key: cord-0980385-fvz7yf4o
authors: Ameen, Ismail Gad; Ali, Hegagi Mohamed; Alharthi, M.R.; Abdel-Aty, Abdel-Haleem; Elshehabey, Hillal M.
title: Investigation of the dynamics of COVID-19 with a fractional mathematical model: a comparative study with actual data
date: 2021-02-19
journal: Results Phys
DOI: 10.1016/j.rinp.2021.103976
sha: cbd2bdbb0e9000ba3fd1703f0a93dec11955a55a
doc_id: 980385
cord_uid: fvz7yf4o

One of the greatest challenges facing the humankind nowadays is to confront that emerging virus, which is the Coronavirus (COVID-19), and therefore all organizations have to unite in order to tackle that the transmission risk of this virus. From this standpoint, the scientific researchers have to find good mathematical models that do describe the transmission of such virus and contribute to reducing it in one way or another, where the study of COVID-19 transmission dynamics by mathematical models is very important for analyzing and controlling this disease propagation. Thus, in the current work, we present a new fractional-order mathematical model that describes the dynamics of COVID-19. In the proposed model, the total population is divided into eight classes, in addition to three compartments used to estimate the parameters and initial values. The effective reproduction number ([Formula: see text]) is derived by next generation matrix (NGM) method and all possible equilibrium points and their stability are investigated in details. We used the reported data (from January 23, 2020, to November 21, 2020) from the National Health Commission (NHC) of China to estimate the parameters and initial conditions (ICs) which suggested for our model. Simulation outcomes demonstrate that the fractional order model (FOM) represents behaviors that follow the real data more accurately than the integer-order model. The current work enhances the recent reported results of Zu et al. published in THE LANCET (doi:10.2139/ssrn.3539669).

According to what was published in the World Health Organization (WHO) [1] , Coronavirus disease (COVID-19) is a newly discovered infectious disease and is a new strain that has not been previously specified in humans. The COVID-19 virus transmitted through closed contact and droplets of saliva or discharge from the nose when an infected person coughs or sneezes at close 5 range. The symptoms of this disease appear in the form of coughing, sore throat, fever, headache, breathing difficulties, fatigue and diarrhea [2, 3, 4] . In critical cases, the infected patient has severe pneumonia which leads to death. As a result, the elderly and those with a sick history like diabetes, cardiovascular disease, hypertension, cancer and chronic respiratory disease are more likely to reach critical cases. Till now there are no specific treatments or clear vaccines for COVID-19. However, 10 there are many ongoing clinical trials evaluating potential treatments.

The outbreak of COVID-19 started since 31 December 2019, as the Health Committee of Wuhan Province in China received 27 cases of viral pneumonia, including 7 critical cases. After that, the outbreak of this disease started in different parts of China and different countries such as the United States of America, Singapore, Thailand, South Korea, Mexico and some regions in Europe, where 15 the WHO monitored on 23 January 2020, more than 571 confirmed cases with 17 deaths in China and various countries. As of 6 February 2020, around 28276 cases, of which 3863 are in critical condition, and 565 deaths had been reported. For that, COVID-19 has received considerable global attention and the WHO released a wide range of interim guidance for all countries on how they can get prepared for coping with this emergency. For more information on the precautionary measures 20 and protocols used to confront this global epidemic, we recommend viewing the following references [5, 6, 7, 8, 9] . The fast track in which a virus has spread and the rapid growth in the number of infected cases has led to a global alert for governments, local health organizations and the WHO to take action to control this disease. Within these procedures, a public awareness campaign is being carried out using TV stations, posters and newspapers. Sterilize most public and vital places 25 by spraying with sterile materials. In addition to quarantining people who have direct or indirect contact with infected cases of this virus, either by quarantined in their homes or in quarantined hospitals and strict monitoring of migrants and so on.

One of the important efforts to face COVID-19 is to found a well-mathematical model. Certainly, mathematical models for infectious disease can help forecast the probable path of an epidemic, and 30 detect the most promising and realistic strategies for containing it [10, 11, 12, 13, 14] . Moreover, mathematical models can simulate the impacts of diseases by different ways such as how the disease influences the interactions between cells in a single patient (within-host models), how it spreads across several geographically separated populations (metapopulation models) and how it spreads within and between individuals, such as those used to predict the COVID-19 outbreak. There are 35 a few research efforts done to construct mathematical models to study COVID-19 in the form of a system of ordinary differential equations (ODEs), which relied on estimating the initial values and parameters of the model on the data reported by global and national public health (see, e.g.

[7, 15, 16, 17, 18, 19, 20, 21, 22] and some references therein). Since several decades ago, a new branch of mathematics called fractional calculus (FC) appeared which represents a generalization of 40 classical integer order for differentiation and integration. Recently, FC attracted much attention of researchers and became an active research field and by using it, many promising ideas were modeled and proposed in various scientific fields [23, 24, 25, 26, 27, 28, 29, 30, 31] .

There are several different kinds of definitions for fractional differential operators (FDOs) in the literature such as Caputo, Riemann-Liouville, Jumarie, Hadamard, Gröunwald-Letnikov, Baleanu and others (see e.g. [32, 33, 34, 35, 36] ). In this paper, we have used Caputo fractional operator which is the most common one within physicists and scientists, it has a key advantage that the fractional derivative of constants are equal to zero. The significance of using the FDOs due to is eligible for capturing memory effects because of their non local nature. Therefore, FDOs are an appropriate tool to describe biological and epidemic models to predict the spread of diseases, 50 controlling of the transmission of these diseases and so much more [37, 38, 39, 40, 41, 42] . Since the emergence of COVID-19, many researchers have been dedicated to their efforts to forecasting the inflection point and terminating this disease in order to assist policymakers concerning the different actions that have been taken by different governments, and among these efforts is to provide mathematical models in order to understand the nature and transmission of this epidemic 55 and design effective strategies to control it. A numerous of fractional-order mathematical models have developed and studied by many researchers to analyse the spreading outbreak of COVID-19 such as, in [43] , proposed a fractional dynamic system for the COVID-19 epidemic contain eight population classes, five of them describe the infected cases depending on the detection and appearance of symptoms. The transmission of COVID-19 in Wuhan China modeled by a fractional 60 mathematical model depended on Caputo-Fabrizio fractional derivative has been investigated [44] , which split the population to five classes, susceptible, exposed, infected, recovered and concentration of COVID-19 in the surrounding environment. They used Adamas-Bashforth numerical scheme to solve this model and give their numerical simulations. The Caputo fractional-order derivative has used in a mathematical model to describe COVID-19 epidemic in [45] , where the individuals are 65 divided into five groups, susceptible, exposed, symptomatic infected, asymptomatic infected and removed (recovered and death) individuals. Also, they conducted a comparison between the results of the fractional-order model and the integer-order model with the real data which reported from around the world from January 22 to April 11 and from this comparison, they concluded that the values derived from the fractional derivative are closer to the real data, and have a less relative error. Motivated by the investigations mentioned above, specially the work of Zu et al [15] , and the 75 current situation of COVID-19, the main contribution of the present work is to find a good strategy to trace the Pandemic trend and reduce the transmission risk based on the fractional mathematical model. First, we have simulated the proposed model with it's fractional order based on the reported parameters in [15] , from which we conclude the need of using fractional order and re-estimate the parameters again. Then, Simulations of the proposed model in it's fractional order with the new 80 estimated parameters are presented together with the real data. The organization of this paper as follows. In Section 2, we formulate the FOM for COVID-19. In Section 3, we discuss the equilibrium points (EPs) and analyzed their stability with the help of the effective reproduction number. Section 4 is devoted to give numerical simulations for the proposed model and an adequate explanation of our results with various values of the fractional order and comparing it with the real 85 data. Summarizing the results of this paper will be provided in Section 5.

The mathematical model considered in [15] describes COVID-19 as a system of ODEs. Here, we introduced a more generalized model that is governed by a system of fractional differential equations (FDEs) with Caputo fractional derivative of order 0 < α ≤ 1, which is defined as [29, 34] 

where f is a given function and Γ(.) denotes the gamma function. It is known

where the total population N is divided into eight components, namely; S describes the susceptible individuals in the free environment, L be the latent individuals, L ρ be the traced latent individuals, P be the suspected individuals, D be the diagnosed individuals, S ρ characterizes the traced susceptible individuals who had direct contact with diagnosed or suspected individuals, I be the infectious individuals in the free environment and R be the recovered individuals. In addition, we took into consideration the cumulative number of confirmed cases X, the cumulative number of suspected cases Y and the cumulative number of deaths Z. The meaning of the parameters and the ICs for the FOM are given in Table 1 .

100 Table 1 : Meaning and values of the parameters in the FOM (1)-(11) as well as the ICs.

Parameter Description Value Ref.

ρ

The quarantined rate of close contacts 0.2432 Fitted β

The transmission rate 0.0977 Fitted [15] c i , i = 1, 2, 3 Positive real constants to compute C r 0.0393, 17.263, 0.118 Fitted k The transfer rate from S(ξ) to P (ξ) 1.7718e-04 Fitted

The release rate from S ρ (ξ) to S(ξ) 1/14 [15, 58] ǫ The transfer rate from L(ξ) to I(ξ) 1/5.2 [15, 59] δ The death rate due to infection 0.0021 Fitted γ The recovery rate from D(ξ) to R(ξ) 0.0425 Fitted

The initial value of L(ξ) 7.6322e+03 Fitted

The initial value of S ρ (ξ) 591.8880 Fitted

The initial value of P (ξ) 2.7832e+03 Fitted

In this section, we explore the stability for the FOM by considering the disease free equilibrium, the effective reproduction number R 0 and the endemic equilibrium.

(i) A disease-free equilibrium (DFE) point:

We shall use only the Eqs.(1)-(7) of the FOM to find the EPs. The model equilibria is obtained 105 here by assuming

by solving Eqs. (12) , then the DFE for the FOM is

Following [60] , in order to derive the expression of R 0 , the choice of the necessary computations of the matrices F and V , which is epidemiologically correct, are given as

Then, the spectral radius of F V −1 is the required effective reproduction number of the FOM 110 which is given by

Theorem 3.1. The equilibrium Ξ 0 of the system (1)-(7) is asymptotically stable if R 0 < 1.

Proof. We compute the Jacobian matrix at DFE Ξ 0 as follow:

By calculating the eigenvalues of J Ξ0 , we have χ 1 = −λ < 0, χ 2 = −k 2 < 0 and the rest eigenvalues are given as follows:

where

The last two eigenvalues are obtained through the following quadratic equation:

where

From Eq. (16) and Eq. (18), we can observe that

• For asymptotically stable, it must be B 1 > 0 in Eq. (15) , which means that (k 1 + k 4 )(γ + δ) > kk 4 .

• The coefficients A 2 , B 2 of Eq. (17) have positive signal whenever R 0 < 1.

• The stability of the DFE depends on the signal of B 1 .

Remark 3.1. If all coefficients of polynomials (15) and (17) have the same signal (positive), then the eigenvalues have negative real part (see, e.g. [61] ).

Consequently, the DFE is asymptotically stable, if B 1 > 0 and R 0 < 1.

(ii) An endemic equilibrium point:

We denote the endemic equilibrium point by Ξ * , which is given when there is an infection I * .

Ξ * = (S * , S * ρ , L * , L * ρ , I * , P * , D * ),

where the values of S * , S * ρ , L * , L * ρ , P * , D * are obtained as follows:

Now, we end this section by proving the following theorem of the stability of Ξ * when the Theorem 3.2. If R 0 > 1, then the unique positive endemic equilibrium Ξ * of the system (1)-(7) is marginally stable.

Proof. The Jacobian matrix at Ξ * is given by

where ̥ 1 , ̥ 2 , ̥ 3 are given by Eqs. (13) . Then the eigenvalues of J Ξ * are κ 1 = 0, κ 2 = −λ < 130 0, κ 3 = −k 2 < 0, κ 4 = − ǫ(k 2 + ǫ) β 1 (k 3 + δ) + ǫ + k 3 + δ < 0 and the eigenvalues κ 5 , κ 6 are also negative, where these eigenvalues can be obtained by solving a quadratic equation w.r.t κ, which has the same coefficients of Eq. (15) . Briefly, we have one of the eigenvalues is zero and the others are negative, whenever R 0 > 1, thus the FOM will be marginally stable. Table 1 . Multi step Adams-Bashforth-Moulton is adapted for this purpose. Indeed, this method has been widely used and its 140 accuracy and convergence have been studied well in [62, 63, 64] .

In Figs. 1 and 2 , we simulate the FOM (1)-(11) using the same estimated ICs and parameters reported in [15] . The values of the reported data of group I are shown in blue diamond-shaped and those of group II are in red circle-shaped, whereas the green square-shaped represents those in group III. The estimated parameters in [15] were obtained based on the reported data from the 145 NHC of China for the period from 23 rd of January to 13 rd of February 2020 (Group I in Table 2 for the fractional derivatives are simulated in Figs. 1 and 2 , from which one could find that as the time increases the model should be considered in its fractional form. This result make the benefits of considering our proposed model as in (1)-(11). Moreover, by examine Fig. 2 , we concluded that the reported parameters in [15] , for the days from 23 rd of January to 13 rd of February 2020 (Group I in Table 2 ), have to be re-estimated using more real data. Then, data of group I together with 155 those of group II have been used in order to re-estimate the new parameters based on the same method as in [15] . These new parameters and ICs are reported in Table 1 , and they are used for the remaining simulations.

As in Section 3, the R 0 of COVID-19 is given by Eq. (14), which is approximated to equal 2.252 on January 23, 2020 (t = 0) in case of using the new re-estimated parameters from Table 2 . The 160 influence of this number along time is illustrated in Fig. 3 for both the current results and those reported in [15] . From Fig. 3 , there is a bit quantitative difference in the values of the reproduction number obtained from the current study and that of [15] . it should be noted the difference comparing this plot with that of Fig. 2(a) .

The influence of the existing suspected cases along time for various values of α is illustrated in Fig.   185 7. As time involves, the number of the existing suspected cases decreases for all the values of α. It is found that when α ≥ 0.973 (as seen in the second zoom in of the plot), one could gain a better prediction of the number of the existing suspected cases along time. In addition, simulation of the cumulative suspected cases along time is plotted in Fig. 8 for various values of α. From those green values, it seems that when time evolves the best value of α, for this case, converges to 1.0. of α is given in Fig. 10 . The actual data are close to the simulated curve corresponding to law values of α, till day 16 April 2020. In the next day, 17 April 2020, there is a big jump on the actual data corresponding to the death of 1290 person, which, from that day and on, coincide with the simulation of the model for α = 0.983. This behaviour in the actual data is going to affect the behaviour of the commutative number of the recovered cases shown in Fig. 11 . As it can be seen 200 from that figure, there is a decay in the commutative actual data corresponding to death numbers.

Comparing the resulting simulations of Fig. 11 with those of Fig. 2 

In the present work, we managed to propose a COVID Elshehabey; Writing -review editing: all authors.

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