamic, and AI models, as well as in the use of numerous technologies. Although epidemiological models help to assess the dynamics of progression, they make various assumptions and rely on the knowledge of various specific factors. To the best of our knowledge, no attempts to overcome these restrictions have yet been made, with no work being published that uses the characteristics of mathematical models to create a computational model (programmed in Python) to predict the growth of the COVID-19 population and to approximate epidemic evolution. Therefore, it is necessary to construct prediction techniques for frequent diseases, as well as to further expand prediction methodologies.
The objective of the present study is to address this need.
The intended contributions of this work are as follows: • We aim to create a new computational model similar to the traditional mathematical model based on a make blobs generator, which generates a feature matrix and corresponding discrete targets, and is used to demonstrate clustering.
• The proposed computational and mathematical models have the same objective: to study the transmission dynamics of COVID-19 and to simulate its propagation within a population of individuals. We will investigate several rates that make up our model to better understand how they affect the virus's pace of spread. This modelling will be implemented with the Plotly library [25].
• The computational model will then be compared to the SEIQHRV mathematical model, which belongs to the family of compartmental models in epidemiology.
The applicability of the proposed model will be assessed by simulating a population of 100,000 individuals. The information they provide and the situations in which the models are used show that the predictions produced by the two models are similar.
An overview of the paper's structure is provided below: The first two sections cover the description of the two models. We begin by analysing the qualitative properties of the SEIQHRV mathematical model, including the bornitude of solutions and the disease-free and endemic equilibrium points. We then establish the conditions for the local asymptotic stability of the disease-free and endemic equilibrium points, before explaining the computational model. After that, we discuss the experimental findings in the results and discussion section, demonstrating how both approaches (mathematical and computational) affected the behaviour of the different compartments. Finally, we describe a sensitivity analysis of the infection parameters, which we conducted in order to determine the impact of each parameter.

METHODS
Using previous knowledge of the COVID-19 pandemic, we propose a SEIQHRV model and a computational model to estimate the virus's epidemic spread. The computational model we propose differs from the traditional SEIQHRV model but produces relatively similar results to the mathematical model. In this section, we will provide a brief description of the two proposed models.

SEIQHRV model.
We employ a mathematical formulation built on differential equations to examine the transmission dynamics of COVID-19. The model separates the entire population into seven classes: susceptible individuals (S), exposed individuals (E), infected individuals who were not hospitalised (I), hospitalised individuals (H), quarantined individuals (Q), and recovered individuals (R). In addition, the model includes a compartment for vaccinated individuals (V ). Individuals in the exposed class (E) are considered to be incubating; that is, even if they have no symptoms and have not tested positive for the disease, they are still contagious. The typical recommendation is for class (I) infected patients who tested positive but only have mild to moderate symptoms to self-quarantine at home rather than being admitted to hospital. People in class (H) who tested positive and are in hospital are more vulnerable (e.g. the elderly and those with underlying health problems). The parameters are described in Table 1 with initial conditions: The following schema illustrates a detailed description of our model (2.1): The signification of these parameters and variables is shown in

Boundedness and positivity of solutions.
In this section, we will proof the boundedness of the solutions of our model (2.1).
Lemma 2.1. The solutions of the system (2.1), which belongs in R 7 + , are non-negative and uniformly bounded.
Using the theory of differential equation [26,27], we obtain In order to find the equilibrium points, the equations of our system (2.1) must equal zero.
It is clear that system (2.1) always has an infection-free equilibrium, as denoted by P 0 = , and which is given by P 0 ( Λ d+a s , 0, 0, 0, 0, 0, a s Λ d(d+a s ) ).

Basic reproduction number.
The reproduction number R 0 for the dynamics of disease is calculated using the next-generation After calculation, the spectral radius of the next generation matrix p(FV −1 ) is shown by

Endemic equilibrium and local Stability.
First, we find the coordinates of the endemic equilibrium point, after which we will analyse the local stability of the two equilibrium points. Proof. For the equilibrium points of model (2.1), we have Then we get: Now we analyse the local stability of two equilibrium points P 0 and P * .
The Jacobian matrix associated with the model (2.1) is defined by Proof. We calculate the eigenvalues of the Jacobian matrix at the equilibrium point P 0 in order to analyse its local stability. Then, the characteristic equation of P 0 ( Λ d , 0, 0, 0, 0, 0, a s Λ d(d+a s ) ) is: Then, the eigenvalues of matrix (2.5) to the equilibrium point P 0 are reel roots, so : If s > 0 and R 0 < 1, then λ 5 + λ 6 < 0 and λ 5 λ 6 > 0. So λ 5 < 0 and λ 6 < 0. The proof is completed.
Theorem 2.3. The endemic equilibrium point P * is locally asymptotically stable if R 0 > 1 a 1 , a 2 > 0 and a 1 a 2 > a 0 is realised.
Proof. The eigenvalues of the Jacobian matrix associated with the equilibrium point P * are calculated in order to study its local stability.
From the Jacobian matrix J * (2.5), the characteristic equation at P * is as follows: Using (2.4), we get: Then, P * is locally asymptotically stable if R 0 > 1, a 1 , a 2 > 0, and a 1 a 2 > a 0 are realised.

Description of computational model.
In this section, we present a computational model (created in Python), designed specifically to predict the evolution of COVID-19-infected individuals over time based on a sample of random individuals. During the simulation, several parameters are taken into account. A population of 100.000 individuals is chosen to provide a reasonable execution time. We assume that the entire population is likely to be contaminated. The distribution of these individuals is given by the parameter population variance, which varies between 0 and 2 approximately. The larger this parameter is, the wider the distribution of the population.
Then, the parameter radius contamination corresponds to the area in which an infected individual can spread the virus to an uninfected individual, with a probability named infectiousness (β 1 ; rate of individuals who become exposed without confirmation of infection and β 2 ; rate of individuals who become exposed with confirmation of infection). After being infected, there are three possibilities: an infected person can be hospitalised with a probability η, confined with a probability δ , or recover immediately with a probability µ. An individual can also be hospitalised in case of severity with a rate of ε and the passage of a hospitalised person recovered with a rate of r. These categories reflect the range of possible states of the population in the majority of real cases. We have considered the occurrence of vaccinations during the simulation, which relates to susceptible and recovered individuals with rates of a s and a r , respectively.
A mortality rate d is imposed on all the categories of the individuals in our model.
The computational model is based on the make blobs generator, which produces a feature matrix, and the corresponding discrete targets. This generator creates multi-class datasets by assigning to each class one or more groups of normally distributed points. Make blobs allows better control of the centres and standard deviations of each group and is used to demonstrate clustering [28]. The simulations in this section are compiled using the Plotly library [25]. See the steps description diagram 2, which describes the core logic of the proposed model. (1) Step 1 Parameters initialisation (2) Step 2 (a) Generated isotropic Gaussian blobs for clustering, created using make blobs.
(b) list S: A two-dimensional data structure, i.e. data is aligned in a tabular fashion in rows and columns. list S represents the coordination of each individual.
(c) A dictionary that holds the data of the curve and the coordination of each individual in certain compartments.
(d) Random selection of the first exposed individual from the susceptible individual group.
(e) Define distance between two points of the Cartesian plane.
(f) Thus chosing a random coordinate (individual) from the list 'list S' to be the first infected person.
(h) Update of the compartment S. (3) Step 3 (a) Clustering of compartments of individuals (data frame), each with its own probability. To provide a deeper insight, we present the clustering process of the first two compartments (see Figure 3) (b) The same process applies to the remaining individuals.
(4) Step 4 (a) In our case, we used the Plotly library only to manage multiple plots but used matplotlib to draw the curves (using scatter function).

RESULTS AND DISCUSSION
In this section, we illustrate the numerical simulations and explore the two approaches that we adopted (mathematical and computational) to make predictions and obtain graphical representations. Here, using the parameter values in Tables 2 and 3, which were mostly borrowed from the literature [13], we perform some simulations to numerically illustrate our theoretical results. Then, we evaluate the impact of two parameters (transmission rate; infection rate of infected and exposed individuals) on the transmission dynamics of the disease.
We vary the basic reproduction rate R 0 of our mathematical model (2.1), whose expression is as follows: . We consider the following two cases: Case 1: R 0 = 0.9504 < 1  The curves for the seven compartments of daily COVID-19 cases are reported for both models (see Figures 4 and 5 ). In this case, the fact that R 0 < 1 indicates that the risk of infection is relatively low over the 30-day period, due to the containment of individuals that greatly limits inter-human and inter-environmental contact and effectively weakens all routes of virus transmission, leading to fewer infected individuals, which in turn weakens the spread of the virus. Thus, the peak number of people affected is relatively low. We can therefore say that the epidemic is under control on a global scale.
With the values of the parameters indicated in the table below (Table 3)

Model validation:
Let us first observe the results (the graphs) for the mathematical model SEIQHRV and the computational model created using Python by taking two values of R 0 , one lower and the other higher than 1. For the two cases of simulations carried out, the values of the common parameters are identical for the two models (mathematical and computational). We can see that the curves of the individuals S, E, I, Q, H, and V are relatively similar in these two simulations for both values of R 0 . However, the curve of recovered individuals R is a little more 'volatile' in our computational model. We will validate the COVID-19 model by using the mathematical model as a reference model. Among the seven curves found in the results of the two models, we will take the daily infected cases for the two cases (R 0 ¡1) and (R 0 ¿1). The graphs and the residuals of the fit are presented in Figures 8 and 9, where it can be seen that the residuals are small and random. Thus, we can conclude that the fit is reasonably good [29].

Simulation with different infection rates.
We will study the impact of certain parameters (β 1 , β 2 ) on the different states. To do this, we vary the infection rate β 1 between 0.001, 0.0030, and 0.005. The figure 10 shows the impact of the infection rate β 1 on all compartments. Over the first two days, the increase in the β 1 rate leads to an increase in exposed cases E, after which it rapidly decreases. Infected I, confined Q, and recovered R increase over the first four, six, and eight days, respectively. Thereafter, they decrease rapidly. Hospitalised cases increase in line with the increase of the β 1 rate. In this figure 10, we can see that the β 1 rate has a weak impact on susceptible S and vaccinated V cases.  Figure 11 shows the impact of the infection rate β 2 on the seven compartments. From this we can see that β 2 has almost the same impact as β 1 . Indeed, over the first three days, the β 2 rate rapidly increases in line with the number of exposed cases E, before quickly falling.
Infected I, quarantined Q, and recovered R increase over the first four, seven, and nine days, respectively, and then decrease rapidly. The curves of hospitalised cases, meanwhile, are almost linear for a long period, before converging to the steady state. The difference lies in the speed of convergence at which the threshold is reached; that is, the higher the infection rate of the exposed cases β 2 , the faster the threshold (the steady state) is reached. Thus, we can conclude that the β 2 rate has a negligible impact on the susceptible S and vaccinated V cases. This confirms the theoretical solutions because it shows that they will converge to the equilibrium point (see equation (2.4)).
We conclude that the higher the infectivity (β 1 and β 2 ), the higher the maximum number (the peak) of infected cases. After that peak is reached, though, the number of infected cases falls rapidly.
The sensitivity analysis demonstrates the effect of each parameter on the spread of the disease. It enables us to understand which variables significantly influence the threshold of R 0 . In particular, sensitivity indices measure the proportional variation of a variable when a parameter changes. If this variable is differentiable with respect to the parameter a, the sensitivity index is defined as follows: [30]: where a is the contribution with respect to R 0 . For the parameters β 1 et β 2 , we have: β 1 α+β 2 (δ +µ+η+d) and S R 0 β 1 = β 2 (δ +µ+η+d) β 1 α+β 2 (δ +µ+η+d) . For example, using the values from Table 2, we obtain S R 0 In this study, we presented a computational model designed to improve our understanding of how a virus (COVID-19) behaves in a population, how specific parameters influence its spread, and how the evolution of the disease outbreak can be estimated. Furthermore, we presented a simulation of the model applied to a set of random individuals, before evaluating and validating our model by comparing these results with those of the proposed mathematical model (2.1).
In addition, we analysed the dynamic evolution of the disease over time by varying some parameters. Finally, we compared the curves of the results of each case with the curves obtained using the SEIQHRV mathematical model. This research aimed to validate predictions regarding the effect of specific parameters on the behaviour of COVID-19 and in so doing improve our understanding of the virus.
In our analysis, we observed that the model we created is easy to manipulate when tested and that the curves it produced are relatively similar for the two models in the different cases studied. Hence, the reliability of the model was confirmed, meaning that these results can be used to aid in the decision-making process of specialists in the field of epidemiology.
Indeed, the results of the compartmental models have already been exploited in some studies by medical doctors [31] and epidemiologists. It is reasonable to suppose, then, that the computational model can be easily manipulated in various studies that are focused only on the results (numerical simulation) without any need for the complicated mathematical calculations of the mathematical models, especially in the fields that rely on this type of results (the estimation of the evolution of individuals in each compartment in the course of time) to be exploited. The use of this kind of method will facilitate the work of specialists and allow health officials to more effectively determine precautionary and suitable measures of action to combat the epidemic.
This approach can be applied in future research to model several infectious diseases that are currently modeled using compartmental mathematical models.
To the best of our knowledge, this report is the first to use a Python model that can be easily applied to study the dynamic evolution of COVID-19 in a way similar to that achieved by

CONCLUSION
In order to help researchers build new, easy-to-use, and efficient prediction techniques for common diseases, as well as to extend prediction methodologies and overcome the technical limitations of mathematical models, we created a computational model (programmed in Python) that can be easily used to predict the growth of the COVID-19 population and to estimate the evolution of the epidemic.
It should be noted, however, that this epidemiological study remains highly theoretical in a deeply realistic field. In other words, even if the results and observations obtained by the Python model appear to be relatively close to reality (mathematical model), it should be borne in mind that these models do not take into account certain key factors, such as the different types of vaccination, the quarantine of individuals immediately after they are infected, or even the movement of individuals in the population. Nevertheless, such a study does help us to understand how a virus behaves in a population, measure the influence of certain factors on the virus's propagation, and to estimate the evolution of the epidemic.
Thus, these kinds of models are crucial because performance, speed, and simplicity are essential metrics in improving forecasting methods without the need for sophisticated equipment and resources. Mathematical modeling approaches are rapidly becoming better at facilitating epidemiological studies in the current pandemic, which may represent a step toward a developed forecasting system, which could form rapid predictions, be beneficial in eliminating the risk of such pandemics ever starting in the first place, and offer tremendous opportunities to harmonize sustainable development goals.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.