
This article discusses the problem of estimation with step stress partially accelerated life tests using Type-II progressively censored samples. The lifetime of items under use condition follows the two-parameters inverted Kumaraswamy distribution. The maximum likelihood estimates for the unknown parameters are computed numerically. Using the property of asymptotic distributions for maximum likelihood estimation, we constructed asymptotic interval estimates. The Bayes procedure is used to calculate estimates of the unknown parameters from symmetrical and asymmetric loss functions. The Bayes estimates cannot be obtained explicitly, therefor the Lindley's approximation and the Markov chain Monte Carlo technique are used to obtaining the Bayes estimates. Furthermore, the highest posterior density credible intervals for the unknown parameters are calculated. An example is presented to illustrate the methods of inference. Finally, a numerical example of March precipitation (in inches) in Minneapolis failure times in the real world is provided to illustrate how the approaches will perform in practice.
Citation: Manal M. Yousef, Rehab Alsultan, Said G. Nassr. Parametric inference on partially accelerated life testing for the inverted Kumaraswamy distribution based on Type-II progressive censoring data[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 1674-1694. doi: 10.3934/mbe.2023076
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This article discusses the problem of estimation with step stress partially accelerated life tests using Type-II progressively censored samples. The lifetime of items under use condition follows the two-parameters inverted Kumaraswamy distribution. The maximum likelihood estimates for the unknown parameters are computed numerically. Using the property of asymptotic distributions for maximum likelihood estimation, we constructed asymptotic interval estimates. The Bayes procedure is used to calculate estimates of the unknown parameters from symmetrical and asymmetric loss functions. The Bayes estimates cannot be obtained explicitly, therefor the Lindley's approximation and the Markov chain Monte Carlo technique are used to obtaining the Bayes estimates. Furthermore, the highest posterior density credible intervals for the unknown parameters are calculated. An example is presented to illustrate the methods of inference. Finally, a numerical example of March precipitation (in inches) in Minneapolis failure times in the real world is provided to illustrate how the approaches will perform in practice.
Coronavirus disease 2019, COVID-19, is a new infectious disease that emerged at the end of 2019. Within three months, became a global epidemic [1]. As of December 26, 2021, the number of confirmed cases around the world had reached 278,714,484 [2]. The epidemic affected various aspects of life, including the economic, health and social aspects.
Several researchers have discussed the disease dynamics and characteristics of COVID-19 patients in different regions of the world. Individuals infected with COVID-19 may or may not have symptoms that range from mild to severe. Individuals who experience a fever, dry cough, fatigue and headaches as mild symptoms or acute respiratory distress syndrome as a severe symptom [3] are called symptomatic individuals. Individuals who do not show any signs and their infection is confirmed through real-time reverse transcriptase-polymerase chain reaction (RT-PCR), i.e., a type of diagnostic test for COVID-19, are called asymptomatic individuals [4]. These patients can also transmit the disease to other individuals [5,6].
The presence of asymptomatic patients has led researchers to focus on studying this class of patients. Some research has described the clinical characteristics of asymptomatic and symptomatic patients of COVID-19 [7,8]. Kim et al. [7] determined the spread of asymptomatic cases in South Korea. Of 213 infected individuals with COVID-19, 41 (19.2%) were asymptomatic. They concluded that as much as one-fifth of infected individuals with COVID-19 are asymptomatic. Therefore, strict social distancing must be applied to prevent the transmission of the disease through these individuals. AlJishi et al. [8] described the spread of COVID-19 at the epicenter of its spread in Saudi Arabia, in the eastern region of the kingdom. They concluded that most COVID-19 cases were asymptomatic and returned from travel. In addition, Alsofayan et al. [9] emphasized the need to give attention to asymptomatic individuals and health care workers because they contribute to the spread of the disease.
Several studies have proposed a mathematical model that considers asymptomatic and symptomatic infected individuals [10,11,12,13,14,15,16,17,18]. A mathematical model was proposed in [10] to discover the effect of asymptomatic infected people and estimate the number of beds needed in hospitals and intensive care units in Saudi Arabia. They concluded that asymptomatic people are more effective in increasing the number of infections than symptomatic people. However, the increase in COVID-19 testing and social distancing reduces the infection by asymptomatic individuals. Sun and Weng [15] established a mathematical model to discuss the effect of asymptomatic and imported patients in the Jiangsu and Anhui provinces in China. Their analysis found that asymptomatic patients are more dangerous than imported patients. Moreover, asymptomatic patients can rapidly cause an outbreak of the disease without strict preventive measures.
Serhani and Labbardi [16] analyzed two mathematical models to study the spread of COVID-19 in Morocco without and with containment of the pandemic. They found that the free equilibrium point is asymptotically stable when the containment coefficient exceeds the basic reproduction number R0. Furthermore, increasing the containment rate decreases the number of asymptomatic and symptomatic individuals. Huo et al. [17] studied the spread of COVID-19 and the effect of non-pharmaceutical interventions in Wuhan. They estimated that 20% of patients remain asymptomatic during infection, and that their ability to transmit is around 70% of the symptomatic patients' ability.
Moreover, in [13,14], the authors analyzed mathematical models for COVID-19 in Saudi Arabia. The models assumed the transmission of infection is due to the environment and occurs through contact with exposed, asymptomatic and symptomatic individuals. Alzahrani et al. [13] dealt with a fractional model. Alqarni et al. [14] used a classical epidemic model and concluded that the cases in Saudi Arabia would decrease if contact with exposed individuals and the environment decreases.
With the increasing number of infections and the contribution of asymptomatic individuals to the transmission of the disease, there have been efforts in various countries to constrain the epidemic. Several precautions have been imposed to keep susceptible individuals away from infected individuals. Studies have demonstrated the effectiveness of preventive measures, such as lockdowns [19,20,21], social distancing [22,23,24], quarantining [12,16] and isolation [25,26]. Nevertheless, we need measures to detect asymptomatic patients, particularly at the beginning of the disease and before treatment is available, to limit its spread and reduce the number of deaths, an example of such a measure is diagnosis testing [27].
Al-Salti et al. [18] developed a mathematical model to analyze the optimal control strategies, especially diagnosis and quarantining, for the spread of COVID-19 in the Sultanate of Oman. It was concluded that diagnosis and quarantining are the most important strategies for controlling the disease. Similarly, in [11], Ahmed et al. confirmed the effectiveness of the diagnosis in reducing the infected cases. In addition, Khan et al. [28] studied the effectiveness of Saudi Arabia's experience in producing the diagnosis of COVID-19. From April 14 to mid-July of 2020, the Saudi's diagnostic program for COVID-19 was conducted in three stages [28,29]. The first stage was performing random diagnostic tests on people in crowded places such as workers' housing, which included 807 locations from April 16 to May 5, 2020 [30]. The second stage was conducting diagnostic tests in specific health centers through an application (Mawid) from May 3 to May 19, 2020. The third stage was conducting rapid tests at medical centers through drive-in vehicles, which started on May 29, 2020 [31,32]. Also, the government of Saudi Arabia has continued to expand the diagnostic procedures for COVID-19 [33,34,35,36].
This paper describe the study of the impact of detecting asymptomatic COVID-19 patients through random diagnostic testing in Saudi Arabia. Random diagnosis means diagnosing COVID-19 for suspected individuals at random, whether they show symptoms or not. In other words, officials in the Ministry of Health go to homes and diagnose individuals in places where there are factors that contribute to the spread of the disease, such as overpopulation.
We extend our model in [37] to include the asymptomatic COVID-19 patients and analyze the impact of random diagnostic tests. Therefore, this model contains three control measures that have been implemented in Saudi Arabia: lockdown, social distancing and random diagnostic testing. The analysis of this model attempts to determine if it is possible to rely on random diagnosis while reducing or eliminating lockdown and social distancing. This paper is organized as follows. In Section 2, we formulate the model and prove that it is well posed. In Section 3, we demonstrate the qualitative analysis, including the equilibrium points, basic and control reproduction numbers, and the local and global stability of the equilibrium points. Section 4 presents the numerical analysis, which includes the fitting of the model to COVID-19 cases in Saudi Arabia and the estimation of the model parameters. Furthermore, we describe numerical experiments that were conducted to confirm the qualitative analysis results and investigate the sensitivity analysis for the control reproduction number. Finally, we analyze different scenarios for control strategies, random diagnostic testing, lockdown and social distancing.
The model was formulated to examine the effect of the random diagnostic tests for asymptomatic patients on the spread of COVID-19 in Saudi Arabia. The Saudi population, N, comprises five compartments: susceptible individuals, S, exposed individuals, E, asymptomatic infected individuals (with no symptoms), Ia, symptomatic infected individuals (with symptoms), Is, and recovered individuals, R. Individuals move from the susceptible compartment to the exposed compartment after interacting with infected individuals, symptomatic or asymptomatic, at the transmission rates β1 and β2, respectively. The control measures, i.e., the lockdown (ρ(t)∈(0,1]) and the social distancing (SD(t)∈[0,1)), impact the transmission rates β1 and β2. However, the random diagnostic testing for asymptomatic individuals affects only β2. The function ε(t)∈[0,1] represents the effectiveness of the random diagnostic testing for Ia. If ε=1, then the random diagnostic test was administered to all asymptomatic individuals in the community. Conversely, no random diagnostic test is performed for Ia when ε=0. After completing the incubation period, 1/γ, individuals in the exposed compartment move to the asymptomatic compartment with the probability θ, where θ∈(0,1); and the rest of the individuals (1−θ) move to the symptomatic compartment. We assume that individuals in the exposed compartment may or may not show symptoms by the end of the incubation period. Also, individuals in the asymptomatic compartment can transmit the disease more than individuals in the symptomatic compartment, that is, β2>β1. Moreover, individuals die due to COVID-19 only from the symptomatic compartment at a rate d. Infected individuals, asymptomatic and symptomatic, recover and gain immunity to COVID-19 at a rate δ. The natural death rate for each compartment is μ, and η is the natural birth rate.
Figure 1 displays the model's dynamics. We express the model mathematically with the following nonlinear system of ordinary differential equations:
dSdt=η−ρ(1−SD)[β1SIs+β2(1−ε)SIa]−μS,dEdt=ρ(1−SD)[β1SIs+β2(1−ε)SIa]−(γ+μ)E,dIadt=γθE−(δ+μ)Ia,dIsdt=γ(1−θ)E−(δ+d+μ)Is,dRdt=δIa+δIs−μR, | (2.1) |
where all parameters belong to the interval (0,1] and N=S(t)+E(t)+Ia(t)+Is(t)+R(t).
We start the analysis by proving that the state variables of Model (2.1) are epidemiologically meaningful, that is, non-negative and bounded.
Theorem 1
If (S,E,Ia,Is,R)∈R5≥0, then the set
Ω={(S,E,Ia,Is,R)∈R5≥0:0≤N≤ημ} |
is positively invariant for Model (2.1).
Proof.
Let (S(0),E(0),Ia(0),Is(0),R(0))∈Ω. We have
dSdt|S=0=η>0,dEdt|E=0=ρ(1−SD)[β1SIs+β2(1−ε)SIa]≥0,forall S, Ia, Is≥0,dIadt|Ia=0=γθE≥0,forall E≥0,dIsdt|Is=0=γ(1−θ)E≥0,forall E≥0,dRdt|R=0=δIa+δIs≥0,forall Ia, Is≥0. |
Thus, all non-negative solutions remain non-negative for t≥0. By adding all of the equations of Model (2.1), we get
dNdt=η−dIs−μN≤η−μN. |
Multiplying the above inequality by the integrating factor (eμr), we have
ddr[eμrN(r)]≤ηeμr. |
By integrating both sides over the interval [0,t], we obtain
N(t)≤ημ+[N(0)−ημ]e−μt. |
Hence,
limt→∞Sup[N(t)]≤ημ. |
Therefore, for t≥0, the solutions of Model (2.1) are non-negative and bounded. Thus, Ω is positively invariant.
We qualitatively investigate Model (2.1). First, we determine the equilibrium points and the expression of the control reproduction number. Second, we examine the local and global stability of the equilibrium points.
The equilibrium points of the model are determined by setting the rates for Model (2.1) to zero:
η−ρ(1−SD)[β1SIs+β2(1−ε)SIa]−μS=0,ρ(1−SD)[β1SIs+β2(1−ε)SIa]−(γ+μ)E=0,γθE−(δ+μ)Ia=0,γ(1−θ)E−(δ+d+μ)Is=0,δIa+δIs−μR=0. | (3.1) |
System (3.1) produces two steady-state solutions: the COVID-19 free equilibrium point, P0=(η/μ,0,0,0,0), which always exists, and the COVID-19 endemic equilibrium point, P1=(S1,E1,Ia1,Is1,R1), where
S1=ημRc,Ia1=ηγθ(γ+μ)(δ+μ)(1−1Rc),E1=η(γ+μ)(1−1Rc),Is1=ηγ(1−θ)(γ+μ)(δ+d+μ)(1−1Rc),R1=ηγ(δθd+δ2+δμ)μ(γ+μ)(δ+μ)(δ+d+μ)(1−1Rc). |
Here,
Rc=ρ(1−SD)β1(1−θ)γημ(γ+μ)(δ+d+μ)+ρ(1−SD)β2θ(1−ε)γημ(γ+μ)(δ+μ). |
The endemic equilibrium P1 exists only if Rc>1.
Control reproduction number. We use the next-generation matrix method [38] to find the control reproduction number. Let the infected compartments O=(E,Ia,Is)T; then, the exposed class, asymptomatic class and symptomatic class equations of Model (2.1) can be rewritten as ˙O=F(O)−V(O), where
F=[ρ(1−SD)[β1SIs+β2(1−ε)SIa]00] |
and
V=[(γ+μ)E−γθE+(δ+μ)Ia−γ(1−θ)E+(δ+d+μ)Is]. |
Evaluating the Jacobian matrix of F and V at the COVID-19 free equilibrium point P0, we get, respectively,
F=[0ρ(1−SD)β2(1−ε)ημρ(1−SD)β1ημ000000] |
and
V=[γ+μ00−γθδ+μ0−γ(1−θ)0δ+d+μ]. |
The next-generation matrix is
FV−1=[K1ρ(1−SD)β2(1−ε)ημ(δ+μ)ρ(1−SD)β1ημ(δ+d+μ)000000], |
where
K1=ρ(1−SD)β1(1−θ)γημ(γ+μ)(δ+d+μ)+ρ(1−SD)β2θ(1−ε)γημ(γ+μ)(δ+μ). |
Hence, the control reproduction number is the spectral radius of the matrix FV−1, that is,
Rc=ρ(1−SD)β1(1−θ)γημ(γ+μ)(δ+d+μ)+ρ(1−SD)β2θ(1−ε)γημ(γ+μ)(δ+μ). | (3.2) |
Also, we can rewrite Expression (3.2) as follows: Rc=ρ(1−SD)(Rs+(1−ε)Ra), where Rs gives the secondary cases of COVID-19 by one symptomatic individuals, and Ra gives the secondary cases of COVID-19 by one asymptomatic individuals.
The terms in Rc are explained further as follows. First, for Rs, the term that expresses the incidence of new infections by symptomatic individuals is β1SIs. Thus, the number of secondary cases by one symptomatic individual (Is=1) in a population containing only susceptible individuals is β1S0, where S0=η/μ. Moreover, 1/(δ+d+μ) represents the average time spent by one symptomatic individual in the symptomatic compartment. Also, γ(1−θ)/(γ+μ) is the proportion of newly symptomatic individuals that survived the incubation period. Second, for Ra, the term that represents the incidence of new infections by asymptomatic individuals is β2SIa. Therefore, the number of secondary cases by one asymptomatic individual (Ia=1) in a population containing only susceptible individuals is β2S0, where S0=η/μ. Further, 1/(δ+μ) expresses the average time spent by one asymptomatic individual in the asymptomatic compartment. Again, γθ/(γ+μ) is the proportion of newly asymptomatic individuals that survived the incubation period.
We employ the linearization method [39] to examine the local stability of the equilibrium points of Model (2.1).
Theorem 2
The COVID-19 free equilibrium point P0 is locally asymptotically stable if Rc<1.
Proof.
The Jacobian matrix of Model (2.1) evaluated at P0 yields
J(P0)=[−μ0−ρ(1−SD)β2(1−ε)S0−ρ(1−SD)β1S000−(γ+μ)ρ(1−SD)β2(1−ε)S0ρ(1−SD)β1S000γθ−(δ+μ)000γ(1−θ)0−(δ+d+μ)000δδ−μ]. |
By solving the characteristic equation ∣J(P0)−λI∣=0, we obtain the eigenvalues λ1,2=−μ, and λ3,4,5 are roots of the following equation:
λ3+a1λ2+a2λ+a3=0, |
where
a1=γ+d+2δ+3μ,a2=(δ+μ)(δ+d+μ)+(γ+μ)(δ+μ)[1−ρ(1−SD)(1−ε)Ra]+(γ+μ)(δ+d+μ)[1−ρ(1−SD)Rs],a3=(γ+μ)(δ+μ)(δ+d+μ)(1−Rc). |
We use the Routh-Hurwitz criteria [39] to determine the signs of the remaining eigenvalues λ3,4,5. They are negative if a1>0,a3>0 and a1a2−a3>0. Clearly, a1>0. Also, a3>0 if Rc<1. Moreover,
a1a2−a3=2(γ+μ)(δ+μ)(δ+d+μ)+(δ+μ)2(δ+d+μ)+(δ+μ)(δ+d+μ)2+((γ+μ)2(δ+d+μ)+(γ+μ)(δ+d+μ)2)[1−ρ(1−SD)Rs]+((γ+μ)2(δ+μ)+(γ+μ)(δ+μ)2)[1−ρ(1−SD)(1−ε)Ra]. |
If Rc=ρ(1−SD)[Rs+(1−ε)Ra]<1, then ρ(1−SD)Rs<1 and ρ(1−SD)(1−ε)Ra<1. Thus, a1a2−a3>0. Hence, P0 is locally asymptotically stable if Rc<1.
Theorem 3
The COVID-19 endemic equilibrium point P1 is locally asymptotically stable if Rc>1.
Proof.
The Jacobian matrix of Model (2.1) evaluated at P1 yields
J(P1)=[J110−ρ(1−SD)β2(1−ε)S1−ρ(1−SD)β1S10J21−(γ+μ)ρ(1−SD)β2(1−ε)S1ρ(1−SD)β1S100γθ−(δ+μ)000γ(1−θ)0−(δ+d+μ)000δδ−μ], |
where
J11=−ρ(1−SD)[β1Is1+β2(1−ε)Ia1]−μ,J21=ρ(1−SD)[β1Is1+β2(1−ε)Ia1]. |
The characteristic equation, ∣J(P1)−λI∣=0, has the eigenvalues λ1=−μ, and λ2,3,4,5 are the roots of the following equation:
λ4+a1λ3+a2λ2+a3λ+a4=0, | (3.3) |
where
a1=μRc+γ+d+2δ+3μ,a2=μ(δ+d+μ)Rc+μ(γ+δ+2μ)Rc+(δ+μ)(δ+d+μ)+(γ+μ)(δ+μ)[1−ρ(1−SD)(1−ε)RaRc]+(γ+μ)(δ+d+μ)[1−ρ(1−SD)RsRc],a3=μ(γ+μ)(δ+μ)(δ+d+μ)Rc+μ(δ+μ)(δ+d+μ)Rc+ηγρ(1−SD)[β1(1−θ)+β2θ(1−ε)] (1−1Rc),a4=μ(γ+μ)(δ+μ)(δ+d+μ)Rc. |
Clearly, a1>0 and a4>0. Also, since Rc=ρ(1−SD)(Rs+(1−ε)Ra)>1, then a2>0 and a3>0. By the Descartes' rule [40], the characteristic equation given by Eq (3.3) has no positive roots since there is no change in the signs of the coefficients. When substituting for (λ) by (−λ) in Eq (3.3), the signs of the coefficients change as follows:
λ4−a1λ3+a2λ2−a3λ+a4=0. |
The number of sign changes is four; therefore, the characteristic Eq (3.3) has four negative eigenvalues. Hence, P1 is locally asymptotically stable if Rc>1.
We utilize the Lyapunov and Krasovkii–LaSalle stability theorems [41,42,43] to examine the global stability of the equilibrium points of Model (2.1). Also, we use the function W(u)=u−1−lnu, which is a positive function, in the following proofs.
Theorem 4
The COVID-19 free equilibrium point P0 is globally asymptotically stable if Rc<1.
Proof.
Define the Lyapunov function L0(S,E,Ia,Is,R) as follows:
L0=S0W(S/S0)+E+(γ+μ)γ[ρ(1−SD)(1−ε)θRa+(δ+d+μ)(θd+δ+μ)(1−Rc)]Ia+(γ+μ)γ[ρ(1−SD)(1−θ)Rs+(δ+μ)(θd+δ+μ)(1−Rc)]Is+(γ+μ)(δ+μ)(δ+d+μ)γδ(θd+δ+μ)(1−Rc)R. |
If Rc<1, then L0 is positive-definite since L0(S,E,Ia,Is,R)>0 for all (S,E,Ia,Is,R)∈Ω and L0(P0)=0. Computing the time derivative of L0 along the solutions of Model (2.1), we get
dL0dt=(1−S0S)[η−ρ(1−SD)[β1SIs+β2(1−ε)SIa]−μS]+[ρ(1−SD)[β1SIs+β2(1−ε)SIa]−(γ+μ)E]+(γ+μ)γ[ρ(1−SD)(1−ε)θRa+(δ+d+μ)(θd+δ+μ)(1−Rc)][γθE−(δ+μ)Ia]+(γ+μ)γ[ρ(1−SD)(1−θ)Rs+(δ+μ)(θd+δ+μ)(1−Rc)][γ(1−θ)E−(δ+d+μ)Is]+(γ+μ)(δ+μ)(δ+d+μ)γδ(θd+δ+μ)(1−Rc)[δIa+δIs−μR]. |
Since S0=η/μ, then η=μS0. Collecting terms, we get
dL0dt=(1−S0S)(μS0−μS)−μ(γ+μ)(δ+μ)(δ+d+μ)γδ(θd+δ+μ)(1−Rc)R−[(γ+μ)−θ(γ+μ)(ρ(1−SD)(1−ε)θRa+(δ+d+μ)(θd+δ+μ)(1−Rc))−(1−θ)(γ+μ)(ρ(1−SD)(1−θ)Rs+(δ+μ)(θd+δ+μ)(1−Rc))]E−[(γ+μ)(δ+μ)γ(ρ(1−SD)(1−ε)θRa+(δ+d+μ)(θd+δ+μ)(1−Rc))−ρ(1−SD)β2(1−ε)S0−(γ+μ)(δ+μ)(δ+d+μ)γ(θd+δ+μ)(1−Rc)]Ia−[(γ+μ)(δ+d+μ)γ(ρ(1−SD)(1−θ)Rs+(δ+μ)(θd+δ+μ)(1−Rc))−ρ(1−SD)β1S0−(γ+μ)(δ+μ)(δ+d+μ)γ(θd+δ+μ)(1−Rc)]Is. |
After simplifications by using expressions of Ra and Rs, we obtain
dL0dt=−μ(S−S0)2S−μ(γ+μ)(δ+μ)(δ+d+μ)γδ(θd+δ+μ)(1−Rc)R. |
If Rc<1, then dL0/dt≤0 for all S,R>0. Also, dL0/dt=0 when S(t)=S0, and R(t)=0. Applying the Krasovkii-Lasalle theorem, we suppose that
I0={(S(t),E(t),Ia(t),Is(t),R(t)):dL0dt=0}, |
and M0 is the largest invariant subset of I0, where all elements in it satisfy S(t)=S0 and R(t)=0. Then, from the fifth equation of Model (2.1), we get
dRdt=0=δ(Ia+Is)⟹Ia(t)=0 and Is(t)=0. |
Substituting this into the third equation of Model (2.1), we have
dIadt=0=γθE⟹E(t)=0. |
Hence, M0={P0}; thus, the equilibrium P0 is globally asymptotically stable if Rc<1.
Theorem 5
The COVID-19 endemic equilibrium point P1 is globally asymptotically stable if Rc>1.
Proof.
Define the Lyapunov function L1(S,E,Ia,Is,R) as follows:
L1=S1W(S/S1)+E1W(E/E1)+ηρ(1−SD)β2(1−ε)μ(δ+μ)Rc Ia1W(I/Ia1)+ηρ(1−SD)β1μ(δ+d+μ)Rc Is1W(I/Is1). |
Clearly, L1(S,E,Ia,Is,R) is a positive semi-definite function since L1≥0 for all (S,E,Ia,Is,R)∈Ω and L1(S1,E1,Ia1,Is1,R1)=0. The time derivative of L1 along the solutions of Model (2.1) is given by
dL1dt=(1−S1S)[η−ρ(1−SD)[β1SIs+β2(1−ε)SIa]−μS]+(1−E1E)[ρ(1−SD)[β1SIs+β2(1−ε)SIa]−(γ+μ)E]+(1−Ia1Ia)ηρ(1−SD)β2(1−ε)μ(δ+μ)Rc[γθE−(δ+μ)Ia]+(1−Is1Is)ηρ(1−SD)β1μ(δ+d+μ)Rc[γ(1−θ)E−(δ+d+μ)Is]. |
From the equilibrium described by Eq (3.1) for P1, we have
η=ρ(1−SD)[β1S1Is1+β2(1−ε)S1Ia1]+μS1,(γ+μ)E1=ρ(1−SD)[β1S1Is1+β2(1−ε)S1Ia1]. |
Then,
dL1dt=−μ(S−S1)2S+3ρ(1−SD)[β1S1Is1+β2(1−ε)S1Ia1]+[γ(1−θ)ρ(1−SD)β1S1(δ+d+μ)+γθρ(1−SD)β2(1−ε)S1(δ+μ)−(γ+μ)]E−ρ(1−SD)β1S1Is1S1S−ρ(1−SD)β2(1−ε)S1Ia1S1S−ρ(1−SD)β1SIsS1Is1E1S1Is1E−ρ(1−SD)β2(1−ε)SIaS1Ia1E1S1Ia1E−γθρ(1−SD)β2(1−ε)S1(δ+μ) E1Ia1EIa1E1Ia1Ia−γ(1−θ)ρ(1−SD)β1S1(δ+d+μ) E1Is1EIs1E1Is1Is. | (3.4) |
Now, we substitute for S1=η/μRc in the coefficient of E in Eq (3.4) and use the formula of Rc. Moreover, we substitute the values of E1, 1/Ia1 and 1/Is1 in the following terms:
(γθE1(δ+μ)Ia1)ρ(1−SD)β2(1−ε)S1Ia1EIa1E1Ia=ρ(1−SD)β2(1−ε)S1Ia1EIa1E1Ia,(γ(1−θ)E1(δ+d+μ)Is1)ρ(1−SD)β1S1Is1EIs1E1Is=ρ(1−SD)β1S1Is1EIs1E1Is. |
After simplifying Eq (3.4), we obtain
dL1dt=−μ(S−S1)2S+ρ(1−SD)β2(1−ε)S1Ia1(3−S1S−SIaE1S1Ia1E−EIa1E1Ia)+ρ(1−SD)β1S1Is1(3−S1S−SIsE1S1Is1E−EIs1E1Is). |
Since the arithmetic mean is greater than or equal to the geometric mean, we have
3−S1S−SIaE1S1Ia1E−EIa1E1Ia≤0,3−S1S−SIsE1S1Is1E−EIs1E1Is≤0. |
Hence, dL1/dt≤0 for all S,E,Ia,Is>0 and dL1/dt=0 when S(t)=S1, E(t)=E1, Ia(t)=Ia1 and Is(t)=Is1. Applying the Krasovkii-Lasalle theorem, we consider the set
I1={(S(t),E(t),Ia(t),Is(t),R(t)):dL1dt=0}, |
and M1 is the largest invariant subset of I1, where all elements satisfy S(t)=S1, E(t)=E1, Ia(t)=Ia1 and Is(t)=Is1; it remains to be proven that R(t)=R1. Assume that (S(t),E(t),Ia(t),Is(t),R(t)) is a solution to Model (2.1) belonging to the set M1; thus, we have
dRdt=δ(Ia1+Is1)−μR. | (3.5) |
Solving Eq (3.5) by using the integrating factor method, we obtain
R(t)=R1+(R(0)−R1)e−μt. | (3.6) |
Note that, δ(Ia1+Is1)/μ=R1. From Eq (3.6), as time increases, R(t) approaches R1, that is, limt→∞R(t)⟶R1. The solution (S(t),E(t),Ia(t),Is(t),R(t)) will stay at the set M1; hence, M1={P1}. Therefore, the equilibrium P1 is globally asymptotically stable if Rc>1.
Here, we fit Model (2.1) to the actual data of COVID-19 cases in Saudi Arabia and estimate the parameters that make it compatible with reality. Also, we perform numerical experiments to demonstrate the agreement with the qualitative results. Furthermore, we discuss the sensitivity analysis for the control reproduction number, Rc. Finally, we examine different scenarios for the control measures in the model, namely, lockdown, social distancing and the detection of asymptomatic individuals through random diagnostic testing.
We estimate the parameters of Model (2.1) to validate it with the actual data of infected cases in Saudi Arabia. The data of the infected cases were taken from the COVID-19 dashboard of the Saudi Ministry of Health for the period of March 12, 2020 to September 23, 2020 [45].
The health care workers in the Saudi Ministry of Health have applied random diagnostic testing for asymptomatic and symptomatic individuals in dense regions. The random diagnostic tests were done between April 16, 2020 and May 3, 2020, for 42,765 individuals. The number of confirmed cases (positive test result) was 7,776 (18.18%) individuals [28]. We estimate that the effectiveness of the random diagnostic tests for the asymptomatic individuals, ε, to be approximately zero because of the insufficient information about the number of asymptomatic individuals at the time of implementing these tests. Also, the period for these tests is small compared to the time interval of the data considered from the COVID-19 dashboard.
The parameters η, μ,δ, d, ρ, SD and γ have the same values as in [37]. As for the values of the remaining parameters of Model (2.1), we estimate the value of β2 to be greater than β1 based on the model's assumption, which is similar to [10,13].
The parameter values are presented in Table 1. In Table 2, we display the values of lockdown, ρ, social distancing, SD and the corresponding values of Rc. These values were taken from [37], where Saudi Arabia implemented different levels of lockdown and social distancing, which led to dividing the study period into seven phases.
Parameter | Description | Value | Unit | Source |
N | Population of Saudi Arabia | 34218169 | Individual | [44] |
η | Birth rate | 1250 | Individual×Day−1 | [37] |
μ | Natural death rate | 3.6529×10−5 | Day−1 | [37] |
β1 | Transmission rate by Is | 0.0554×10−7 | (Individual×Day)−1 | Estimated |
β2 | Transmission rate by Ia | 1.0596×10−7 | (Individual×Day)−1 | Estimated |
γ | Incubation rate | 1/6 | Day−1 | [9] |
θ | Probability of becoming Ia | 0.54 | − | Estimated |
ε | Effectiveness of random diagnostic testing for Ia | 0 | − | Estimated |
δ | Recovery rate | 3.2772×10−1 | Day−1 | [37] |
d | Death rate due to COVID-19 | 2.3724×10−1 | Day−1 | [37] |
Phase | Time | ρ | SD | Rc |
Phase 1 | t1=[12,22] | 0.85 | 0.15 | 4.4267 |
Phase 2 | t2=[22,25] | 0.75 | 0.30 | 3.2166 |
t3=[25,28] | 0.65 | 0.45 | 2.1904 | |
t4=[28,36] | 0.60 | 0.55 | 1.6543 | |
Phase 3 | t5=[36,56] | 0.55 | 0.55 | 1.5164 |
Phase 4 | t6=[56,67] | 0.55 | 0.60 | 1.3479 |
t7=[67,83] | 0.55 | 0.69 | 1.0446 | |
Phase 5 | t8=[83,88] | 0.40 | 0.80 | 0.4902 |
Phase 6 | t9=[88,92] | 0.65 | 0.70 | 1.1947 |
t10=[92,112] | 0.75 | 0.70 | 1.3785 | |
Phase 7 | t11=[112,120] | 0.80 | 0.76 | 1.1764 |
t12=[120,130] | 0.80 | 0.80 | 0.9803 | |
t13=[130,144] | 0.80 | 0.82 | 0.8823 | |
t14=[144,169] | 0.85 | 0.83 | 0.8853 | |
t15=[169,179] | 0.85 | 0.82 | 0.9374 | |
t16=[179,193] | 0.90 | 0.82 | 0.9926 | |
t17=[193,204] | 0.95 | 0.85 | 0.8731 | |
t18=[204,207] | 0.95 | 0.88 | 0.6985 |
Model (2.1) was solved numerically by using the MATLAB package ode45 with the following initial values: S(0)=34813577, E(0)=150, Ia(0)=95, IS(0)=44 and R(0)=1. We assumed that each initial value for the exposed class was set to be greater than those for the infected classes, as in [13,23,46]. Also, each initial value of the asymptomatic class was set to be greater than those for the symptomatic class based on the results in [8], which demonstrated that most of the COVID-19 cases at the beginning of the spread in Saudi Arabia were asymptomatic and returning from imported. Moreover, Li et al. [6] estimated the undocumented infections (infections with mild, limited or loss of symptoms) to be 86% of the infections in China before imposing travel restrictions on January 23, 2020.
The results of fitting Model (2.1) to the COVID-19 infected cases (symptomatic and asymptomatic) in Saudi Arabia from March 12, 2020 to September 23, 2020 are illustrated in Figure 2, which shows good agreement.
This section demonstrates the agreement between the numerical solution of Model (2.1) and the qualitative analysis offered in Section 3. We conducted numerical simulations of the model for different initial values in the feasible set to show the tendency of the solution curves to equilibrium P0 if Rc<1, or equilibrium P1 if Rc>1. We start by rescaling the state variables in the model. Let
S=ˉSN, E=ˉEN, Ia=¯IaN, Is=¯IsN, R=ˉRN. | (4.1) |
Thus, the rescaled Model (2.1) becomes (omitting the bar onward)
dSdt=μ−ρ(1−SD)ημ[β1SIs+β2(1−ε)SIa]−μS,dEdt=ρ(1−SD)ημ[β1SIs+β2(1−ε)SIa]−(γ+μ)E,dIadt=γθE−(δ+μ)Ia,dIsdt=γ(1−θ)E−(δ+d+μ)Is,dRdt=δ(Ia+Is)−μR. | (4.2) |
Note that we have used the limiting value of N in the model, i.e., N=η/μ. We numerically solve Model (4.2) with parameters chosen to satisfy the stability conditions from the qualitative analysis, and with the following different initial conditions belonging to the set Ω:
IC1: S(0)=0.8, E(0)=0.1, Ia(0)=0.05, Is(0)=0.02, R(0)=0.01,
IC2: S(0)=0.6, E(0)=0.2, Ia(0)=0.08, Is(0)=0.07, R(0)=0.05,
IC3: S(0)=0.4, E(0)=0.3, Ia(0)=0.12, Is(0)=0.10, R(0)=0.07.
Experiment1: Assume the parameters of Model (4.2) have the following values: μ=0.04, η=1250, γ=0.167, β1=0.0554×10−4, β2=1.0596×10−4, ε=0.50, θ=0.54, δ=3.2772×10−1, d=2.3724×10−1, ρ=0.5 and SD=0.75. Thus, Rc=0.2585<1. Therefore, we expect that the solution curves of the model tend to the COVID-19 free equilibrium point P0=(1, 0, 0, 0, 0). This is evident in Figure 3; the numerical solutions eventually reach P0 for different initial conditions. Accordingly, COVID-19 diminishes.
Experiment2: In this experiment, we increased the value of the parameter ρ and decreased the values of the parameters ε and SD. That is to say, the control measures of lockdown, social distancing and random diagnostic testing were reduced. We assume the parameters of Model (4.2) have the following values: μ=0.04, η=1250, γ=0.167, β1=0.0554×10−4, β2=1.0596×10−4, ε=0.10, θ=0.54, δ=3.2772×10−1, d=2.3724×10−1, ρ=0.85 and SD=0.30. Therefore, Rc=2.1639>1. Hence, we expect the solution curves of the model to approach the COVID-19 endemic equilibrium point P1. This is displayed in Figure 4; the numerical solutions eventually tend to P1=(0.4621, 0.1039, 0.0255, 0.0132, 0.3170) for different initial conditions. Consequently, COVID-19 remains at a specific percentage.
We conclude from the experiments that the numerical results are in good agreement with the qualitative results.
To determine which parameters influence the spread of COVID-19, we investigated the sensitivity of the control reproduction number for Model (2.1). The sensitivity of Rc was examined analytically by evaluating ∂Rc/∂P, where P=(η,β1,β2,ρ,SD,γ,δ,ε,θ,d,μ). The changes in Rc corresponding to one parameter at a time are as follows:
∂Rc∂ε=−ηγρ(1−SD)β2θμ(γ+μ)(δ+μ)<0,∂Rc∂β1=ηγρ(1−SD)(1−θ)μ(γ+μ)(δ+d+μ)>0,∂Rc∂β2=ηγρ(1−SD)θ(1−ε)μ(γ+μ)(δ+μ)>0,∂Rc∂d=−ηγρ(1−SD)β1(1−θ)μ(γ+μ)(δ+d+μ)2<0,∂Rc∂SD=−ηγρμ(γ+μ)[β1(1−θ)(δ+d+μ)+β2θ(1−ε)(δ+μ)]<0,∂Rc∂η=γρ(1−SD)μ(γ+μ)[β1(1−θ)(δ+d+μ)+β2θ(1−ε)(δ+μ)]>0,∂Rc∂γ=ηρ(1−SD)(γ+μ)2[β1(1−θ)(δ+d+μ)+β2θ(1−ε)(δ+μ)]>0,∂Rc∂ρ=ηγ(1−SD)μ(γ+μ)[β1(1−θ)(δ+d+μ)+β2θ(1−ε)(δ+μ)]>0,∂Rc∂δ=−ηγρ(1−SD)μ(γ+μ)[β1(1−θ)(δ+d+μ)2+β2θ(1−ε)(δ+μ)2]<0,∂Rc∂θ=ηγρ(1−SD)μ(γ+μ)[β2(1−ε)(δ+μ)−β1(δ+d+μ)]=ηγρ(1−SD)β2μ(γ+μ)(δ+μ)(1−ω)>0when ω<1(<0 when ω>1),∂Rc∂μ=−ηγρ(1−SD)(γ+2μ)μ2(γ+μ)2[β1(1−θ)(δ+d+μ)+β2θ(1−ε)(δ+μ)]−ηγρ(1−SD)μ(γ+μ)[β1(1−θ)(δ+d+μ)2+β2θ(1−ε)(δ+μ)2]<0, | (4.3) |
where ω=ε+(β1(δ+μ)/β2(δ+d+μ)). From Eq (4.3), we see that an increase in ε, SD, δ, d or μ leads to a decrease in Rc. Conversely, an increase in β1, β2, ρ, γ or η leads to an increase in Rc. Finally, the sensitivity of Rc with respect to θ depends on the value of ω. If ω>1, then Rc decreases, and if ω<1, Rc increases. Figure 5 illustrates these results.
The normalized sensitivity index (elasticity) of Rc with respect to the model parameters P is defined as follows [47]:
ΓPRc=∂Rc∂PPRc. | (4.4) |
By applying the formula in Eq (4.4), we get
ΓηRc=1,ΓρRc=1,ΓγRc=μγ+μ,ΓSDRc=−SD1−SD,ΓθRc=β2θ(1−ε)(δ+d+μ)−β1θ(δ+μ)β2θ(1−ε)(δ+d+μ)+β1(1−θ)(δ+μ),Γβ1Rc=β1(1−θ)(δ+μ)β1(1−θ)(δ+μ)+β2θ(1−ε)(δ+d+μ),Γβ2Rc=β2θ(1−ε)(δ+d+μ)β1(1−θ)(δ+μ)+β2θ(1−ε)(δ+d+μ),ΓεRc=−β2εθ(δ+d+μ)β1(1−θ)(δ+μ)+β2θ(1−ε)(δ+d+μ),ΓdRc=−dβ1(1−θ)(δ+μ)β1(1−θ)(δ+μ)(δ+d+μ)+β2θ(1−ε)(δ+d+μ)2,ΓδRc=−δ(δ+μ)(δ+d+μ)[β1(1−θ)(δ+μ)2+β2θ(1−ε)(δ+d+μ)2β1(1−θ)(δ+μ)+β2θ(1−ε)(δ+d+μ)],ΓμRc=−(γ+2μ)(γ+μ)−μ(δ+μ)(δ+d+μ)[β1(1−θ)(δ+μ)2+β2θ(1−ε)(δ+d+μ)2β1(1−θ)(δ+μ)+β2θ(1−ε)(δ+d+μ)]. |
Table 3 exhibits the elasticity of Rc with respect to the model parameters. The parameter values were taken from Table 1, in addition to ρ=0.80, SD=0.60 and ε=0.40. Any positive value for the elasticity in Table 3 means that the contribution of the parameter to Rc is to increase its value. For instance, Γβ2Rc=0.9587 means that an increase in β2 by 1% will increase Rc by 0.9587%. On the contrary, any negative value for the elasticity means that the contribution of the parameter to Rc is to decrease its value. For example, ΓεRc=−0.6391 means that an increase in ε by 1% will decrease Rc by 0.6391%. We notice that the most sensitive parameters are β2, ρ, SD and ε. Therefore, the increase in imposing social distancing and lockdown measures and detecting asymptomatic individuals through random diagnostic testing reduces the spread of the disease.
Parameter P | Value | Sensitivity index ΓPRc |
η | 1250 | 1 |
β1 | 0.0554×10−7 | 0.0413 |
β2 | 1.0596×10−7 | 0.9587 |
γ | 0.167 | 0.0002 |
θ | 0.54 | 0.9103 |
δ | 3.2772×10−1 | −0.9826 |
d | 2.3724×10−1 | −0.0173 |
μ | 3.653×10−5 | −1.0003 |
ρ | 0.80 | 1 |
ε | 0.40 | −0.6391 |
SD | 0.10 | −0.1111 |
0.30 | −0.4286 | |
0.50 | −1 | |
0.70 | −2.3333 | |
0.90 | −9 | |
0.99 | −99 |
Model (2.1) examines some of the policies that have been applied in Saudi Arabia to control COVID-19, namely, the effectiveness of random diagnostic testing on the asymptomatic class (ε), lockdown (ρ) and social distancing (SD). We executed several scenarios to investigate the impact of these policies by simulating Model (2.1) for infected cases. Figure 2 shows the actual data in Saudi Arabia and the fitting curve for Model (2.1) for infected cases from March 12, 2020 to September 23, 2020. We solve Model (2.1) numerically with the following initial values: S(0)=34813577, E(0)=150, Ia(0)=95, Is(0)=44, R(0)=1 and the parameter values given in Table 1. The parameters ε, ρ and SD took different values for each scenario. We describe different scenarios in Table 4. For the fitting values of ρ and SD, refer to Table 2.
Scenario | ε | ρ | SD |
1st | Different values | Fitting values | Fitting values |
2nd | Different values | No lockdown (ρ=1) | No social distancing (SD=0) |
3rd | Different values | Different values | Fitting values |
4th | Different values | Fitting values | Different values |
5th | Different values | 0.85 | Different values |
6th | Different values | 0.75 | Different values |
7th | Different values | Fitting values | Fitting values |
for three time periods |
The results of applying the scenarios (1st−7th) are summarized in Tables 5–11 and illustrated in Figures 6–12. We compared the scenarios with the peak value of the infected cases in the fitting of Model (2.1), that is, 59,830, which occured on Day 123.
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
Fitting values | Fitting values | 0.01 | 122 | 52,010 | −13.07% |
0.02 | 122 | 45,140 | −24.55% | ||
0.03 | 122 | 39,120 | −34.61% | ||
0.04 | 122 | 33,830 | −43.45% | ||
0.10 | 121 | 13,770 | −76.98% | ||
0.20 | 68 | 4,159 | −93.04% | ||
0.30 | 58 | 1,556 | −97.39% | ||
0.40 | 37 | 705 | −98.82% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
1 | 0 | 0.10 | 53 | 4,728,000 | +7802.39% |
0.20 | 57 | 4,420,000 | +7287.60% | ||
0.30 | 62 | 4,057,000 | +6680.88% | ||
0.40 | 69 | 3,607,000 | +5928.75% | ||
0.50 | 79 | 3,042,000 | +4984.41% | ||
0.60 | 97 | 2,332,000 | +3797.71% | ||
0.70 | 133 | 1,431,000 | +2291.78% | ||
0.80 | 266 | 391,700 | +554.68% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
1 | Fitting values | 0.10 | 84 | 1,163,000 | +1843.84% |
0.20 | 112 | 630,200 | +953.31% | ||
0.30 | 115 | 195,700 | +227.09% | ||
0.40 | 113 | 27,360 | −54.27% | ||
0.85 | Fitting values | 0.10 | 113 | 454,000 | +658.81% |
0.20 | 115 | 127,900 | +113.77% | ||
0.30 | 113 | 22,410 | −62.54% | ||
0.40 | 77-83 | 4,208 | −92.96% | ||
0.75 | Fitting values | 0.10 | 115 | 110,600 | +84.85% |
0.20 | 113 | 23,550 | −60.63% | ||
0.30 | 83 | 5,273 | −91.18% | ||
0.40 | 68 | 1,514 | −97.46% | ||
0.55 | Fitting values | 0.10 | 70 | 2,882 | −95.18% |
0.20 | 68 | 1,175 | −98.03% | ||
0.30 | 58 | 474 | −99.20% | ||
0.40 | 30 | 244 | −99.59% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
Fitting values | 0.40 | 0.10 | 124 | 2,314,000 | +3767.62% |
0.20 | 138,139 | 2,035,000 | +3301.30% | ||
0.30 | 160 | 1,764,000 | +2848.35% | ||
0.40 | 194 | 1,441,000 | +2308.49% | ||
Fitting values | 0.50 | 0.10 | 150 | 1,874,000 | +3032.21% |
0.20 | 171 | 1,629,000 | +2622.71% | ||
0.30 | 203 | 1,434,000 | +2296.79% | ||
0.40 | 247 | 1,121,000 | +1773.64% | ||
Fitting values | 0.60 | 0.10 | 198 | 1,439,000 | +2305.15% |
0.20 | 229 | 1,271,000 | +2024.35% | ||
0.30 | 279 | 902,300 | +1408.11% | ||
0.40 | 377 | 506,400 | +746.39% | ||
Fitting values | 0.70 | 0.10 | 302 | 776,200 | +1197.34% |
0.20 | 387 | 480,900 | +703.77% | ||
0.30 | 572 | 213,800 | +257.34% | ||
0.40 | 1402, 1408 | 28,830 | −51.81% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
0.85 | 0.40 | 0.10 | 88 | 2,674,000 | +4369.33% |
0.20 | 98 | 2,284,000 | +3717.48% | ||
0.30 | 114 | 1,847,000 | +2987.08% | ||
0.40 | 138 | 1,357,000 | +2168.09% | ||
0.85 | 0.50 | 0.10 | 106 | 2,052,000 | +3329.72% |
0.20 | 121,122 | 1,666,000 | +2684.56% | ||
0.30 | 145 | 1,247,000 | +1984.24% | ||
0.40 | 185,186 | 801,700 | +1239.96% | ||
0.85 | 0.60 | 0.10 | 141 | 1,310,000 | +2089.54% |
0.20 | 168,169 | 957,200 | +1499.87% | ||
0.30 | 216 | 599,300 | +901.67% | ||
0.40 | 320 | 265,700 | +344.09% | ||
0.85 | 0.70 | 0.10 | 240 | 488,600 | +716.64% |
0.20 | 331,332 | 246,000 | +311.16% | ||
0.30 | 600 | 60,790 | +1.60% | ||
0.40 | 12 | 139 | −99.76% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
0.75 | 0.40 | 0.10 | 100 | 2,246,000 | +3653.97% |
0.20 | 113 | 1,859,000 | +3007.14% | ||
0.30 | 133,134 | 1,430,000 | +2290.11% | ||
0.40 | 167 | 967,500 | +1517.08% | ||
0.75 | 0.50 | 0.10 | 123 | 1,630,000 | +2624.39% |
0.20 | 144 | 1,259,000 | +2004.30% | ||
0.30 | 178 | 867,000 | +1349.11% | ||
0.40 | 242 | 475,400 | +694.58% | ||
0.75 | 0.60 | 0.10 | 172 | 924,900 | +1445.88% |
0.20 | 214,215 | 608,800 | +917.55% | ||
0.30 | 297 | 310,600 | +419.13% | ||
0.40 | 550 | 75,830 | +26.74% | ||
0.75 | 0.70 | 0.10 | 344 | 226,300 | +278.23% |
0.20 | 585-588 | 64,690 | +8.12% | ||
0.30 | 12 | 139 | −99.76% | ||
0.40 | 12 | 139 | −99.76% |
Event | ε1 | ε2 | ε3 | Peak (day) | Cases | Percentage Change |
Ⅰ | 0.10 | 0 | 0 | 123 | 38,480 | −35.68% |
Ⅱ | 0.10 | 0.10 | 0 | 124 | 20,890 | −65.08% |
Ⅲ | 0 | 0.10 | 0.10 | 121 | 21,920 | −63.36% |
Ⅳ | 0.20 | 0 | 0 | 124 | 23,650 | −60.47% |
Ⅴ | 0.20 | 0.20 | 0 | 125 | 6,539 | −89.07% |
Ⅵ | 0 | 0.20 | 0.20 | 68 | 11,470 | −80.82% |
Ⅶ | 0.10 | 0.20 | 0 | 125 | 10,870 | −81.83% |
Ⅷ | 0.10 | 0.20 | 0.10 | 121 | 7,132 | −88.07% |
Ⅸ | 0.20 | 0.10 | 0 | 124,125 | 12,680 | −78.80% |
Ⅹ | 0.20 | 0.10 | 0.10 | 121 | 8,326 | −86.08% |
Ⅺ | 0 | 0.20 | 0.10 | 121 | 11,480 | −80.81% |
The 1st scenario examines the impact of detecting asymptomatic individuals by applying random diagnostic testing in Saudi Arabia. At the same time, the lockdown and social distancing are kept as the fitting values. Figure 6 illustrates that the peak of infected cases decreases with increasing ε. For instance, when detecting only 2% of the asymptomatic individuals (ε=0.02) by random diagnostic testing, the cases will reach 45,140 on Day 122. Compared with actual data, the number of cases decreased by 24.55% (see Table 5). This indicates the importance of applying the random diagnostic testing to find asymptomatic individuals and reduce the cases.
The 2nd scenario was applied to determine whether it is possible to rely only on detecting asymptomatic individuals through random diagnostic testing without imposing social distancing and a lockdown. Figure 7 shows that the infected cases will far surpass the actual data. For example, if the detection of asymptomatic cases reaches 60% (ε=0.60), the peak of cases will be 2,332,000 on Day 97 (see Table 6). That is an increase of 3797.71% relative to the peak value of the actual data. We concluded that detecting asymptomatic individuals is insufficient without imposing social distancing and lockdown.
The 3rd scenario analyzes four different values for the level of lockdown (ρ=1, 0.85, 0.75, 0.55) the values for social distancing were kept as the fitting values. When ρ=1, there is no lockdown implementation. The remaining values of ρ correspond to the lockdown level undertaken by Saudi Arabia in Phase 1, Phase 2 and Phase 4, respectively [37].
Figure 8 shows that increasing the lockdown level leads to a decrease in the infected cases. Note that increasing the lockdown level means decreasing the value of ρ. Moreover, increasing the lockdown level requires a smaller value for detecting asymptomatic individuals through random diagnostic testing to reach case numbers lower than the actual data. To explain further, when the lockdown level is equal to 1, 0.85, 0.75 or 0.55, we need to apply a 40, 30, 20 or 10% level of random diagnostic testing, respectively, to detect the asymptomatic individuals (see Table 7).
The 4th scenario investigates the impact of social distancing while keeping the value of ρ equal to the fitting values. Figure 9 illustrates that increasing the application of social distancing delays peak days; however, only when SD=0.70 and ε=0.40 does the cases reach a value that is lower than the actual data by 51.81%, i.e., 28,830 (see Table 8).
The 5th and 6th scenarios examine the impact of imposing a lockdown at low levels and relying on social distancing and random diagnostic testing. We let ρ=0.85 and 0.75 for the 5th and 6th scenarios, respectively. Figures 10 and 11 show that it is possible to reach a lower number of infected cases than in the actual data if SD=0.70, where ε=0.40 for the case where ρ=0.85 (see Table 9), and ε=0.30 for the case where ρ=0.75 (see Table 10). This indicates that enforcing low lockdown levels is effective with a high level of social distancing implementation and 30% or more of the detection of asymptomatic individuals through random diagnostic testing.
The 7th scenario was designed to determine the effect of different values of ε through the phases. We assume that the value of ε is ε1 in Phase 1 to Phase 2, ε2 in Phase 3 to Phase 5 and ε3 in Phase 6 to Phase 7 (see Table 11). Figure 12(a) and (b) show that performing random diagnostic tests in the middle phases (Phase 3–5) and the beginning phases (Phase 1 and 2), rather than the last phases (Phase 6 and 7), gives better results. Furthermore, applying the tests in all phases reduces the number of cases, as seen in Figure 12(c).
We conclude from the analysis of the 1st to the 6th scenarios that applying random diagnostic testing alone is inadequate without imposing social distancing and lockdown. The importance of using random diagnostic testing to find asymptomatic individuals is apparent when lockdown and social distancing measures are enforced; only then does the number of cases decline. An increase in the level of lockdown decreases the required percentage of random diagnostic tests to lower the number of cases. However, an increase in social distancing does not have the same effect as a lockdown; a high percentage of random diagnostic tests is needed to reduce the number of cases. On the other hand, implementation of lower levels of lockdown is possible only when a high level of social distancing and a high percentage of random diagnostic tests are imposed.
As for the 7th scenario, we deduce that performing the random diagnostic tests in all phases is effective in lowering the number of cases. However, applying the random diagnostic tests in the early and middle phases is sufficient to reduce the cost of conducting these tests.
This study was purposed to develop a mathematical model to examine the impact of random diagnostic testing on COVID-19 patients in the presence of a lockdown and social distancing in Saudi Arabia. Qualitative and numerical analyses were applied to the model. The model was well posed and had two equilibrium points. The COVID-19 free equilibrium existed and was locally and globally asymptotically stable if Rc<1. Conversely, the COVID-19 endemic equilibrium existed and was locally and globally asymptotically stable if Rc>1.
The model was validated by using the data from the COVID-19 dashboard of the Saudi Ministry of Health spanning March 12, 2020 to September 23, 2020. In addition, the numerical experiments performed using the model showed the consistency of the numerical solutions with the qualitative analysis.
Furthermore, the sensitivity analysis for Rc revealed that the most influential parameter in terms of increasing Rc was the transmission rate due to contact with asymptomatic individuals (β2). However, the control parameters, i.e., the lockdown (ρ), social distancing (SD) and the effectiveness of random diagnostic testing for asymptomatic individuals (ε), played a significant role in decreasing Rc. Finally, we analyzed different scenarios numerically for the control strategies applied in Saudi Arabia.
We concluded that applying random diagnostic testing to detect asymptomatic individuals in the presence of a lockdown and social distancing significantly reduced the cases. On the other hand, it was impossible to rely on random diagnostic testing without imposing a lockdown and social distancing. Moreover, implementing a lockdown at low levels required increased social distancing levels and a high percentage of random diagnostic testing.
The authors declare that there is no conflict of interest.
[1] |
E. K. Al-Hussaini, A. H. Abdel-Hamid, Bayesian estimation of the parameters, reliability and hazard rate functions of mixtures under accelerated life tests, Commun. Stat. Simul. Comput., 33 (2004), 963–982. https://doi.org/10.1081/SAC-200040703 doi: 10.1081/SAC-200040703
![]() |
[2] |
E. K. Al-Hussaini, A. H. Abdel-Hamid, Accelerated life tests under finite mixture models, J. Stat. Comput. Simul., 76 (2006), 673–690. https://doi.org/10.1080/10629360500108087 doi: 10.1080/10629360500108087
![]() |
[3] |
A. M. Abd-Elfattah, A. S. Hassan, S. G. Nassr, Estimation in step-stress partially accelerated life tests for the Burr type XⅡ distribution using type I censoring, Stat. Methodol., 5 (2008), 502–514. https://doi.org/10.1016/j.stamet.2007.12.001 doi: 10.1016/j.stamet.2007.12.001
![]() |
[4] |
M. M. Mohie El-Din, S. E. Abu-Youssef, N. S. A. Ali, A. M. Abd El-Raheem, Estimation in step-stress accelerated life tests for Weibull distribution with progressive first-failure censoring, J. Stat. Appl. Probab., 3 (2015), 403–411. http://dx.doi.org/10.1155/2015/319051 doi: 10.1155/2015/319051
![]() |
[5] |
A. E. B. A. Ahmad, A. A. Soliman, M. M. Yousef, Bayesian Bayesian estimation of exponentiated Weibull distribution under partially accelerated life tests, Bull. Malays. Math. Sci. Soc., 39 (2016), 227–244. https://doi.org/10.1007/s40840-015-0170-9 doi: 10.1007/s40840-015-0170-9
![]() |
[6] |
A. A. Ismail, Likelihood Inference for a step stress partially accelerated life test model with type-I progressive hybrid censored data from Weibull distribution, J. Stat. Comput. Simul., 84 (2016), 2486–2494. https://doi.org/10.1080/00949655.2013.836195 doi: 10.1080/00949655.2013.836195
![]() |
[7] |
B. Liu, Y. Shi, J. Cai, R. Wang, Reliability analysis of masked data in adaptive step-stress partially accelerated lifetime tests with progressive removal, Commun. Stat. Theory Methods, 46 (2017), 6174–6191. https://doi.org/10.1080/03610926.2015.1122058 doi: 10.1080/03610926.2015.1122058
![]() |
[8] | S. A. Lone, A. Rahman, A. Islam, Step stress partially accelerated life testing plan for competing risk using adaptive Type-I progressive hybrid censoring, Pak. J. Stat., 33 (2017), 237–248. |
[9] |
M. Nassar, S. G. Nassr, S. Dey, Analysis of Burr type XⅡ distribution under step stress partially accelerated life tests with Type I and adaptive Type Ⅱ progressively hybrid censoring schemes, Ann. Data Sci., 4 (2017), 227–248. https://doi.org/10.1007/s40745-017-0101-8 doi: 10.1007/s40745-017-0101-8
![]() |
[10] |
S. G. Nassr, N. Elharoun, Inference for exponentiated Weibull distribution under constant stress partially accelerated life tests with multiple censored, Commun. Stat. Appl. Methods, 26 (2019), 131–148. https://doi.org/10.29220/CSAM.2019.26.2.131 doi: 10.29220/CSAM.2019.26.2.131
![]() |
[11] |
A. S. Hassan, S. G. Nassr, S. Pramanik, S. S. Maiti, Estimation in constant stress partially accelerated life tests for Weibull distribution based on censored competing risks data, Ann. Data Sci., 7 (2020), 45–62. https://doi.org/10.1007/s40745-019-00226-3 doi: 10.1007/s40745-019-00226-3
![]() |
[12] |
Ç. ÇETİNKAYA, Estimation in step-stress partially accelerated life tests for the power Lindley distribution under progressive censoring, Gazi Univ. J. Sci., 34 (2021), 579–590. https://doi.org/10.35378/gujs.682499 doi: 10.35378/gujs.682499
![]() |
[13] |
I. Alam, A. Ahmed, Inference on maintenance service policy under step-stress partially accelerated life tests using progressive censoring, J. Stat. Comput. Simul., 92 (2022), 813–829. https://doi.org/10.1080/00949655.2021.1975282 doi: 10.1080/00949655.2021.1975282
![]() |
[14] |
Y. Wang, W. Wang, Y. Tang, A Bayesian semiparametric accelerate failure time mixture cure model, Int. J. Biostat., 2021 (2021). https://doi.org/10.1515/ijb-2021-0012 doi: 10.1515/ijb-2021-0012
![]() |
[15] |
A. Xu, S. Zhou, Y. Tang, A unified model for system reliability evaluation under dynamic operating conditions, IEEE Trans. Reliab., 70 (2019), 65–72. https://doi.org/10.1109/TR.2019.2948173 doi: 10.1109/TR.2019.2948173
![]() |
[16] |
C. Luo, L. Shen, A. Xu, Modelling and estimation of system reliability under dynamic operating environments and lifetime ordering constraints, Reliab. Eng. Syst. Saf., 218 (2022), 108136. https://doi.org/10.1016/j.ress.2021.108136 doi: 10.1016/j.ress.2021.108136
![]() |
[17] |
M. M. Yousef, S. A. Alyami, A. F. Hashem, Statistical inference for a constant-stress partially accelerated life tests based on progressively hybrid censored samples from inverted Kumaraswamy distribution, PloS One, 17 (2022), e0272378. https://doi.org/10.1371/journal.pone.0272378 doi: 10.1371/journal.pone.0272378
![]() |
[18] | N. Balakrishnan, R. Aggarwala, Progressive Censoring: Theory, Methods, and Applications, Birkhauser, Boston, 2000. |
[19] | N. Balakrishnan, E. Cramer, The Art of Progressive Censoring, Birkhauser, New York, 2014. https://doi.org/10.1007/978-0-8176-4807-7 |
[20] |
F. Zhang, X. Shi, H. K. T. Ng, Information geometry of the exponential family of distributions with progressive Type-Ⅱ censoring, Entropy, 23 (2021), 687. https://doi.org/10.3390/e23060687 doi: 10.3390/e23060687
![]() |
[21] |
S. Dey, A. Elshahhat, Analysis of Wilson‐Hilferty distribution under progressive Type‐Ⅱ censoring, Qual. Reliab. Eng. Int., 2022 (2022). https://doi.org/10.1002/qre.3173 doi: 10.1002/qre.3173
![]() |
[22] | A. M. Abd AL-Fattah, A. A. EL-Helbawy, G. R. AL-Dayian, Inverted Kumaraswamy distribution: Properties and estimation, Pak. J. Stat., 33 (2017), 37–61. |
[23] |
M. Mohie El-Din, M. Abu-Moussa, On estimation and prediction for the inverted Kumaraswamy distribution based on general progressive censored samples, Pak. J. Stat. Oper. Res., 14 (2018), 717–736. https://doi.org/10.18187/pjsor.v14i3.2103 doi: 10.18187/pjsor.v14i3.2103
![]() |
[24] |
A. M. Daghistani, B. Al-Zahrani, M. Q. Shahbaz, Relations for moments of dual generalized order statistics for a new inverse Kumaraswamy distribution, Pak. J. Stat. Oper. Res., 15 (2019), 989–997. https://doi.org/10.18187/pjsor.v15i4.3079 doi: 10.18187/pjsor.v15i4.3079
![]() |
[25] |
K. Bagci, T. Arslan, H. E. Celik, Inverted Kumarswamy distribution for modeling the wind speed data: Lake Van, Turkey, Renewable Sustainable Energy Rev., 135 (2021), 110110. https://doi.org/10.1016/j.rser.2020.110110 doi: 10.1016/j.rser.2020.110110
![]() |
[26] |
F. Noor, S. Masood, M. Zaman, M. Siddiqa, R. A. Wagan, I. U. Khan, et al., Bayesian analysis of inverted Kumaraswamy mixture model with application to burning velocity of chemicals, Math. Prob. Eng., 2021 (2021). https://doi.org/10.1155/2021/5569652 doi: 10.1155/2021/5569652
![]() |
[27] | V. Bagdonavicius, M. Nikulin, Accelerated Life Models: Modeling and Statistical Analysis. Chapman and Hall/CRC Press, Boca Raton, Florida, 2021. |
[28] | H. R. Varian, A Bayesian approach to real estate assessment, in Variants in Economic Theory: Selected Works of H. R. Varian, Edward Elgar Publishing, (2000), 144–155. |
[29] |
M. Doostparast, S. Deepak, A. Zangoie, Estimation with the lognormal distribution on the basis of records, J. Stat. Comput. Simul., 83 (2013), 2339–2351. https://doi.org/10.1080/00949655.2012.691973 doi: 10.1080/00949655.2012.691973
![]() |
[30] |
D. V. Lindley, Approximate Bayesian methods, Trabajos de Estadistica y de investigacin operativa, 31 (1980), 223–245. https://doi.org/10.1007/BF02888353 doi: 10.1007/BF02888353
![]() |
[31] |
M. Doostparast, M. G. Akbari, N. Balakrishna, Bayesian analysis for the two-parameter Pareto distribution based on record values and times, J. Stat. Comput. Simul., 81 (2011), 1393–1403. https://doi.org/10.1080/00949655.2010.486762 doi: 10.1080/00949655.2010.486762
![]() |
[32] | N. Balakrishnan, R. A. Sandhu, A simple simulational algorithm for generating progressive Type-Ⅱ censored samples, Am. Stat., 49 (1985), 229–230. |
[33] |
D. Hinkley, On quick choice of power transformations, J. R. Stat. Ser. C, 26 (1977), 67–69. https://doi.org/10.2307/2346869 doi: 10.2307/2346869
![]() |
[34] | M. H. Chen, Q. M. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graphical Stat., 8 (1999), 69–92. |
Parameter | Description | Value | Unit | Source |
N | Population of Saudi Arabia | 34218169 | Individual | [44] |
η | Birth rate | 1250 | Individual×Day−1 | [37] |
μ | Natural death rate | 3.6529×10−5 | Day−1 | [37] |
β1 | Transmission rate by Is | 0.0554×10−7 | (Individual×Day)−1 | Estimated |
β2 | Transmission rate by Ia | 1.0596×10−7 | (Individual×Day)−1 | Estimated |
γ | Incubation rate | 1/6 | Day−1 | [9] |
θ | Probability of becoming Ia | 0.54 | − | Estimated |
ε | Effectiveness of random diagnostic testing for Ia | 0 | − | Estimated |
δ | Recovery rate | 3.2772×10−1 | Day−1 | [37] |
d | Death rate due to COVID-19 | 2.3724×10−1 | Day−1 | [37] |
Phase | Time | ρ | SD | Rc |
Phase 1 | t1=[12,22] | 0.85 | 0.15 | 4.4267 |
Phase 2 | t2=[22,25] | 0.75 | 0.30 | 3.2166 |
t3=[25,28] | 0.65 | 0.45 | 2.1904 | |
t4=[28,36] | 0.60 | 0.55 | 1.6543 | |
Phase 3 | t5=[36,56] | 0.55 | 0.55 | 1.5164 |
Phase 4 | t6=[56,67] | 0.55 | 0.60 | 1.3479 |
t7=[67,83] | 0.55 | 0.69 | 1.0446 | |
Phase 5 | t8=[83,88] | 0.40 | 0.80 | 0.4902 |
Phase 6 | t9=[88,92] | 0.65 | 0.70 | 1.1947 |
t10=[92,112] | 0.75 | 0.70 | 1.3785 | |
Phase 7 | t11=[112,120] | 0.80 | 0.76 | 1.1764 |
t12=[120,130] | 0.80 | 0.80 | 0.9803 | |
t13=[130,144] | 0.80 | 0.82 | 0.8823 | |
t14=[144,169] | 0.85 | 0.83 | 0.8853 | |
t15=[169,179] | 0.85 | 0.82 | 0.9374 | |
t16=[179,193] | 0.90 | 0.82 | 0.9926 | |
t17=[193,204] | 0.95 | 0.85 | 0.8731 | |
t18=[204,207] | 0.95 | 0.88 | 0.6985 |
Parameter P | Value | Sensitivity index ΓPRc |
η | 1250 | 1 |
β1 | 0.0554×10−7 | 0.0413 |
β2 | 1.0596×10−7 | 0.9587 |
γ | 0.167 | 0.0002 |
θ | 0.54 | 0.9103 |
δ | 3.2772×10−1 | −0.9826 |
d | 2.3724×10−1 | −0.0173 |
μ | 3.653×10−5 | −1.0003 |
ρ | 0.80 | 1 |
ε | 0.40 | −0.6391 |
SD | 0.10 | −0.1111 |
0.30 | −0.4286 | |
0.50 | −1 | |
0.70 | −2.3333 | |
0.90 | −9 | |
0.99 | −99 |
Scenario | ε | ρ | SD |
1st | Different values | Fitting values | Fitting values |
2nd | Different values | No lockdown (ρ=1) | No social distancing (SD=0) |
3rd | Different values | Different values | Fitting values |
4th | Different values | Fitting values | Different values |
5th | Different values | 0.85 | Different values |
6th | Different values | 0.75 | Different values |
7th | Different values | Fitting values | Fitting values |
for three time periods |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
Fitting values | Fitting values | 0.01 | 122 | 52,010 | −13.07% |
0.02 | 122 | 45,140 | −24.55% | ||
0.03 | 122 | 39,120 | −34.61% | ||
0.04 | 122 | 33,830 | −43.45% | ||
0.10 | 121 | 13,770 | −76.98% | ||
0.20 | 68 | 4,159 | −93.04% | ||
0.30 | 58 | 1,556 | −97.39% | ||
0.40 | 37 | 705 | −98.82% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
1 | 0 | 0.10 | 53 | 4,728,000 | +7802.39% |
0.20 | 57 | 4,420,000 | +7287.60% | ||
0.30 | 62 | 4,057,000 | +6680.88% | ||
0.40 | 69 | 3,607,000 | +5928.75% | ||
0.50 | 79 | 3,042,000 | +4984.41% | ||
0.60 | 97 | 2,332,000 | +3797.71% | ||
0.70 | 133 | 1,431,000 | +2291.78% | ||
0.80 | 266 | 391,700 | +554.68% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
1 | Fitting values | 0.10 | 84 | 1,163,000 | +1843.84% |
0.20 | 112 | 630,200 | +953.31% | ||
0.30 | 115 | 195,700 | +227.09% | ||
0.40 | 113 | 27,360 | −54.27% | ||
0.85 | Fitting values | 0.10 | 113 | 454,000 | +658.81% |
0.20 | 115 | 127,900 | +113.77% | ||
0.30 | 113 | 22,410 | −62.54% | ||
0.40 | 77-83 | 4,208 | −92.96% | ||
0.75 | Fitting values | 0.10 | 115 | 110,600 | +84.85% |
0.20 | 113 | 23,550 | −60.63% | ||
0.30 | 83 | 5,273 | −91.18% | ||
0.40 | 68 | 1,514 | −97.46% | ||
0.55 | Fitting values | 0.10 | 70 | 2,882 | −95.18% |
0.20 | 68 | 1,175 | −98.03% | ||
0.30 | 58 | 474 | −99.20% | ||
0.40 | 30 | 244 | −99.59% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
Fitting values | 0.40 | 0.10 | 124 | 2,314,000 | +3767.62% |
0.20 | 138,139 | 2,035,000 | +3301.30% | ||
0.30 | 160 | 1,764,000 | +2848.35% | ||
0.40 | 194 | 1,441,000 | +2308.49% | ||
Fitting values | 0.50 | 0.10 | 150 | 1,874,000 | +3032.21% |
0.20 | 171 | 1,629,000 | +2622.71% | ||
0.30 | 203 | 1,434,000 | +2296.79% | ||
0.40 | 247 | 1,121,000 | +1773.64% | ||
Fitting values | 0.60 | 0.10 | 198 | 1,439,000 | +2305.15% |
0.20 | 229 | 1,271,000 | +2024.35% | ||
0.30 | 279 | 902,300 | +1408.11% | ||
0.40 | 377 | 506,400 | +746.39% | ||
Fitting values | 0.70 | 0.10 | 302 | 776,200 | +1197.34% |
0.20 | 387 | 480,900 | +703.77% | ||
0.30 | 572 | 213,800 | +257.34% | ||
0.40 | 1402, 1408 | 28,830 | −51.81% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
0.85 | 0.40 | 0.10 | 88 | 2,674,000 | +4369.33% |
0.20 | 98 | 2,284,000 | +3717.48% | ||
0.30 | 114 | 1,847,000 | +2987.08% | ||
0.40 | 138 | 1,357,000 | +2168.09% | ||
0.85 | 0.50 | 0.10 | 106 | 2,052,000 | +3329.72% |
0.20 | 121,122 | 1,666,000 | +2684.56% | ||
0.30 | 145 | 1,247,000 | +1984.24% | ||
0.40 | 185,186 | 801,700 | +1239.96% | ||
0.85 | 0.60 | 0.10 | 141 | 1,310,000 | +2089.54% |
0.20 | 168,169 | 957,200 | +1499.87% | ||
0.30 | 216 | 599,300 | +901.67% | ||
0.40 | 320 | 265,700 | +344.09% | ||
0.85 | 0.70 | 0.10 | 240 | 488,600 | +716.64% |
0.20 | 331,332 | 246,000 | +311.16% | ||
0.30 | 600 | 60,790 | +1.60% | ||
0.40 | 12 | 139 | −99.76% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
0.75 | 0.40 | 0.10 | 100 | 2,246,000 | +3653.97% |
0.20 | 113 | 1,859,000 | +3007.14% | ||
0.30 | 133,134 | 1,430,000 | +2290.11% | ||
0.40 | 167 | 967,500 | +1517.08% | ||
0.75 | 0.50 | 0.10 | 123 | 1,630,000 | +2624.39% |
0.20 | 144 | 1,259,000 | +2004.30% | ||
0.30 | 178 | 867,000 | +1349.11% | ||
0.40 | 242 | 475,400 | +694.58% | ||
0.75 | 0.60 | 0.10 | 172 | 924,900 | +1445.88% |
0.20 | 214,215 | 608,800 | +917.55% | ||
0.30 | 297 | 310,600 | +419.13% | ||
0.40 | 550 | 75,830 | +26.74% | ||
0.75 | 0.70 | 0.10 | 344 | 226,300 | +278.23% |
0.20 | 585-588 | 64,690 | +8.12% | ||
0.30 | 12 | 139 | −99.76% | ||
0.40 | 12 | 139 | −99.76% |
Event | ε1 | ε2 | ε3 | Peak (day) | Cases | Percentage Change |
Ⅰ | 0.10 | 0 | 0 | 123 | 38,480 | −35.68% |
Ⅱ | 0.10 | 0.10 | 0 | 124 | 20,890 | −65.08% |
Ⅲ | 0 | 0.10 | 0.10 | 121 | 21,920 | −63.36% |
Ⅳ | 0.20 | 0 | 0 | 124 | 23,650 | −60.47% |
Ⅴ | 0.20 | 0.20 | 0 | 125 | 6,539 | −89.07% |
Ⅵ | 0 | 0.20 | 0.20 | 68 | 11,470 | −80.82% |
Ⅶ | 0.10 | 0.20 | 0 | 125 | 10,870 | −81.83% |
Ⅷ | 0.10 | 0.20 | 0.10 | 121 | 7,132 | −88.07% |
Ⅸ | 0.20 | 0.10 | 0 | 124,125 | 12,680 | −78.80% |
Ⅹ | 0.20 | 0.10 | 0.10 | 121 | 8,326 | −86.08% |
Ⅺ | 0 | 0.20 | 0.10 | 121 | 11,480 | −80.81% |
Parameter | Description | Value | Unit | Source |
N | Population of Saudi Arabia | 34218169 | Individual | [44] |
η | Birth rate | 1250 | Individual×Day−1 | [37] |
μ | Natural death rate | 3.6529×10−5 | Day−1 | [37] |
β1 | Transmission rate by Is | 0.0554×10−7 | (Individual×Day)−1 | Estimated |
β2 | Transmission rate by Ia | 1.0596×10−7 | (Individual×Day)−1 | Estimated |
γ | Incubation rate | 1/6 | Day−1 | [9] |
θ | Probability of becoming Ia | 0.54 | − | Estimated |
ε | Effectiveness of random diagnostic testing for Ia | 0 | − | Estimated |
δ | Recovery rate | 3.2772×10−1 | Day−1 | [37] |
d | Death rate due to COVID-19 | 2.3724×10−1 | Day−1 | [37] |
Phase | Time | ρ | SD | Rc |
Phase 1 | t1=[12,22] | 0.85 | 0.15 | 4.4267 |
Phase 2 | t2=[22,25] | 0.75 | 0.30 | 3.2166 |
t3=[25,28] | 0.65 | 0.45 | 2.1904 | |
t4=[28,36] | 0.60 | 0.55 | 1.6543 | |
Phase 3 | t5=[36,56] | 0.55 | 0.55 | 1.5164 |
Phase 4 | t6=[56,67] | 0.55 | 0.60 | 1.3479 |
t7=[67,83] | 0.55 | 0.69 | 1.0446 | |
Phase 5 | t8=[83,88] | 0.40 | 0.80 | 0.4902 |
Phase 6 | t9=[88,92] | 0.65 | 0.70 | 1.1947 |
t10=[92,112] | 0.75 | 0.70 | 1.3785 | |
Phase 7 | t11=[112,120] | 0.80 | 0.76 | 1.1764 |
t12=[120,130] | 0.80 | 0.80 | 0.9803 | |
t13=[130,144] | 0.80 | 0.82 | 0.8823 | |
t14=[144,169] | 0.85 | 0.83 | 0.8853 | |
t15=[169,179] | 0.85 | 0.82 | 0.9374 | |
t16=[179,193] | 0.90 | 0.82 | 0.9926 | |
t17=[193,204] | 0.95 | 0.85 | 0.8731 | |
t18=[204,207] | 0.95 | 0.88 | 0.6985 |
Parameter P | Value | Sensitivity index ΓPRc |
η | 1250 | 1 |
β1 | 0.0554×10−7 | 0.0413 |
β2 | 1.0596×10−7 | 0.9587 |
γ | 0.167 | 0.0002 |
θ | 0.54 | 0.9103 |
δ | 3.2772×10−1 | −0.9826 |
d | 2.3724×10−1 | −0.0173 |
μ | 3.653×10−5 | −1.0003 |
ρ | 0.80 | 1 |
ε | 0.40 | −0.6391 |
SD | 0.10 | −0.1111 |
0.30 | −0.4286 | |
0.50 | −1 | |
0.70 | −2.3333 | |
0.90 | −9 | |
0.99 | −99 |
Scenario | ε | ρ | SD |
1st | Different values | Fitting values | Fitting values |
2nd | Different values | No lockdown (ρ=1) | No social distancing (SD=0) |
3rd | Different values | Different values | Fitting values |
4th | Different values | Fitting values | Different values |
5th | Different values | 0.85 | Different values |
6th | Different values | 0.75 | Different values |
7th | Different values | Fitting values | Fitting values |
for three time periods |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
Fitting values | Fitting values | 0.01 | 122 | 52,010 | −13.07% |
0.02 | 122 | 45,140 | −24.55% | ||
0.03 | 122 | 39,120 | −34.61% | ||
0.04 | 122 | 33,830 | −43.45% | ||
0.10 | 121 | 13,770 | −76.98% | ||
0.20 | 68 | 4,159 | −93.04% | ||
0.30 | 58 | 1,556 | −97.39% | ||
0.40 | 37 | 705 | −98.82% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
1 | 0 | 0.10 | 53 | 4,728,000 | +7802.39% |
0.20 | 57 | 4,420,000 | +7287.60% | ||
0.30 | 62 | 4,057,000 | +6680.88% | ||
0.40 | 69 | 3,607,000 | +5928.75% | ||
0.50 | 79 | 3,042,000 | +4984.41% | ||
0.60 | 97 | 2,332,000 | +3797.71% | ||
0.70 | 133 | 1,431,000 | +2291.78% | ||
0.80 | 266 | 391,700 | +554.68% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
1 | Fitting values | 0.10 | 84 | 1,163,000 | +1843.84% |
0.20 | 112 | 630,200 | +953.31% | ||
0.30 | 115 | 195,700 | +227.09% | ||
0.40 | 113 | 27,360 | −54.27% | ||
0.85 | Fitting values | 0.10 | 113 | 454,000 | +658.81% |
0.20 | 115 | 127,900 | +113.77% | ||
0.30 | 113 | 22,410 | −62.54% | ||
0.40 | 77-83 | 4,208 | −92.96% | ||
0.75 | Fitting values | 0.10 | 115 | 110,600 | +84.85% |
0.20 | 113 | 23,550 | −60.63% | ||
0.30 | 83 | 5,273 | −91.18% | ||
0.40 | 68 | 1,514 | −97.46% | ||
0.55 | Fitting values | 0.10 | 70 | 2,882 | −95.18% |
0.20 | 68 | 1,175 | −98.03% | ||
0.30 | 58 | 474 | −99.20% | ||
0.40 | 30 | 244 | −99.59% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
Fitting values | 0.40 | 0.10 | 124 | 2,314,000 | +3767.62% |
0.20 | 138,139 | 2,035,000 | +3301.30% | ||
0.30 | 160 | 1,764,000 | +2848.35% | ||
0.40 | 194 | 1,441,000 | +2308.49% | ||
Fitting values | 0.50 | 0.10 | 150 | 1,874,000 | +3032.21% |
0.20 | 171 | 1,629,000 | +2622.71% | ||
0.30 | 203 | 1,434,000 | +2296.79% | ||
0.40 | 247 | 1,121,000 | +1773.64% | ||
Fitting values | 0.60 | 0.10 | 198 | 1,439,000 | +2305.15% |
0.20 | 229 | 1,271,000 | +2024.35% | ||
0.30 | 279 | 902,300 | +1408.11% | ||
0.40 | 377 | 506,400 | +746.39% | ||
Fitting values | 0.70 | 0.10 | 302 | 776,200 | +1197.34% |
0.20 | 387 | 480,900 | +703.77% | ||
0.30 | 572 | 213,800 | +257.34% | ||
0.40 | 1402, 1408 | 28,830 | −51.81% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
0.85 | 0.40 | 0.10 | 88 | 2,674,000 | +4369.33% |
0.20 | 98 | 2,284,000 | +3717.48% | ||
0.30 | 114 | 1,847,000 | +2987.08% | ||
0.40 | 138 | 1,357,000 | +2168.09% | ||
0.85 | 0.50 | 0.10 | 106 | 2,052,000 | +3329.72% |
0.20 | 121,122 | 1,666,000 | +2684.56% | ||
0.30 | 145 | 1,247,000 | +1984.24% | ||
0.40 | 185,186 | 801,700 | +1239.96% | ||
0.85 | 0.60 | 0.10 | 141 | 1,310,000 | +2089.54% |
0.20 | 168,169 | 957,200 | +1499.87% | ||
0.30 | 216 | 599,300 | +901.67% | ||
0.40 | 320 | 265,700 | +344.09% | ||
0.85 | 0.70 | 0.10 | 240 | 488,600 | +716.64% |
0.20 | 331,332 | 246,000 | +311.16% | ||
0.30 | 600 | 60,790 | +1.60% | ||
0.40 | 12 | 139 | −99.76% |
ρ | SD | ε | Peak (day) | Cases | Percentage Change |
0.75 | 0.40 | 0.10 | 100 | 2,246,000 | +3653.97% |
0.20 | 113 | 1,859,000 | +3007.14% | ||
0.30 | 133,134 | 1,430,000 | +2290.11% | ||
0.40 | 167 | 967,500 | +1517.08% | ||
0.75 | 0.50 | 0.10 | 123 | 1,630,000 | +2624.39% |
0.20 | 144 | 1,259,000 | +2004.30% | ||
0.30 | 178 | 867,000 | +1349.11% | ||
0.40 | 242 | 475,400 | +694.58% | ||
0.75 | 0.60 | 0.10 | 172 | 924,900 | +1445.88% |
0.20 | 214,215 | 608,800 | +917.55% | ||
0.30 | 297 | 310,600 | +419.13% | ||
0.40 | 550 | 75,830 | +26.74% | ||
0.75 | 0.70 | 0.10 | 344 | 226,300 | +278.23% |
0.20 | 585-588 | 64,690 | +8.12% | ||
0.30 | 12 | 139 | −99.76% | ||
0.40 | 12 | 139 | −99.76% |
Event | ε1 | ε2 | ε3 | Peak (day) | Cases | Percentage Change |
Ⅰ | 0.10 | 0 | 0 | 123 | 38,480 | −35.68% |
Ⅱ | 0.10 | 0.10 | 0 | 124 | 20,890 | −65.08% |
Ⅲ | 0 | 0.10 | 0.10 | 121 | 21,920 | −63.36% |
Ⅳ | 0.20 | 0 | 0 | 124 | 23,650 | −60.47% |
Ⅴ | 0.20 | 0.20 | 0 | 125 | 6,539 | −89.07% |
Ⅵ | 0 | 0.20 | 0.20 | 68 | 11,470 | −80.82% |
Ⅶ | 0.10 | 0.20 | 0 | 125 | 10,870 | −81.83% |
Ⅷ | 0.10 | 0.20 | 0.10 | 121 | 7,132 | −88.07% |
Ⅸ | 0.20 | 0.10 | 0 | 124,125 | 12,680 | −78.80% |
Ⅹ | 0.20 | 0.10 | 0.10 | 121 | 8,326 | −86.08% |
Ⅺ | 0 | 0.20 | 0.10 | 121 | 11,480 | −80.81% |