A Dynamical Modelling of the Epidemic Diseases to Assessing the Rates of Spread of COVID-19 in Saudi Arabia: SEIQR Model

A model of critical epidemic dynamics for the emergence of the new coronavirus COVID-19 is being established in this paper. A new approach to the assessment and control of the COVID-19 epidemic is given with the SEIQR pandemic model. This paper uses real knowledge on the distribution of COVID-19 in Saudi Arabia for mathematical modeling and dynamic analyses. The reproductive number and detailed stability analysis are provided in the SEIQR model dynamics. In a Jacobian method of linearization, we will address the domain of the solution and the equilibrium situation based on the SEIQR model. The equilibrium and its importance have been proven, and a study of the stability of the equilibrium free from diseases has been implemented. The reproduction number was evaluated in accordance with its internal parameters. The Lyapunov theorem of stability has proven the global stability of the current model's equilibrium. The SEIQR model was contrasted by comparing the results based on the SEIQR model with the real COVID-19 spread data in Saudi Arabia. Numerical evaluation and predictions were given. The results indicate that the SEIQR model is a strong model for the study of the spread of epidemics, such as COVID-19. At the end of this work, we implemented an optimum protocol that can quickly stop the spread of COVID-19 among the Saudi populations. The key solution to slowing COVID-19 transmission is to stay home and bring sick persons as far as possible in a remote location or in a safe place. Ultimately, it is vital to offer safe and adequate treatment to ill people, and to avoid them, medications, tones, and nutrients should be provided to non-infected persons.


Introduction
As COVID-19 outbreaks continue, the number of infections steadily increases. This is due to the presence of many factors that increase the complexities of COVID-19 infection and create barriers to disease management. Since scientists and researchers all over the world are trying to establish a vaccine or a cure for the outbreak to control such pandemics in the future, from a medical engineering framework, an infectious disease can be well known and understood through the use of mathematical models. This idea began in 1927. After that, many different mathematical models have been constructed for various diseases and infections. For some essential studies, we refer to [1][2][3][4][5][6][7][8] .
To explain transmission dynamics and estimate domestic and global disease spread based on data recorded from December 31, 2019, to January 28, 2020, Wu et al. 9 have implemented the Susceptible Exposed Infectious Recovered Model (SEIR). They also found that COVID-19 had a fundamental reproductive number of approximately 2.68. Read et al. 10  To determine the scale of the disease outbreak in Wuhan, Iman 12 carried out calculational modeling of the possible epidemic tracks with an emphasis on human-to-human transmissions.
Its findings suggest that controls must be efficiently controlled by well over 60% of the transmission. To analyze and forecast the infectivity of the new coronavirus, Guo et al. 13 developed a deep learning algorithm. They found that two animal hosts of this virus were bats and minks. Most of the models illustrate the significant role of a direct transmission mechanism between humans and humans in the outbreak, as demonstrated by the fact that many individuals infected in the Wuhan area have no interaction and the number of infections has been growing rapidly and spreading across the Chinese provinces and over 20 people 14 . There is a relatively long incubation period in many infected individuals, so they do not show symptoms and have not been aware of their infection for 10-14 days. Over time, the disease can easily be spread by direct exposure to other people. On the other hand, the published models have not, to date, taken into account the environmental position of COVID-19 transmission. Several other modeling studies for the COVID-19 outbreak have also been carried out 6,8,[15][16][17][18][19][20][21][22][23] .
Statistical epidemiology is based on the dynamics of health and disease and related population factors. The presence of a pathogenic microbial agent identifies an infectious disease as a clinically obvious disease. For modeling purposes, four forms of transmission are characterized: straightforward, if the causative disease agent is individual; vector, if the causative agent is transmitted from a vector to a person; natural, if the touch of a pathogen infects the human via the environment; and vertical, if the disease agent is transmitted from mother to child at birth. Airborne and personal diseases are generally known to be transmitted directly where transmission occurs through contact between individuals and others. 24 .
Mathematical modeling of infectious diseases is important and critical with the advent of HIV epidemics. Since then, several models for investigating infectious diseases have been developed, studied, and applied. Mathematical modeling currently applies enormously to public health and mathematics. 3,15,18,23,25 .
In the emergence of HIV epidemics, the mathematical modeling of infectious diseases is significant. Since then, numerous models have been developed, studied, and applied for the investigation of infectious diseases 26 . Farman et al. studied the stability and control of glucose insulin glucagon system in human 27 . Farman et al. 26 discussed the dynamical behavior of fractional-order cancer model with vaccine strategy. Gondim and Machado 28 introduced the optimal quarantine strategies for the COVID-19 pandemic in a population with a discrete age structure. Davies et al. 29 studied the age-dependent effects in the transmission and control of COVID-19 epidemics.
The goal of this paper is to construct a new COVID-19 vital dynamical model that is more applicable to cases in any country through mathematical analysis of the model in question by using a system of similar models with different considerations and new in / outflows between population divisions. In addition, this paper presents a new formula that explores the sensitivity of a reproduction number. The mechanisms of virus transmission by humans are to be discovered. Another aim is to investigate and learn the optimal procedures, controls, and techniques to minimize the outbreak substantially.

Formulation of a coronavirus disease (SEIQR model)
During the spread of COVID-19 in any country, the population can be divided into five vital dynamic subpopulations or five groups, which are represented in Fig. 1 and can be described as follows 4,6,16,23,29-31 : The main group ( ) St is dedicated to the healthy people but who may get the disease population. For certain diseases, the infected person may not become infectious immediately, but the latent phase is not contagious. It takes time for the pathogen to replicate and develop itself in the new host. In general, the exposed (latent) cycle follows the sensitive process 4,6,18,20,30 .

Thus, the group ( )
Et is dedicated to the exposed population or individuals who are infected but not yet infectious.
The group ( ) Itis devoted to the population who are confirmed infected (individuals who have contracted the disease and are now sick with it and infected individuals are also infectious).

The group ( )
Qt is dedicated to the quarantined population (separated from the general population even in their houses).

The group ( )
Rtis defined as the recovered population (individuals who have recovered and cannot contract the COVID-19 again), as in Fig. 1. For any group, the outflow based on the natural death rate is defined by the nonnegative rate 1 d .
The total population size is ( ) Nt, which is defined as 4,6,16,22,23,25,32 : Starting with group ( ) St, we have two outflows; a population flows out to the exposed

dS
The group of exposed ( ) Et has only one inflow , while it has four outflows.
The first outflow is the population that flows out to group ( ) All inflows and outflows are shown in the flowchart in Fig. 1, and the five groups can be converted into equations to formulate the following system of first-order ordinary nonlinear differential equations 4,6,16,22,23,25,32 : Proof: It follows from equation (2) that where I

 =
It can be re-written in the following form 3 Thus, we obtain Then, we get In similar manner, it can be shown that ( )

Theorem 2 (the domain of solutions)
All the solutions of the model structure that initiate in 5 + are bounded inside the region Proof: By differentiating both sides of equation (1), we obtain Substituting from the model (2)-(6), we obtain From theorem 1, we have + ; hence, the following inequality is valid: Then, we obtain ( ) ( ) Then, when t →we obtain the solution which completes the proof 3,6,25 .
The number 0  is called the reproduction number (RBN), which takes the form 3,6,25 : Then, if 0 1  the system has a unique endemic equilibrium 8 : individual converted to an exposed individual) 6,8 .

Reproduction number by using the Jacobian matrix
To obtain the reproduction number 0  by using the Jacobian matrix method, we consider that the disease-free equilibrium (DFE) of the SEIQR model is acquired by setting (19)- (23). Hence, we obtain DFE in form 0 The Jacobian matrix of the SEIQR model takes the following form: First, we will linearize the first two equations by using the Jacobian method. The first two equations have a disease-free equilibrium (DFE) situation when 00 Hence, we consider Then, we have By substituting from the equilibrium position, we obtain Hence, the system of nonlinear equations (2) and (3) has been converted to the following linear system 6 : For the complete system at equilibrium, the stability of the disease-free equilibrium (DFE) is given by the Jacobian matrix: By calculating the characteristic equation given by and the remaining roots are the solution to the following equation: The roots of the above equation after inserting 0  will take the forms: The formulas (44) generate the following cases 6 : Thus, from (41), we obtain the following condition of equilibrium: Thus, the condition (45) is the only condition of the equilibrium of the SEIQR model. Therefore, the unique equilibrium condition of the SEIQR model is The reproduction number (RBN) 0 is also unique 6 .

Local sensitivity analysis of RBN
Local sensitivity analysis is a sensitivity analysis that examines the change in the output values that results from a change in one input value (parameter) 6 .
The sensitivity or elasticity of quantity G concerning parameter p is given by: The sensitivity of G concern p is positive if G is increasing concerning p and negative if G is decreasing concerning p.
Applying formula (49) into reproduction number 0  , which takes the form 6 : It means that a 1% increase in each one (   )   1  1  3  2  2  2 , , , , , dd     will produce ( )  where  is a parameter that will be determined later, and * 1 S d  = .
The equation (59) The second possibility is 1 x  ; then, the two terms are non-positive. Thus, Therefore, by the Lyapunov theorem, the disease-free equilibrium is globally asymptotically stable for the system of the SEIQR model in all.

Solutions for the system of the SEIQR model
We assume the initial conditions of the SEIQR system in (38), (39), and (4)-(6) to take the form Consequently, we can obtain the other functions ,, S E I , and Q .

Model verification and predictions
To verify the SEIQR model, we will apply it to the real data regarding the COVID-19 outbreak in Saudi Arabia. COVID-19 has been in Saudi Arabia since March 3, 2020. Cases continued to be discovered in small numbers until the beginning of April, and then the number of detected cases increased. Therefore, we decided in this study to consider April 1, 2020, as the real beginning of the spread of the COVID-19 epidemic in Saudi Arabia.
We used tables of statistics issued from the Saudi Ministry of Health 33 and the daily official statement issued by the ministry as well as Wikipedia 34 , which also depends on the ministry's website and other websites that would announce these statistics.
Another source of these data is the "Saudi Centre for Disease Prevention and Control 35 ." We used the official website of the General Statistics Authority of Saudi Arabia for more information about the kingdom's population, mortality rate, and population growth rate.
To study the spread of COVID-19 in Saudi Arabia before June 13, 2020, we will represent the curve of the number of daily infections and the time series curve of the total number of infections, as shown in Figs. 2 and 3:  and reached an accumulated amount of 122,259 infections on June 13, 2020. Therefore, we will use these data through the present SEIQR model to discern whether there is a convergence between the model results and the real data 33-35 .

Applying the SEIQR model to Saudi Arabia data of the spread of Covid-19
According to the official data of Saudi Arabia, we have the following initial data, which are considered the initial conditions of the system based on the SEIQR model, as in Table 1 33-35 :  . Some of the other parameters have been calculated, estimated, or assumed, as in Table 2. The estimated data has been calculated by using the most powerful methods, however, is calibration or curve fitting. Curve fitting is the process of identifying the parameters of a curve, or mathematical function, that has the best fit to a series of data points. MAPLE software has been used for the fitting curves and to estimate the parameters we need.    Now, we will predict the spread of COVID-19 in Saudi Arabia based on the current data and parameters with the same rates without any change in the procedures and considering that everything will continue as it is. We will illustrate the results of the total number of infections by applying the SEIQR model for the next three months, starting from April 1, 2020, and ending on October 18, 2020. As shown in Fig. 6, the curves and results show whether the number of infections will be reduced and whether the spread of COVID-19 will continue to be unstable. The curves have been established by using the same three values of the two parameters and 0  .
The figure shows that the spread of COVID-19 will continue with an unstable situation without being slowed and that the number of daily infections will rise to very high numbers.
Thus, we will later describe the best practices for this situation (best protocol) to control the spread of the COVID-19.
The other parameters change within its suitable range, making all its significant private effects, even the value of the reproduction number 0  higher or smaller than one.

The current state and how to stop the spread of Covid-19 in Saudi Arabia
Now, we are in the most critical part of the assessment of the current situation and evaluate what needs to take place later in Saudi Arabia to control the COVID-19 spread. Therefore, in this section, we will apply the SEIQR model to analyze the current situation with new initial conditions and different values of the system parameters according to the current state. We will consider June 14, 2020, as the fresh start, and we will renew all the initial conditions in Table   I. The number of infections on this day was ( ) 0 4223 I = 33-35 . We will keep the values of the parameters 1 d and 2 d as it is without any change, while the other parameters will take the values in Table 3:   Table 3. It is noted in figure 8 that the spreading of COVID-19 in Saudi Arabia passed through its peak point on 16-18 July 2020, which agrees with the actual data; after that, the spread has slowed down and kept this attitude until the current days, and the reproduction number takes the value 0 0.1 1  =  which means the situation is stable According to this curve, we can also see that the number of daily infections on October 18, 2020, will be approximately 600 persons/day, and we can predict that the spreading situation will go to a more stable position and better state.

The ideal protocol to halt the spread Covid-19 in Saudi Arabia
To obtain the ideal situation, which can help us break the spread COVID-19 in Saudi Arabia, we must start implementing the following protocols and procedures (see figure 9): 1. Decrease the value of the transmission rate from the susceptible population to infected but not detected by testing the population to be in the following interval 2. Increase the value of the transmission coefficient from an infected population but not detected by testing to a quarantine population 1  to be 1 0.2   , which means expanding the detection work and the need to isolate infected people in compulsory quarantine areas as an example.
3. Increase the value of the transmission coefficient from the confirmed detected population by testing to a quarantine population 2  to be 2 0.01   , which means we must help the confirmed infected population, which they need to be in the quarantine zone.  Five measures are included in the optimal procedure, and guidance has been comprehensive in helping delay the spread of COVID-19 in Saudi Arabia. Prevention is safer than recovery, one of the key targets in this procedure.
The main approach to slowing down the transmission of COVID-19 is to remain home and to put sick individuals in a distant location or a protected place as far as possible.
In order to evaluate the reported infections, we need more reliable and effective diagnostic methods. In addition, awareness raising on ways in which the infection can be confirmed, the disease symptoms and ways.
Ultimately, efficient and appropriate care of sick patients must be given, and medicines, tones, and nutrients must be distributed to non-infected individuals to prevent them.