key: cord-0990057-ah534eec authors: Raslan, W. E. title: Fractional Mathematical Modeling and Epidemic Prediction of COVID-19 in Egypt date: 2021-02-16 journal: Ain Shams Eng. J. DOI: 10.1016/j.asej.2020.10.027 sha: 8bab34786afee592a7f77f87df0ae8c4cffbfaf2 doc_id: 990057 cord_uid: ah534eec In this work the concept of fractional derivative is used to improve a mathematical model for the spread of the novel strain of corona virus disease COVID-19 in Egypt. We establish the dynamics model to predict the transmission of COVID-19. The results predicted by the model show a good agreement with the actual reported data. The effect of precautionary measures on the behavior of the model was studied and it was confirmed that the quarantine period should be long enough to achieve the desired result. In December 2019, the first confirmed infection case of coronavirus is reported in China [1] . In a short time later the disease has rapidly spread around to most countries of the world. The World Health Organization declared that the rapid spread of this virus represents an unprecedented challenge, which requires cooperation and collaboration of all countries of the world [2] . The main challenge lies in the ability of countries to provide the necessary health care to the large number expected to be infected in a short time. Egyptian Ministry of Health and Population announced the first case in Egypt on February 14 for a Chinese citizen [3] . In March 5 2020, the first case of coronavirus among citizens inside the country was reported. The Egyptian government has taken several gradual precautionary measures since the announcement of the emergence of the virus to control the spread, such as imposing a curfew for certain hours and imposing limited partial isolation of some places that witness an increase in the number of infections in addition to the relatively expanded examination of contacts of confirmed cases. In general there are three different categories of the methods used to study the spread of infection diseases. The first is to establish a mathematical model of a dynamical system to describe the evolution of the disease; the second is to use statistical methods and the third is using machine learning expert methods [4] . The second and third methods need a large number of data to be effective. Mathematical modeling plays a vital role in predicting the health and environmental impacts of epidemics and diseases for animals and humans [5] [6] [7] The importance of using mathematics in transmission and spread of epidemics was born in 1766 in the article of Bernoulli that describe the influence of smallpox on the average expected life period using mathematical table analysis [8] . In 1902, Ronald Ross used mathematics to investigate the feasibility of alternative strategies for malaria [9] . In 1927, Kermack and McKendrick illustrated the dynamics of infections spreads using a system of differential equations [10] that are called SIR models. In mathematical modeling, the population is divided into compartments in accordance to the state of their health, such as susceptible (S), infected (I), and recovered (R) as in SIR model. Other states of the population linked with control policies such as quarantined (Q) are also used. In this work we shall consider formulation of the model using fractional derivatives in the Caputo sense definition [11] [12] [13] . Other fractional operators such as the Caputo-Fabrizio and Atangana-Baleanu will be discussed in the future and compared with the Caputo one [14] [15] . Fractional derivatives have been used efficiently to fine tune many existing models of physical and natural phenomena [16] [17] . In the field of Bio-Modeling, fractional derivatives were used to generalize the Hodgkin-Huxley model [18] . The model using fractional derivatives gave more accurate results than the model using classical derivatives. In the field of physical science, fractional derivatives were used to model viscoelastic substances with good agreement with experimental results [19] [20] [21] . Also, Sherief et al [22] have constructed a successful fractional model for thermoelasticity (see also [23] [24] ). The main advantage behind the use of fractional derivatives stems from its memory effect. This means that its definition as an integral over past times makes the effects of the stimuli somewhat retarded not instantaneous as in the classical models. This is in accord with what is found in nature. After the spread of the Corona pandemic, a few models have emerged that use fractional differentiation to model the corona propagation such that the paper by M.A. Khan, A. Atangana, in this paper the author consider the model formulation by applying Atangana-Baleanu fractional derivative operator [6] . They applied their model in Wuhan China considering the sea food marketr as the main source of infection. All the parameter that they have used were either values from the literature or estimated from the Wuhan city of China. In this paper we develop a fractional model using Caputo fractional operator that adapts well to COVID-19 in Egypt to support efforts to control proliferation at rates that health care can handle. We consider the reported cases in the interval from March 10, 2020 to May 12, 2020. According to the known characteristics of COVID-19 and the treatment and where k 1 is the ratio that reflects the rate of infection from an asymptomatically person symptoms one and is the infection rate that can be approximated by the The rate at which highly infectious people will be transferred to confirmed cases 3 k Transition rate from exposed to infected class 2 k Transition rate from exposed to isolation 4 k The proportion of recovering peoples if being admitted to isolation ward As soon as a person is infected, the proportion of becoming in I is , which  means that the proportion of becoming in Q equal [28] . There are a number of (1 )   numerical methods that are used to solve fractional differential equations such as those in [29] [30] [31] [32] . We use Laplace transform technique to solve the model as follows: Taking the Laplace transform with parameter z defined by the relation where , , , and are considered the initial conditions taken in March 10, 2020. Solving the system of equations (10) and then using numerical inverse Laplace transform to find the unknown functions in the time domain. The inversion method uses Fourier expansion techniques. The details can be found on [33] . We now apply our model to study the COVID-19 epidemic in Egypt. We use : The transition rate from exposed to infected class and : The proportion of becoming symptomatic The average number of days in Expose = 5 days and of 20 % all them would be transferred to I class. In figure 2 , the graph shows a good agreement of the trend of daily real confirmed cases with that estimated by FSEQHIR model at α equal 0.97. The effect of fractional parameter α in fine tuning the model is illustrated in figure3. Figure 4 represents the infection rate according to the relation (8) . Finally, in figure 5 , the cumulative of the modeling confirmed cases are drawn to predict the total numbers of cases. We have proposed a new fractional mathematical model to predict the spread of epidemics in Egypt. The proposed model predicts a significant increase in the number of cases after two months, as in figure 2 . Accordingly, a number of measures to control proliferation must be taken such as the application of a comprehensive ban, isolation, and social divergence policies for a specific period of time. To show the effect of this precautionary policies, we have prepared an imaginary model, as in figure 6 , to control the spread of the disease after reaching near 500 confirmed cases daily. The change in the model depends on expeditiously expanding the isolation procedures for suspects or potential exhibitors, as well as increasing the duration of curfews. We see from Figure 7 that the suggested modified model predicts reaching a maximum number of cases per day and then these numbers decrease monotonically after that. 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