ANALYSIS OF DENGUE FEVER OUTBREAK BY GENERALIZED FRACTIONAL DERIVATIVE

In this paper, we use the generalized fractional derivative in order to study the fractional differential equation associated with a fractional Gaussian model. Moreover, we propose new properties of generalized differential and integral operators. As a practical application, we estimate the order of the derivative of the fractional Gaussian models by solving an inverse problem involving real data on the dengue fever outbreak.


INTRODUCTION
Fractional calculus was successfully used to model a broad range of phenomena that occur in physics, economy and science. For example, it has been observed that the time or space evolution of many physical processes can be more precisely described when derivatives of fractional order are introduced. In practice, and in many occasions, all that needs to be done is replacing the ordinary derivative in an equation by a derivative of fractional order. Also, the most interesting part is that this is not the result of chance; on the contrary, there is a strong mathematical foundation for it and there are many applications, as several general studies show (cf. Refs. [1][2][3][4][5][6][7][8][9][10]. For a complementary study on the recent developments in the field of the fractional calculus as well as its applications, see Refs. 11 and 12. Fractional operators keep track of the history of the process being studied; this feature allows modeling the nonlocal and distributed responses that commonly appear in natural and physical phenomena. On the other hand, one has to recognize that these fractional derivatives D α show some drawbacks. To overcome some of these and other difficulties, Khalil et al. 13 came up with an interesting idea that generalizes the familiar limit definition of the derivative and allows to introduce successfully a conformable fractional derivative (see Definition 3 below); more recently, a nonconformable fractional derivative is introduced in Ref. 14. In this way, a new direction in fractional calculus was opened, which has shown to be interesting from a theoretical viewpoint and useful in the applications.
According to World Health Organization (WHO), vector-borne diseases account for more than 17% of all infectious diseases, and cause more than 700,000 deaths each year. Among them, more than 3.9 billion people in more than 128 countries are at the risk of dengue. Vector-borne diseases are undoubtedly one of the main health challenges worldwide, given the presence of disease vectors in almost all countries. The efforts of the researchers have focused on the detection and subsequent treatment of infections, however, research and development of technologies that contribute to the prevention of infections have not been sufficient. Understanding the dynamics of vectors (mosquito Aedes aegypti) and the population infected with dengue has the potential to generate early warnings for the prevention and containment of the spread of the disease.
Qureshi and Atangana 15 present a comparative study of the dynamics of the dengue fever outbreak under three fractional order differential operators; Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu. The true field data for the dengue fever outbreak was used. In this approach, the fractional derivative accomplishes a better fit for the curves of human and mosquito populations but they did not analyze the results associated with the solution of the inverse problems in the local or global approaches.
This paper has been organized as follows: In Sec. 2, we present some fundamental concepts from fractional calculus needed for the convenience of readers. Section 3 is devoted to develop the main results presented in this work. In particular, the lineal-fractional conformable derivative equations are studied using the previous results, obtaining results both for the general solution and for the stability of the solutions. In Sec. 4, a practical application of the Gaussian model is made when solving an inverse problem that involves real data of dengue fever outbreak. Section 5 is devoted to conclusions.

PRELIMINARIES
In Ref. 16 is defined the following generalized fractional derivative, which generalizes the conformable fractional derivative in Ref. 13 for the order 0 < α ≤ 1. Definition 1. Given an interval I, f : I → R, α ∈ R + and a positive continuous function T (t, α) where · represents the upper integer part function.
If T (t, α) = t α −α , then we obtain the following particular case of G α T , defined in Ref. 13. Note that T (t, α) = t α −α = 1 for every α ∈ N. Definition 2. Let I be an interval I ⊆ (0, ∞), f : I → R and α ∈ R + . The conformable derivative G α f of f of order α at the point t ∈ I is defined by We know from the classical calculus that if f is defined in a neighborhood of the point t, and there exists D n f (t), then Therefore, if α = n ∈ N and f is smooth enough, then Definition 2 coincides with the classical definition of the nth derivative. In Ref. 13 is defined a conformable derivative in the following way.
The following results in Refs. 13 and 16 contain some basic properties of the derivative G α T .
Theorem 4. Let I be an interval I ⊆ R, f : I → R and α ∈ R + . We have the following: Theorem 5. Let I be an interval I ⊆ R, f, g : I → R and α ∈ R + . Assume that f, g are G α Tdifferentiable functions at t ∈ I. Then the following statements hold: .

NEW RESULTS
The following result shows an advantage of our definition over some previous definitions of fractional derivatives: a general rule in order to compute iterated derivatives.

t) exists, and we have in this case
where P n = p n (T 1 , . . . , T n , f , . . . , f (n−1) ), p n is a polynomial on (n 2 + 3n − 2)/2 variables which is homogeneous of degree n + 1, and Proof. We will proceed by induction on n. For Fractals 2020.28. Downloaded from www.worldscientific.com by 201.221.123.37 on 10/13/21. Re-use and distribution is strictly not permitted, except for Open Access articles.
P. Bosch et al. exists, since T (t, α 1 ) is differentiable, Theorems 4 and 5 give Note that in this case we have is a polynomial of four variables and homogeneous of degree 3.
Assume now that the induction hypothesis holds for n − 1 and let us prove it for n.
By induction hypothesis we have where p n−1 is a polynomial on (n 2 + n − 4)/2 variables which is homogeneous of degree n, Note that Q n−1 and P n−1 are differentiable, since there exist the derivatives ofT 1 , . . . ,T n−1 , f , . . . , f (n−2) and p n−1 is a polynomial. Note also that Q n−1 (t) and P n−1 (t) are homogeneous polynomials of degrees n − 1 and n, respectively. P n−1 has (n 2 . . , f (n−1) ) and the set of variables of Q n−1 is a subset of the variables of P n−1 .
Then, we have Since α n ∈ (0, 1], then, by Theorems 4 and 5, we exists, then Q n−1 (t)f (n−1) (t) + P n−1 (t) exists, and since Q n−1 and P n−1 are differentiable and Q n−1 > 0, there exists Since Q n−1 and P n−1 are homogeneous polynomial of degrees n − 1 and n, respectively, then is a homogeneous polynomial of degree n + 1 on (n 2 + 3n − 2)/2 (the set of variables of P n−1 and T (t, α n )), and this finishes the proof.
We now provide a more explicit formula for the previous result. Recall that we denote by D the usual derivative.
Assume now that the induction hypothesis holds for k − 1 ≥ 1, i.e. b k−1,m (t) exists for 1 ≤ m ≤ n − k + 1, and let us prove that b k,m (t) exists for By hypothesis, T (j) (t, α k ) exists for 1 ≤ j ≤ m ≤ n−k and since m ≤ n−k < n−k+1 then m+1−j ≤ m+1 ≤ n−k +1 and the induction hypothesis gives The formula trivially holds for k = 0 and 1 ≤ m ≤ n − 1.
Assume now that the induction hypothesis holds for n − 1 ≥ 1 and let us prove it for n. Induction hypothesis and Theorem 4 give since 1 ≤ n − (n − 2) − 1, and the formula holds for n.
As a particular case from the above results, and considering the kernel function T (t, α) = t 1−α for α ∈ (0, 1], the following corollary is immediately obtained.
Let I be an interval, a, t ∈ I and α ∈ R. Following Ref. 13, the integral operator J α T,a is defined for every locally integrable function f on I as for every continuous function f on I and a, t ∈ I.
Theorem 12 in the following contains some elementary facts on the integral operator J α T,a .
Theorem 12. Let I be an interval, a, b ∈ I and α ∈ R. Let f, g be locally integrable functions on I, and k 1 , k 2 ∈ R. We have the following: Given a, b ∈ I (b > a), let us denote by F a,b the usual inner product in L 2 [a, b],

] with respect to the inner product F a,b is the operator
Proof. We obtain, by using integration by parts, a,b (g)). Let Then there exists h > 0 such that (4) has a unique solution on [t 0 − h, t 0 + h] ∩ I. Furthermore, if Ω = R n and F is in (C, Lip) on J × R n for each compact interval J ⊆ I, then (4) has a unique solution on I.
The study of boundedness of solutions of a differential equation, either fractional or not, plays an important role in qualitative theory. In addition, the qualitative behavior of solutions is very important in applied research. Using the previous propositions one can check that.
Let I be an interval with [t 0 , ∞) ⊆ I and α ∈ (0, 1]. Let p be a continuous function on [t 0 , ∞). Then the initial value problem has a unique solution We have the following: Proof. The initial value problem (8) with y(t 0 ) = y 0 has a unique solution Since J α T,t 0 (p) is a continuous function, if lim inf t→∞ J α T,t 0 (p)(t) > −∞, then there exists a constant M with J α T,t 0 (p)(t) ≥ M for every t ∈ [t 0 , ∞). Hence, |y(t)| ≤ |y 0 |e −M for every t ∈ [t 0 , ∞). Therefore, for any ε > 0, if we choose y 0 with |y 0 | < εe M , then |y(t)| < ε for every t ∈ [t 0 , ∞), and so, the trivial solution is stable.

THE GAUSSIAN MODEL INVOLVING THE GENERALIZED FRACTIONAL DERIVATIVE
This section is framed in the context of the fractional Gaussian model presented in the following, with b, c ∈ R + and α ∈ (0, 1]: A broad range of dynamic systems arising in physical problems are depicted by this model. For a more complete study of the Gaussian model, we will consider some particular cases of interest, in the kernel of definition of the derivative G. Thus, we can consider (among others) the following cases: Case (a): T (t, α) = e (1−α)t . The solution of the system (10) is The solution of the system (10) is Note that if α → 1 in (12) and (11), then the solution of the ordinary differential equation The fractional Gaussian model is studied by using Caputo fractional derivative, through its numerical relation with Grünwald-Letnikov fractional derivative (see Ref. 20). Figure 1 shows the curves associated with a direct-type problem for a Gaussian model, with the following values of the parameters: a = 1, b = 1, c = 1 and α = 0.85, for the approaches namely ordinary derivative (black ), fractional conformable derivative according to T (t, α) = e (1−α)t (red), Khalil et al. (blue) and Caputo fractional derivative (green).
An observation equation can be related to the model: where each y i corresponds to the ith observed value under uncertainty from a solution of (10) 2040038-7 According to Bayesian statistical inversion theory, and provided that all information available has been assimilated into the model, the solution to an inverse problem is given by the subsequent distribution of the quantity of interest. For the model defined in (10), the parameter of interest is θ = (a, b, c, α). Next, some prior distributions are proposed, where G(r, λ) denotes the Gamma distribution with shape parameter r and rate parameter λ, U is the continuous uniform distribution on the interval (0, 1). The parameters introduced in the prior distributions are called hyperparameters. As mentioned before, the prior distributions have been defined in order to reflect the information already known about the possible values of the parameters of interest. In the model (10), a, b and c are positive and α ∈ (0, 1]. Taking for granted the prior independence of the parameters, the joint prior distribution can be written as where p (a | r a , λ a ), p(b | r b , λ b ), p(c | r c , λ c ) are p(α) are defined in (17)- (20), respectively. Let y = (y 1 , y 2 , . . . , y n ) denote the i.i.d. observed data at times (t 1 , t 2 , . . . , t n ) from the model defined by (10) and (16), the likelihood function is given by where f (t i ), i = 1, . . . , n, is a solution of (10). Then, applying Bayes' theorem, the posterior distribution of the parameters of interest is given by Adopting a loss quadratic function, the Bayesian point estimation is the posterior mean ofθ B , which is given byθ Markov chain Monte Carlo (MCMC) simulations are employed to work with (24). Among the most For this study, we used data on the dengue fever from the work by Quereshi and Atangana 15 and the kernel function used to define the conformable derivative was T (t, α) = e (1−α)t . Figure 2 shows the adjustments corresponding to the observations (black points) associated with dengue fever, according to the different fractional approaches.
The errors in the adjustments according to the different approaches are the following: Khalil et al.

CONCLUSIONS
Fractional calculus is successfully used to model a broad range of problems, since the evolution of many physical processes can be more precisely described with fractional derivatives. We study some general properties of fractional Gaussian model with the conformable fractional derivative. In particular, we used a generalized conformable fractional derivative [G α T with T (t, α) = e (1−α)t ] in order to study a Gaussian model. Taking into account an experimental dataset, we solve an inverse problem to estimate the order of the involved fractional derivative. From the above results, it is inferred that the conformable approach proposed by us minimizes the error in the adjustments in relation to the other models studied.