Abstract

Since the first confirmed case of SARS-CoV-2 coronavirus (COVID-19) on March 02, 2020, Saudi Arabia has not reported quite a rapid COVD-19 spread as seen in America and many European countries. Possible causes include the spread of asymptomatic COVID-19 cases. To characterize the transmission of COVID-19 in Saudi Arabia, a susceptible, exposed, symptomatic, asymptomatic, hospitalized, and recovered dynamical model was formulated, and a basic analysis of the model is presented including model positivity, boundedness, and stability around the disease-free equilibrium. It is found that the model is locally and globally stable around the disease-free equilibrium when . The model parameterized from COVID-19 confirmed cases reported by the Ministry of Health in Saudi Arabia (MOH) from March 02 till April 14, while some parameters are estimated from the literature. The numerical simulation showed that the model predicted infected curve is in good agreement with the real data of COVID-19-infected cases. An analytical expression of the basic reproduction number is obtained, and the numerical value is estimated as .

1. Introduction

As of April 22, 2020, more than 12772 cases and 114 deaths of coronavirus disease 2019 (COVID-19) caused by the SARS-CoV-2 virus had been confirmed in the Kingdom of Saudi Arabia. Since March 04 [1], control measures have been implemented within Saudi Arabia to control the spread of the disease. Isolation of confirmed cases and contact tracing are crucial parts of these measures, which are common interventions for controlling infectious disease outbreaks [2–4]. For example, the severe acute respiratory syndrome (SARS) outbreak and the Middle East respiratory syndrome (MERS) were controlled through tracing suspected cases and isolating confirmed cases because the majority of transmission occurred concurrent or after symptom onset [3–5].

However, it is unknown if transmission of COVID-19 can occur before symptom onset, which could decrease the effectiveness of isolation and contact tracing [2, 3, 5].

In this paper, the impact of asymptomatic COVID-19 cases on the spread of the disease is considered using a modified version of the susceptible-exposed-infected-recovered (SEIR) dynamical model, along with the data of COVID-19 cases reported daily by the Ministry of Health in Saudi Arabia (MOH). Other main objectives of this paper include obtaining an analytical expression and numerical estimation of the basic reproduction number of COVID-19 in Saudi Arabia and estimating the maximum required number of hospital beds and intensive care units (ICU). The rest of this study is organized as follows: The model establishment is presented in Section 2. Basic analysis of the model, including local and global stability results of the disease-free equilibrium, is explored in Section 3. The model fitting to daily reported cases and estimation of parameters is given in Section 4. Numerical results and discussion are presented in Section 5. Finally, brief concluding remarks are given in Section 6.

2. Model Establishment

The Saudi population will be divided into six categories: susceptible , exposed , symptomatic , asymptomatic , hospitalized , and recovered individuals (). Individuals move from the susceptible compartment to the exposed compartment after interacting with infected individuals with transmission rates , , and as shown in Figure 1. COVID-19 is known to have an incubation period, from 2 to 14 days, between exposure and development of symptoms [6, 7]. After this period, exposed individual transits from the compartment to either compartment at a rate or compartment at a rate . An individual could move from compartment to at a rate if they show symptoms. Once an individual becomes infected with the coronavirus that causes COVID-19, that individual develops immunity against the virus with a rate or the individual will be hospitalized with a rate of or dies because of the disease with a rate of . When an individual becomes hospitalized, that individual receives treatment and develops immunity against the virus with a rate or dies because of the disease with a rate .

Figure 1 

Schematic diagram of the compartment model. The arrows, except the black ones, represent a progression from one compartment to the next.

As shown in Figure 1, the model has six compartments; therefore, a discrete dynamical system consisting of six nonlinear differential equations is formed as follows: where and is the Saudi population at time . The next-generation matrix is used in the next section to derive an analytical expression of the basic reproduction number , for the compartmental model above. Calculating is a useful metric for assessing the transmission potential of an emerging COVID-19 in Saudi Arabia.

3. Basic Analytical Results

In this section, the positivity and boundedness of the proposed model solution (6) and stability of the model around the disease-free equilibrium is investigated. Furthermore, an analytical expression of the basic reproduction number is established.

3.1. Positivity and Boundedness

To show that the model (6) is biologically well-behaved or epidemiologically meaningful, we must show that all its state variables are nonnegative and bounded for .

Lemma 1. For and initial values , where , the solution of the model (6) is nonnegative if they exist.

Proof. Assume , then it follows from the first equation of (6) as

It can be written as

Hence, which can be found as below

Hence, is nonnegative for . In a similar way, it can be shown that , , , , and for .

Lemma 2. is the positively invariant region of the model (6) with nonnegative initial conditions in .

Proof. Suppose holds for , then we get

It is obvious that

Thus, . Furthermore, if . This shows that solutions of the model (6) point towards . Hence, is positively invariant and solutions of (6) are bounded.

3.2. Basic Reproduction Number

It can be easily verified that system (6) always has a disease-free equilibrium (DFE), which we will denote by , and it is given by

Next, we investigate an important concept in epidemiology which is the basic reproduction number, defined as “the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual” [8]. The next-generation method is used to calculate ; more details about the method are in [8]. The system (6) can be rewritten as follows: where and , or more explicitly

The Jacobian matrices of and evaluated at the solution are given by

Direct calculations show that where

Denoting the identity matrix by , the characteristic polynomial of the matrix is given by where

The solutions are given by

Therefore, the reproduction number for the model (6) is given by

For simplicity, let us denote

In order to interpret the basic reproduction number , we split the above expression of as follows: where

The first and second terms in show the probability of the total population becoming symptomatic upon infection multiplied by mean symptomatic, exposed, and hospitalized infectious periods multiplied by symptomatic and hospitalized contact rates. The first and second terms in show the probability of the total population becoming asymptomatic multiplied by mean exposed, symptomatic, asymptomatic, and hospitalized infectious periods multiplied by symptomatic, asymptomatic, and hospitalized contact rates. Furthermore, the first and second terms in show the probability of the total population becoming asymptomatic multiplied by mean exposed, symptomatic, asymptomatic, and hospitalized infectious periods multiplied by asymptomatic and hospitalized contact rates. The obtained analytical expression of shows that asymptomatic as well as symptomatic cases of COVID-19 could drive the growth of the pandemic in Saudi Arabia.

3.3. Stability of Disease-Free Equilibrium

In this section, the local and global stability of the model (6) around the disease-free equilibrium is explored. The epidemiological implication of the local stability result of DFE is that a small influx of COVID-19 infection cases will not generate a COVID-19 outbreak if while the global stability result of DFE indicates that any influx of COVID-19 infection cases will not generate a COVID-19 outbreak if .

Theorem 3. The DFE of the system (6) is locally asymptotically stable if and unstable otherwise.

Proof. The Jacobian matrix obtained at the DFE is as follows:

Clearly, the eigenvalues and are negative real numbers. The remaining eigenvalues can be obtained through the following equation: where where and is obviously positive. We can rewrite as

Clearly if , then is positive. Similarly, we can show that and are all positive if . Further, it is easy to show the remaining Routh-Hurtwiz condition for the fourth-order polynomial (27). Thus, the DFE is locally asymptotically stable if .

The global stability of DFE of the COVID-19 transmission model is studied in the following result. where for , are positive constants. Differentiating the function with respect to and using the solutions of system (6), we obtain

Theorem 4. The DFE of the system (6) is globally asymptotically stable if and unstable otherwise.

Proof. Let us consider the following Lyapunov function, described in [9]:

Now choosing