key: cord-0881508-tafxs782 authors: Omar, Othman A.M.; Elbarkouky, Reda A.; Ahmed, Hamdy M. title: Fractional Stochastic Models for COVID-19: Case Study of Egypt date: 2021-03-12 journal: Results Phys DOI: 10.1016/j.rinp.2021.104018 sha: 002d4ded6c4a6ba7fecd9d6452dd2dba27cd8efd doc_id: 881508 cord_uid: tafxs782 In this paper, COVID-19 dynamics are modelled with three mathematical dynamic models, fractional order modified SEIRF model, stochastic modified SEIRF model, and fractional stochastic modified SEIRF model, to characterize and predict virus behavior. By using Euler method and Euler-Murayama method, the numerical solutions for the considered models are obtained. The considered models are applied to the case study of Egypt to forecast COVID-19 behavior for the second virus wave which is assumed to be started on 15 November 2020. Finally, comparisons between actual and predicted daily infections are presented. Mathematical modeling approach is a strong key tool in order to handle different infectious diseases [1] [2] [3] [4] [5] [6] [7] [8] [9] . Many real-world problems in science and engineering can be modeled by stochastic differential equations [10] [11] [12] . Fractional differential equations have been widely applied in many fields such as physics, chemical, fluid dynamic and epidemic models [13, 14] . The main challenge according studying infectious diseases is the way to predict the disease behavior and how many people will be infected in the future. Different researchers from different interdisciplinary such as applied mathematics and data science have been working on studying these types of predictions so it becomes a great area of research. Such studies are now focusing on describing COVID-19 dynamics as it became an epidemic disease. Dynamic models are used to model viral waves and time of actions to calculate the total number of infections and deaths. In all studies, COVID-19 transmission rates are varied from one country to another and similarly the virus dynamics as it depends on the behaviors of individuals, temperature, relative air humidity and wind speed. Based on these predictions and studies, governments can take more suitable actions to mitigate human and economic losses. On the other hand, the classic susceptible exposed infectious recovered model (SEIR) is adopted model for characterizing the epidemic of COVID-19 outbreak in different countries. The extension of classical SEIR model with delays is another routine to simulate the incubation period and the period before recovery [15] . In [16] , authors used numerical approaches and logistic modelling technique to make complete analysis for COVID-19. In [17] , authors made a fractional order modified SEIR model of pandemic diseases and apply it on COVID-19 to predict virus spreading behavior in Pakistan and Malaysia. In [18] , authors made predictions about Coronavirus transmission dynamics in the African countries. The model parameters (protection rate, infection rate, average incubation time, average quarantined time, cure rate, and mortality rate) are selected using Metropolis-Hasting (MH) parameter optimization method [19] . Further study for modelling COVID-19 daily confirmed cases in Egypt and Iraq by using Gaussian fitting model and logistic model is reached [20] [21] [22] . The virus dynamic behavior is modelled with a new SEIR model. One of the important quantities should be calculated during modelling virus dynamics is the basic reproduction number. This value helps to eliminate the diseases and expect number of secondary infections produced by an infected individual when all individuals are susceptible to infections. In [23] , authors made a multi-strain modified SEIR epidemic model for COVID-19 and wrote a complete analysis on how to control the value of the reproduction number for next upcoming virus peaks. Based on the above-mentioned works, by using Euler method and Euler-Murayama method, we study the predictions of COVID-19 second wave in Egypt through developing different dynamic mathematical models. In particular, the fractional order dynamic model, stochastic dynamic model and the fractional order stochastic dynamic model. Finally, comparisons between actual and predicted daily infections are presented. Secondary data for Egypt first viral wave are carefully collected from references [24, 25] . The second virus wave of Egypt is assumed started on 15 November 2020. This paper is prepared as follows. In Section 2, we review some essential facts from fractional calculus and stochastic analysis. In Section 3, we describe COVID-19 dynamical models. In Section 4, we apply dynamical models on case study of Egypt and compare between actual and predicted daily infections. In the end, conclusions and future work are written. Definition 2.1. Riemann-Liouville fractional integral operator of order for a function can be defined as [26] : Definition 2.2. Caputo derivative of order with the lower limit zero for a function can be written as [27] : , where . A standard one-dimensional Wiener process is a stochastic process indexed by { ( ), > 0 } nonnegative real numbers t with the following properties [28] [29] [30] : (1) with probability 1. ( 0 ) = 0 (2) The function is continuous on t. → ( ) (3) If then and are independent. (4) For Ɐ , all increments , are normally distributed with mean 0 and variance i.e., represents the stochastic Wiener process . One of the simplest numerical approximations for the SDE is the Euler-Maruyama method. If we truncate Ito's formula of the stochastic Taylor series after the first order terms, we obtain the Euler-Maruyama method as follows: with the initial value . ( +1) = ( ) + ( ( )) ∆ + ( ( )) ∆ , i = 0, 1, 2, . . . , N -1 ( 0 ) = 0 Based on the epidemiological feature of COVID-19 and the several strategies imposed by the government, with different degrees, to fight against this pandemic, we extend the classical SEIR model [23] to describe the transmission of COVID-19. In particular, we partitioned the population into five classes, denoted by S, E, I, R and F, where S represents the daily Susceptible individuals; E is the daily Exposed individuals; I the daily Infected individuals, which have not yet been treated. Finally, R and I are the daily Recovered and daily Deaths. All proposed model satisfy the following assumptions: (1) All involved transmission rates, for such models, are positive and varying with time. (2) Natural birth and death rate are not considered. (3) Symptomatic patients are only who transmit the virus. (4) Susceptible individuals can move into infected class without passing by exposed class. Caputo fractional derivative [31] of order , , is applied on our SEIRF modelling Equations (1-5) . Through varying the value of , different viral wave peaks can be reached to describe virus dynamics with more than one scenario. Taking equals 1 results in generating first order modified SEIRF so it is a special case of the fractional order modified SEIRF model. Through modelling Equations (1-5), Daily Susceptible (S) enter the Exposed class (E) due to contact between exposed individuals and infected individuals. Susceptibles move to Exposed class with daily transmission rate and into infected class with daily transmission rate . Exposed (E) enter the Infected class 1 ( ) 2 ( ) (I) with daily transmission rate . A part of infected individuals moves to Recovered class (R) with daily cure rate ( ) and the remaining part moves to Deaths class (F) with daily death rate . All rates are assumed dynamic rates ɤ( ) ( ) and varying with time. where is the average number of closed contacts between susceptible individuals per infected individuals per day and 1 is the average number of closed contacts between susceptible individuals per exposed individuals per day. The 2 assumed model dynamic rates are defined as: Where is the incubation period and is a controlling parameter decreasing from country to another according to 0 how precautionary measures are followed. and are the initial values of dynamic system rates. 10 , 20 , 0 ,ɤ 0 In this model, COVID-19 dynamic behavior is modelled using equations of the first order modified SERIF model, equals 1, after adding white noise terms satisfying Wiener process properties to all modelling equations. These additions can make modelling equations more realistic in describing COVID-19 viral waves. The new added diffusion terms are as indicated in Equations (11) (12) (13) (14) (15) . All virus dynamic rates are taking the same form as in the first order modified SEIRF model. where are respectively the diffusion coefficients for Susceptible, Exposed, Infected, Recovered and , , , , Deaths individuals. This dynamic modelling approach authorize an additional novelty and dimension of the article. Modelling equations are firstly constructed using the fractional order modified SERIF model of order . After that new white noise terms satisfying Wiener process properties are added. The new added diffusion terms are different in shape than used in the first order stochastic modified SEIRF model to reach more probabilistic scenarios covering wide range of probable viral wave dynamics. Dynamical system equations after adding the new stochastic terms are as indicated in Equations (16) (17) (18) (19) (20) (21) (22) . During creating the white noise terms, balance in equations is assumed which means that if all system equations are mathematically added then both the drift and diffusion terms will vanish as in Equation (21). = ɤ( ) ( ) = ( ) ( ) 1 + 2 = 1 (22) where is the exposed-infected diffusion coefficient and is the infected-recovered diffusion coefficient. is 1 the fractional portion of infected-recovered diffusion stochastic term affecting on recovered individuals and similarly is the remaining fractional portion of infected-recovered diffusion stochastic term affecting on deaths individuals. 2 Fractional order SEIRF models can be solved numerically using many techniques and the used technique here is Euler's Method [16] . In case of stochastic SEIRF models, Euler-Murayama method is used to solve system numerically [28, 30] . After that, All dynamic models can be applied on the Egyptian case study to predict number of Exposed, Infected, Recovered and Deaths. To estimate dynamic models parameters, first wave transmission rates are used for trial and error estimations based on data in references [24, 25, 32] . In all dynamic models, the second virus wave is assumed that is started in Egypt on 15 November 2020 (day 0). All dynamic models are applied on the case study of Egypt to reach different and more realistic scenarios for In case of expected Susceptible class, Figure 1 shows this class variation with days through using different dynamic models. All SEIRF models, except the first order stochastic model, give very near curve shapes. The daily Sucepitable individuals are decreasing with time till forecasting day no. 80 and then they settles with time. The first order stochastic model gives a swinging curve shape of Susceptible class with time. Figure 2 describes daily expected Exposed individuals under using all SEIRF dynamic models. In case of fractional order dynamic models, when the fractional order increases from 0.96 to 1 the curve peak decreases. Similarly the case in the fractional stochastic dynamic model. Stochastic models not only give different curve peaks but also different time of actions. In case of the stochastic Predicted scenarios using SEIRF dynamic models are compared with their actual values from [32] . The comparison made for both actual daily infected individuals and daily deaths with their predicted ones. Comparisons are set every 7 days starting from prediction day no. 7 till prediction day no. 56. In comparisons, it is assumed that virus incubation period is 14 days, so comparisons are made each half incubation period. Prediction day no. 46 is also considered in comparisons as it is the actual second viral wave peak [32] . From indicated results in Tables (3, 4 In this paper, we proposed novel general modified dynamic models to evaluate COVID-19 pandemic. The fractional derivative was used to improve the mathematical modelling of COVID-19 under formulating fractional dynamic models to suggest some possible realistic scenarios. These scenarios will be helpful for public health officials to eradicate this contagious disease. Through generated dynamic models, daily Susceptible (S), Exposed (E), Infected Adding white noise to these models results in a novel fractional order stochastic dynamic model. After that, all proposed models are applied on the case study of Egypt to predict different scenarios of the second virus wave started on 15 November 2020. Numerical solution of the proposed models is valid through using Euler's method and Euler-Maruyama method then simulations are made using MATLAB software. From results, both the stochastic modified SEIRF model and the fractional order stochastic modified SEIRF model make daily curves more realistic through adding diffusion terms with different predicted peaks and time of actions. Finally, it has been found out that the reached models are very useful for the Egyptian government to control the COVID-19 outbreak for the upcoming months. Our future work will be focused on fractional stochastic differential equation SEIRF epidemic model with vaccination. Othman A.M. Omar: Methodology of dynamic modelling, data Collection, analysis, software, text writing. Reda A. Elbarkouky: Conceptualization, analysis, supervision. Hamdy M. Ahmed: Data collection, conceptualization, figures preparation, supervision. All authors: Reviewing manuscript. Authors declare that they have no conflict of interests. 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