Fractional Order Model for the Spread of Leptospirosis

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1 International Journal of Mathematical Analysis Vol. 8, 214, no. 54, HIKARI Ltd, Fractional Order Model for the Spread of Leptospirosis Moustafa El-Shahed Department of Mathematics, Faculty of Art and Sciences Qassim University, P.O. Box 3771 Qassim, Unizah 51911, Saudi Arabia Copyright c 214 Moustafa El-Shahed. 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. Abstract This paper deals with the fractional order for the spread of Leptospirosis. The non-local property of Leptospirosis epidemic model presented by fractional order differential equation makes the model to be more realistic compare to the analogues integer order, which lacks this property. The stability of disease free and positive fixed points is studied. We show that the model introduced in this paper has non negative solutions. AdamsBashforthMoulton algorithm have been used to solve and simulate the system of differential equations. Mathematics Subject Classification: 92B5, 93A3, 93C15 Keywords: Leptospirosis, SIR model; Fractional order; Stability; Numerical method 1. Introduction Leptospirosis is a bacterial disease that affects both humans and animals. Humans become infected through direct contact with the urine of infected animals or with a urine-contaminated environment. The bacteria enter the body through cuts or abrasions on the skin, or through the mucous membranes of the mouth, nose and eyes. Person-to-person transmission is rare. In the early stages of the disease, symptoms include high fever, severe headache, muscle pain, chills, redness of the eyes, abdominal pain, jaundice, haemorrhages in the

2 2652 Moustafa El-Shahed skin and mucous membranes, vomiting, diarrhoea, and rash [26]. There are ten different types of Leptospira that cause disease in humans. In the developing world the disease most commonly occurs in farmers and poor people who live in cities. In the developed world it most commonly occurs in those involved in outdoor activities in warm and wet areas of the world [5, 15, 2, 24]. Mathematical modeling can provide valuable insights into the biological and epidemiological properties of infectious diseases as well as the potential impact of intervention strategies employed by health organizations worldwide. Solutions to systems of differential equations which model disease transmission are of particular use and importance to epidemiologists who wish to study effective means to slow and prevent the spread of disease [5]. The mathematical formulation and dynamical sketch of Leptospirosis has been studied by several authors. Pongsuumpun et al. [23] represents mathematical model and considered some real data for numerical simulation. A simple deterministic model for the spread of leptospirosis in Thailand can be found in [25]. In their work, they represented the rate of change for both rats and human population. The human population is further divided into two main groups Juveniles and adults. Zaman [27] considered the real data presented in [25] to study the dynamical behavior and role of optimal control theory. Pimpunchat et al. [21] proposed a modification of the SIR model[25]. The dynamical interaction between leptospirosis infected vector and human population is studied by Zaman et al. [28]. In their work, they presented global dynamics and bifurcation analysis. They also showed the numerical simulations for different values of the interaction parameter. In recent decades, the fractional calculus and fractional differential equations have attracted much attention and increasing interest due to their potential applications in science and engineering [14, 22]. In this paper, we consider the fractional order model for Leptospirosis diseases. We give a detailed analysis for the asymptotic stability of the model. Adams Bashforth-Moulton algorithm have been used to solve and simulate the system of differential equations. 2. Fractional calculus For the convenience of the reader, we present here the necessary definitions and properties from fractional calculus theory. These definitions and properties can be found in the literature [6, 7, 14, 16, 17, 22]. Definition 2.1. [22].The Riemann-Liouville fractional integral of order α > of a function f : (, ) R is defined by provided the integral exists. I α t f(t) = 1 Γ(α) t a (t s) α 1 f(s)ds,

3 Fractional calculus model 2653 Definition 2.2. [22].The fractional derivative of a continuous function f : [, ) R in the Caputo sense is defined as D α t f(t) = 1 Γ(n α) t f (n) (s) ds, n 1 < α < n, n Z+ (t s) α n+1 Lemma 2.3. [6]. Suppose that m(t) C p (R +, R)satisfies D α t m(t) θ m(t) + d, m(t ) = m, t t, where λ, d R. Then one has m(t) m(t ) E α (θ(t t ) α ) + d (t t ) α E α,α+1 (θ(t t ) α ), where E α,β (z) is the mittag-leffler function with two parameters. Lemma 2.4. [14]. When α >, then E α,β (z) has an asymptotic behavior at infinity for < α < 2 E α,β (z) = p k=1 z k Γ(β αk) + O(z 1 p ), ( z, α π 2 arg z π), Lemma 2.5. [6]. Let < α < 1 and λ <. Then E α,α (λ t α ) and E α,α+1 (λ t α ) tend monotonically to zero as t. Definition 2.6. [16, 17].The constant x is an equilibrium point of Caputo fractional dynamic system D α t x(t) = f(t, x), if and only if f(t, x ) =. Remark 2.7. When α (, 1), it follows that the Caputo fractional-order system D α t x(t) = f(t, x) has the same equilibrium points as the integer-order system x (t) = f(t, x). Lemma 2.8. If D α t x(t) and x(), < α < 1, then x(t). According to the properties of the fractional derivatives and Lemma 2.8, one obtain the comparison theorem of the fractional derivatives [7]. Theorem 2.9. [7] Suppose that < α < 1 and D α t v(t) D α t w(t) on R +. If v() w(), then v(t) w(t)on R Model formulation The total population sizes for the humans hosts and animal vectors are denoted by N h and N a, respectively. The human population N h is divided into the epidemiological subclasses: susceptible, infected and recovered denoted by S h, I h and R h, respectively. Thus, N h = S h + I h + R h. The vector population is denoted by N a consists of two classes, that is susceptible S a and infected I a

4 2654 Moustafa El-Shahed, and N a = S a + I a. The model consists of a system of non-linear differential equation is given by ds h dt = A µ h S h β h I a S h + λ h R h, di h dt = β h I a S h (µ h + δ h + γ h )I h, dr h dt = γ h I h (µ h + λ h ) R h, (3.1) ds a dt = B γ a S a β a S a I h, di a dt = β a S a I h (γ a + δ a ) I a, where A and B are the recruitment rate of human and vector population respectively. µ h is the natural death rate of human population and δ h is the rate immune individuals become susceptible S h again and the infectious human die due to disease at vector populations at the rate of δ h. β h is the rate of transmission of leptospirosis from an infected vector to a susceptible human, varying with rain fall. λ h is the rate immune individuals become susceptible again. γ a is the natural death rate of vector population. The infectious vector die due to disease at vector populations at the rate of δ a. β a is the rate of transmission of leptospirosis from an infected vector to a susceptible vector, varying with rain fall. Fractional order models are more accurate than integer-order models as fractional order models allow more degrees of freedom. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. The presence of memory term in such models not only takes into account the history of the process involved but also carries its impact to present and future development of the process. Fractional differential equations are also regarded as an alternative model to nonlinear differential equations. In consequence, the subject of fractional differential equations is gaining much importance and attention. For some recent work on fractional differential equations, see [6, 7, 12, 16, 17, 14, 22]. Now we introduce fractional order to the ODE model (3.2). The new system is described by the following set of fractional order differential equations: D α t S h = A µ h S h β h I a S h + λ h R h, D α t I h = β h I a S h (µ h + δ h + γ h )I h, D α t R h = γ h I h (µ h + λ h ) R h, (3.2) D α t S a = B γ a S a β a S a I h, D α t I a = β a S a I h (γ a + δ a ) I a,

5 Fractional calculus model 2655 where Dt α is the Caputo fractional derivative. Because model (3.2) 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 non-negative for all time t (see, for instance, [5, 11] Invariant regions. The fractional order Leptospirosis model (3.2) will be analyzed in a biologically-feasible region as follows. The system (3.2) is split into two parts, namely the human population (N h ; with N h = S h + I h + R h ) and the vector population (N a ; with N a = S a + I a ). Consider the feasible region D = D h D a R 3 + R 2 +, with D h = {(S h, E h, I h ) R 3 + : S h + I h + R h A µ h }, D a = {(S a, I a ) R 2 + : S a + I a B γ v } The following steps are done to establish the positive invariance of D (i.e., solutions in D remain in D for all t > ). Adding the first three equations and the last two equations of the model (3.2) gives Dt α N h (t) = A µ h N h (t) δ h I h (t), Dt α (3.3) N a (t) = B γ a N a (t) δ a I a (t). The fractional order of the humans and animals populations is given in equation (3.4), it follows that D α t N h (t) A µ h N h (t), D α t N a (t) B γ a N a (t). Lemma (2.3) can then be used to show that and N h (t) A t α E α,α+1 ( µ h t α ) + N h () E α,1 ( µ h t α ), N a (t) B t α E α,α+1 ( γ a t α ) + N a () E α,1 ( γ a t α ). (3.4) Base on Lemma (2.4), one can observe that N h (t) A µ h and N a (t) B γ a. Thus, the region D is positively-invariant. Hence, it is sufficient to consider the dynamics of the flow generated by (3.2) in D. In this region, the model can be considered as been epidemiologically and mathematically well-posed [13]. Thus, every solution of the basic model (3.1) with initial conditions in D remains in D for all t >. This result is summarized below. Lemma 3.1. The region D = D h D a R 3 + R 2 + is positively-invariant for the basic model (3.2) with non-negative initial conditions in R 5 +. In the following, we will study the dynamics of system (3.2).

6 2656 Moustafa El-Shahed 4. Equilibrium Points and Stability In the following, we discuss the stability of the commensurate fractional order dynamical system: D α t x i = f i (x 1, x 2,..., x m ), α (, 1), 1 i m. (4.1) Let E = (x 1, x 2,..., x m) be an equilibrium point of system (4.1)andx i = x i + θ i, where θ i is a small disturbance from a fixed point. Then Dt α θ i = Dt α x i = f i (x 1 + θ 1, x 2 + θ 2,..., x m + θ m ) f θ i (E) f 1 x 1 + θ i (E) f 2 x θ i (E) m x m. System(4.2) can be written as: (4.2) D α t θ = Jθ, (4.3) where θ = (θ 1, θ 2,..., θ m ) T and J is the Jacobian matrix evaluated at the equilibrium points. Using Matignon s results [19], it follows that the linear autonomous system (4.3) is asymptotically stable if arg(λ) > α π 2 is satisfied for all eigenvalues of matrix J at the equilibrium point E = (x 1, x 2,..., x m). One remarks that the given theoretical results make clear that the stability condition for fractional order systems differs from the well-known condition for integer order systems. In particular, the left half-plane (stable region) for integer-order systems maps into the angular sector arg(λ) > α π 2 in the case of fractional-order systems, indicating that the stable region becomes larger and larger when the value of fractional-order α is decreased. To evaluate the equilibrium points let Dt α S h =, Dt α I h =, Dt α R h =, Dt α S a =, Dt α I a =. ( ) A Then E = µ h, B γ a,. Denote a basic reproduction number R = β h β a A B µ h µ a (µ h + δ h + γ h )(γ a + δ a ). It means the average new infections produced by one infected individual during his lifespan when the population is at E. By (3.2), a positive equilibrium E 1 = (S 1 h, I1 h, R1 h, S1 a, I 1 a) satisfies S 1 h = η 1η 3 Ω 1 β a Ω 2, I 1 h = η 1η 2 η 3 γ a µ h β a Ω 2 (R 1), R 1 h = η 1η 2 η 3 γ h µ h γ a β a Ω 2 (R 1) S 1 a = Ω 2 β h Ω 1, I 1 a = η 1η 2 η 3 γ a µ h β h η 3 Ω 1 (R 1), where Ω 1 = η 2 β a A + γ a Φ 1, Ω 2 = β h B Φ 1 + Φ 2, Φ 1 = (δ h + µ h ) (λ h + µ h ) + γ h µ h, Φ 2 = η 1 η 2 η 3 µ h, η 1 = µ h + δ h + γ h, η 2 = µ h + λ h, η 3 = γ a + δ a The Jacobian matrix J(E ) for system given in (3.2) evaluated at the disease free equilibrium is as follows:

7 Fractional calculus model 2657 µ h λ h Aβ h µ h Aβ η 1 h µ h J(E ) = γ h η 2 Bβa γ v γ v Bβ a γ v η 3 Theorem 4.1. The disease free equilibrium point E is locally asymptotically stable if R < 1 and is unstable if R > 1. Proof.The disease free equilibrium is locally asymptotically stable if all the eigenvalues, λ i, i = 1, 2, 3, 4, 5 of the Jacobian matrix J(E )satisfy the following condition [1, 2, 3, 1, 19]: arg(λ i ) > α π 2. (4.4) The eigenvalues of the characteristic equation of J(E ) are λ 1 = µ h, λ 2 = γ a, λ 3 = η 2. The other two roots are determined by the quadratic equation λ 2 + (η 1 + η 3 ) λ + η 1 η 3 (1 R ) =. Hence E is locally asymptotically stable if R < 1 and is unstable if R > 1. We now discuss the asymptotic stability of the endemic (positive) equilibrium of the system given by (3.2). The Jacobian matrix J(E 1 ) evaluated at the endemic equilibrium is given as: J(E 1 ) =. β h Ia 1 µ h λ h β 2 Sh 1 β h Ia 1 η 1 β h Sh 1 γ h η 2 β a Sa 1 β a Ih 1 γ v β a Sa 1 β a Ih 1 η 3 The characteristic equation of J(E 1 ) is: λ 5 + a 1 λ 4 + a 2 λ 3 + a 3 λ 2 + a 4 λ + a 5 =, where a 1 = γ v + Ψ + µ h + β 3 Rh + β 2Ia, a 2 = Ψγ v + β 2 ((Ia Rh S a Sh )β 3 + Ia (Ψ + γ v )) + + (Ψ + γ v ) µ h + Rh β 3 (Ψ + µ h ) a 3 = γ v + η 1 η 2 η 3 + ( + Ψγ v ) µ h + Rh β 3 ( + Ψµ h ) + β 2 Λ a 4 = η 1 η 2 η 3 (γ v + µ h ) + γ v µ h + β 2 (Iaψ + β 3 IaR h Φ 3 β 3 SaS h Θ) + R h β 3 ( µ h + η 1 η 2 η 3 ) a 5 = η 1η 2 η 3 γ a µ h (Aβ aγ h +γ aφ 1 )(R 1) Ω 1 = η 1 η 2 + η 1 η 3 + η 2 η 3, Ψ = η 1 + η 2 + η 3, Θ = η 2 µ h + γ v η 2 + η 2 µ h Λ = Ia (Φ 3 + Ψγ v ) + β 3 (IaR hψ SaS h (γ v + η 2 + µ h )), ψ = η 3 Φ 1 + γ v Φ 3. Following [3], a necessary condition for arg(λ) > α π is a 2 5 >. Then one has the following theorem: Theorem 4.2. The endemic equilibrium point E 1 is locally asymptotically stable if R > 1 and is unstable if R < 1..

8 2658 Moustafa El-Shahed If one take A = 1.6, µ h =.34, µ a =.36, δ h =.1, γ h =.3, δ a =.94, B = 1.2, β h =.98, β a =.78, λ h =.67, γ h =.7 γ v =.17, In this case the endemic equilibrium point E 1 = (25.73, , , , ) is local asymmetrically stable where R = > 1 and the eigenvalues are λ 1 = , arg(λ 1 ) = π > α π λ 2 = , arg(λ 2 ) = π > α π, 2 λ 3 =.34, arg(λ 3 ) = > α π, 2 λ 4 = i, arg(λ 4 ) = > α π λ 5 = i, arg(λ 5 ) = π > α π 2 If one take A=1.6 µ h =.34, µ a =.36, δ h =.1,γ h =.3, δ a =.94, B=1.2, β h =.98, β a =.78, λ h =.67,γ h =.7, γ a =.417. In this case the equilibrium point E =(47.588,,2.8777,,) is local asymmetrically stable where R < 1 and the eigenvalues are λ 1 =.45161, λ 2 =.417, λ 3 =.3467, λ 4 =.34, λ 5 = , 2, 5. Numerical methods and simulations Since most of the fractional-order differential equations do not have exact analytic solutions, 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 system (3.2), 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 [8, 9, 18] Dt α y(t) = f(t, y(t)), t T, y (k) () = y, k k =, 1, 2,...m 1, where m = [α], This equation is equivalent to the Volterra integral equation y(t) = m 1 k= y (k) t k k! + 1 Γ(α) t (t s) α 1 f(s, y(s))ds. (5.1) Diethelm et al. used the predictor-correctors scheme [8, 9], based on the Adams-Bashforth-Moulton algorithm to integrate Eq. (5.1). By applying this scheme to the fractional-order model for childhood diseases, and setting h = T, t N n = nh, n =, 1, 2,,..., N Z +, Eq. (5.1) can be discretized as follows [8, 9, 18]:

9 S h(n+1) = S h() + I h(n+1) = I h() + R h(n+1) = R h() + S a(n+1) = S a() + I a(n+1) = I a() + where Fractional calculus model 2659 ( ) A µ h S p h(n+1) β h S p h(n+1) Ip a(n+1) + λ h R p h(n+1) + n Γ(α+2) j= a ( ) j,n+1 A µh S h(j) β h S h(j) I a(j) + λ h R h(j), hα Γ(α+2) h α hα Γ(α+2) hα Γ(α+2) hα Γ(α+2) hα Γ(α+2) ( ) β h S p h(n+1) Ip a(n+1) η 1I p h(n+1) + h α n Γ(α+2) j= a ( j,n+1 βh S h(j) I a(j) η 1 I h(j), ) ( ) γ h I p h(n+1) η 2 R p h(n+1) + h α n Γ(α+2) j= a ( ) j,n+1 γh I h(j) η 2 R h(j), ( ) B γ a S p a(n+1) β a S p a(n+1) Ip h(n+1) + h α n Γ(α+2) j= a ( ) j,n+1 B γa S a(j) β a S a(j) I a(j), ( ) β a S p a(n+1) Ip h(n+1) η 3 I p h(n+1) + h α n Γ(α+2) j= a ( ) j,n+1 βa S a(j) I a(j) η 3 I a(j), S p h(n+1) = S h() + 1 n Γ(α) j= b ( ) j,n+1 A µh S h(j) β h S h(j) I a(j) + λ h R h(j), I p h(n+1) = I h() + 1 n Γ(α) j= b ( j,n+1 βh S h(j) I a(j) η 1 I h(j), ), R p h(n+1) = R h() + 1 n Γ(α) j= b ( ) j,n+1 γh I h(j) η 2 R h(j), S p a(n+1) = S a() + 1 n Γ(α) j= b ( ) j,n+1 B γa S a(j) β a S a(j) I a(j) I p a(n+1) = I a() + 1 n Γ(α) j= b ( ) j,n+1 βa S a(j) I a(j) η 3 I a(j) a j,n+1 = n α+1 (n α)(n + 1), j =, (n j + 2) α+1 + (n j) α+1 2(n j + 1) α+1 1 j n, 1 j = n + 1, b j,n+1 = hα α ((n j + 1)α (n j) α ), j n. 6. Conclusions In this paper, we consider the fractional order model for Leptospirosis. We have obtained a stability condition for equilibrium points. We have also given a numerical example and verified our results. Following [21], the spreading of leptospirosis has two states: the disease-free state and the endemic state. The occurrence of a state depends on the basic reproduction number R. If R < 1, then the disease- free state will occur but if R > 1 then the endemic state will occur as shown in Figures 1-2. Figures 3-5 show that the lower values of α increase the time to convergence to the disease free state and endemic positive equilibrium point. From the numerical results in Figure 6, it is clear that the number of infected humans initially increases before decreasing to the endemic state while the number of recovered humans decreases to the endemic state. The numerical results show that the approximate solutions depend continuously on the fractional derivative α. The time to convergence to the disease free state is longer than the time to convergence to the endemic

10 266 Moustafa El-Shahed state because the amount of time before the disease disappears is longer than that of the endemic state. This reflects what would happen in the real world. 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. The transformation of a classical model into a fractional one makes it very sensitive to the order of differentiation α : a small change in α may result in a big change in the final result. References [1] E. Ahmed, A. M. A. El-Sayed, E. M. El-Mesiry and H. A. A. El-Saka; Numerical solution for the fractional replicator equation, IJMPC, 16 (25), [2] E. Ahmed, A. M. A. El-Sayed, H. A. A. El-Saka; On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and Chen systems, Physics Letters A, 358 (26), [3] E. Ahmed, A. M. A. El-Sayed, H. A. A. El-Saka; Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl, 325 (27), [4] Amercan Public Health Association; Leptospirosis, Control of Communicable Disease Manual, 17 (2), [5] R. M. Anderson; R. M. May; Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, [6] S. K. Choi, B. Kang, and N. Koo; Stability for Caputo fractional differential systems,abstract and Applied Analysis, Article ID (214), [7] S. K. Choi and N. Koo; The monotonic property and stability of solutions of fractional differential equations,nonlinear Analysis: Theory, Methods and Applications, 74 (211), [8] K. Diethelm, N. J. Ford; Analysis of fractional differential equations,j Math Anal Appl, 265 (22), [9] K. Diethelm, N. J. Ford, A.D. Freed; A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn, 29 (22), [1] Y. Ding, H. Ye; A fractional-order differential equation model of HIV infection of CD4+T -Cells,Mathematical and Computer Modeling, 5 (29), [11] E. H. Elbasha, A. B. Gumel; Analyzing the dynamics of an SIRS vaccination model with waning natural and vaccine-induced immunity, Trends in Parasitology, 12 (211), [12] M. Elshahed and A. Alsaedi; The Fractional SIRC Model and Influenza A,Mathematical Problems in Engineering, Article ID (211), [13] H. W. Hethcote; The mathematics of infectious diseases, SIAM Review, 42 (2),

11 Fractional calculus model 2661 [14] A. A. Kilbas.; H. M. Srivastava.; and J. J. Trujillo.; Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, The Netherlands,24 (26). [15] C. H. Koutis ; Special Epidemiology, Technological Educational Institute of Athens. Athens,(27). [16] Y. Li, Y. Chen, and I. Podlubny; Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability,computers and Mathematics with Applications, 59 (21), [17] Y. Li, Y. Chen, and I. Podlubny; Mittag-Leffler stability of fractional order nonlinear dynamic systems,automatica, 45 (29), [18] C. Li, C. Tao; On the fractional Adams method, Computers and Mathematics with Applications,,58 (29), [19] D. Matignon; Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems and Applications, Multiconference, vol. 2, IMACS, IEEE-SMC Proceedings, Lille, France, 2 (1996), [2] A. J. McBride, D. A.Athanazio, M. G. Reis and A. I. Ko; Leptospirosis, Current opinion in infectious diseases, 18 (25), [21] B. Pimpunchat, G. C. Wake, C. Modchang, W. Triampo, A. M. Babylon; Mathematical Model of Leptospirosis: Linearized Solutions and Stability Analysis, Applied Mathematics,, 4(213), [22] I. Podlubny; Fractional Differential Equations, Academic Press, New York, NY, USA (1999). [23] P. Pongsuumpun, T. Miami and R. Kongnuy; Age Structural Transmission Model for Leptospirosis, The 3rd International Symposium on Biomedical Engineering, Bangkok, 1-11 November 28, (28), [24] A. Slack; Leptospirosis, Australian family physician, 39 (21), [25] W. Triampo, D. Baowan, I. M. Tang, N. Nuttavut, J.Wong-Ekkabut and G. Doungchawee; A Simple Deterministic Model for the Spread of Leptospirosis in Thailand, International Journal of Biological and Life Sciences, 2 (26), [26] WHO; Human Leptospirosis: Guidance for Diagnosis, Surveillance and Control, World Health Organization, Geneva (213). [27] G. Zaman; Dynamical Behavior of Leptospirosis Disease and Role of Optimal Control Theory, International Journal of Mathematics and Computation, 7 (21), [28] G. Zaman, M. A. Khan, S. Islam, M. I. Chohan and I. H. Jung; Modeling Dynamical Interactions between Leptospirosis Infected Vector and Human Population, Applied Mathematical Sciences, 6 (212), Received: October 21, 214; Published: November 24, 214

12 2662 Moustafa El-Shahed S h t h t R h t Sa t a t t Fig.1, A = 1.6, µ h =.34, µ a =.36, δ h =.1, γ h =.3, δ a =.94, B = 1.2, β h =.98, β a =.78, λ h =.67, γ h =.7 γ v =.17, α = 1

13 Fractional calculus model 2663 S h t h t R h t S a t a t t Fig.2, A = 1.6, µ h =.34, µ a =.36, δ h =.1, γ h =.3 δ a =.94, B = 1.2, β h =.98, β a =.78, λ h =.67, γ h =.7 γ v =.417, α = 1

14 2664 Moustafa El-Shahed S h t h t v t t Fig.3, A = 1.6, µ h =.34, µ a =.36, δ h =.1, γ h =.3, δ a =.94, B = 1.2, β h =.98, β a =.78, λ h =.67, γ h =.7 γ v =.17, α =.99

15 Fractional calculus model 2665 S h t h t v t t Fig.4, A = 1.6, µ h =.34, µ a =.36, δ h =.1, γ h =.3, δ a =.94, B = 1.2, β h =.98, β a =.78, λ h =.67, γ h =.7 γ v =.17, α =.9

16 2666 Moustafa El-Shahed S h t h t v t t Fig.5, A = 1.6, µ h =.34, µ a =.36, δ h =.1, γ h =.3, δ a =.94, B = 1.2, β h =.98, β a =.78, λ h =.67, γ h =.7 γ v =.17, α =.8

17 Fractional calculus model 2667 a t Fig.6, A = 1.6, µ h =.34, µ a =.36, δ h =.1, γ h =.3, δ a =.94, B = 1.2, β h =.98, β a =.78, λ h =.67, γ h =.7 γ v =.17, α = 1,.9,.8,