key: cord-342591-6joc2ld1
authors: Higazy, M.
title: Novel Fractional Order SIDARTHE Mathematical Model of The COVID-19 Pandemic
date: 2020-06-13
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.110007
sha: 
doc_id: 342591
cord_uid: 6joc2ld1

Nowadays, COVID-19 has put a significant responsibility on all of us around the world from its detection to its remediation. The globe suffer from lockdown due to COVID-19 pandemic. The researchers are doing their best to discover the nature of this pandemic and try to produce the possible plans to control it. One of the most effective method to understand and control the evolution of this pandemic is to model it via an efficient mathematical model. In this paper, we propose to model the COVID-19 pandemic by fractional order SIDARTHE model which not appear in the literature before. The existence of a stable solution of the fractional order COVID-19 SIDARTHE model is proved and the fractional order necessary conditions of four proposed control strategies are produced. The sensitivity of the fractional order COVID-19 SIDARTHE model to the fractional order and the infection rate parameters are displayed. All studies are numerically simulated using MATLAB software via fractional order differential equation solver.

All the world states' governments introduce a big effort and vital measures to eliminate the outbreak of COVID-19 [22] . COVID-19 is a new progeny of coronavirus, SARS-CoV-2 and firstly detected in Wuhan, China [52, 57] . In the few months after discovering it, the number of patients were increasing exponentially. The taken measures against COVID-19 until the day of writing these words didn't prevent the growth of infected cases around the globe. The World

Health Organization situation report published in 25 May 2020 informed that 5304772 cases as total cases and 342029 deaths around the globe [54] .

Using mathematical model to predict the epidemics is very useful in order to understand the nature of the epidemic and to design an efficient strategies to control it [4, 8, 14, 26, 27] .

It is common to study the humanitarian diffusion of epidemics via SIR or SEIR models [11, 13, 33, 48] . Various models have been proposed to model and study COVID-19 pandemic.

Taking into account the risk understanding and the accumulative issue of cases, COVID-19 pandemic has been modeled by Lin et al. via extending SEIR model [38] where S signifies the susceptible, E signifies the exposed, I signifies the infected and R signifies the removed cases.

In [3] , Anastassopoulou et al. have suggested the SIR model in the discrete time mode taking into account the dead cases. In [9] by Casella, SIR model is expanded to study the delays effect and to compare the policies of containment. In [56] by Wu et al., the COVID-19 severity has been estimated using the dynamics of transition. In [25, 34] , random transition models have been studied. In [12] , the general multi-group SEIRA model was represented and numerically tested for modelling the diffusion of COVID-19 between a non-homogeneous population. The basic mathematical tool used to model several epidemics is differential equations in various modes (ordinary, fractional, with delay, randomly detected or partial) [27] [28] [29] . 4 Many research efforts have been widely done to control the outbreaks of epidemics via optimal control [20, [44] [45] [46] [47] . The optimal control idea is to look for the utmost powerful plan that decreases the rate of infection to a possible minimum limit with optimal minimum cost of circulating a treatment or preventative inoculation [39, [44] [45] [46] [47] 51] . These plans may include treatments, inoculation with vaccines, social distances, educational programs [6, 10 ] .

Studying the epidemiological diseases mathematically become very important [6, 7, 16, 17, 37, 40] . The literature has several studies to control for example models of HIV [24] , dengue fever [2] , tuberculosis [50] , delayed SIR [1] and delayed SIRS [29, 58] . The fractional order differential equations add an extra dimensions in the study of dynamics of epidemiological models. Therefore the fractional version of many epidemical models have been investigated as in [29] [30] [31] [32] , [39, 41] and [53] .

Here, a new epidemiological fractional mathematical model for the COVID-19 epidemic is proposed as an extension of the classical SIR model, similar to that introduced by Gumel et al.

for SARS in [23] and as a generalization of SIDARTHE model proposed in [21] . As explained in Section 3, in SIDARTHE model, the infected cases are classified into five different classes depending on the detection and the appearance of symptoms [21] .

In this work, we consider the fractional order SIDARTHE model and then we derive the fractional order necessary conditions for existence of a stable solution. In addition, we study an optimal control plans for the fractional order SIDARTHE model via four control strategies that include the availability of vaccination and existence of treatments for the infected detected three population fraction phases. Applying the fractional order differential equations numerical solver using MATLAB software, we show the dynamics of the state variables of the model and display the effect of changing the fractional derivative order on the system response. Also, the effect of changing the infection rates on the fractional order SIDARTHE model's state 5 variables. We also implement the optimal control strategies numerically for the fractional order SIDARTHE model.

The remaining parts of the paper are organized as follows. In Section 2, Preliminaries and basic definition of the fractional derivative are introduced. Describing COVID-19 epidemic SIDARTHE fractional mathematical model is introduced in Section 3. The details of the optimal control strategy and its implementation are given in Section 4. Numerical simulations of the uncontrolled fractional order SIDARTHE model, the effects of changing the fractional derivative order on the system response and the effects of changing the infection rates are all given in Section 5. Numerical simulations of the controlled fractional order SIDARTHE model and the effects of applying the proposed control strategies are represented in Section 6. The concluding remarks are put in Section 7 followed by the list of cited references.

Many definitions of fractional order derivatives exist such as Riemann-Lioville's derivative, Grunwald Letnikov's derivative, Caputo's derivative, Caputo-Fabrizio, Atangana-Baleanu, etc. The interested reader can consult for example [42, 43] and the references therein for more details about fractional order defintions with applications. We have used Caputo's definition throughout the paper.

Caputo fractional derivative operator q  of order q (see [18, 30, 39, 42, 43, 49] ) is defined as:

symbolizes the Gamma function. For more details about the basic definitions and characteristics of fractional derivatives see [18, 30, 39, 49] .

Notation: For numerical simulations, the predictor-corrector PECE method of Adams-Bashforth-Moulton type described in details in [15, 19] has been utilized and programmed with MATLAB software. 

Giulia Giordano et al. in [21] modeled the COVID-19 epidemic via SIDARTHE and compare its response with the real data in Italy. SIDARTHE distinguishes between determined infected cases and undetermined infected cases and between various degrees of illness (DOI).

In SIDARTHE COVID-19 epidemic model, the total population is partitioned into eight phases of malady as recorded in the Table 1 . Figure 1 shows the interaction graph (  ) between the eight phases of malady. In Figure 1 , the susceptible population partition is partitioned into four sub-phases to show the hidden sub-phases of the susceptible population partition , namely . The detailed interaction digraph in Figure 1 is named  that may be studied using graph theory tools to discover more features about the model. Infected (symptomless, undetermined) population fraction.

Diagnosed (infected, symptomless, determined) population fraction.

Ailing (infected, with symptoms undetermined) population fraction.

Recognized (infected, with symptoms, determined) population fraction.

Threatened (infected, with life-menacing symptoms, determined) population fraction.

Healed (recuperate) population fraction.

Extinct (died out) population fraction. Table 1 where the sub-phases SI, SD, SA, SR are appeared.

The COVID-19 epidemic SIDARTHE is modeled mathematically by eight ordinary differential equations [21] .

The deterministic characteristic is essential in modeling the epidemics transition phenomena; the fractional derivative is very useful in modeling the epidemics transition systems because they consider the memory effect and the universal properties of the system, that are primary in the deterministic feature. The system is said to have a memory effect if its future states depend on its current states and the history of the states, and the fractional operator has this memory effect feature so it is very helpful in modeling the COVID-19 diffusion model.

Here, as recorded in equations (3.1) to (3.8) , the dynamics of the population in each phase with time is described with eight fractional order ( ) differential equations: [21] using the real data. Figure 1 shows the impact of each different phases of epidemic graphically. The SIDARTHE Covid-19 model parameters have the following real meaning:

 a signifies the rate of infection as a result of contacting among a susceptible case and an infected case.  b signifies the rate of infection as a result of contacting among a susceptible case and a diagnosed case.  c signifies the rate of infection as a result of contacting among a susceptible case and an ailing case.  d signifies the rate of infection as a result of contacting among a susceptible case and a recognized case.  e signifies the detection probability rate of infected symptomless cases.  θ signifies the detection probability rate of infected with symptoms cases.  z signifies the rate of probability at which an infected case is not conscious of becoming infected.  h signifies the rate of probability at which an infected case is conscious of becoming infected.  m signifies the rate at which undetermined infected case develops life-menacing signs.

 v signifies the rate at which the determined infected case develops life-menacing signs.  τ signifies the death rate (for infected cases with life-menacing signs).  g, k, x, r and σ signify the rate of healing for the five phases of infected cases.

For more details about the model choices see [21] and the references cited there. 9 

From equations (3.1) to (3.8) and since the states and are sink vertices in the model graph  ( see Figure 1 ), then they are considered as accumulative state variables that rely only on their own starting situations and the other state variables.

Since summing up all equations (3.1) to (3.8) gives zero as a result then the system is compartmentalized and shows the conservation property of mass : as can be directly proved,

Which implies that the total population (the sum of all state variables) is constant. Let (3.10) be the state variables vector. Since the state variables signify the population fractions, we can suppose that ∑ , such that 1 signifies the total population, dead are included. ; ; ; ; 20)   3  3  3  3  31  32   3  3  2  33  3  34   3  3  3  3  35  3  36  3  37   3 3 38 0 ( ) ; ; ; 

Where ( ) are all positive constants. Then from (3.19) to (3.26) , each of the eight functions agree with the Lipschitz condition [18, 39] . With respect to the eight arguments, then all eight functions are absolutely continuous.

In this area, the existence of the fractional order SIDARTHE model's optimal control is investigated and then the Hamiltonian of the optimal control problem is constructed in order to produce the optimal control necessary requirements.

Compute the optimal values of vaccination and treatment strategies that would maximize the Healed population phase ( ) and minimize the determined infected phases ( ) and susceptible ( ) population phases. In addition, the charges of utilizing the vaccination and treatment methods are minimized. Then the optimal control problem of the following form is considered (see for example [1, 2, 5, 6, 10, 20, 24, 25, 34, 37, [44] [45] [46] [47] .

That obeys the constraints 

The control function represents the vaccination strategy applied on the susceptible 

The four constants 1 2 3 ,, c c c and 4 c are the cost correspond to utilizing each control function.

The uncontrolled system (put 0 i u  in (4.2)). For certain initial conditions:

that their sum equal one, it is obvious that the final values of the state variables approach to an equilibrium:

 which means that the phenomenon of epidemic is finished (see [21] ).

The possible equilibrium points of the system are given by ( , 0, 0, 0, 0, 0, , )

The incidence function of the system is:

In the following result, the existence of the optimal control is proved. Proof. In order to prove the optimal control quadruple 

               Since 1 S I D A R T H E        

This subsection records the needed conditions for the optimal control to be exist. The necessary conditions are computed here via constructing the Hamiltonian (  ) and satisfying the maximum basis of Pontryagin [5] . Let us denote by is the optimal controls and be the related optimal population phases fractions.

Consequently, there exists such that the necessary conditions for the optimal control to be exist are produced by (see for example [5] ):

From (4.4) and (4.6), the optimization constraints can be found as: 

From (4.7), we can find that 

From the state variables system given in (4.2), the co-state conditions are:

Which can be simplified to produce the following co-state system: 

The transversality conditions leads to The collection of permissible controls U is convex. 17 The effect of applying the different control strategies will be simulated numerically in Section 6.

In this fifth section, we solve the fractional order SIDARTHE model numerically utilizing the predictor-corrector PECE method of Adams-Bashforth-Moulton type described in details in [15, 19] .

The parameters' values used for the numerical simulation are estimated from the Italian real life statistics published in [21] where:

The total population are taken 100 Million and the initial values of the different population phases after normalization are (let be the total population):

In the following, we display the results of the numerical simulation of the uncontrolled SIDARTHE model. Figure 2 with different fractional derivative order . Figure 9 displays the phase plane of state variables: total infected ( ) and susceptible cases (S(t)) with different fractional derivative order . Figure 10 displays the phase plane of state variables: total infected ) and

healed cases (H(t)) with different fractional derivative order .

From the results and the figures mentioned here, we can state that decreasing the fractional derivative order decreases the number of each population phase (except Susceptible 19 population fraction as expected) and flatten the curves also delays reaching the maximum in each population phase. 23 Figure 10 . The phase plane of state variables: Total infected ) and Healed cases (H(t)) with different fractional derivative order .

In this subsection, we show the effect of changing certain system parameters on different population phases at day 60 with various fractional derivative order. Figure 

The mean rate of separation or contraction of tiny phase-space disturbances of a dynamical system beginning from near starting points is metered by the Lyapunov exponents (LEs) [55] . Thus, they can be utilized to study the stability of dynamical systems and to examine sensitive reliance on starting conditions, that implies, the presence of hidden chaotic dynamics. It is important to check the epidemic transition models if it is chaotic or not via calculating LEs. Corresponding techniques for the LEs calculation and their distinction are studied, e.g., in [35, 36] .

The studied system represented by equations: (3.1) -(3.8) is stable [21] . Here, we confirm its stability for different values of fractional derivative order via plotting the relationship between the fraction derivative order and the eight Lyapunov exponents of the system. Figure   23 shows that all system eight Lyapunov exponents are negative with different fraction derivative order as time approaches infinity (here, time is taken 1500). Figure 24 shows the 31 dynamics of the system eight Lyapunov exponents with time. In Figure 24 -(a) to 24-(d) with different fraction derivative order, the system eight Lyapunov exponents approaches negative end which confirm the system stability with different fractional derivative orders. For more details about Lyapunov exponents see [55] . Here, we use the method in [13] for calculating the fractional order Lyapunov exponents. 

This research has been carried out to the analysis of an eight dimension fractional -order SIDARTHE COVID-19 mathematical model. In this 8-D CVID-19 mathematical model, the infected population fraction is partitioned into five different population fractions:

. It is the first time to study such model with fractional order. The existence of stable solution of the fractional order SIDARTHE model is proved. The fractional order necessary conditions for a four optimal control strategies are implemented. In addition, the system dynamics displayed via the fraction order numerical solver by MATLAB software with different fractional orders and the effects of changing the infection rates parameters are presented in this manuscript with different fractional orders. The effects of changing the fractional order on the system Lyapunov exponent are also displayed. The dynamics of the system are presented before and after control. From our study, we can state that decreasing the fractional derivative order decreases the number of cases in all population fraction phases and delays the maximum plus changing the value of the fractional derivative order has no effect on the stability of the system since its all Lyapunov exponents still negative. The proposed 38 fractional order COVID-19 SIDARTHE model predicts the evolution of COVID-19 epidemic and try to help in understanding the impact of different plans to limit the diffusion of this epidemic with different values of the fractional order. Our results confirm the importance of decreasing the infection rates. Decreasing the infection rates include taking various actions like insure the social distance, closing the airports, closing all teaching authorities, random testing the asymptomatic cases and contact tracing. The author hope that the COVID-19 study using the proposed model continues. And via utilizing the real data the optimum fractional order can be estimated.

No competing of interest.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. 

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In this section, we show numerically, the effect of applying the four control strategies studied in Section 3. Figure 25 , (c) . Figure 29 shows the effect of the control strategies: (vaccination) and 33 (treatment of the Diagnosed population phase ) on the different population phases at day 70 with fractional derivative order . Figure 30 shows the effect of the control strategies:(vaccination) and (treatment of the Diagnosed population phase ) on the different population phases at day 70 with fractional derivative order . Figure