key: cord-1036923-l55kk33p
authors: Omar, Othman A.M.; Elbarkouky, Reda A.; Ahmed, Hamdy M.
title: Fractional Stochastic Modelling of COVID-19 under Wide Spread of Vaccinations: Egyptian Case Study
date: 2022-02-07
journal: Alexandria Engineering Journal
DOI: 10.1016/j.aej.2022.02.002
sha: b6331154eb18159c5e50532190b0d14d9e8b2286
doc_id: 1036923
cord_uid: l55kk33p

This work predicts the dynamics of the COVID-19 under widespread vaccination to anticipate the virus's current and future waves. We focused on establishing two population-based models for predictions: the fractional-order model and the fractional-order stochastic model. Based on dose efficacy, which is one of the main imposed assumptions in our study, some vaccinated people will probably be exposed to infection by the same viral wave. We validated the generated models by applying them to the current viral wave in Egypt. We assumed that the Egyptian current wave began on 10th September 2021. Using current actual data and varying our models’ fractional orders, we generate different predicted wave scenarios. The numerical solution of our models is obtained using the fractional Euler method and the fractional Euler Maruyama method. At the end, we compared the current predicted wave under a high vaccination rate with the previous viral wave. Through this comparison, the vaccination control effect is quantified.

All countries are moving as fast as possible to vaccinate their population to limit daily infections and deaths and 20 make viral waves more smooth. Vaccination is assumed to be a worldwide mass process with a large number of units 21 needed. It is estimated that 20 different manufacturers will produce around 1010 doses of COVID-19 vaccination by the 22 end of 2021, which should be ready for global distribution [1, 2] . About 68% of the population worldwide is interested 23 in taking the COVID-19 vaccination. That means about 3.7 × 109 adult people need to be vaccinated [3] . The 24 mathematical models suggest that doses will not be enough until 2023 due to limited sources and manufacturing capacity 25 for population coverage [4] . In many recent studies , mathematical epidemiological modelling is used to predominantly 26 develop mass-action models and suitable tools for analyzing COVID-19 dynamics [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] . Different mathematical 27 models are used for this purpose as the transmission rates and virus behaviour depend on individuals' precautionary 28 measures, daily vaccination rates, and vaccination efficiencies [15] [16] [17] [18] [19] [20] [21] . In [5, 6] , a complete analysis of the actual 29 number of deaths due to COVID-19 was studied by the authors using a new class of probability density functions and 30 the exponentiated transformation of the Gumbel type-II distribution. The authors of [8] used numerical approaches and 31 logistic modelling techniques to conduct a thorough analysis of COVID-19.In [9, 10] , COVID-19 daily confirmed cases 32 in Egypt and Iraq are modelled using a Gaussian fitting model and a logistic model is reached. In [11] , the authors model 33 COVID-19 daily confirmed cases in different countries using Fourier and sine wave fitting models. 34 On the other hand, the classic susceptible exposed infectious recovered model (SEIR), using integer order 35 derivatives, is the adopted model for characterizing the epidemic of the COVID-19 outbreak in different countries. The 36 extension of the classical SEIR model with delays is another routine to simulate the incubation period and the period 37 before recovery [12] . In [13] , the authors created a multi-strain SEIR epidemic model for COVID-19 and wrote a 38 complete analysis on how to control the value of the reproduction number for the next upcoming virus peaks. Fractional 39 order deterministic and stochastic differential equations are also used for developing dynamical SEIR models to study time delays for COVID-19 behaviour estimation. In [39] , the authors constructed a stochastic mathematical model to 48 investigate virus temporal dynamics in Oman. They divided the total population into four classes and considered vaccine 49 effects in their model. In [40] , the authors generated a fractional-order stochastic model to optimize daily vaccinations 50 in Saudi Arabia. 51 The aim of this paper is to develop two dynamical epidimic models, the fractional-order model and the fractional-order 52 stochastic model, to describe COVID-19 behaviour and wide vaccination spread using Caputo fractional derivative and 53 additive white noise. The main contribution of this is to justify the complete effect of vaccines, through different 54 vaccines types, number of doses, and efficiencies, on viral wave shape and peak through affecting the wave reproduction 55 number and then applying them to the Egyptian case study to verify the vaccination effectiveness on the fourth viral 56 wave in Egypt which is assumed to start on 10 th September 2021. 57 This paper is prepared as follows. In Section 2, we provide the used mathematical preliminaries in our wok. In Section 58 3, we describe COVID-19 dynamical models. In Section 4, we apply the dynamical models to the Egyptian case study 59 and compare predicted daily infections, deaths and vaccinations with their actual values. In the end, conclusions and 60 future work are written. 

Recently, fractional differential equations have gained much attention since fractional-order system response 63 ultimately converges to the integer-order system response. The advantages of fractional derivatives are that they have a 64 greater degree of flexibility in the model and provide an excellent instrument for the description of the properties of 65 various practical processes and dynamical systems. For high accuracy, fractional derivatives are then used to describe 66 the dynamics of our generated epidemiological dynamic models which will describe COVID-19 virus behaviour.

Definition 2.1. Suppose that ϕ>0, t>a, α, a, t ∈ R. The fractional operator (D t ϕ ), defined as [41, 42] : 

the system (2) has a unique solution on the interval [43] .

To find the solution of Equation (2) 

The general formula for the fractional Euler method to solve Equation (2) numerically is defined as:

Convergency of the fractional Euler method is detailed described in [27] . The fractional-order stochastic model driven by Caputo fractional derivative of order ϕ indicated in definition 2.4 can 99 be solved numerically using the fractional-order Euler Maruyama scheme indicated in [28] . Applying fractional-order 100 Maruyama scheme to Equation (4) on the interval , it yields:

for ∈ ( , ( + 1) ] , = 0,1,…, -1,

where n is the total time discretization and belongs to the set of positive integer numbers. is described in detail in [28] . 

S, E and I respectively symbolize the main daily susceptible, exposed and infected individuals. V 1i and V 2i are the daily 111 first and second vaccinated individuals using vaccination type i. E 1i and I 1i are the daily first-vaccinated exposed and 112 first-vaccinated infected individuals while E 2i and I 2i are the daily second-vaccinated exposed and second-vaccinated 113 infected individuals. Finally, R and D are the daily recovered and daily deaths. All proposed models satisfy the following 114 assumptions:

(1) All involved transmission rates are with positive values and vary with time.

(2) Symptomatic patients are only those who transmit the virus.

(3) Susceptible individuals can move into infected classes without passing by exposed classes. first dose efficiency (ζ 1i ) will affect directly increasing the cure rate to be (1+ ζ 1i )ɤ(t) and decrease the death rate to be

The daily susceptible people (S) transfer to the exposed first vaccinated class (E 1i ) and the infected first 135 vaccinated class (I 1i ) with daily transmission rates (1-ζ 1i ) β 1 (t) and (1-ζ 1i ) β 2 (t) respectively when nearly contact occurs 136 between them. 13 The remaining part of the first vaccinated individuals moves to the second vaccinated class (V 2i ) with a constant 138 daily rate (δ 2i ) in case of two-dose vaccines. In one-dose vaccines, the remaining part of the first vaccinated individuals 139 will move to the recovered class at a daily rate (ζ 1i-full /a ); where a is the number of days to make sure that fully 140 vaccinated people are recovered. We assumed that a small part of the exposed class of first vaccinated will by mistake 141 take the second dose so moved to the second vaccinated class with a daily rate β 4 (t). The lower part of the second 142 vaccinated people is assumed that will pass through the same track of infection with the second vaccinated exposed 143 class (E 2i ) and second vaccinated infected class (I 2i ) with lower rates β 3 (t) and α E2 (t) respectively. While the larger part 144 of the second vaccinated class will move to the recovered class with the daily rate (ζ 2i /a). The second dose efficiency 145 (ζ 2i ) will affect directly increasing the cure rate to be (1+ ζ 2i )ɤ(t) and decrease the death rate to be (1-ζ 2i ) μ(t). Similarly, 146 the daily susceptible people (S) move to the exposed class of second vaccinated (E 2i ) and the infected class of second 147 vaccinated (I 2i ) with lower daily transmission rates (1-ζ 2i ) β 1 (t) and (1-ζ 2i ) β 2 (t) respectively. By adding the vaccination effect, the fractional-order dynamical model in [7, 44, 45] is described by the following 151 equations, where 0< ϕ< 1:

where c 1 , c 2 , c 3 and c 4 are respectively the daily average numbers of closed contacts for susceptible per exposed, 164 susceptible per infected, first vaccinated per exposed first vaccinated and second vaccinated per exposed first vaccinated. With different ϕ values, we can obtain different viral wave characteristics and peaks. The assumed model dynamic rates 166 are defined as:

where t 0 is the virus incubation period, t 1 is the time after which the first vaccinated started to enter the first vaccinated 171 infected class and t 2 is the time after which the second vaccinated started to enter the second vaccinated infected class.

The controlling parameter (j) has a variable value changed from one country to another by following the precautionary 173 measures. The system initial rates values are β 10 , β 20 , β 30 , β 40 , α 0 , γ 0 , and μ 0 . 17 We consider the region to prove the existence and uniqueness criterion of the solution of our fractional- ( 1 ( )

Proof: Assume that all transmission rates and vaccination efficiencies are positive bounded coefficients such that 195 and hence:

where . = max { 1 , 2 , 3 , ..., 9 } 223 The Lipschitz condition is satisfied by the function B(X) with respect to . contradicts to the assumption . Hence for any , we have . Similarly, we can prove that 

The disease-free equilibrium points are given by С 0 ( 0 , 0 , 0 , 1 0 , 1 0 , 1 0 , 2 0 , 2 0 , 2 0 , 0 , 0

The endemic equilibrium points is denoted by ) = ( , 0, 0, 0, 0, 0, 0, 0,0,0,0). С * ( * , * , * , 1 * , 1 * , 1 * , 2 * , 2 * , 2 * , 256 and can be obtained by solving the system of Equations from (6) to (16) simultaneously, taking into * , * ) 257 consideration the fact Since the С * ( * , * , * , 1 * , 1 * , 1 * , 2 * , 2 * , 2 * , * , * ) ≠ (0, 0, 0, 0, 0, 0, 0, 0,0,0,0).

dimension of the proposed model is much higher, therefore, it is very difficult to discuss the stability of the proposed 259 model concerning equilibrium points. Thus, we will study the stability analysis based on the basic reproduction number

. The basic reproduction number ( ) is computed by using the next-generation matrix approach as:

For more details about the next-generation matrix method see [46] . 

Fractional-order stochastic model. 26 Stochastic models are even used to discuss complex systems with a high degree of randomness due to applied ( 2 + 0.5 2 ) 2 ( ) -2 ( ) 2 2 ( ) ) + 3 2 2 ( ) 2 ( ) where is the exposed-main infected diffusion coefficient. and are the exposed-first vaccinated and 1 2 286 exposed-second vaccinated diffusion coefficients. 

Hence, the fractional stochastic model is stable to 0, when 313

Using both the fractional deterministic and stochastic models, we can study the vaccination effect on countries by 318 comparing currently predicted waves using these two models and previous waves before vaccines. For real comparisons, 319 the actual number of vaccination doses and types should be considered because different vaccinations have a different 320 number of doses and efficiencies. In our models, we preferred that vaccination will not limit infections completely but 321 it will reduce the infection rate according to vaccine efficiency and the number of shots. To check our two generated 322 models, we will apply them to Egypt fourth virus wave which is assumed to start on 10 th September 2021 (day 0). Egypt Table 1 . There exists other used vaccines types but with very low daily rates because of 331 not receiving large numbers of them. We applied our two generated models on the fourth viral wave of Egypt to predict 332 the total number of Exposed, Infected, Recovered and Deaths under valid vaccine doses. All transmission rates are 333 assumed that be all controlled under vaccines and don't exceed their values in the Egyptian second viral wave [7] . We used the transmission rate values of the previous wave in Ref. [7] as maximum values to estimate the current 337 wave transmission rates. To begin making trial and error estimations and reach estimated coefficients in Table 1 models' parameters are assumed and estimated by trial and error based on the second viral wave transmission rates [7] . 354 We used Euler's method with step time equals 0.001 days to solve the different fractional-order deterministic models.

After that, we applied stochastic fractional-order models of orders 0.98 and 0.96 to reach more realistic wave dynamics.

The solution of the stochastic fractional-order models is justified using the Euler-Murayama method with a number of 357 paths equals 20 and discretization of time equals 0.002 days, then the average path solution is taken. We used MATLAB 358 program version 2018b to simulate our models' dynamics. . Daily susceptible, main exposed and main infected using different dynamical models.

Daily Susceptible class variation with time under using different dynamic models is as presented in Figure 2a .

The class is decreasing with time and has minimum values when predicted using the fractional deterministic model of 366 order 0.94. Figure 2b shows the predicted values of the daily main exposed class. The high vaccination rate at the fourth Figure 5 . Daily first vaccinated, first vaccinated exposed and first vaccinated infected using Sinopharm type.

The estimated curves of daily first vaccinated people using AstraZeneca type, using different deterministic and 382 stochastic models, are as indicated in Figure 3 . The highest estimated curve values are reached using the stochastic 383 fractional model of order 0.96 with a peak daily vaccination rate equals 126000 doses in viral wave day no. 27. This is 384 mainly based on our assumptions that the AstraZeneca type will represent 43% of valid doses. Considering Astrazeneca 385 type first dose efficiency, a part of first vaccinated people will be exposed to the virus and then move to the infection 386 area before taking the second dose. Reached results for the estimated first vaccinated exposed and infected due to using 387 AstraZeneca type are presented in Figure 3 . The peak of the daily first vaccinated exposed will take values between Figure 4 indicates the estimated curves for daily first vaccinated people using Johnson and Johnson type using 395 different deterministic and stochastic models. The maximum estimated peak for the daily vaccinated people using this 396 type will equal 42700 in day no. 45 using the deterministic model of order 0.94. Because this type is a one-shot type 397 with high efficiency, then the numbers of daily exposure and infections taken this type of vaccine will be limited. As 398 shown in Figure 4 , the estimated peak range of daily exposure using this type will be between 204 and 305 persons 399 while the daily peak on infections will be between 7 and 12 people. Based on these curves, the one-shot vaccine type is 400 more effective in controlling and reducing daily exposure and daily infections than the two-shot vaccine type. In the 401 case of using the Sinopharm vaccine type, Figure 5 shows the assumed daily first vaccinated curves using different 402 dynamical models. The estimated maximum peak of daily first vaccinated will be 114000 doses using the fractional 403 model of order 0.94. Because this vaccination type has a first dose efficiency higher than the Aztrazeneca type the peak 404 of estimated daily exposure and infected individuals using this type will be lower. Using results in Figure 5 , the 405 estimated peak range of daily exposure using this type will be between 821 and 1195 persons while the daily peak on 406 infections will be between 72 and 101 people. 407 408 Figure 6 . Daily second vaccinated, second vaccinated exposed and second vaccinated infected using AstraZeneca type. 

In this paper, we developed a fractional dynamical model and stochastic fractional dynamical model to evaluate deaths and using the stochastic model of order 0.96, the current wave predicted peak will be decreased from 64 to 58 455 people. Based on the proposed models' predicted scenarios of daily exposure, infections and deaths, the public health 456 officials can move on to eradicate this contagious disease considering the negative aspects of vaccines with low 457 efficiency in producing daily infections again. In our future work, we will expand our models to study the next waves 458 after increasing the number of doses shots and considering reinfection of the recovered individuals. 

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