On fractional order dengue epidemic model

Research Article On Fractional Order Dengue Epidemic Model Hamed Al-Sulami,1Moustafa El-Shahed,1,2 Juan J. Nieto,1,3 and Wafa Shammakh1 1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2Department of Mathematics, Faculty of Art and Sciences, Qassim University, P.O. Box 3771, Unaizah-Qassim 51911, Saudi Arabia 3Departamento de Analisis Mateatico, Facultad de Matematicas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain Correspondence should be addressed to Hamed Al-Sulami; [email protected] Received 17 April 2014; Accepted 23 June 2014; Published 18 August 2014 Academic Editor: J. A. Tenreiro Machado Copyright © 2014 Hamed Al-Sulami et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper deals with the fractional order dengue epidemic model. The stability of disease-free and positive fixed points is studied. Adams-Bashforth-Moulton algorithm has been used to solve and simulate the system of differential equations. 1. Introduction Dengue is a major public health problem in tropical and subtropical countries. It is a vector-borne disease transmitted by Aedes aegypti and Aedes albopictus mosquitoes. Four different serotypes can cause dengue fever. A human infected by one serotype, when recovers, gains total immunity to that serotype and only partial and transient immunity with respect to the other three. Dengue can vary from mild to severe. The more severe forms of dengue include shock syndrome and dengue hemorrhagic fever (DHF). Patients who develop these more serious forms of dengue fever usually need to be hospitalized. The full life cycle of dengue fever virus involves the role of the mosquito as a transmitter (or vector) and humans as the main victim and source of infection. Preventing or reducing dengue virus transmission depends entirely on the control of mosquito vectors or interruption of human vector contact [1,2]. In this paper we study the fractional order dengue epidemic model. The stability of equilibrium points is studied. Numerical solutions of this model are given. We like to argue that fractional order equations are more suitable than integer order ones in modeling biological, economic, and social systems (generally complex adaptive systems) where memory effects are important. Adams-Bashforth-Moulton algorithm hasbeenusedtosolveandsimulatethesystemofdifferential equations. 2. Model Derivation Esteva and Vargas [3] developed a dengue fever transmission model by assuming that, once a person recovers from the disease, he or she will not be reinfected by the disease. The model also assumes that the host population 𝑁ℎis constant, that is, the death rate and the birth rate equal 𝜇𝐻.Thehostvector model for the dengue transmission of Esteva and Vargas [3]isasfollows: 𝑑𝑆ℎ 𝑑𝑡 =𝐴−𝛽ℎ𝑏 𝑁ℎ𝑆ℎ𝐼V−𝜇ℎ𝑆ℎ, 𝑑𝐼ℎ 𝑑𝑡=𝛽ℎ𝑏 𝑁ℎ𝑆ℎ𝐼V−(𝜇ℎ+𝛾)𝐼ℎ, 𝑑𝑅ℎ 𝑑𝑡 =𝛾𝐼ℎ−𝜇ℎ𝑅ℎ, 𝑑𝑆V 𝑑𝑡=𝐵−𝛽V𝑏 𝑁ℎ𝑆V𝐼ℎ−𝜇V𝑆V, 𝑑𝐼V 𝑑𝑡=𝛽V𝑏 𝑁ℎ𝑆V𝐼ℎ−𝜇V𝐼V, (1) where 𝐴is the recruitment rate of the host population, 𝐵is the recruitment rate of the vector population, Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 456537, 6 pages http://dx.doi.org/10.1155/2014/456537 2Mathematical Problems in Engineering 𝑆ℎis the number of susceptible in the host population, 𝐼ℎis the number of infective in the host population, 𝑅ℎis the number of immunes in the host population, 𝑁Vis the vector population, 𝑆Vis the number of susceptible in the vector population, 𝐼Visthenumberofinfectiveinthevectorpopulation, 𝜇Vis the death rate in the vector population, 𝛽ℎis the transmission probability from vector to host, 𝛽Vis the transmission probability from host to vector, 𝛾is the recovery rate in the host population, 𝑏isthebitingrateofthevector. The notion of fractional calculus was anticipated by Leibniz, one of the founders of standard calculus, in a letter written in 1695. Recently great considerations have been made to the models of FDEs in different aria of researches. The most essential property of these models is their nonlocal property which does not exist in the integer order differential operators. We mean by this property that the next state of a model depends not only upon its current state but also upon all of its historical states. There are many definitions of fractional derivatives [4,5]. Perhaps the best-known is the Riemann-Liouvile definition. The Riemann-Liouville derivative of order 𝛼is defined as RL𝐷𝛼 0+𝑓(𝑡)=1 Γ(𝑛−𝛼)(𝑑 𝑑𝑡)𝑛∫𝑡 0𝑓(𝑠) (𝑡−𝑠)𝛼−𝑛+1 𝑑𝑠, 𝑛=[𝛼]+1, (2) where Γ() is the gamma function and 𝑛is an integer. An alternative definition was introduced by Caputo as follows, which is a sort of regularization of the Riemann-Liouville derivative: 𝐷𝛼 𝑡𝑓(𝑡)=1 Γ(𝑛−𝛼)∫𝑡 0𝑓(𝑛) (𝑠) (𝑡−𝑠)𝛼−𝑛+1 𝑑𝑠. (3) Pooseh et al. [6] introduced the notion of fractional derivative in the sense of Riemann-Liouville to reformulate the dynamics of the classical model (1) in terms of fractional derivatives. They applied a recent approximate technique to obtain numerical solutions to the fractional model. The system in this paper will be in the sense of Caputo fractional derivative by the following set of fractional order differential equations: 𝐷𝛼 𝑡𝑆ℎ=𝐴−𝛽ℎ𝑏 𝑁ℎ𝑆ℎ𝐼V−𝜇ℎ𝑆ℎ, 𝐷𝛼 𝑡𝐼ℎ=𝛽ℎ𝑏 𝑁ℎ𝑆ℎ𝐼V−(𝜇ℎ+𝛾)𝐼ℎ, 𝐷𝛼 𝑡𝑅ℎ=𝛾𝐼ℎ−𝜇ℎ𝑅ℎ, 𝐷𝛼 𝑡𝑆V=𝐵−𝛽V𝑏 𝑁ℎ𝑆V𝐼ℎ−𝜇V𝑆V, 𝐷𝛼 𝑡𝐼V=𝛽V𝑏 𝑁ℎ𝑆V𝐼ℎ−𝜇V𝐼V. (4) Because model (4) monitors the dynamics of human populations, all the parameters are assumed to be nonnegative. Furthermore, it can be shown that all state variables of the model are nonnegative for all time 𝑡≥0(see, for instance, [7–9]). Lemma 1. The closed set Ω={(𝑆ℎ,𝐼ℎ,𝑅ℎ,𝑆V,𝐼V)∈𝑅5 +:𝑆ℎ+ 𝐼ℎ+𝑅ℎ=𝐴/𝜇ℎ,𝑆V+𝐼V=𝐵/𝜇V}is positively invariant with respect to model (4). Proof. The fractional derivative of the total population, obtained by adding all the equations of model (4), is given by 𝐷𝛼 𝑡𝑁ℎ(𝑡)=𝐴−𝜇ℎ𝑁ℎ(𝑡).(5) The solution to (5)isgivenby𝑁ℎ(𝑡) = 𝑁ℎ(0)𝐸𝛼,1(−𝜇ℎ𝑡𝛼)+𝐴𝑡 𝛼𝐸𝛼,𝛼+1(−𝜇ℎ𝑡𝛼),where𝐸𝛼,𝛽 is the Mittag-Leffler function. Considering the fact that the Mittag-Leffler function has an asymptotic behavior [4,10], 𝐸𝛼,𝛽 (𝑧)∼−𝜔 ∑ 𝑘=1 𝑧−𝑘 Γ(𝛽−𝛼𝑘)+𝑂(|𝑧|−1−𝜔), (|𝑧|󳨀→∞,𝛼𝜋 2<󵄨󵄨󵄨󵄨arg (𝑧)󵄨󵄨󵄨󵄨≤𝜋). (6) One can observe that 𝑁ℎ(𝑡) → 𝐴/𝜇ℎas 𝑡→∞.The proofofvectorpopulationcaseiscompletelysimilartothat of host population and is therefore omitted. One can observe that 𝑁V(𝑡)→𝐵/𝜇V. Therefore, all solutions of the model with initial conditions in Ωremain in Ωfor all 𝑡>0. Thus, region Ωis positively invariant with respect to model (4). Inthefollowing,wewillstudythedynamicsofsystem(4). 3. Equilibrium Points and Stability To evaluate the equilibrium points let 𝐷𝛼 𝑡𝑆ℎ=0, 𝐷𝛼 𝑡𝐼ℎ=0, 𝐷𝛼 𝑡𝑅ℎ=0, 𝐷𝛼 𝑡𝑆V=0, 𝐷𝛼 𝑡𝐼V=0. (7) Mathematical Problems in Engineering 3 Then 𝐸0=(𝐴/𝜇ℎ,0,0,(𝐵/𝜇V)0).By(4), a positive equilibrium 𝐸1=(𝑆1 ℎ,𝐼1 ℎ,𝑅1 ℎ,𝑆1 V,𝐼1 V)satisfies 𝑆1 ℎ=𝑁ℎ𝜇V(𝐴𝑏𝛽V+𝑁ℎ(𝛾+𝜇ℎ)𝜇V) 𝑏𝛽V(𝑏𝐵𝛽ℎ+𝑁ℎ𝜇ℎ𝜇V), 𝐼1 ℎ=𝐴𝑏2𝐵𝛽ℎ𝛽V−𝑁2 ℎ𝜇ℎ(𝛾+𝜇ℎ)𝜇2 V 𝑏𝛽V(𝛾+𝜇ℎ)(𝑏𝐵𝛽ℎ+𝑁ℎ𝜇ℎ𝜇V), 𝑅1 ℎ=𝐴𝑏2𝐵𝛾𝛽ℎ𝛽V−𝛾𝑁2 ℎ𝜇ℎ(𝛾+𝜇ℎ)𝜇2 V 𝑏𝛽V𝜇ℎ(𝛾+𝜇ℎ)(𝑏𝐵𝛽ℎ+𝑁ℎ𝜇ℎ𝜇V), 𝑆1 V=𝑁ℎ(𝛾+𝜇ℎ)(𝑏𝐵𝛽ℎ+𝑁ℎ𝜇ℎ𝜇V) 𝑏𝛽ℎ(𝐴𝑏𝛽V+𝑁ℎ(𝛾+𝜇ℎ)𝜇V), 𝐼1 V=𝐴𝑏2𝐵𝛽ℎ𝛽V−𝑁2 ℎ𝜇ℎ(𝛾+𝜇ℎ)𝜇2 V 𝑏𝛽ℎ𝜇V(𝐴𝑏𝛽V+𝑁ℎ(𝛾+𝜇ℎ)𝜇V). (8) The Jacobian matrix 𝐽(𝐸0)for the system given in (4)evaluated at the disease-free equilibrium is as follows: 𝐽(𝐸0)=( ( ( ( ( ( ( ( ( ( ( ( −𝜇ℎ000− 𝐴𝑏𝛽ℎ 𝑁ℎ𝜇ℎ 0−𝛾−𝜇 ℎ00𝐴𝑏𝛽ℎ 𝑁ℎ𝜇ℎ 0𝛾−𝜇 ℎ00 0− 𝑏𝐵𝛽V 𝑁ℎ𝜇V0−𝜇 V0 0𝑏𝐵𝛽V 𝑁ℎ𝜇V00−𝜇 V ) ) ) ) ) ) ) ) ) ) ) ) .(9) Theorem 2. The disease-free equilibrium 𝐸0is locally asymptotically stable if 𝑅0<1and is unstable if 𝑅0>1. Proof. The disease-free equilibrium is locally asymptotically stable if all the eigenvalues, 𝜆𝑖,𝑖=1,2,3,4,5of the Jacobian matrix 𝐽(𝐸0)satisfy the following condition [11–14]: 󵄨󵄨󵄨󵄨arg (𝜆𝑖)󵄨󵄨󵄨󵄨>𝛼𝜋 2.(10) The eigenvalues of the Jacobian matrix 𝐽(𝐸0)are 𝜆1= −𝜇ℎ,𝜆2=−𝜇 ℎ,and𝜆3=−𝜇 V; the other two roots are determined by the quadratic equation 𝜆2+𝜆(𝛾+𝜇ℎ+𝜇V)+𝜇V(𝛾+𝜇ℎ)(1−𝑅0)=0, (11) where 𝑅0=(𝐴𝐵𝑏2𝛽ℎ𝛽V)/(𝜇ℎ𝜇2 V𝑁2 ℎ(𝛾+𝜇ℎ)).Hence𝐸0is locally asymptotically stable if 𝑅0<1and is unstable if 𝑅0>1. The quantity 𝑅∗ 0=√𝑅0is called the basic reproductive number of the disease, since it represents the average number of secondary cases that one case can produce if introduced into a susceptible population. We now discuss the asymptotic stability of the endemic (positive) equilibrium of the system given by (4). The Jacobian matrix 𝐽(𝐸1)evaluated at the endemic equilibrium is given as 𝐽(𝐸1)( ( ( ( ( ( ( ( ( ( ( ( −𝑏𝛽ℎ𝐼1 V 𝑁ℎ−𝜇ℎ00 0− 𝑏𝑆1 ℎ𝛽ℎ 𝑁ℎ 𝑏𝛽ℎ𝐼1 V 𝑁ℎ−𝛾−𝜇ℎ00𝑏𝑆1 ℎ𝛽ℎ 𝑁ℎ 0𝛾−𝜇 ℎ00 0− 𝑏𝑆1 ℎ𝛽V 𝑁ℎ0− 𝑏𝛽V𝐼1 ℎ 𝑁ℎ−𝜇V0 0𝑏𝑆1 V𝛽V 𝑁ℎ0𝑏𝛽V𝐼1 ℎ 𝑁ℎ−𝜇V ) ) ) ) ) ) ) ) ) ) ) ) . (12) The characteristic equation of 𝐽(𝐸1)is (𝜆+𝜇ℎ)(𝜆+𝜇V)(𝜆3+𝑎1𝜆2+𝑎2𝜆+𝑎3)=0, (13) where𝑎1=𝛾+𝜇V+𝑀2(𝐾𝜃+𝑀1𝜇V) 𝐾𝜃𝑀1𝑁ℎ𝜇V, 𝑎2=(𝑀2 2+𝐾𝜃𝑀1𝑁2 ℎ𝜇ℎ𝜇V(𝜃+𝜇V) +𝑀2𝑁ℎ(𝐾𝜃2+(𝐾𝜃+𝑀1(𝜃+𝜇ℎ))𝜇V)) ×(𝐾𝜃𝑀1𝑁2 ℎ𝜇V)−1, 𝑎3=𝑀2(𝑀2+𝑁ℎ(𝐾𝜃+𝑀1𝜇ℎ)𝜇V) 𝐾𝑀1𝑁2 ℎ𝜇V, 𝑀1=𝐴𝑏2𝐵𝛽ℎ𝛽V−𝑁2 ℎ𝜇ℎ(𝛾+𝜇ℎ)𝜇2 V, 𝑀2=𝐴𝑏𝛽V+𝑁ℎ(𝛾+𝜇ℎ)𝜇V, 𝐾=𝑏𝐵𝛽ℎ+𝑁ℎ𝜇ℎ𝜇V, 𝜃=𝛾+𝜇ℎ. (14) If 𝑝(𝑥)=𝑥3+𝑎1𝑥2+𝑎2𝑥+𝑎3.Let𝐷(𝑝)denote the discriminantofapolynomialp(x);then 𝐷(𝑝)=−󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 1𝑎 1𝑎2𝑎30 01𝑎 1𝑎2𝑎3 32𝑎1𝑎200 03𝑎 1𝑎20 0032𝑎 1𝑎2 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 =18𝑎1𝑎2𝑎3+(𝑎1𝑎2)2−4𝑎3𝑎3 1−4𝑎3 2−27𝑎2 3.(15) Following [14–18], we have Proposition 3. Proposition 3. One assumes that 𝐸1exists in 𝑅3 +. (i) If the discriminant of 𝑝(𝑥),𝐷(𝑝),ispositiveandRouth- Hurwitz are satisfied, that is, 𝐷(𝑝)>0,𝑎1>0,𝑎3>0, and 𝑎1𝑎2>𝑎3,then𝐸1is locally asymptotically stable. 4Mathematical Problems in Engineering (ii) If 𝐷(𝑝)<0,𝑎1>0,𝑎2>0,𝑎1𝑎2=𝑎3,and𝛼∈[0,1), then 𝐸1is locally asymptotically stable. (iii) If 𝐷(𝑝)<0,𝑎1<0,𝑎2<0,and𝛼>2/3,then𝐸1is unstable. (iv) The necessary condition for the equilibrium point 𝐸1,to be locally asymptotically stable, is 𝑎3>0. 4. Numerical Methods and Simulations Since most of the fractional order differential equations do not have exact analytic solutions, so approximation and numerical techniques must be used. Several analytical and numerical methods have been proposed to solve the fractional order differential equations. For numerical solutions of the system (4) one can use the generalized Adams-Bashforth- Moulton method. To give the approximate solution by means of this algorithm, consider the following nonlinear fractional differential equation [19]: 𝐷𝛼 𝑡𝑦(𝑡)=𝑓(𝑡,𝑦(𝑡)), 0≤𝑡≤𝑇, 𝑦(𝑘) (0)=𝑦𝑘 0, 𝑘=0,1,2,...,𝑚−1, where 𝑚=[𝛼].(16) This equation is equivalent to Volterra integral equation: 𝑦(𝑡)=𝑚−1 ∑ 𝑘=0𝑦(𝑘) 0𝑡𝑘 𝑘!+1 Γ(𝛼)∫𝑡 0(𝑡−𝑠)𝛼−1𝑓(𝑠,𝑦(𝑠))𝑑𝑠. (17) Diethelm et al. used the predictor-correctors scheme [15, 16,20] based on the Adams-Bashforth-Moulton algorithm to integrate (17). By applying this scheme to the fractional order dengue epidemic model and setting ℎ=𝑇/𝑁,𝑡𝑛=𝑛ℎ,and 𝑛=0,1,2,,...,𝑁∈𝑍+,(17)canbediscretizedasfollows [19]: 𝑆𝑛+1 =𝑆0+ℎ𝛼 Γ(𝛼+2)(𝐴−𝛽ℎ𝑏ℎ 𝑁ℎ𝑆𝑝 𝑛+1𝑌𝑝 𝑛+1 −𝜇ℎ𝑆𝑝 𝑛+1) +ℎ𝛼 Γ(𝛼+2)𝑛 ∑ 𝑗=0𝑎𝑗,𝑛+1 (𝐴−𝛽ℎ𝑏ℎ 𝑁ℎ𝑆𝑗𝑌𝑗−𝜇ℎ𝑆𝑗), 𝐼𝑛+1 =𝐼0+ℎ𝛼 Γ(𝛼+2)(𝛽ℎ𝑏ℎ 𝑁ℎ𝑆𝑝 𝑛+1𝑌𝑝 𝑛+1 −(𝜇ℎ+𝛾)𝐼𝑝 𝑛+1) +ℎ𝛼 Γ(𝛼+2)𝑛 ∑ 𝑗=0𝑎𝑗,𝑛+1 (𝛽ℎ𝑏ℎ 𝑁ℎ𝑆𝑗𝑌𝑗−(𝜇ℎ+𝛾)𝐼𝑗), 𝑅𝑛+1 =𝑅0+ℎ𝛼 Γ(𝛼+2)(𝛾𝐼𝑝 𝑛+1 −𝜇ℎ𝑅𝑝 𝑛+1) +ℎ𝛼 Γ(𝛼+2)𝑛 ∑ 𝑗=0𝑎𝑗,𝑛+1 (𝛾𝐼𝑗−𝜇ℎ𝑅𝑗), 𝑋𝑛+1 =𝑋0+ℎ𝛼 Γ(𝛼+2)(𝐵−𝛽V𝑏ℎ 𝑁ℎ𝑋𝑝 𝑛+1𝐼𝑝 𝑛+1 −𝜇ℎ𝑋𝑝 𝑛+1) +ℎ𝛼 Γ(𝛼+2)𝑛 ∑ 𝑗=0𝑎𝑗,𝑛+1 (𝐵−𝛽V𝑏ℎ 𝑁ℎ𝑋𝑗𝐼𝑗−𝜇V𝑋𝑗), 𝑌𝑛+1 =𝑌0+ℎ𝛼 Γ(𝛼+2)(𝛽V𝑏ℎ 𝑁ℎ𝑋𝑝 𝑛+1𝐼𝑝 𝑛+1 −𝜇ℎ𝑌𝑝 𝑛+1) +ℎ𝛼 Γ(𝛼+2)𝑛 ∑ 𝑗=0𝑎𝑗,𝑛+1 (𝛽V𝑏ℎ 𝑁ℎ𝑋𝑗𝐼𝑗−𝜇V𝑌𝑗), (18) where 𝑆𝑝 𝑛+1 =𝑆0+1 Γ(𝛼)𝑛 ∑ 𝑗=0𝑏𝑗,𝑛+1 (𝐴−𝛽ℎ𝑏ℎ 𝑁ℎ𝑆𝑗𝑌𝑗−𝜇ℎ𝑆𝑗), 𝐼𝑝 𝑛+1 =𝐼0+1 Γ(𝛼)𝑛 ∑ 𝑗=0𝑏𝑗,𝑛+1 (𝛽ℎ𝑏ℎ 𝑁ℎ𝑆𝑗𝑌𝑗−(𝜇ℎ+𝛾)𝐼𝑗), 𝑅𝑝 𝑛+1 =𝑅0+1 Γ(𝛼)𝑛 ∑ 𝑗=0𝑏𝑗,𝑛+1 (𝛾𝐼𝑗−𝜇ℎ𝑅𝑗), 𝑋𝑝 𝑛+1 =𝑋0+1 Γ(𝛼)𝑛 ∑ 𝑗=0𝑏𝑗,𝑛+1 (𝐵−𝛽V𝑏ℎ 𝑁ℎ𝑋𝑗𝐼𝑗−𝜇V𝑋𝑗), 𝑌𝑝 𝑛+1 =𝑌0+1 Γ(𝛼)𝑛 ∑ 𝑗=0𝑏𝑗,𝑛+1 (𝛽V𝑏ℎ 𝑁ℎ𝑋𝑗𝐼𝑗−𝜇V𝑌𝑗), 𝑎𝑗,𝑛+1 ={ { { { { { { { { 𝑛𝛼+1 −(𝑛−𝛼)(𝑛+1),𝑗=0, (𝑛−𝑗+2)𝛼+1 +(𝑛−𝑗)𝛼+1 −2(𝑛−𝑗+1)𝛼+1,1≤𝑗≤𝑛, 1, 𝑗=𝑛+1, 𝑏𝑗,𝑛+1 =ℎ𝛼 𝛼((𝑛−𝑗+1)𝛼−(𝑛−𝑗)𝛼), 0≤𝑗≤𝑛. (19) 5. Discussion In this paper, we have considered a fractional calculus model for dengue disease. Following [21], Figure 1 shows that 𝑆ℎ drops significantly in a relatively small period of time. Both 𝐼ℎ and 𝐼Vincrease significantly during the period of 30 days and then eventually oscillate around the endemic state (0.09529, 0.0.00029, and 0.00058). This seems unrealistic in the nature. With constant population of mosquitoes, this fluctuation (in a short period of time) cannot be shown to happen in the nature [21]. As mentioned by [6], Figures 2and 3show that even a simple fractional model may give surprisingly good results. However, the transformation of a classical model into a fractional one makes it very sensitive to the order of differentiation 𝛼:asmallchangein𝛼may result in a big change in the final result. From the numerical results in Figures 2and 3, it is clear that the approximate solutions depend continuously on the fractional derivative 𝛼. Mathematical Problems in Engineering 5 10000 8000 6000 4000 2000 0 020 40 60 80 100 Sh(t) Ih(t) I(t) t Figure 1: 𝑆ℎ(𝑡),𝐼ℎ(𝑡),and𝐼V(𝑡)for 𝛼=1and 𝜇ℎ=0.0000457;𝜇V= 0.25;𝑏=0.5;𝛽ℎ=0.75;𝛽V=1;𝛾=0.1428;𝑁ℎ=10000;𝐵=5000. 020 40 60 80 100 0 2000 4000 6000 8000 10000 Sh(t) Ih(t) I(t) t Figure 2: 𝑆ℎ(𝑡),𝐼ℎ(𝑡),and𝐼V(𝑡)for 𝛼=0.9and 𝜇ℎ=0.0000457; 𝜇V=0.25;𝑏=0.5;𝛽ℎ=0.75;𝛽V=1;𝛾=0.1428;𝑁ℎ=10000; 𝐵=5000. 020 40 60 80 100 0 2000 4000 6000 8000 10000 Sh(t) Ih(t) I(t) t Figure 3: 𝑆ℎ(𝑡),𝐼ℎ(𝑡),and𝐼V(𝑡)for 𝛼=0.8and 𝜇ℎ=0.0000457; 𝜇V=0.25;𝑏=0.5;𝛽ℎ=0.75;𝛽V=1;𝛾=0.1428;𝑁ℎ=10000; 𝐵=5000. 20 40 60 80 100 2000 4000 6000 8000 t 𝛼=1 𝛼 = 0.9 𝛼 = 0.8 Figure 4: 𝑆ℎ(𝑡)for 𝛼=1,0.9,0.8and 𝜇ℎ=0.0000457;𝜇V=0.25; 𝑏=0.5;𝛽ℎ=0.75;𝛽V=1;𝛾=0.1428;𝑁ℎ=10000;𝐵=5000. 20 40 60 80 100 2000 4000 6000 8000 t 𝛼=1 𝛼 = 0.9 𝛼 = 0.8 Figure 5: 𝐼ℎ(𝑡)for 𝛼=1,0.9,0.8,𝜇ℎ=0.0000457;𝜇V=0.25;𝑏=0.5; 𝛽ℎ=0.75;𝛽V=1;𝛾=0.1428;𝑁ℎ=10000;𝐵=5000. The approximate solutions 𝑆ℎ(𝑡),𝐼ℎ(𝑡),and𝐼V(𝑡)are displayed in Figures 4and 5with different values of 𝛼.In each figure three different values of 𝛼are considered. When 𝛼=1,system(4) is the classical integer-order system (1). In Figure 4,thevariationof𝑆ℎ(𝑡)versus time 𝑡is shown for different values of 𝛼=1,0.9,0.8by fixing other parameters. It is revealed that 𝑆ℎdoes not drop significantly in a relatively small period of time for small values. Figure 5 depicts 𝐼ℎ(𝑡) versus time 𝑡.Asmentionedby[22,23], one should note that although the equilibrium points are the same for both integer order and fractional order models the solution of the fractional order model tends to the fixed point over a longer period of time. One also needs to mention that when dealing with real life problems, the order of the system can be determined by using the collected data. 6Mathematical Problems in Engineering Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant no. 3-130/1433HiCi. The authors, therefore, acknowledge with thanks DSR technical and financial support. References [1] H. S. Rodrigues, M. T. Monteiro, and D. F. M. Torres, “Sensitivity analysis in a dengue epidemiological model,” Conference Papers in Mathematics,vol.2013,ArticleID721406,7pages,2013. [2] WHO, Dengue: Guidelines for Diagnosis, Treatment, Prevention and Control, World Health Organization, Geneva, Switzerland, 2nd edition, 2009. [3] L. Esteva and C. 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