Study on the SEIQR model and applying the epidemiological rates of COVID-19 epidemic spread in Saudi Arabia

This article attempts to establish a mathematical epidemic model for the outbreak of the new COVID-19 coronavirus. A new consideration for evaluating and controlling the COVID-19 outbreak will be constructed based on the SEIQR Pandemic Model. In this paper, the real data of COVID-19 spread in Saudi Arabia has been used for the mathematical model and dynamic analyses. Including the new reproductive number and detailed stability analysis, the dynamics of the proposed SEIQR model have been applied. The local sensitivity of the reproduction number has been analyzed. The domain of solution and equilibrium based on the SEIQR model have been proved using a Jacobian linearization process. The state of equilibrium and its significance have been proved, and a study of the integrity of the disease-free equilibrium has been carried out. The Lyapunov stability theorem demonstrated the global stability of the current model equilibrium. The SEIQR model has been numerically validated and projected by contrasting the results from the SEIQR model with the actual COVID-19 spread data in Saudi Arabia. The result of this paper shows that the SEIQR model is a model that is effective in analyzing epidemic spread, such as COVID-19. At the end of the study, we have implemented the protocol which helped the Saudi population to stop the spread of COVID-19 rapidly.


Introduction
With the continuation of COVID-19 outbreaks, the number of infections is gradually growing, and this is because several factors increase the severity of COVID-19 infection and build obstacles to the management of diseases. Since scientists and researchers around the world are trying to set up a vaccine or an epidemic cure for control of such pandemics in the future, an infectious disease can be well identified and understood using mathematical models from a medical engineering context. This idea originated in 1927. Afterward, several different mathematical models for various diseases and infections were created. We refer to some critical studies (Gao et al., 2020;Goyal et al., 2019;Kumar et al., 2019;Martcheva, 2015a;Shah et al., 2020;Van den Driessche & Watmough, 2002). To define the dynamics of the transmission and to estimate the national and global spread of this disease, Wu et al. have implemented the Susceptible Exposed Infectious Recovered Model (SEIR) based upon the data recorded from December 31, 2019, to January 28, 2020. They also found that COVID-19 had a simple reproductive number of approximately 2.68 (Wu et al., 2020). Read et al. (Read et al., 2020) have reported a value of 3.1 for the basic reproductive number based on data fitting of the SEIR model, using an assumption of Poisson-distributed daily time increments. Imai et al. (Imai et al., 2020) developed a deterministic compartmental model that involved the disease's clinical development, human epidemiological status, and engagement levels. The authors find that the reproductive control number may be as large as 6.47. That interaction techniques such as simplified touch tracing and quarantine would efficiently minimize the number of reproductive controls and the risk of transmission (Tang et al., 2020a). To predict the severity of the disease outbreak, Imai et al. (Imai et al., 2020) carried out a computer simulation of possible infectious tracks in Wuhan with an emphasis on communications between individuals. Their findings suggest that control measures must block more than 60% of transmission to avoid the outbreak. (Imai et al., 2020). Guo et al. also developed a deep learning algorithm to evaluate the infectivity of the new coronavirus and to predict its future hosts. Their findings showed that maybe two animal hosts of this virus are bats and minks (Guo et al., 2020). Most of these models have highlighted the critical role played by direct, human-to-human transmission in this epidemic. They have shown that the majority of those infected have no interaction with the market in Wuhan and that the number of infections has risen steadily, and that the disease has spread to every province in China and more than 20 people. Many infected persons have a relatively long incubation period so that they are unaware of their infection for 10e14 days. They can easily spread the disease to others by direct contact during this period. On the other hand, the models published to date have not taken into consideration the role of the environment in COVID-19 transmission. Various further modeling research for the COVID-19 outbreak has also been carried out (Benvenuto et al., 2020;Chen et al., 2020;Din et al., 2020;Kucharski et al., 2020;Mangoni & Pistilli, 2020a;Nadim et al., 2020;Peng et al., 2020a;Rabajante, 2020;Read et al., 2020;Wang et al., 2020;Wu et al., 2020;Yang & Wang, 2020a).
Mathematical epidemiology is the topic of research at the population level of trends of health and disease. An infectious disease is characterized by the presence of a pathogenic microbial agent as a clinically obvious disorder. For modeling purposes, we define four different forms of transmission: directly, when the triggering pathogen is a person-to-person transmission of the pathogen; vector if the causative agent is a vector transmitted to a human; environmentally, if a person gets infected through interaction with an environmental pathogen; and vertically, if the pathogen is transmitted from mother to child by birth. Airborne and personal diseases are commonly considered to be transmitted directly when transmission takes place via any interaction between one person and another (Martcheva, 2015b). Mathematical modeling of infectious diseases is relevant and critical in the emergence of HIV epidemics. Since then, several models have been developed, studied, and applied to investigate the spread of infectious diseases. Mathematical modeling today makes a significant contribution to mathematics and public health (Hethcote, 2000;Martcheva, 2015b). The classic SEIR model is commonly used and accepted in many countries to assess the outbreak of the COVID epidemic. Since the mathematical model can draw simple and straightforward conclusions on the COVID-19 epidemic, a cascade of the SEIR models has been developed to explain the mechanisms of its infection source transmission, storage, and hosts for humans. (Chen et al., 2020;Youssef et al., 2020b, c). Babaei et al.(Babaei et al., 2021) constructed a mathematical model to examine the effect of quarantine on the spread of coronavirus.
This paper aims to construct a new COVID-19 model that is more applicable to cases in any country through mathematical analysis of the given model using a background of similar models with different considerations and further outflows between populations. Another aim is to research and learn the perfect procedures, controls, and techniques that could reduce the outbreak.

Formulation of a novel coronavirus disease (SEIQR model)
The population can be divided into five dynamic sub-populations or five groups that are represented in Fig. 1; during the propagation of COVID-19 in any region, and the following could be described: The primary group SðtÞ is devoted to the susceptible population (individuals who are healthy but are able to contract the disease). For several diseases, the infected person may not become instantly contagious, but it is also called the latent phase. The pathogen needs time to spread and settle in the new host. The exposed time (latent) typically follows the sensitive point.
Group EðtÞ is also devoted to the exposed population or persons who are infected but not yet infectious. Group IðtÞ is intended for the confirmed infected population (individuals who contracted and are now ill, as well as infected persons).
The group Q ðtÞ is dedicated to the quarantined population (hospitalized or isolated from the general population) (Maier & Brockmann, 2020).
The parameter a 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 L > 0 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 non-negative rate d 1 .
The total population size is NðtÞ, which is defined as (Maier & Brockmann, 2020;Mangoni & Pistilli, 2020b;Martcheva, 2015b;Pal et al., 2020;Peng et al., 2020b;Yang & Wang, 2020b;Youssef et al., 2020a): Starting with the group SðtÞ, we have two outflows; a population flows out to the exposed group EðtÞ by the rate aSðtÞ (each one in SðtÞ can transfer the infection to aSðtÞ, so the total number of outflow is equal to multiple aSðtÞEðtÞ, and the outflow of the natural death isd 1 S. The group of exposed EðtÞ has only one inflow aEðtÞSðtÞ, while it has four outflows. The first outflow is the population that flows out to the group Q ðtÞ by the rate of transmission b 1 . The second outflow is the population that flows out to the recovery group directly without needing treatment by transmission rate of recovery s 3 . The third outflow is a population that flows out to the infected group IðtÞ with the transmission rate of infected r, and the fourth outflow is the population that experiences natural death by the transmission rate d 1 (Din et al., 2020;Maier & Brockmann, 2020;Mangoni & Pistilli, 2020b;Martcheva, 2015b;Pal et al., 2020;Peng et al., 2020b;Yang & Wang, 2020b;Youssef et al., 2020a).
For the group of confirmed infected population IðtÞ, we have only one inflow, which comes from the group EðtÞ, with the transmission rate r, while it has three outflows of population. The first outflow is the population that must go to the quarantine area Q ðtÞ by the transmission rate b 2 , and the second outflow comes from the population in which treatment has succeeded; individuals in this population can go out to the recovery group RðtÞ by recovery transmission rates 2 . The last outflow from the infected group is the total death, which comes from natural death by transmission rate d 1 and death due to the COVID-19 virus by transmission rate of mortality d 2 .
For the recovery population RðtÞ, three inflows exist, and only one outflow. The first inflow comes from the quarantine area Q ðtÞ by transmission rate of recovery s 1 , the second inflow is the population that comes out from the infected group by transmission recovery rate s 2 , and the third inflow is the population that flows out from the exposed area directly by transmission recovery rate s 3 . The only outflow from the recovering group is the death by natural transmission rate of mortality d 1 . For the quarantine group Q ðtÞ, two inflows b 1 EðtÞ and b 2 IðtÞ, and two outflows are present. The first outflow is the population flow out to the recovery group RðtÞ with transmission rate s 1 , while the second outflow is the total death, which comes from the natural death by transmission rate of death d 1 , and by the transmission rate of death due to the COVID-19 virus d 2 .
All inflows and outflows have been 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 non-linear differential equations (Din et al., 2020;Maier & Brockmann, 2020;Mangoni & Pistilli, 2020b;Martcheva, 2015b;Pal et al., 2020;Peng et al., 2020b;Yang & Wang, 2020b;Youssef et al., 2020a): We can simplify the above equations to be in the forms: Theorem 1. (all solutions are definite positive) Each solution of the SEIQR model with its initial condition is a subset in the interval ½0; ∞Þ and fSðtÞ; EðtÞ; IðtÞ; RðtÞ; Q ðtÞg ! 0 for all values 0 t < ∞.
Proof: All the right-hand sides of the SEIQR model are completely continuous and locally Lipschitzian on ℝ. The solutions fSðtÞ; EðtÞ; IðtÞ; RðtÞ; Q ðtÞg with their initial conditions exist and are unique in the interval ½0; ∞Þ (Martcheva, 2015a).

Theorem 2. (the domain of solutions)
All the solutions of the model structure that initiate in ℝ 5 þ are bounded inside the region j defined by. j ¼ ðS; E; I; Q ; RÞ2ℝ 5 : 0 NðtÞ L d1 t/∞ Proof: By differentiating both sides of equation (1), we get Substituting from the model (7)e (11), we obtain From Theorem 1, we have d 2 ðQ ðtÞ þIðtÞÞ ! 0 ; hence, the following inequality is valid: Then, we have Then, when t/∞ we get the solution NðtÞ3 0; L d1 , which completes the proof.

The equilibrium of the SEIQR model
To determine the equilibrium of this model, we set all the derivatives equal to zero and solve the system as follows (Martcheva, 2015a): which gives From equation (25), we have From equation (24) when Es0, we have Substituting from equations (28) and (29) into equation (23), we get Substituting from equation (30) into equation (28), we obtain Substituting from equations (30) and (32) into equation (27), we get Substituting from equations (30), (32) and (33) in equation (26), we obtain Thus, at disease-free equilibrium (DFE) < 0 ¼ 1 gives (23) and (29), which agrees with the domain of solution in (21).
When < 0 ¼ 0, there is no transmission, where a ¼ 0:0. 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) (Martcheva, 2015b).
The current proposed SEIQR model introduces a new and different reproduction number that is more sensitive to more parameters than other SEIQR models.

Achieving equilibrium by applying a Jacobian matrix
To get the reproduction number < 0 by using a Jacobian matrix method, we consider that the disease-free equilibrium (DFE) of the model SEIQR is acquired by setting (23)-(27). Hence, we obtain DFE in the 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 Hence, the system of non-linear equations (7) and (8)

Condition of equilibrium (Hartman-Grobman theorem)
The Hartman-Grobman theorem says that the solutions of a square system of non-linear ordinary differential equations (7)e(10) in a neighbourhood of a steady-state look "qualitatively" similar to the solutions of the linearized system near the . This result holds only when the equilibrium is hyperbolic; that is when none of the eigenvalues of the matrix J E0 have zero real part (Martcheva, 2015b). Thus, from equation (45) we obtain the following condition of equilibrium: The uniqueness of equilibrium condition If the matrix J E0 is obtained from the linearization and is the Jacobian evaluated at equilibrium DFEðE 0 Þ ¼ L d1 ; 0; 0; 0; 0 , the condition J E0 s0 means that the equilibrium is isolated, which means there is a disk around it that does not contain other equilibria (Martcheva, 2015b). Hence, from equation (44), we have J E0 ¼ Thus, condition (46) 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 ¼ aL d1ε1 is also unique (Martcheva, 2015b).

d1
< 0 which gives a L d1ε1 < 1, i.e., the stability condition takes the form: and the instability condition is: Local sensitivity analysis of RBN ð< 0 Þ Local sensitivity analysis is that examines the change in the output values that result from a change in one input value (parameter) (Youssef et al., 2020b, c).
The sensitivity or elasticity of quantity G concerning the parameter p is given by (Martcheva, 2015b): The sensitivity of G concern H is positive if G is increasing concerning H and negative if G is decreasing concerning H. Applying formula (53) into the reproduction number < 0 which takes the form: means that a 1% increase in each one of ðb 1 ; s 3 ; r; % a decrease in < 0 , respectively. From relation (55), § a <0 ¼ 1 it implies that a 1% increase a will produce a rise of 1% in < 0 (Martcheva, 2015a).

Global stability of equilibria of the SEIQR model (Lyapunov stability theorem)
One of the most used 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 VðxÞ is globally positively definite and radially unbounded, and its time derivative is globally negative, VðxÞ < 0 for all xsx * then the equilibrium x * is globally stable for the autonomous system x 0 ¼ f ðxÞ, and VðxÞ is called a Lyapunov function (Martcheva, 2015b).

Proof:
We will consider the SEIQR model on the space of the first three variables only ðS; E; IÞ. If the disease-free equilibrium for the first three equations is globally stable, then ðQ ;RÞ/0, and the disease-free equilibrium for the full SEIQR model is globally stable.
We construct the Lyapunov function on ℝ 3 þ in the following form (Martcheva, 2015b): where g is a parameter will be determined later, and S * ¼ L d1 .
equation (60) shows that, at the disease-free equilibrium The first term is positive for any value of S=S * , and the other two terms are also non-negative, so. V > 0 Now, we take the derivative of equation (60), we obtain: Substituting from the first three equations of the SEIQR model and using equation (29), we obtain We choose g ¼ 1 ε1 , then we have: Since < 0 < 1 then, the last term is non-positive.
For the first term, consider d1S L ¼ y, then the term inside the brackets takes the form y þ 1 y À 2 ¼ ðyÀ1Þ 2 y > 0, and now we have two possibilities. The first one is at the equilibrium point, where we have S ¼ S * ¼ L d1 , and it gives y ¼ 1, then the first term completely vanishes, then we have the last term only, which is already non-negative. Thus, V 0 < 0.
The second possibility is ys1, then the two terms are non-positive. Thus V 0 < 0.
Therefore, by the Lyapunov theorem, the disease-free equilibrium is globally asymptotically stable for the system of the SEIQR model in all (Martcheva, 2015b).

Solutions for the system of the SEIQR model
After linearization of the system of the SEIQR model, the system takes the form: We assume the initial conditions of the above system take the form: fSðtÞ; EðtÞ; IðtÞ; Q ðtÞ; RðtÞgj t¼0 ¼ fSð0Þ; Eð0Þ; Ið0Þ; Q ð0Þ; Rð0Þg To solve the above system, we start to solve equation (66) as follows: Then, by substituting from equation (71) into equation (67), we get dIðtÞ dt þ ε 3 IðtÞ ¼ rEð0Þe g 1 t (72) By solving the above equation, we get the infection function in the form: IðtÞ ¼ ðIð0Þ À g 2 Þe Àε3t þ g 2 e g 1 t (73) where g 2 ¼ rEð0Þ g 1 þε3 . We can solve the first equation of the system by using equation (71) dSðtÞ After solving the above equation, we get By inserting equations (71), (73) and (75) into equations (68) and (69), we obtain a system of two ordinary differential equations on RðtÞ and Q ðtÞ, which has been solved by using MAPLE software. We could not write the final forms of the two functions RðtÞ and Q ðtÞ because it contains many long terms.

Results
To verify the model SEIQR, 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 (Health, 2020) and the daily official statement issued by the ministry as well as Wikipedia (Saudi_Arabia, 2020), 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 (Control, 2020)." 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, 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. Fig. 2 shows that the number of daily cases on April 1, 2020, was 157 infections, and it reached 4919 infections on June 15, 2020, and between the two numbers, the curve passed through many up and down variations. After June 15, 2020, the number of daily infections has taken a decreasing curve with some changes up and down until February 17, 2021. Fig. 3 shows that the total number of infections at the same interval started with 157 infections and reaches an accumulated amount of 139,506 infections on June 15, 2020, and an accumulated amount 372,200 number of infections on February 17, 2021. Therefore, we will use these data through the present model SEIQR to discern whether there is a convergence between the model results and the real data.

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 initial data, which is considered the initial conditions of the system based on the SEIQR model, as in Table 1 (Control, 2020;Health, 2020;Saudi_Arabia, 2020): The total number of new births of Saudi children and new residents Lz2300 person/day and the rate of natural deaths is approximately 1030 people/day, which results in d 1 z3 Â 10 À5 . Some of the other parameters have been calculated, estimated, or assumed, as in Table 2.
By using MAPLE software, we get the results that indicate the number of daily infections as outcomes of the SEIQR model in the interval from April 1, 2020, to June 15, 2020. The value of the parameter a in that interval (the rate of transmission from susceptible population to infected in Saudi Arabia) was a ¼ 2:64 Â 10 À9 , and the reproduction number RBN ð< 0 Þ was < 0 ¼ 1:262 > 1. In other words, the transmission rate at which the susceptible individual converted to an exposed individual was higher than one, which means the spread of COVID-19 was unstable within the studied interval from April 1, 2020, to June 15, 2020.   After June 15, 2020, the value of the parameter a in that interval was a ¼ 1:18 Â 10 À9 , and the reproduction number RBN ð< 0 Þ was < 0 ¼ 0:42 < 1. In other words, the transmission rate at which the susceptible individual converted to an exposed individual was smaller than one, which means the spread of COVID-19 was stable within the studied interval. Fig. 4 shows the number of daily infections based on the SEIQR model against the real data in Saudi Arabia between 4/1/ 2020 and 6/15/2020 with the value of the rate of transmission from susceptible populations to infected in Saudi Arabia a ¼ 2:64 Â 10 À9 which gives the value of RBN < 0 ¼ 1:262 and between 6/16/2020 and 2/17/2021 with the value of the rate of transmission from susceptible populations to infected in Saudi Arabia a ¼ 1:18 Â 10 À9 which gives the value of RBN < 0 ¼ 0:42. It is noted that the curve which comes as results from the SEIQR model work as trends to the curve that belong to the real data. It makes the results due to applying the SEIQR model are close to the real data.
To illustrate the convergence between the results of the proposed model SEIQR and the real results, we displayed Fig. 5 in which the cumulatively infected numbers within the same interval 4/1/2020 and 2/17/2021. This figure proves the success of the proposed SEIQR model.

Discussions
According to the SEIQR model results and the real situation, we can conclude what happened and what makes us could halt the spread of COVID-19 in Saudi Arabia: Table 2 The values of parameters in SEIQR (Control, 2020;Din et al., 2020;Gerberry & Milner, 2009;Health, 2020;Jumpen et al., 2009;Khan and Atangana, 2020;Pal et al., 2020;Peng et al., 2020b;Saudi_Arabia, 2020;Tang et al., 2020b 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 a 1:18 Â 10 À9 .
2. Increase the value of the transmission coefficient from infected but not detected by testing population to a quarantine population b 1 to be b 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 rate from the quarantine population to the recovery population s 1 to be s 1 ! 0:02, which means that we must apply the successful treatment on the quarantine population and help them to recover. 4. Increase the value of the transmission rate from infected and not detected individuals by testing the population to recovery s 3 to be s 3 ! 0:05 by using a successful treatment. 5. Increase the value of infected but not detected individuals by checking the population to infected population for treatment r to be r 1 ! 0:05, which means we have to offer the more effective and accurate methods of diagnosis to find out the confirmed infections.

Conclusion
In this work, we developed a new mathematical epidemic SEIQR model for the outbreak of the new COVID-19 coronavirus. This pandemic model offers a new method for evaluating and handling the COVID-19 epidemic. In Saudi Arabia, actual COVID-19 data have been used to validate the findings of this new model. The results show that the SEIQR model is a useful model for studying the spread of epidemics in Saudi and other countries, such as COVID-19.
The current model introduced a new and different reproduction number which is more sensitive to more parameters than the past SEIQR models.
Five steps were the perfect procedure, and thorough advice was implemented to help the population of Saudi Arabia slow the spread of COVID-19. Prevention is one of the key targets of this procedure rather than treatment.
The other main problem that helps to delay COVID-19 spread is to remain at home and to keep sick people in an isolated area or a protected location.
Finally, we need a safe and effective treatment of people with infections and vitamins, tonics, and supplements to protect non-infected people.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.