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

A new 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 SEIRQ 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 SEIRQ model dynamics. In a Jacobian method of linearization, we will address the domain of the solution and the equilibrium situation based on the SEIRQ 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 SEIRQ model was contrasted by comparing the results based on the SEIRQ model with the real COVID-19 spread data in Saudi Arabia. Numerical evaluation and predictions were given. The results indicate that the SEIRQ 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.


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 registered a value of 3.1 for the fundamental reproductive number based on the SEIR model data adaptation, assuming that the daily time spent by Poisson increases. Tang et al. 11 suggested a deterministic compartmental model that included clinical disease progression, individual epidemiological status, and intervention steps. The authors found that the number of reproductive controls can be up to 6.47 and that engagement techniques, such as simplified traceability accompanied by insulation and quarantine, may minimize reproductive control numbers and risk of transmission effectively.
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. 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,[14][15][16][17][18][19][20][21][22] .
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. 23 .
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, 14, 17, 22, 24 . 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. Today, mathematical modeling is extremely important for public health and mathematics. 4,6,15,22 .
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 novel coronavirus disease (SEIRQ 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 4,6,15,22,25 : The main group is dedicated to the vulnerable (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 4,6,17,19,25 .
Thus, the group is dedicated to the exposed population or individuals who are infected but not yet infectious.
The group is 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 is dedicated to the quarantined population (separated from the general population even in their houses).
The group is defined as the recovered population (individuals who have recovered and cannot contract the COVID-19 again), as in Fig. 1. The parameter is defined as the transmission rate from a susceptible population to infected but not detected by the testing population. We consider the net inflow of the susceptible population at a non-negative rate per unit value of time (comprising new births and new residents).
For any group, the outflow based on the natural death rate is defined by the nonnegative rate .
The total population size is , which is defined as 4,6,15,21,22,24,26 : (1) Starting with group , we have two outflows; a population flows out to the exposed group by the rate (each one in can transfer the infection to ), so the total number of outflows is equal to multiple ), and the outflow of the natural death is The group of exposed has only one inflow , while it has four outflows.
The first outflow is the population that flows out to group by the rate of transmission . The second outflow is the population that flows out to the recovery group directly without needing treatment by transmission rate of recovery . The third outflow is a population that flows out to the infected group with the transmission rate of infected , and the fourth outflow is the population that experiences natural death by the transmission rate 4,6,15,21,22,24,26 .
For the group of confirmed infected population , we have only one inflow, which comes from the group , with the transmission rate , while it has three outflows of population. The first outflow is the population that must go to the quarantine area by the , and the second outflow comes from the population in which treatment has succeeded; individuals in this population can go out to the recovery group by recovery transmission rate . The last outflow from the infected group is the total death, which comes from natural death by transmission rate and death due to the COVID-19 virus by transmission rate of mortality .
For the recovery population , three inflows exist, and only one outflow. The first inflow comes from the quarantine area by transmission rate of recovery , the second inflow is the population that comes out from the infected group by transmission recovery rate , and the third inflow is the population that flows out from the exposed area directly by transmission recovery rate . The only outflow from the recovering group is death by the natural transmission rate of mortality .
For the quarantine group , two inflows and two outflows are present.
The first outflow is the population flow out to the recovery group with transmission rate , while the second outflow is the total death, which comes from natural death by transmission rate of death and by the transmission rate of death due to the COVID-19 virus .
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 4,6,15,21,22,24,26 : where , , and . From equation (2) where , we obtain the following inequality: which gives after the solution (8) Hence, it is a nonnegative function for all values .
From equation (3), we have (9) which gives after the solution Hence, it is a nonnegative function for all values .
In similar manners, from the other equations in the model system, we have Hence, are nonnegative functions for all values of that complete the proof.

Theorem 2 (the domain of solutions)
All the solutions of the model structure that initiate in are bounded inside the region defined by 6 . Proof: By differentiating both sides of equation (1), we obtain (14) Substituting from the model (2)-(6), we obtain (15) From theorem 1, we have ; hence, the following inequality is valid: Then, we obtain (17) Then, when we obtain the solution which completes the proof 3, 6, 24 .
The number is called the reproduction number (RBN), which takes the form 3, 6, 24 : (32) Then, if the system has a unique endemic equilibrium [7]: and .
Thus, the system has a unique disease-free equilibrium when and has a unique endemic equilibrium when 6 .
( ) , it can be interpreted as the number of secondary cases or the new infection rate (transmission rate at which the susceptible individual converted to an exposed individual) 6 .

Achieving equilibrium by applying a Jacobian matrix
To obtain the reproduction number by using the Jacobian matrix method, we consider that the disease-free equilibrium (DFE) of the SEIRQ model is acquired by setting in equations (20)- (24). Hence, we obtain DFE in form 6 .
The Jacobian matrix of the SEIRQ 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 and .
Hence, we consider that 6 : 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 and the remaining roots are the solution to the following equation:  The roots of the above equation after inserting will take the forms: The formulas (45) generate the following cases 6 : 1-If , then we have in which the disease-free equilibrium is locally asymptotically stable.
2-If , then we have in which the endemic equilibrium is locally asymptotically unstable.
3-If , then we have in which the disease-free equilibrium is locally asymptotically unstable.

Condition of equilibrium (Hartman-Grobman theorem)
The Hartman-Grobman theorem states that the solutions of a square system of nonlinear ordinary differential equations (2)-(5) in a neighborhood of a steady-state look "qualitatively" similar to the solutions of the linearized system near the point . This result holds only when the equilibrium is hyperbolic, that is, when none of the eigenvalues of the matrix have zero real part 6 .
Thus, from (42), we obtain the following condition of equilibrium:

The uniqueness of equilibrium condition
If the matrix is obtained from the linearization and is the Jacobian evaluated at equilibrium , the condition means that the equilibrium is isolated, which means there is a disk around it that does not contain other equilibria 6,17,22,24 .
Hence, from (41), we have (47) which gives Thus, the condition (46) is the only condition of the equilibrium of the SEIRQ model. Therefore, the unique equilibrium condition of the SEIRQ model is The reproduction number (RBN) is also unique 6 . 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 .

Local sensitivity analysis of RBN
The sensitivity or elasticity of quantity concerning parameter p is given by: The sensitivity of concern is positive if is increasing concerning p and negative if is decreasing concerning p.
Applying formula (50) into reproduction number , which takes the form 6 : Then, It means that a 1% increase in each one will produce a decrease in , respectively, and a 1% increase in will produce an increase in RBN( ). From relation (52), means that a 1% increase will produce a rise of 1% in 6 .

Global stability of equilibria of the SEIRQ model (Lyapunov stability theorem)
One of the most commonly used functions is the Lyapunov function. Lyapunov functions are scalar functions that may be used to prove the global stability of equilibrium. Lyapunov states that if a function is globally positively definite and radially unbounded and its time derivative is globally negative, then the equilibrium is globally stable for the autonomous system , and is called a Lyapunov function 6 .

Theorem 5 (global stability)
The SEIRQ model is globally stable in disease-free equilibrium under the condition . Proof: We will consider the SEIRQ model on the space of the first three variables only .
It is clear that if the disease-free equilibrium for the first three equations is globally stable, then and the disease-free equilibrium for the full SEIRQ model is globally stable.
We construct the Lyapunov function on in the following form: where is a parameter that will be determined later, and .
The equation (60) shows that at the disease-free equilibrium , .
Now, we have to show that for all .
The equation (60) can be rewritten as follows: (61) The first term is positive for any value of , and the other two terms are also non-negative, so Now, we take the derivative of equation (60); we obtain (62) Substituting from the first three equations of the SEIRQ model and using the equation (26) completely vanishes, and then we have the last term only, which is already non-negative. Thus, . The second possibility is ; then, the two terms are non-positive. Thus, .
Hence, we have for every 6 .
Therefore, by the Lyapunov theorem, the disease-free equilibrium is globally asymptotically stable for the system of the SEIRQ model in all.

Solutions for the system of the SEIRQ model
We assume the initial conditions of the SEIRQ system in (39), (40), and (4) Consequently, we can obtain the other functions , and .

Model verification and predictions
To verify the SEIRQ 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.  website and other websites that would announce these statistics.
Another source of these data is the "Saudi Centre for Disease Prevention and Control 29 ." 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:  Therefore, we will use these data through the present SEIRQ model to discern whether there is a convergence between the model results and the real data 27-29 .

Applying the SEIRQ 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 SEIRQ model, as in Table 1 27-29 :  . Some of the other parameters have been calculated, estimated, or assumed, as in Table 2.  Moreover, the reproduction number RBN is . In other words, the transmission rate at which the susceptible individual converted to an exposed individual is higher than one, which means the spread of COVID-19 is unstable. curves that come as results from the SEIRQ model work as three trends to the curves belong to the real data, which makes the results due to applying the SEIRQ model close to the actual data.
To illustrate the convergence between the results of the proposed SEIRQ model and the real results, we displayed Fig. 5, which shows the cumulatively infected numbers within the same interval referred to earlier. It is noted that the curve of the real data is set between the three cases of the SEIRQ model with the mentioned values of parameters. 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 SEIRQ 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 0 and a Â be unstable. The curves have been established by using the same three values of the two parameters and .
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 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 SEIRQ 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 [27][28][29] . We will keep the values of the parameters and 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 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 to be , 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 to be , which means we must help the confirmed infected population, which they need to be in the quarantine zone.
4. Increase the value of the transmission rate from the quarantine population to the recovery zone to be , which means we must apply a successful treatment on the quarantine area and help them recover. to be by applying a successful treatment for the confirmed infected population and help them recover without needing to go to the quarantine zone.
6. Increasing the value of the transmission rate from infected and undetected populations to the recovery zone directly to be by using a successful treatment and supplying with vitamins, health awareness, social spacing, and applying the principle of prevention are better than curing.
7. Increase the value of infected but not detected individuals by checking population to infected population for treatment to be , which means we have to offer the more effective and accurate methods of diagnosis to determine the confirmed infections. Moreover, raising awareness about ways to identify the disease and the symptoms and ways of confirming the infection. 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.
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.