key: cord-0962883-8xdd3x27
authors: Mahdy, A.M.S.; Higazy, M.; Gepreel, K. A.; El-dahdouh, A.A.A.
title: Optimal control and bifurcation diagram for a model nonlinear fractional SIRC
date: 2020-06-30
journal: nan
DOI: 10.1016/j.aej.2020.05.028
sha: cb207bc2990e297bc8ec91ef2e74742c87a7d14d
doc_id: 962883
cord_uid: 8xdd3x27

Abstract In this article, the optimal control for nonlinear SIRC model is studied in fractional order using the Caputo fractional derivative. Graph signal flow is given of the model and simulated by Simulink/Matlab which helps in discussing the topological structure of the model. Dynamics of the system versus certain parameters are studied via bifurcation diagrams, Lyapunov exponents and Poincare maps. The existence of a uniformly stable solution is proved after control. The obtained results display the activeness and suitability of the Mittag Generalized-Leffler function method (MGLFM). The approximate solution of the fractional order SIRC model using MGLFM is explained by giving the figures of solutions before and after control. Also, we plot the 3D relationships with different alpha (fractional order) which display the originality and suitability of the results.

Studiyng the notion of disease discovery and diffusion mathematically was done for the first time in [2, 46] . The discovery of diseases and epidemics using mathematical models have got a great attention of many researchers [1] [2] [3] [4] [5] [6] [7] [8] [9] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] 33, 35, [40] [41] [42] [43] [44] [49] [50] [51] [52] [53] .

Mathematical models are considered as a great tool in studying the sawing and controlling of diseases epidemic. Recognition the propagation manner of diseases infectious in society, zones, and nations is very helpful in reducing the propagation of these diseases [18] [19] [20] [21] [22] .

Several researchers have interested in applying the new notions of derivatives to solve some of the ordinary cases [13] . In [40] , via Atangana-Baleanu fractional derivative, the the polluted lakes system's dynamics were investigated. Also, the fractal-fractional model was solved utilizing a novel proposed technique. In addition, they proved the existence and uniqueness of their results. In [41] , via Atangana-Baleanu fractional derivative, the authors investigated the dynamics of the competition between rural and commercial banks. The authors of [42] described the mathematical modeling and dynamical behavior of a novel coronavirus (2019-nCoV) in utilizing the fractional derivatives. The authors in [43] presented the dynamics of a fractional SIR model with a generalized incidence rate using two differential derivatives, which are the Caputo and the Atangana-Baleanu. the system of Indonesian rural and commercial banks, in the period 2004-2014, has been investigated and solved using Caputo-Fabrizio concept in [44] .

In the current place, the fraction SIRC model is given and we display some important rules and mathematical notions of the fractional calculus concepts needed for the next improvement. Definition 1. Caputo fractional derivative operator D g of order g [3] [4] [5] [6] [7] [8] [13] [14] [15] 30, 31, [55] [56] [57] is defined as

x À t ð Þ gÀqþ1 dt; x > 0; g > 0;

q À 1 < g 6 q; q 2 N;

where Cð:Þ is the Gamma function.

The Caputo fractional derivative operator is linear operator D g l 1 g 1 ðtÞ þ l 2 g 2 ðtÞ ð Þ ¼ l 1 D g g 1 ðtÞ þ l 2 D g g 2 ðtÞ;where l 1 and l 2 are arbitrary constants.

Here is some illustrative examples of Caputo fractional derivative operator:

If K is constant function, thenD g K ¼ 0;

for n 2 N and n < g d e;

Cðnþ1ÀgÞ t nÀg ; for n 2 N and n P g d e;

( where g d e is the ceiling function to imply the shortest integer larger than or equal to g and N ¼ 0; 1; 2; ::: f g : Notice that for g 2 R; the Caputo fractional differential operator completely agrees with the familiar differential operator of integer order.

For more details about the basic definitions and characteristics of fractional derivatives, see [13, 55] .

Flu is transmitted by an infection that can be of three distinct sorts, to be specific A, B, and C [45] . Among these, the epidemiologically infection A is the most significant one for people since it can recreate its properties with properties of strains living in creature populace, for example, feathered creatures, swine, ponies [46, 47] . Throughout the most recent two decades, various pandemic systems for foreseeing the propagation of flu among people populace have been suggested dependent on the old-style contaminated evacuated SIR system created by Kermack and McKendrick [48] . In [19] , the authors have suggested the SIRC system by collecting an extra room C, that was called a cross-immune room, to the SIR system. This cross-immune room C represents an middle status among the well susceptible S and the well-isolated R room. The dynamical actions of this system have been analysezed numerically [20, 21] . The authors in [18] used a Chebyshev spectral technique to present the fractional SIRC system. The authors in [22] found the two equilibrium points and study the system dynamics.

Here, the fractional-order model of the SIRC [18, 21] is studied that has the following form:

with given initial condition:

where D a is derivative Caputo fractional 0 < a 6 1 and following are the used symbols:

The variables

The concept The model (1) with initial conditions must be split up of each other and confirm the law N t

The paper is built in 7 sections. We discuss the fractional optimal control for SIRC modeling and we prove the existence of a uniformly stable solution after control in Section 2. In Section 3, a graph of signal flow is proposed for the model which helps in understanding the structure of the system from graph theory point of view. In Section 4, we give the numerical implementation to show the efficiency of using FMGLM. Section 5, we display a simulation of this model using FMGLM before and after control and construct its Simulink simulation scheme. Section 6 represents the dynamics of the system versus certain parameters via bifurcation diagrams, Lyapunov exponents and Poincare maps. We give the conclusions, in Section 7.

Here, the fractional optimal control for SIRC system is to be discussed:

According to Refs. [26] [27] [28] [29] , taking into account the state model written in Eqs. , in R 4 , functions of admissible control can be represented as:

where T f is the final time, v 1 ð:Þ is the average of headway from infective to recovered per year and v 2 ð:Þ average of headway from recovered to susceptible per year. v 1 ð:Þ and v 2 ð:Þ are the control functions, The L 1 norm k x k 1 ¼ max i x i j j is a function space. This definition of the L 1 norm is equivalent to taking the limit as p ! 1 of the L p norm.

The objective functional is defined as follows (quadratics are the control variables)

where A 1 ; A 2 and A 3 represent the measure of infectious, an average of headway from infective to recovered per year and an average of headway from recovered to susceptible per year respectively. Find the optimal controls v 1 ð:Þ and v 2 ð:Þ is the main task in FOCPs, in order to minimize the selected fitness function:

ð2:4Þ

where n i ¼ n S; I; R; C; v 1 ; v 2 ; t ð Þ ; i ¼ 1; 2; 3; 4: The following initial conditions are satisfied:

To design the FOCP, taking into account the new fitness function as [26] [27] [28] [29] :

The Hamiltonian for the objective (cost) functional (2-6) and the control fractional order SIRC model (2-1) is given as follows:

From (2-6) and (2) (3) (4) (5) (6) (7) (8) , the necessary and sufficient conditions of FOPC can be derived (see [26] [27] [28] [29] [36] [37] [38] [39] ) as follows:

additionally,

where k i ; i ¼ 1; 2; 3; 4 are the Lagrange multipliers. Eqs. (2) (3) (4) (5) (6) (7) (8) (9) and (2-10) represent the necessary conditions in terms of the Hamiltonian of the FOPC.

The following theorem holds our result:

If v 1 and v 2 are optimal controls with corresponding state S à ; I à ; R à and C à then there exist adjoint variables k à i ; i ¼ 1; 2; 3; 4 satisfies the following:

Applying the conditions in the text theorem and applying Eq. (2.9) see, [26] [27] [28] [29] , we get the following four equations that can be written as follows:-

(ii) Transversality conditions:

moreover, by applying Eq. (2.10), the control functions v à 1 ; v à 2 are given as follows:

Optimal control and bifurcation diagram

Proof. The co-state system Eqs. (2-13)-(2-16) are found from Eq. (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) where the Hamiltonian H Ã is given by

Further, the condition in Eq. (2-12) also satisfied, and the optimal control written in Eqs. Putting v à i ; i ¼ 1; 2 in (2-1), the following state system can be found as:

For more details about fractional optimal control see the Ref. [26] [27] [28] [29] [36] [37] [38] [39] [49] [50] [51] 54, 56, 58] .

The existence of the control system (2-13)-(2-16) is to be proved here, the same can be found in [5, 7, 52] as follows: Let

We have, at X: 

The signal flow between the system's states is proposed as shown in Fig. 1 , the signal flow graph is indicated by G ! which characterized by a couple of two sets: the vertices set vð G ! Þ and the edges set Eð G ! Þ, in which every vertex corresponds to a state variable and named by its symbol in (1-1). So, The cardi- which helps in understanding the structure of the system (1-1) from graph theory point of view. For more about the graph of signal flow, see for instance [16, 17, [32] [33] [34] [35] .

Here in this section, we resolve and study the model (1-1) using the FMGLM [9-12,33], Let S ¼ P 1 n¼0 a n t na Cðnaþ1Þ ;

f n t na Cðraþ1Þ :

We have, Caputo fractional derivative

a nþ1 t na Cðnaþ1Þ ; D a IðtÞ ¼ 

Using Eqs. (4-1) and in the model (1-1), as follows:

n¼0 a n t na

Àðl þ dÞ Cðnaþ1Þ þ l a n Cðnaþ1Þ þ b f n À c b n Cðnaþ1Þ n o t na ¼ l;

Cðnaþ1Þ À d d n Cðnaþ1Þ þ b f n 1 þ lþc ð Þe n Cðnaþ1Þ n o t na ¼ 0:

At n ¼ 0 in Eq. (4-3), we find:

ð4-4Þ

Then system (4-4) become P 1 n¼1 a nþ1

Cðnaþ1Þ þ l a n Cðnaþ1Þ þ b f n À c b n Cðnaþ1Þ n o t na ¼ 0;

Cðnaþ1Þ À b f n À rb f n 1 þ lþh ð Þb n Cðnaþ1Þ

In Eq. (4-5), t na is not to equal zero so the coefficient that equal zero and we get the recurrence relationship from which we calculate the constants a n ; c n ; e n ; f n ; g n ; n ¼ 1; 2; 3; :::; 1.

ÞÀð l þ cÞ e n :

At n = 1

Optimal control and bifurcation diagram

At the same way, we find a 3 ; b 3 ; d 3 and e 3 . . . Setting (4) (5) (6) and (4) (5) (6) (7) in (4-2), we have the settling in the infinite series:

Cðaþ1Þ þ a 2 t 2a Cð2aþ1Þ þ a 3 t 3a Cð3aþ1Þ þ Á Á Á ;

Cðaþ1Þ þ e 2 t 2a Cð2aþ1Þ þ e 3 t 3a Cð3aþ1Þ þ Á Á Á :

ð4-8Þ

Here, in Fig. 2 , the fractional SIRC model's solution is shown before control at fraction derivative order, a ¼ 1; Sð0Þ ¼ 0:8; Ið0Þ ¼ 0:1; Rð0Þ ¼ 0:05; and Cð0Þ ¼ 0:05: Figs. 3-6, show the dynamics of S(t), I(t), R(t) and C(t) of the approximate solution of fractional order SIRC model before control with differenta. In Fig. 7 , the originality and proper of the results are clear in view of the figure using a 3D plot. In Fig. 8 , we display a simulation of this model using Simulink. The diagram of Simulink is very important because it shows the dependency on the model states on each other. Fig. 2 The dynamics of S(t), I(t), R(t) and C(t). before control at a ¼ 1. Optimal control and bifurcation diagram 7 Fig. 6 The dynamics of C(t) before control with different a. Optimal control and bifurcation diagram 9

Here, in Figs. 9-12, we show the behavior of approximate solution of S(t), I(t), R(t) and C(t) after control at a ¼ 1: 9 The dynamics of S(t) after control. Fig. 10 The dynamics of S(t) after control. Fig. 11 The dynamics of R(t) after control. Fig. 12 The dynamics of C(t) after control. Fig. 13 The dynamics of S(t) after control.

Optimal control and bifurcation diagram 11 Fig. 14 The dynamics of I(t) after control. Fig. 15 The dynamics of R(t) after control. Fig. 16 The dynamics of C(t) after control. Fig. 17 The dynamics of S(t) after control. Fig. 18 The dynamics of I(t) after control. Fig. 19 The dynamics of R(t) after control.

Optimal control and bifurcation diagram 13 Fig. 20 The dynamics of C(t) after control. 

In this essay, we study the fractional optimal control for nonlinear SIRC modeling using the Caputo fractional derivative. The model is presented by the graph of signal flow and simulated using Simulink/Matlab. FMGLM is to find the approximate solutions of the model. We discuss the existence of uniformly stable solution after control. We have presented the behavior of approximate solution by giving the figures Fig. 23 The various variables' 3D plots after control. before and after control. Dynamics of the system versus certain parameters are studied via bifurcation diagram, Lyapunov exponents and Poincare maps. The results are consistent with that obtained in [21] .

The information about this research is ready from the authors upon request.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 

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We are thankful to the referees and the editor for the neat reading of the paper and for their comments and proposal, which have amended the paper.

No funding.

 Fig. 29 Poincare map of this model.