Analysis of dengue model with fractal-fractional Caputo–Fabrizio operator

In this work, we study the dengue dynamics with fractal-factional Caputo–Fabrizio operator. We employ real statistical data of dengue infection cases of East Java, Indonesia, from 2018 and parameterize the dengue model. The estimated basic reduction number for this dataset isR0 ≈ 2.2020. We briefly show the stability results of the model for the case when the basic reproduction number isR0 < 1. We apply the fractal-fractional operator in the framework of Caputo–Fabrizio to the model and present its numerical solution by using a novel approach. The parameter values estimated for the model are used to compare with fractal-fractional operator, and we suggest that the fractal-fractional operator provides the best fitting for real cases of dengue infection when varying the values of both operators’ orders. We suggest some more graphical illustration for the model variables with various orders of fractal and fractional.


Dengue model transmission
In this present section, we describe a host-vector model for dengue transmission. The host-vector model is divided into three mosquito populations, susceptible (S m ), exposed (E m ), and infectious (I m ), and five human (host) populations, susceptible (S h ), exposed (E h ), infectious (I h ), hospitalized and/or notified infectious (P h ), and recovered (R h ). Thus, the total human population, denoted by N h , is given as N h = S h + E h + I h + P h + R h , where the total dynamics of mosquitos, denoted by N m , is N m = S m + E m + I m . The system of differential equations that describes the host-vector model is written as (2) In model (1) above, the populations of vectors and humans are denoted by Π m and Π h . The natural death rate of mosquitoes and humans are given by μ m and ν h . The parameter τ measures the biting rate per mosquito per person. The transmission probability from an infected human to susceptible mosquitos, and the transmission probability from infected mosquitos to susceptible humans are shown respectively by α h and α v . The intrinsic incubation period of mosquitos is given by β m , and that of the humans is shown by β h . The human infected cases that are notified and get hospitalization increase at a rate ψ h . The infected human recover naturally at a rate ω h while the hospitalized individuals get recovery at a rate σ h . The hospitalized individuals die due to disease at a rate τ h . The complete details of the parameter descriptions of the dengue model (1) are presented briefly in Table 1.

Positivity and boundedness of the solution
In order to show that the dengue model given by (1) is epidemiologically meaningful, we have to prove that the associated state variables of the model stay nonnegative. It can also Natural death rate of human be explained that the solution of the dengue model with nonnegative initial conditions will remain nonnegative for every time greater than zero. We have the following lemma.
Then the solutions of the model given by (1) are nonnegative for every time t > 0. Further, Proof Consider t 1 = sup{t > 0 : G(t) > 0}. So, t 1 > 0. The first equation of the dengue model (1) leads to the following: with , then the above equation (3) becomes The equation given by (4) can be expressed further as follows: Hence, so that Similar steps can be followed for the rest of the equations of the dengue model (1) that is, G(t) > 0 for every t > 0. In order to show the second part of the result, note that 0 < . Now, summing the mosquito and human compartments leads to the following: which is the required claim.
Next, we show the invariant regions for the given dengue model (1). Consider the feasible We have the following results for this feasible region. (1) with the nonnegative initial conditions in (2).

Lemma 2 The region given by
Proof The summation of the mosquito and human populations of the dengue model (1) leads to

Preliminaries
This section explores the related results regarding fractal-fractional calculus, the model construction in fractal-fractional Caputo-Fabrizio operator and model stability results. We consider first the basic of fractal-fractional (FF) calculus by following the literature in [34,35].

Definition 1 If a function g(t)
is continuous and fractally differentiable over the given interval (a 1 , a 2 ) with order θ 2 , then the definition of the FF derivative of g(t) with order θ 1 in Riemann-Liouville sense with exponentially decaying kernel is given by: with θ 1 > 0, θ 2 ≤ m ∈ N and M(0) = M(1) = 1.

Definition 2 For a function g(t)
which is continuous and fractally differentiable over the given interval (a 1 , a 2 ), the definition of FF integral of g(t) with order θ 1 and exponentially decaying kernel is given by:

Fractal-fractional dengue model
In the present subsection, we apply the fractal-fractional operator in the sense of Caputo-Fabrizio to the dengue model (1) described above. And we have where θ 1 and θ 2 respectively represent the fractional and fractal order. The model (10) described above for the dengue infection using the fractal-fractional Caputo-Fabrizio operator is used further to obtain its numerical solution by using a novel numerical procedure in the following section.

Stability analysis of the disease-free case
This section explores the stability results for the dengue model given at the disease-free equilibrium (DFE) at E 0 . We set the right-hand side of the dengue model (10) equal to zero and obtain the following expressions: The stability of DFE at E 0 can be analyzed by using the next generation matrix of the dengue model (10). Considering the infected compartments in the dengue model (10), namely E m , I m , E h , I h , P h , and following the instructions given in [36], the matrices F and V are obtained as follows: The required basic reproduction number of the given model is obtained through the spectral radius of the matrix R 0 = ρ(FV -1 ), which is given by the following expression: In the following theorem, we show that the dengue model given by (10) is locally asymptotically stable at E 0 . We give the following result: Theorem 1 The dengue model given by (10) at E 0 is locally asymptotically stable whenever R 0 < 1.
Proof In order to prove the given theorem, we need to obtain the Jacobian matrix by evaluating the model (10) at E 0 , and we have It can be seen from the above matrix J(E 0 ) that the eigenvalues -μ m , -ν h , -ν h , and -π 4 are obviously negative, while the remaining four eigenvalues with negative real parts can be obtained through the following equation: The coefficients given by i for i = 1, 2, . . . , 4 are obviously positive for i , i = 1, 2, 3 while 4 can be positive or negative based on the value of R 0 . For the DFE case, the value of the basic reproduction number should be less than 1, so the last coefficient is positive when R 0 < 1. So, for all the coefficients i for i = 1, 2, . . . , 4 to be positive, they should satisfy the Rough-Hurtwiz criterion, which is easy to be satisfied, for the conditions supplied, Thus, the condition of Rough-Hurtwiz criterion ensures the local asymptotic stability of the dengue model given by (10) at E 0 .

Endemic equilibria and their stability
This subsection presents the endemic equilibria of the dengue model (10) where Using the expressions given by (11) in (12), this leads to the following: Here, g 1 > 0, g 3 depends on the sign of R 0 , and is positive when R 0 < 1 and negative when R 0 > 1. We establish the following result: Theorem 2 The dengue model given by (10) has the following properties: (i) If g 3 < 0 and R 0 > 1, then there exists a unique endemic equilibrium; (ii) If g 2 < 0 and g 3 = 0, then we have a unique endemic equilibrium; (iii) If g 3 > 0, g 2 < 0, and their discriminant is positive, then two endemic equilibria exist; and (iv) there are no equilibria otherwise.
It can be seen from the first point (i) of Theorem 2 that for the case of R 0 > 1, we clearly have a unique positive endemic equilibrium. Theorem 2(iii) shows the possible existence of the backward bifurcation when R 0 < 1.

Novel solution procedure for fractal-fractional model
For the numerical solution of the fractal-fractional dengue model (10), we present a novel procedure here that is based on the Adams-Bashforth technique. In order to have numerical scheme for the fractal-fractional model (10), we first express the model (10) in the following form: We apply the CF integral on equation (13), which leads to the following: We are presenting now a novel approach for the above model and use the approach at t n+1 . We have the following: At t n+1 , we have the following: The following is obtained by taking the difference between consecutive terms: -

Parameter estimation of dengue model
In the present section, we estimate the parameters of the model (10) for the case when θ 1 = θ 2 = 1 of dengue fever cases in East Java, Indonesia, for the year 2018. Based on data obtained from the East Java provincial health office, it was reported that the incidence rate of Dengue Hemorrhagic Fever (DHF) in East Java in 2016 was 64.8 per 100,000 people, an increase compared to 2015 which was 54.18 per 100,000 people, while the incidence rate of DHF in 2017 was 20 per 100,000 people [37]. Although in 2017 the number of cases of DHF decreased compared to the previous year, the awareness of the surge in cases in the next year needs to be improved.  Table 2. The result of fitting the model (10) to the actual data of dengue incidence is displayed in Fig. 1. Using the parameter values in Table 2, the basic reproduction number in East Java is R 0 ≈ 2.2020. The parameters listed in Table 2 are used further to obtain the graphical results for the fractal-fractional model (10).

Simulation results
In this subsection, we apply the novel approach considered above for the fractal-fractional model of dengue in the sense of Caputo-Fabrizio operator to illustrate its numerical results graphically. Throughout these simulation results, the subgraphs in Figs. 2, 4, 6, 8, which represent respectively the dynamics of susceptible, exposed, and infected mosquitos and the susceptible humans, while the subgraphs in Figs. 3, 5, 7, 9 show respectively the components of the exposed, infected, hospitalized, and recovered humans.    where (a) exposed humans, (b) infected humans, (c) hospitalized humans, (d) recovered humans   We use the estimated and fitted parameters in Table 2 for the simulation results and the units are taken in months. We present the graphical results using the fractal and fractional order parameters θ 1 and θ 2 in different scenarios. Initially, we choose θ 1 as varying, fix the fractal order θ 2 and obtain the graphical results shown in Figs. 2 and 3. We obtain Figs. 4 and 5 by choosing the fractal order as varying and fixing the fractional order. Figures 6 and 7 are obtained by setting equal values to the fractional and fractal orders, θ 1 and θ 2 . Figures 7 and 9 are presented for different orders of fractal and fractional parameters simultaneously. We observe from these numerical results that by varying the fractal and fractional orders, the dynamics of infected compartments in both humans and mosquitos decrease much faster than that when using integer order equations, either by varying only fractal order or only fractional orders. Thus, it is concluded that the fractalfractional operator provides a better understanding for the epidemic disease model with real data.

Conclusions
We obtained a dengue fever model in the framework of fractal-fractional operator. We obtained the basic reproduction number for the proposed dengue model equal to R 0 ≈ 2.2020 for the real cases of East Java, Indonesia, for 2018. We presented the model stability results and found that the model is locally asymptotically stable at the disease-free case when R 0 < 1. Then, the application of fractal-fractional operator in the sense of Caputo-Fabrizio was applied to the model, and we obtained a generalized model. The latter model was then used to present a novel numerical procedure and solution. The numerical re-sults for the fractal and fractional orders have been compared with the real cases of dengue fever, and we have found that, by varying the values of the fractal and fractional orders arbitrary, the best fitting has been obtained. Moreover, we used the fractal and fractional order parameter values and presented numerous graphical results for the model. We fixed the fractal order and varied the fractional order, showing the graphical results. We also fixed the fractional order and varied the fractal order to obtain the numerical results. Similarly, we varied both fractal and fractional orders equally and unequally and obtained many graphical results. Based on these numerical results, we came to a conclusion that varying both fractal and fractional orders provides the best results for the minimization of infected compartments in mosquitoes and humans and for the increase of the noninfected compartments of humans and mosquitos by decreasing the fractal and fractional orders. From the present analysis of the dengue infection model, we suggest to the readers that the application of fractal-fractional operators to a real life problem provides better results than using the ordinary order.