COVID-19 deterministic and stochastic modelling with optimized daily vaccinations in Saudi Arabia

Abstract

In this paper, we investigate the stochastic nature of the COVID-19 temporal dynamics by generating a fractional-order dynamic model and a fractional-order-stochastic model. Initially, we considered the first and second vaccination doses as multiple vaccinations were initiated worldwide. The concerned models are then tested for the Saudi Arabia second virus wave, which is assumed to start on 1st March 2021. Four daily vaccination scenarios for the first and second dose are assumed for 100 days from the wave beginning. One of these scenarios is based on function optimization using the invasive weed optimization algorithm (IWO). After that, we numerically solve the established models using the fractional Euler method and the Euler-Murayama method. Finally, the obtained virus dynamics using the assumed scenarios and the real one started by the government are compared. The optimized scenario using the IWO effectively minimizes the predicted cumulative wave infections with a 4.4 % lower number of used vaccination doses.

Introduction

Mathematical modelling is a powerful tool for predicting and controlling epidemics. Many mathematical models have demonstrated their ability to predict daily infections and deaths with pinpoint accuracy [1], [2]. The main two mathematical models used for this object are statistical models and dynamical models. Deterministic dynamic models and stochastic dynamic models are both used for COVID-19 outbreak modelling. A complete analysis of the actual number of deaths due to COVID-19 was studied by the authors in [3], [4] using a new class of distributions and the exponentiated transformation of the Gumbel type-II distribution. One of the best deterministic dynamic models working on controlling COVID-19 diffusion and transmission is the Susceptible – Exposed – Infectious – Recovered (SEIR) epidemic model [5], [6], [7]. In [8], the authors created a multi-strain epidemic model for COVID-19 spread. In [9], the authors obtained the numerical solutions of the non-linear fractional-order Susceptible – Infectious – Recovered (SIR) epidemic model by using different methods. In [10], the authors generated a modified fractional-order epidemic model which partitioned the population into seven classes and applied it to the real data of COVID-19 for Wuhan city. Since the COVID-19 transmission differs everywhere and it has a stochastic effect, the authors in [11] established a fractional-order stochastic dynamical model for COVID-19 to describe the second virus wave behaviour in Egypt. In [12], the authors constructed a stochastic mathematical model to investigate virus temporal dynamics in Oman. They divided the total population into four classes and considered vaccine effects in their model.

Some research papers studied COVID-19 in Saudi Arabia, such as, in [13] the authors conducted a detailed global analysis of new modelling dynamics for COVID-19 data. In [14], the authors made predictions about COVID-19 dynamics in Saudi Arabia's first virus wave using the SEIR model. They found that the number of infected cases will peak by 22nd May 2020, and the cumulative number of infected patients is predicted to reach 70,321. In [15], a Susceptible – Exposed – Symptomatic – Asymptomatic – Hospitalized – Recovered dynamical model was formulated to characterize the transmission of COVID-19 in Saudi Arabia. Then the authors established the critical analysis for model positivity, boundedness, and stability. In [16], the authors evaluated the COVID-19 spread in Saudi Arabia through modelling, simulation, and analysis. In [17], the authors established a modified multi-stage SEIR model to predict the probable new COVID-19 wave in Saudi Arabia after lifting all travel restrictions on 3rd January 2021. In [18], the authors extended the SEIR model by considering a fixed value of daily vaccinations to simulate the novel COVID-19 wave spreading in Saudi Arabia.

In this work, we predict and evaluate the impact of the second pandemic wave of COVID-19 on Saudi Arabia by developing two dynamic models, the fractional-order dynamic model, and the fractional-order-stochastic dynamic model. In both models, different vaccination strategies are examined to limit virus transmission. The optimal strategy for first and second doses is obtained using the invasive weed optimization algorithm to minimize cumulative wave infections.

This article is prepared as follows. In Section 2, we introduce our dynamical models and their assumptions. In Section 3, we describe numerical algorithms to solve the considered models. In Section 4, the models are tested for the Saudi Arabia case study under variable vaccination scenarios. In the end, the paper is concluded in Section 5.

Materials and methods

Based on several plans carried out by the Saudi Arabian government to fight against COVID-19 at different levels, we generalize the susceptible – exposed – infectious – recovered – deaths (SEIRF) model in [11] to describe virus transmission. The expanded models consider the first and second vaccination doses. We divided the total population (N) into seven classes, implied by (S), (V₁), (V₂), (E), (I), (R), and (F), where (S) is the daily susceptible individuals; (V₁) and (V₂) are the cumulative first and second vaccinated people respectively. (E) is the daily exposed, and (I) is the daily infected individuals. Finally, (R) and (F) are the daily recovered and deaths.

In order to obtain the required results, we impose the following assumptions:

• (1)The constants, variables, and parameters are taken as positive values.
• (2)All transmission rates vary with time and have no consideration of natural birth and death rates.
• (3)The total population is subdivided into seven groups.
• (4)Symptomatic individuals mainly form the infected group and represent the major source of disease transmission.
• (5)Susceptibles can move into the infected class without passing through the exposed class.
• (6)With a very low probability, exposed people may take the first or the second vaccination dose.
• (7)First and second vaccinated individuals can transfer to the exposed and infected class.
• (8)First vaccinated individuals can't move into the recovered class without taking the second dose.
• (9)The second vaccination dose will result in permanent immunity for 92% of the fully vaccinated people.

COVID-19 dynamics are modelled with two dynamic mathematical models, which are:

• •Fractional-order SV₁V₂EIRF dynamic model.
• •Fractional-order-stochastic SV₁V₂EIRF dynamic model.

Factional-order SV₁V₂EIRF dynamic model

We used the definition of Caputo fractional derivative of order α [21], 0 < α < 1, and applied it to the SV₁V₂EIRF model in Eqs. (1), (2), (3), (4), (5), (6), (7). According to distinct α values, we can obtain different viral wave peaks to describe virus dynamics in multiple scenarios. As indicated in Fig. 1 , daily susceptible people (S) enter the exposed class (E) due to contact between exposed individuals and infected individuals. Susceptible individuals move into the exposed class with a daily transmission rate β₁(t) and into the infected class with a daily transmission rate β₂(t). The daily first vaccinated people (v_1t) are withdrawn from the susceptible class with probability (p₁) and from the exposed with a low probability (p₁^c). The second daily vaccinated people (v_2t) are pulled from the susceptible class with a probability (p₂) and the exposed class with a low probability (p₂^c). First vaccinated people (V₁) enter the exposed class with a transmission rate of (1-h₁)β₂(t) and the infected class with a transmission rate of (1-h₁)β₁(t); where h₁ is the first vaccine efficiency. Second vaccinated people (V₂) enter the exposed class with a transmission rate of (1-h₂)β₂(t) and the infected class with a transmission rate of (1-h₂)β₁(t); where h₂ is the second vaccine efficiency. Second daily vaccinated people (v_2t) are transformed into the recovered class at a rate of (h₂/a); where a is the number of days to make sure that second daily vaccinated people are recovered. The exposed people (E) enter the infected class (I) at a daily transmission rate α_E(t). A part of infected individuals move to the recovered class (R) with a cure rate ɤ(t), and the residual move to the deaths class (F) with a death rate μ(t).

$$D_t^{\alpha }S=-\frac{{\beta }_1(t)c_1}{N}I(t)S(t)-\frac{{\beta }_2(t)c_2}{N}E(t)S(t)-p_1{v}_{1t}$$ (1)

$$D_t^{\alpha }{V}_1=v_{1t}-p_2{v}_{2t}-\frac{(1-h_1){\beta }_1(t)c_3}{N}I(t)V_1(t)-\frac{(1-h_1){\beta }_2(t)c_4}{N}E(t)V_1(t)$$ (2)

$$D_t^{\alpha }{V}_2=v_{2t}-\frac{(1-h_2){\beta }_1(t)c_5}{N}I(t)V_2(t)-\frac{(1-h_2){\beta }_2(t)c_6}{N}E(t)V_2(t)-\frac{h_2}{a}{v}_{2t}$$ (3)

$$D_t^{\alpha }E=\frac{{\beta }_1(t)c_1}{N}I(t)S(t)+\frac{{\beta }_2(t)c_2}{N}E(t)S(t)+\frac{(1-h_1){\beta }_2(t)c_4}{N}E(t)V_1(t)+\frac{(1-h_2){\beta }_2(t)c_6}{N}E(t)V_2(t)-{\alpha }_E(t)E(t)-p_1^c{v}_{1t}-p_2^c{v}_{2t}$$ (4)

$$D_t^{\alpha }I={\alpha }_E(t)E(t)-\gamma (t)I(t)-\mu (t)I(t)+\frac{(1-h_1){\beta }_1(t)c_3}{N}I(t)V_1(t)+\frac{(1-h_2){\beta }_1(t)c_5}{N}I(t)V_2(t)$$ (5)

$$D_t^{\alpha }R=\gamma (t)I(t)+\frac{h_2}{a}{v}_{2t}$$ (6)

$$D_t^{\alpha }F=\mu (t)I(t)$$ (7)

where c₁ is the mean number of near contacts between the susceptible class and the infected class per day, and c₂ is the mean number of near contacts between the susceptible class and the exposed class per day. c₃ and c₄ are the mean numbers of near contacts between the first vaccinated class and both the infected and exposed classes respectively per day. c₅ and c₆ are the mean numbers of near contacts between the second vaccinated class and both the infected and exposed classes respectively per day. The assumed model dynamic rates are defined as in Eq. (8). β₁(t) and β₂(t) are assumed to have initial constant values β₁₀ and β₂₀ respectively. After the incubation period t_o, β₁(t) and β₂(t) start to exponentially decrease with time through the controlling parameter k which depends on the country's precautionary measures and how these precautions are tracked. The transmission rate α_E(t) from the exposed to infected is assumed to exponentially decay with time with a controlling parameter α₁ and an initial value α₀. The higher value of α₁ means the country is better at controlling virus transmission and can more limit the number of infections. The cure rate ɤ(t) increases with time as a result of following the treatment protocols and providing the necessary medical care for infected people. It has a controlling parameter ɤ₁ and an initial value ɤ₀. The death rate μ(t) has an initial value μ₀ and then assumed exponentially decreases with time with a controlling parameter μ₁ as the infection rate is decreasing and the recovery rate is increasing.

$$\begin{gathered} {\beta }_1\left(t\right)=\left\{\begin{array}{c}{\beta }_{10},t⩽t_0\\ {\beta }_{10}{e}^{-k(t-t_0)},t>t_0\end{array}\right),{\beta }_2\left(t\right)=\left\{\begin{array}{c}{\beta }_{20},t⩽t_0\\ {\beta }_{20}{e}^{-k(t-t_0)},t>t_0\end{array}\right), \\ {\alpha }_E\left(t\right)=a_0{e}^{-{\alpha }_1\left(t\right)},\gamma \left(t\right)={\gamma }_0{e}^{{\gamma }_{1}t},\mu \left(t\right)={\mu }_0{e}^{-{\mu }_1\left(t\right)} \end{gathered}$$ (8)

Fig. 1.

SV₁V₂EIRF dynamic model structure.

Fractional-order-stochastic SV₁V₂EIRF dynamic model

The fact behind the motivation for generating the fractional-order-stochastic dynamic model is the stochastic nature of the COVID-19 dynamics, which differs everywhere and has unpredictable characteristics. We start by adding white noise terms satisfying Wiener process (B_t) properties to the fractional-order (SV₁V₂EIRF) dynamic model to generate the proposed stochastic model as considered in Eqs. (9), (10), (11), (12), (13), (14), (15). The newly added diffusion terms are used to reach more probabilistic positions covering a wide range of probable viral wave dynamics. Two positive diffusion coefficients are assumed to study how the seven classes are stochastically affected by each other. The first diffusion coefficient is the exposed-infected-vaccinated coefficient (σ_EIV) which measures the probabilistic effect of exposed, infected, vaccinated, and susceptible individuals on each other. As vaccinated people are divided into two classes, the first vaccinated and the second vaccinated class, the σ_EIV is stochastically affecting them through the two assumed constant weights b₁ and b₂ respectively. The second diffusion coefficient is the infected-recovered coefficient (σ_IR) which measures the stochastic diffusion effect of the recovered, deaths, and infected individuals on each other. ρ₁ and ρ₂ are constant weights that describe the partial effect of the σ_IR on the recovered and deaths class respectively.

$$D_t^{\alpha }S=-\frac{{\beta }_1(t)c_1}{N}I(t)S(t)-\frac{{\beta }_2(t)c_2}{N}E(t)S(t)-p_1{v}_{1t}-{\sigma }_{\mathit{EIV}}I(t)E(t)V_1(t)\frac{d{B}_t}{\mathit{dt}}$$ (9)

$$D_t^{\alpha }{V}_1=v_{1t}-p_2{v}_{2t}-\frac{(1-h_1){\beta }_1(t)c_3}{N}I(t)V_1(t)-\frac{(1-h_1){\beta }_2(t)c_4}{N}E(t)V_1(t){\text{+ (b}}_1{\sigma }_{\mathit{EIV}}I(t)E(t)V_1(t)-{\sigma }_{\mathit{EIV}}I(t)E(t)V_2(t)\text{)}\frac{d{B}_t}{\mathit{dt}}$$ (10)

$$D_t^{\alpha }{V}_2=v_{2t}-\frac{(1-h_2){\beta }_1(t)c_5}{N}I(t)V_2(t)-\frac{(1-h_2){\beta }_2(t)c_6}{N}E(t)V_2(t)-\frac{h_2}{a}{v}_{2t}{\text{+ b}}_1{\sigma }_{\mathit{EIV}}I(t)E(t)V_2(t)\frac{d{B}_t}{\mathit{dt}}$$ (11)

$$D_t^{\alpha }E=\frac{{\beta }_1(t)c_1}{N}I(t)S(t)+\frac{{\beta }_2(t)c_2}{N}E(t)S(t)+\frac{(1-h_1){\beta }_2(t)c_4}{N}E(t)V_1(t)+\frac{(1-h_2){\beta }_2(t)c_6}{N}E(t)V_2(t)-{\alpha }_E(t)E(t)-p_1^c{v}_{1t}-p_2^c{v}_{2t}{\text{+ b}}_2{\sigma }_{\mathit{EIV}}I(t)E(t)V_1(t)\frac{d{B}_t}{\mathit{dt}}$$ (12)

$$D_t^{\alpha }I={\alpha }_E(t)E(t)-\gamma (t)I(t)-\mu (t)I(t)+\frac{(1-h_1){\beta }_1(t)c_3}{N}I(t)V_1(t)+\frac{(1-h_2){\beta }_1(t)c_5}{N}I(t)V_2(t)-{\sigma }_{\mathit{IR}}I(t)R(t)\frac{d{B}_t}{\mathit{dt}}$$ (13)

$$D_t^{\alpha }R=\gamma (t)I(t)+\frac{h_2}{a}{v}_{2t}+{\rho }_1{\sigma }_{\mathit{IR}}I(t)R(t)\frac{d{B}_t}{\mathit{dt}}$$ (14)

$$D_t^{\alpha }F=\mu (t)I(t)+{\rho }_2{\sigma }_{\mathit{IR}}I(t)R(t)\frac{d{B}_t}{\mathit{dt}}$$ (15)

Vaccination scenarios

Four vaccination scenarios have been established in the first 100 days of the second viral wave, which is expected to begin on 1st March 2021. The second daily vaccinated people (v_2t) are assumed to be one fourth of the daily first vaccinated people (v_1t) in scenarios 1,2 and 3. Using the real vaccination data of Saudi Arabia from 1st March 2021 (day 1) to 19th April 2021 (day 50) [19], [20], approximately 6.25 million vaccine doses were used by the government during that period. We assume another 6.25 million doses are available, with the same random vaccination strategy used by the government from day 1 to day 50, and will be used from day 51 to day 100 to limit daily infections and deaths. The assumed scenarios, which are compared to the Saudi Arabian government's real scenario, are as followed:

• oScenario 1: The first and second daily vaccinations are at a fixed level for 100 days, beginning on day 1 and ending on day 100. The fixed level of the first and second daily vaccinations is equal to 100,000 and 25000, respectively. As a result, the total number of immunization doses given over the span of 100 days is 12.5 million.
• oScenario 2: The first and second daily vaccines decrease gradually from day 1 to day 100. The starting value of the first vaccination is 150,000 doses, which gradually drops by 1000 doses every day. The second daily vaccinated people on day 1 is 37,500 and gradually drops by 250 doses every day. The total number of immunization doses administered over 100 days also equals 12.5 million.
• oScenario 3: The first and second daily vaccinations are increased for 100 days, starting from day 1 to day 100. The initial value of the first daily vaccinated people equals 50,000 doses and increases gradually by 1000 doses daily. The initial value of second daily vaccinated people equals 12,500 and increases gradually by 250 doses daily. So, the total vaccination doses taken for the whole 100 days equals 12.5 million doses.
• oScenario 4: The first and second daily vaccinated people are optimally selected for 100 days, beginning on day 1 and ending on day 100, using the invasive wed optimization algorithm (IWO) [22] to reduce the cumulative number of infections for the full-wave period. The used objective function to minimize the cumulative wave infections for 100 days at second wave beginning is:

$$\text{Minimize}:\sum _{t=1}^{t=100}I\left(t\right)$$ (16)

Subject to: $50000⩽v_{1t}⩽140000,15000⩽v_{2t}⩽35000,$ ${\sum }_{t=1}^{t=100}(v_{1t}+v_{2t})⩽12500000.$

Numerical algorithms

The numerical techniques used in this article are introduced in this section. Firstly, the fractional Euler method [10], [23], used to solve the fractional-order model, is described. Then, we introduced the Euler-Maruyama method [24], [25], [26] which is used to solve the fractional-order-stochastic model. In the end, the optimization algorithm used for optimizing vaccination scenario 4 is presented.

Fractional Euler method

The fractional-order SV₁V₂EIRF dynamic model of order α, obtained in Eqs. (1), (2), (3), (4), (5), (6), (7), is a fractional-order system of differential equations and has the general form indicated in Eq. (17). We used the fractional Euler method [10], [23], preseneted in Eq. (18), to solve the fractional-order system numerically.

$$D_t^{\alpha }\mathit{X}=f(\mathit{X}(t)),{\mathit{X}}_0\text{=}(S_0,V_{10},V_{20}....,F_0)$$ (17)

$$\mathit{X}(t_{i+1})=\mathit{X}(t_i)+\frac{f(\mathit{X}(t_i))}{\mathrm{\Gamma }(\alpha +1)}{(\mathrm{\Delta }t)}^{\alpha }$$ (18)

where $\mathit{X}=(S,V_1,V_2....,F)$ is the dynamic system state vector and Г is the gamma function. The time variable (t) is discretized by j time spacings and a time step Δt. The system solution convergency is checked after j discretizations through the following equation:

$$S^j+v_{1t}^j+v_{2t}^j+E^j+I^j+R^j+F^j=N$$ (19)

Euler-Maruyama method

The fractional-order- stochastic SV₁V₂EIRF model of order α, obtained in Eqs. (9), (10), (11), (12), (13), (14), (15), is a fractional-order stochastic system of differential equations and has the general form indicated in Eq. (20). We can solve Eq. (20) numerically using the iterative formula of the Euler-Maruyama method for $t\in (t_0,T]$ as indicated in Eq. (21) [27].

$$D_t^{\alpha }{X}_i\left(t\right)=b_i(t,\mathit{X}\left(t\right))+{\sigma }_i(t,\mathit{X}\left(t\right))\frac{d{B}_t}{dt},\mathit{X}\left(t_0\right)\text{=}(S_0,V_{10},V_{20}....,F_0),i=1,2,....,7.$$ (20)

$$\begin{gathered} X_i^{\left(n\right)}\left(t\right)=X_{i_0}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{t_0}^t\frac{b_i({\tau }_n\left(s\right),{\mathit{X}}^{\left(n\right)}\left({\tau }_n\left(s\right)\right))}{{(t-s)}^{1-\alpha }}ds+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{t_0}^t\frac{{\sigma }_i({\tau }_n\left(s\right),{\mathit{X}}^{\left(n\right)}\left({\tau }_n\left(s\right)\right))}{{({\rho }_n\left(t\right)-{\tau }_n\left(s\right))}^{1-\alpha }}d{B}_t, \\ {\tau }_n\left(s\right)=\frac{kT}{n}\text{and}{\rho }_n\left(s\right)=\frac{(k+1)T}{n} \\ \text{for}s\in \text{(}\frac{kT}{n},\frac{(k+1)T}{n}],k=0,1,...,n-1 \end{gathered}$$ (21)

where b_i and σ_i are the drift and diffusion functions respectively, while n belongs to the set of positive integer numbers. This equation can be solved step by step at each interval $(\frac{\mathit{kT}}{n},\frac{(k+1)T}{n}]$.

Invasive weed optimization algorithm

Starting on 1st March 2021, the invasive weed optimization algorithm (IWO) will be used to select the optimal values for both first and second daily vaccinated people in Saudi Arabia for 100 days at the second wave beginning. The IWO sequential stages are [22]:

• -Populations Initialization: Seeds with finite numbers are being despread over d-dimensional search space with random positions.
• -Seeds Reproduction: Every seed grows to form a new plant and produces a newer number of seeds.
• -Spatial Dispersal: Random selections of each plant seeds and adaptations to the algorithm are made in this part. The new seeds are being randomly scattered over the search space. The standard deviation of the standard normal random functions will be produced by an initial value (σ_initial) and a final value (σ_final) at every step. A nonlinear alteration modulation index (o) is selected to reach a certain satisfactory performance during simulation.
• -Competitive Exclusion: After maximum number of plants is reached, only the plants with lower fitness can pull out and produce seeds. The process continues in each iteration till maximum iterations is reached. Fig. 2 indicates the IWO flow chart.

Fig. 2.

IWO algorithm flow chart.

Results and discussion

The fractional-order and fractional-order-stochastic models of orders 0.96, 0.98, and 1 are applied to the Saudi Arabia case study to estimate realistic scenarios for each class peak and time during the second wave. We used the actual data for Saudi Arabia from Refs. [19], [20] to get the initial seven class values. Our models' estimated parameters are obtained using trial and error for the first wave transmission rates. All models' transmission rates and estimated coefficients for the second viral wave in Saudi Arabia are obtained as indicated in Table 1 . On 1st March 2021 (day 1), the initial system values for the daily infected, recovered, and deaths were 317, 150, and 6 [19], [20] while Saudi Arabia's total population (N) reached 34.82 million in February 2021. We assumed the initial value of the exposed individuals was 700 people and no vaccination occurred before day 1. The fractional-order SV₁V₂EIRF dynamic model is numerically solved using the fractional Euler's method with a time step of 0.001 days. The average path solution is numerically computed using the Euler-Murayama method with 10 paths and temporal discretization equals 0.001 days for the fractional-order-stochastic SV₁V₂EIRF dynamic model. MATLAB program version 2018b is used for the proposed models' simulation. At first, our two proposed models with the suggested fractional orders are simulated using actual daily vaccination data [19], [20], which is valid for 50 days and assumed repeated for another 50 days. Fig. 3, Fig. 4 indicate the used data for first and second daily doses with actual values in Appendices A and B.

Table 1.

Dynamic SV₁V₂EIRF models estimated parameters for Saudi Arabia.

Parameter	Estimated value	Parameter	Estimated value
β10	0.019	h2	0.92
β20	0.014	a	5
k	0.09	α0	0.015
to	14	α1	0.005
c1	4.1 × 10-6	ɤ0	0.01
c2	6.8	ɤ1	0.001
c3	3	μ0	0.000625
c4	1.5	μ1	0.015
c5	1.5	σEIV	5 × 10-8
c6	1	σIR	9.7 × 10-8
p1	0.0002	b1	0.9
p2	0.9998	b2	0.001
p1c	0.0001	ρ1	0.995
p2c	0.9999	ρ2	0.005
h1	0.5

Fig. 3.

First daily vaccination scenarios and actual one for 100 days.

Fig. 4.

Second daily vaccination scenarios and actual one for 100 days.

The depicted results for the daily estimated infections and deaths using the proposed models of orders 0.96, 0.98, and 1 are as shown in Fig. 5, Fig. 6 The actual daily confirmed infections and deaths [20] are also presented in Fig. 5, Fig. 6 respectively to compare them with their estimated values. Through comparison, the fractional-stochastic model of order 1 and then the fractional model of order 1 are the most accurate dynamic models for describing the actual data of both daily infection and daily deaths. After that came the fractional-stochastic model of order 0.98 and after that came the fractional model of order 0.98. The worst models are the fractional-stochastic model and the fractional model of order 0.96, as they give a very high overestimation of daily infections and deaths. Because of the high accuracy of the fractional-stochastic model and the fractional model of order 1, they will be re-simulated using the four assumed vaccination scenarios shown in Fig. 3, Fig. 4.

Fig. 5.

Daily infected individuals for Saudi Arabia using different dynamical models.

Fig. 6.

Daily deaths individuals for Saudi Arabia using different dynamical models.

Fig. 7, Fig. 8, Fig. 9, Fig. 10 indicate the estimated virus dynamics after applying the assumed vaccine scenarios to the fractional and the fractional-stochastic models of order 1. In the case of the daily predicted susceptible class, Fig. 7 depicts the class variation over time using various validation scenarios. The daily susceptible class decreases with time until forecasting day 100, and then it stabilizes with time at a fixed value. That occurs because the daily vaccination doses are removed from the system after 100 days. From the results, scenario 4 is the best vaccination scenario with the minimum estimated daily stabilization value. The obtained first and second daily doses for scenario 4 using the IWO are indicated in Appendices A and B. Vaccination scenario 3 represents a good scenario and came after vaccination scenario 4, while the worst vaccination scenario is scenario 2. In Fig. 8, the estimated daily exposed individuals are presented. Through variating the vaccination scenarios, different exposed class peaks are reached. The best scenario with the minimum peak is the optimized scenario 4 and after that came scenario 3. The exposed class peak using scenario 2 gives about double the peak value using the optimized scenario 4. The total number of first and second vaccination doses used for 100 days by scenario 4 equals 11.95 million, while each of the remaining three scenarios used 12.5 million doses. That shows how the optimized scenario 4 effectively controls virus dynamics and minimizes the number of daily exposed individuals with a 4.4% lower number of used vaccines than other assumed scenarios.

Fig. 7.

Daily susceptible individuals for Saudi Arabia using different vaccination scenarios.

Fig. 8.

Daily exposed individuals for Saudi Arabia using different vaccination scenarios.

Fig. 9.

Daily infected individuals for Saudi Arabia using different vaccination scenarios.

Fig. 10.

Daily deaths individuals for Saudi Arabia using different vaccination scenarios.

Fig. 9, Fig. 10 depict the estimated daily infections and deaths respectively using the four scenarios. The actual daily confirmed infections and deaths in Saudi Arabia [20] are also presented on the same graphs with their estimated curves. The estimated daily infections' peak reaches 1186 cases on day 65 (4th May 2021) when scenario 1 is applied to the fractional model. Applying the same scenario to the fractional-stochastic model gives a peak value of 1438 on day 79 (18th May 2021). In the case of scenario 2 usage with the fractional model, it makes the estimated daily infections' peak reach 1485 cases on day 69 (8th May 2021). A peak of 2080 cases on day 107 (18th May 2021) was reached when applying the same scenario to the fractional-stochastic model. Using scenario 3, the daily infections' peak reached 906 cases on day 60 (29th April 2021) for the fractional model and a peak of 1295 cases on day 59 (28th April 2021) for the fractional-stochastic model. So, this scenario is more controlled daily infections than scenario 1 and 2. The minimum daily estimated infections was reached using scenario 4. The estimated daily infected class peak will not exceed 840 cases on day 48 (17th April 2021) when applying scenario 4 to the fractional model of order 1. Using the same scenario on the fractional-stochastic model results in a peak of 976 on day 60 (29th April 2021). The optimized scenario 4 is the best vaccination strategy compared to the current random daily vaccination started by the government. From the real data of daily confirmed infections in Fig. 9, it broke the barrier of 993 cases on day 53 (22nd April 2021) and continued increasing with days. If scenario 4 is used, it will limit the virus outbreak with a 4.4% lower number of used vaccination doses. Scenario 3 is also more controlling of daily infections than the current random strategy started by the Saudi government and came after scenario 4.

Conclusions

This paper clarified two dynamic models to evaluate the COVID-19 pandemic using first and second vaccination doses. The fractional-order dynamical model and the fractional-order-stochastic dynamical model have been considered. Through the generated dynamic models, the daily susceptible (S), cumulative first daily vaccinations (V₁), cumulative second daily vaccinations (V₂), exposed (E), infected (I), recovered (R), and deaths (F) can be predicted. The two proposed models were applied to the case study of Saudi Arabia to evaluate the second virus wave outbreak. Four different scenarios for the first and second vaccination doses were assumed to control virus outbreaks for 100 days at the beginning of the wave and compared with the actual random scenario started by the government. From the results of proposed models, the predicted daily infection curves were realistic when compared to the actual daily confirmed infections. Finally, the 4th scenario is the optimal suggested vaccination strategy for the Saudi government compared to the actual random procedure started by the government. It shows its superiority in mitigating and reducing daily and cumulative infections during the second COVID-19 epidemic in the coming months with a 4.4% lower use of the valid immunization doses. In our future work, we will study the effects of different vaccinations on people.

CRediT authorship contribution statement

Othman A.M. Omar: Methodology, Software, Data curation, Formal analysis, Writing - original draft, Writing - review & editing. Yousef Alnafisah: Software, Formal analysis, Supervision, Writing - review & editing. Reda A. Elbarkouky: Conceptualization, Formal analysis, Supervision, Writing - review & editing. Hamdy M. Ahmed: Investigation, Conceptualization, Supervision, Writing - review & editing.

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.

Appendix A. First daily vaccination individuals using the best two scenarios compared to the actual one

Day no.	Date	First daily vaccination doses			Day no.	Date	First daily vaccination doses
		Scenario 3	Scenario 4	Actual			Scenario 3	Scenario 4	Actual
1	1 March 2021	149,000	138,605	28,422	51	20 April 2021	99,000	64,828	28,422
2	2 March 2021	148,000	88,761	32,263	52	21 April 2021	98,000	127,449	32,263
3	3 March 2021	147,000	57,683	43,123	53	22 April 2021	97,000	70,761	43,123
4	4 March 2021	146,000	124,073	59,403	54	23 April 2021	96,000	50,411	59,403
5	5 March 2021	145,000	120,980	66,995	55	24 April 2021	95,000	55,849	66,995
6	6 March 2021	144,000	138,459	63,072	56	25 April 2021	94,000	52,713	63,072
7	7 March 2021	143,000	112,811	59,151	57	26 April 2021	93,000	64,714	59,151
8	8 March 2021	142,000	113,878	69,846	58	27 April 2021	92,000	69,839	69,846
9	9 March 2021	141,000	135,462	80,953	59	28 April 2021	91,000	75,645	80,953
10	10 March 2021	140,000	122,406	73,134	60	29 April 2021	90,000	84,561	73,134
11	11 March 2021	139,000	106,959	67,478	61	30 April 2021	89,000	88,838	67,478
12	12 March 2021	138,000	110,662	79,538	62	1 May 2021	88,000	139,923	79,538
13	13 March 2021	137,000	105,818	83,997	63	2 May 2021	87,000	53,624	83,997
14	14 March 2021	136,000	57,173	96,360	64	3 May 2021	86,000	123,086	96,360
15	15 March 2021	135,000	69,836	91,266	65	4 May 2021	85,000	85,568	91,266
16	16 March 2021	134,000	137,893	86,765	66	5 May 2021	84,000	138,876	86,765
17	17 March 2021	133,000	103,401	95,544	67	6 May 2021	83,000	56,071	95,544
18	18 March 2021	132,000	125,986	93,034	68	7 May 2021	82,000	105,041	93,034
19	19 March 2021	131,000	108,027	79,437	69	8 May 2021	81,000	95,451	79,437
20	20 March 2021	130,000	75,170	98,252	70	9 May 2021	80,000	83,184	98,252
21	21 March 2021	129,000	103,904	102,537	71	10 May 2021	79,000	119,744	102,537
22	22 March 2021	128,000	55,573	106,754	72	11 May 2021	78,000	56,602	106,754
23	23 March 2021	127,000	103,669	111,976	73	12 May 2021	77,000	125,810	111,976
24	24 March 2021	126,000	104,913	114,559	74	13 May 2021	76,000	66,321	114,559
25	25 March 2021	125,000	100,107	119,019	75	14 May 2021	75,000	96,919	119,019
26	26 March 2021	124,000	72,478	122,004	76	15 May 2021	74,000	111,256	122,004
27	27 March 2021	123,000	95,214	106,644	77	16 May 2021	73,000	66,508	106,644
28	28 March 2021	122,000	123,311	100,322	78	17 May 2021	72,000	134,335	100,322
29	29 March 2021	121,000	54,155	97,332	79	18 May 2021	71,000	133,289	97,332
30	30 March 2021	120,000	58,565	88,192	80	19 May 2021	70,000	91,121	88,192
31	31 March 2021	119,000	118,212	82,300	81	20 May 2021	69,000	94,323	82,300
32	1 April 2021	118,000	53,167	82,337	82	21 May 2021	68,000	80,385	82,337
33	2 April 2021	117,000	51,907	89,463	83	22 May 2021	67,000	94,205	89,463
34	3 April 2021	116,000	120,578	93,935	84	23 May 2021	66,000	69,867	93,935
35	4 April 2021	115,000	55,163	93,119	85	24 May 2021	65,000	57,140	93,119
36	5 April 2021	114,000	61,904	93,538	86	25 May 2021	64,000	106,281	93,538
37	6 April 2021	113,000	82,817	106,041	87	26 May 2021	63,000	139,149	106,041
38	7 April 2021	112,000	61,818	123,458	88	27 May 2021	62,000	83,946	123,458
39	8 April 2021	111,000	133,457	126,275	89	28 May 2021	61,000	61,621	126,275
40	9 April 2021	110,000	139,522	134,975	90	29 May 2021	60,000	133,054	134,975
41	10 April 2021	109,000	123,423	134,002	91	30 May 2021	59,000	95,045	134,002
42	11 April 2021	108,000	54,791	137,869	92	31 May 2021	58,000	63,118	137,869
43	12 April 2021	107,000	102,500	137,879	93	1 June 2021	57,000	93,274	137,879
44	13 April 2021	106,000	116,784	127,619	94	2 June 2021	56,000	76,797	127,619
45	14 April 2021	105,000	102,556	108,826	95	3 June 2021	55,000	139,539	108,826
46	15 April 2021	104,000	136,991	100,742	96	4 June 2021	54,000	64,144	100,742
47	16 April 2021	103,000	100,803	87,707	97	5 June 2021	53,000	68,006	87,707
48	17 April 2021	102,000	113,385	89,529	98	6 June 2021	52,000	130,635	89,529
49	18 April 2021	101,000	90,460	91,640	99	7 June 2021	51,000	136,411	91,640
50	19 April 2021	100,000	71,011	96,697	100	8 June 2021	50,000	121,494	96,697

Appendix B. Secondly daily vaccination individuals using the best two scenarios compared to the actual one.

Day	Date	Second daily vaccination doses			Day no.	Date	Second daily vaccination doses
no.		Scenario 3	Scenario 4	Actual			Scenario 3	Scenario 4	Actual
1	1 March 2021	37,250	24,711	9474	51	20 April 2021	24,750	34,735	9474
2	2 March 2021	37,000	28,627	10,754	52	21 April 2021	24,500	20,382	10,754
3	3 March 2021	36,750	24,013	14,374	53	22 April 2021	24,250	32,321	14,374
4	4 March 2021	36,500	16,706	19,801	54	23 April 2021	24,000	34,026	19,801
5	5 March 2021	36,250	32,057	22,331	55	24 April 2021	23,750	29,000	22,331
6	6 March 2021	36,000	22,374	21,024	56	25 April 2021	23,500	18,236	21,024
7	7 March 2021	35,750	20,432	19,717	57	26 April 2021	23,250	32,003	19,717
8	8 March 2021	35,500	19,228	23,282	58	27 April 2021	23,000	29,287	23,282
9	9 March 2021	35,250	15,112	26,984	59	28 April 2021	22,750	19,493	26,984
10	10 March 2021	35,000	30,013	24,378	60	29 April 2021	22,500	18,944	24,378
11	11 March 2021	34,750	19,113	22,492	61	30 April 2021	22,250	26,506	22,492
12	12 March 2021	34,500	31,558	26,512	62	1 May 2021	22,000	33,415	26,512
13	13 March 2021	34,250	30,923	27,999	63	2 May 2021	21,750	26,664	27,999
14	14 March 2021	34,000	20,126	32,120	64	3 May 2021	21,500	21,285	32,120
15	15 March 2021	33,750	23,132	30,422	65	4 May 2021	21,250	23,621	30,422
16	16 March 2021	33,500	32,491	28,921	66	5 May 2021	21,000	15,040	28,921
17	17 March 2021	33,250	19,601	31,848	67	6 May 2021	20,750	20,310	31,848
18	18 March 2021	33,000	22,935	31,011	68	7 May 2021	20,500	15,358	31,011
19	19 March 2021	32,750	31,448	26,479	69	8 May 2021	20,250	17,593	26,479
20	20 March 2021	32,500	16,700	32,750	70	9 May 2021	20,000	28,419	32,750
21	21 March 2021	32,250	30,498	34,179	71	10 May 2021	19,750	31,832	34,179
22	22 March 2021	32,000	22,660	35,584	72	11 May 2021	19,500	29,950	35,584
23	23 March 2021	31,750	17,094	37,325	73	12 May 2021	19,250	24,986	37,325
24	24 March 2021	31,500	22,994	38,186	74	13 May 2021	19,000	34,282	38,186
25	25 March 2021	31,250	26,827	39,673	75	14 May 2021	18,750	17,485	39,673
26	26 March 2021	31,000	28,813	40,668	76	15 May 2021	18,500	22,701	40,668
27	27 March 2021	30,750	28,861	35,548	77	16 May 2021	18,250	33,890	35,548
28	28 March 2021	30,500	22,747	33,440	78	17 May 2021	18,000	25,626	33,440
29	29 March 2021	30,250	28,523	32,444	79	18 May 2021	17,750	15,283	32,444
30	30 March 2021	30,000	15,840	29,397	80	19 May 2021	17,500	25,813	29,397
31	31 March 2021	29,750	27,763	27,433	81	20 May 2021	17,250	22,392	27,433
32	1 April 2021	29,500	18,719	27,445	82	21 May 2021	17,000	23,808	27,445
33	2 April 2021	29,250	19,748	29,821	83	22 May 2021	16,750	26,974	29,821
34	3 April 2021	29,000	27,082	31,311	84	23 May 2021	16,500	34,869	31,311
35	4 April 2021	28,750	21,791	31,039	85	24 May 2021	16,250	34,581	31,039
36	5 April 2021	28,500	20,994	31,179	86	25 May 2021	16,000	18,586	31,179
37	6 April 2021	28,250	22,704	35,347	87	26 May 2021	15,750	16,274	35,347
38	7 April 2021	28,000	19,860	41,152	88	27 May 2021	15,500	22,953	41,152
39	8 April 2021	27,750	19,044	42,091	89	28 May 2021	15,250	15,778	42,091
40	9 April 2021	27,500	28,734	44,991	90	29 May 2021	15,000	21,633	44,991
41	10 April 2021	27,250	25,703	44,667	91	30 May 2021	14,750	33,504	44,667
42	11 April 2021	27,000	25,499	45,956	92	31 May 2021	14,500	19,404	45,956
43	12 April 2021	26,750	28,411	45,959	93	1 June 2021	14,250	29,099	45,959
44	13 April 2021	26,500	22,084	42,539	94	2 June 2021	14,000	32,478	42,539
45	14 April 2021	26,250	19,953	36,275	95	3 June 2021	13,750	20,727	36,275
46	15 April 2021	26,000	24,237	33,580	96	4 June 2021	13,500	19,406	33,580
47	16 April 2021	25,750	25,649	29,235	97	5 June 2021	13,250	22,948	29,235
48	17 April 2021	25,500	29,585	29,843	98	6 June 2021	13,000	16,856	29,843
49	18 April 2021	25,250	19,903	30,546	99	7 June 2021	12,750	34,463	30,546
50	19 April 2021	25,000	22,463	32,232	100	8 June 2021	12,500	20,798	32,232