Dynamics of Ebola Disease in the Framework of Different Fractional Derivatives

In recent years the world has witnessed the arrival of deadly infectious diseases that have taken many lives across the globe. To fight back these diseases or control their spread, mankind relies on modeling and medicine to control, cure, and predict the behavior of such problems. In the case of Ebola, we observe spread that follows a fading memory process and also shows crossover behavior. Therefore, to capture this kind of spread one needs to use differential operators that posses crossover properties and fading memory. We analyze the Ebola disease model by considering three differential operators, that is the Caputo, Caputo–Fabrizio, and the Atangana–Baleanu operators. We present brief detail and some mathematical analysis for each operator applied to the Ebola model. We present a numerical approach for the solution of each operator. Further, numerical results for each operator with various values of the fractional order parameter α are presented. A comparison of the suggested operators on the Ebola disease model in the form of graphics is presented. We show that by decreasing the value of the fractional order parameter α, the number of individuals infected by Ebola decreases efficiently and conclude that for disease elimination, the Atangana–Baleanu operator is more useful than the other two.


Introduction
Ebola caused many deaths in Western Africa, especially in the outbreak of 2014. It includes more than 16 thousand laboratory cases with 70% death cases, which is regarded the deadliest outbreak in history since 1976 with 20 Ebola threats. It is evident that in each outbreak, the first case of infection occurred due to contact with infected animals such as monkeys, fruit bats, etc., which shows the spread of the virus through indirect contact [1]. It is documented in [2] that some percentage of the Ebola-Zaire type survived after two weeks on glass at 4 • C and (10%) on plastic, and on surfaces (3%). Moreover, 0.1% to 1 % of the Ebola virus particle can remain up to 50 days at 4 • C [3]. The survival of the Ebola virus in the environment due to poor sanitary and hygienic conditions considerably become another source of Ebola infection in Africa. In Africa, regions were affected greatly by the Ebola virus outbreak due to their inhabitants being involved in hunting food, being close to the rain-forest, and harvesting forest fruits for food [4,5].
The Ebola disease outbreaks and their transmission have been documented in many articles (see [6][7][8][9][10] and the references therein) and the main focus was to study the human population and the direct transmission. Some models of the type SI, SIR, SEIR, and other types also considered the Definition 1. For a function f : R + → R, then the fractional integral of order α > 0 is given by where Γ shows the Gamma function and α is the fractional order parameter.

Definition 2.
For a function f ∈ C n , then the Caputo derivative with order α is defined as C D α t ( f (t)) = I n−α D n f (t) = 1 Γ(n − α) t 0 f n (z) (t − z) α+n−1 dz, that is defined for the absolute continuous functions and n − 1 < α < n ∈ N. Obviously, C D α t ( f (t)) tends to f (t) as α → 1. Definition 3. [23]. Let z ∈ H 1 (a, b), with b > a, and 0 ≤ α ≤ 1, then the Caputo-Fabrizio derivative can be written as Definition 6. The fractional integral associated with the Atangana-Beleanu derivative is given by: when the fractional order turns to zero, we can obtain the original function.

Theorem 1.
Consider the function f ∈ C[a, b], then the following holds [25]: where f (t) = max a≤t≤b | f (t)|. (10) Further, for the newly derivative the Lipschitz condition can be easily satisfied [25]: Theorem 2. A given fractional differential equation: has the unique solution given by [25]:

Model Formulation
We begin to formulate the Ebola epidemic disease by considering the human population in three compartments, that is, the susceptible individuals, S(t), individuals infected with Ebola virus, I(t), and the individuals recovered from the Ebola virus, R(t). The individuals infected with Ebola and the deceased is D(t) and P(t) is the class for the Ebola virus pathogen in the environment. The model that describes the dynamics of Ebola disease modeled through differential equations is given by where λ = β 1 I + β 2 D + ψP, and the appropriate initial conditions are given by The birth rate of the susceptible individuals is recruited by the rate Λ, while the death rate is given by d. The susceptible individuals become infectious with the effective contact rate β 1 and β 2 with the deceased human individuals. The susceptible are able to attract the disease from the contaminated environment at a rate given by ψ. The death rate of the infected individuals due to Ebola virus is given by a rate δ, while the recovery from infection is φ 1 . The deceased people can be directly buried during funerals at rate ε. At a rate of the environment is contaminated by the Ebola virus. At rates of ξ and θ the infected and deceased individuals, respectively, shed the virus in the environment. The virus decay of the Ebola virus from the population is given by parameter κ.
The sum of the first three equations of the Ebola disease model Equation (14) is given by where N = S + I + R denotes the total alive human population. It should be noted that ε ≤ (d + δ), which is an appropriate condition for the compartment D for which the model becomes relevant, otherwise the deceased human individuals will disappear and the model would be irrelevant. Further, the model given by Equation (14) is well posed and biologically feasible in the region given by where M = (S(t), I(t), R(t), D(t), P(t)).

Ebola Model in the Caputo Sense
The purpose of this section is to apply the proposed three operators on the Ebola disease model Equation (14). Initially, we will apply the Caputo derivative on the Ebola disease model, then, the Caputo-Fabrizio derivative, and finally the Atangana-Baleanu derivative. For each operator we will provide the solution procedure and the discussion on the graphical results in details. So, we start with the Caputo sense.

Equilibrium Points
For the Ebola disease model Equation (18) in the Caputo sense, there is no disease-free equilibrium when > 0 and we have the other equilibrium say, E * = (S * , I * , R * , D * , P * ), we have Using these values in the second equation of the model Equation (18), we have where We have from the coefficient C 1 , Considering the case when = 0, we have and we have a disease-free equilibrium, known as Ebola virus-free equilibrium.

Numerical Procedure for the Ebola Disease Model in the Caputo Sense
In the present subsection, we present the numerical scheme for the solution of the fractional Ebola disease model in the Caputo sense Equation (18). The present scheme that we use for the solution of the fractional Caputo nonlinear ordinary differential equation has been presented in [26,27]. The following procedure is presented Using the fundamental theorem on Equation (20), we obtain thus, at t = t n+1 , n = 0, 1, ..., the following is obtained and From Equations (23) and (22), we have where and Using the Lagrange approximation for the function f (t, z(t)), we have The use of the above expression leads to We have, after further simplification Similarly, Further simplifying, we get We have the final approximate solution for the fractional nonlinear ordinary differential equation by substituting the Equations (30) and (31) into (24), given by The above scheme is used further for the solution of the Ebola disease model in the Caputo sense Equation (18)

Ebola Model in the Caputo-Fabrizio Sense
We can express the model given by Equation (14) in Caputo-Fabrizio derivative as follows: where λ = β 1 I + β 2 D + ψP, and with the initial conditions, and P(0) = P 0 .

Numerical Solution for Caputo-Fabrizio Model
Here we present the numerical solution for the Caputo-Fabrizio model Equation (33) by using the scheme presented [27]. The following steps are taken as specified in one for the solution of Equation (33).

Existence of Solutions for the Atangana-Baleanu Model
It is obvious that the given model Equation (14) shows the dynamics of Ebola disease, which is described by a nonlinear system of differential equations, so it is not possible to obtain their exact solution but the existence of an approximate solution can be very effective if we show that the solution for the Ebola disease model Equation (41) where x(t) = (S, I, R, D, P) represent the vector with state variables S, I, R, D, P and is a continuous vector function and can be defined as follows: The function F can be shown easily to satisfy the Lipschitz condition and can be represented as: Now we have the results for the existence and uniqueness for the Ebola disease model in the Atangana-Baleanu derivative sense. We state and prove the following theorem: Proof. The use of the Atangana-Baleanu fractional integration on model Equation (42) both sides, the following is obtained, Suppose J = (0, T) and the operator Υ : C(J, Then we can write Equation (46) as follows: We have, after applying the supremum norm on J, Obviously, C(J, R 5 ) and the norm . J is a Banach space. Using the operator Equation (48), the following is presented Using the triangular inequality and Lipschitz condition presented in Equation (44) with some simplifications, we have Finally, we have where If the condition given by Equation (45) holds then the operator Υ will be a contraction. Thus, the Banach fixed point theorem ensures that a unique solution for the Ebola disease model in the Atangana-Baleanu form Equation (41) exists, Equation (42).

Numerical Results for the Atangana-Baleanu Model and Simulation Results
In the present subsection we aim to obtain the numerical results for the Ebola disease model in the Atangana-Baleanu form given by Equation (41). First, we provide a numerical scheme in details and then show the graphical results for various values of the fractional order parameter α. The scheme given in [28] will be used to obtain the approximate solution of the Ebola disease model in the Atangana-Baleanu form Equation (41). We write the model Equation (42) after using the fundamental theorem of fractional calculus: At t = t n+1 , n = 0, 1, 2, ..., we have The function F (φ, w(φ)) can be approximated over [t j , t j+1 ], using the interpolation polynomial Substituting in Equation (54) we get After some calculation, we obtain the following: For the Ebola disease model we have the following results: , We using the above scheme for the numerical solution of the Ebola disease model Equation (41)

Conclusions
We presented the dynamics of an Ebola disease model in the framework of fractional calculus. We applied three fractional operators, which are the Caputo, Caputo-Fabrizio, and the Atangana-Baleanu models. Initially, we proposed an epidemic model available literature for Ebola disease and then applied the proposed operators. The Ebola disease model with the Caputo derivative is presented and an effective numerical scheme for the numerical solution was provided. We used many values for the fractional order parameters and obtained the graphical results. The same model is used further and applied to the Caputo-Fabrizio derivative and we then presented a numerical solution for their solution. The solution was obtained and presented in graphical shape with the use of various fractional order parameter values. Then the newly established derivative known as the Atangana-Baleanu derivative was successfully applied to the Ebola disease model. The Ebola disease model in the Atangana-Baleanu sense is used and the uniqueness and existence were presented. Then, we presented a numerical scheme for the solution and presented various graphical results for α. Comparisons of the proposed three operators for various values of the fractional order parameter α = 1, 0.9, 0.7, 0.5, 0.3, 0.1 are presented and discussed. The comparison results show that the Atangana-Baleanu derivative is more helpful for disease elimination by decreasing the value of α, since the population of infected individuals decreased well. The use of three different fractional operators on the Ebola disease model suggests that the fractional order parameter greatly affects disease elimination for the non-integer case when decreasing α. Therefore, we suggest that the application of the various fractional derivatives on the present disease model shows the greater effectiveness of the arbitrary order derivative than that of the integer order model for the case of fractional order parameters.