Journal of the Egyptian Mathematical Society (2014) 22, 50-54

Egyptian Mathematical Society Journal of the Egyptian Mathematical Society

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ORIGINAL ARTICLE

The fractional-order SIS epidemic model with variable population size

H.A.A. El-Saka *

Faculty of Science, Damietta University, 34517 New Damietta, Egypt

Received 29 May 2013; accepted 13 June 2013 Available online 2 August 2013

KEYWORDS

Fractional-order; SIS epidemic model; Variable population size; Stability;

Numerical solutions

Abstract In this work, we deal with the fractional-order SIS epidemic model with constant recruitment rate, mass action incidence and variable population size. The stability of equilibrium points is studied. Numerical solutions of this model are given. Numerical simulations have been used to verify the theoretical analysis.

MATHEMATICS SUBJECT CLASSIFICATION: 37N25; 34D20; 37M05

© 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

1. Introduction

The epidemic models incorporate constant recruitment, disease-induced death and mass action incidence rate.

Some infections do not confer any long lasting immunity. Such infections do not have a recovered state and individuals become susceptible again after infection. This type of disease can be modelled by SIS type. The total population N is divided into two compartments with N = S + I, where S is the number of individuals in the susceptible class, I is the number of individuals who are infectious [1,2].

The use of fractional-orders differential and integral operators in mathematical models has become increasingly wide-

spread in recent years [3]. Several forms of fractional differential equations have been proposed in standard models.

Differential equations of fractional order have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, economic, viscoelasticity, biology, physics and engineering. Recently, a large amount of literature has been developed concerning the application of fractional differential equations in nonlinear dynamics [3].

In this paper, we study the fractional-order SIS 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 modelling biological, economic and social systems (generally complex adaptive systems) where memory effects are important. In Section 2, the equilibrium points and their asymptotic stability of differential equations of fractional order are studied. In Sections 3 and 4, the model is presented and discussed. In Section 5 numerical solutions of the model are given.

Now we give the definition of fractional-order integration and fractional-order differentiation:

* Tel.: +20 57 2403980.

E-mail address: halaelsaka@yahoo.com.

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Definition 1.1. The fractional integral of order ( 2 R+ of the function f(t), t > 0 is defined by

and the fractional derivative of order a 2 (n — 1, n] of f(t), t > 0 is defined by

Daf(t)= r—aDf(t), D = dt •

For the main properties of the fractional-orders derivatives and integrals [4-9].

2. Equilibrium points and their asymptotic stability

Let a 2 (0, 1] and consider the system [10-15]

D>i(t)=fi(yi, y2) D>2(t) =f2(y1, y2)

with the initial values

y1(0)=y01 and y2(0)=y»2-

To evaluate the equilibrium points, let

D>,(t)=0 ) f,(yf, 0; i = 1; 2

from which we can get the equilibrium points yf, ye2q. To evaluate the asymptotic stability, let

yM—yf + 8,(t),

so the equilibrium point (y1q,y2q) is locally asymptotically stable if both the eigenvalues of the Jacobian matrix A f f

evaluated at the equilibrium point satisfies (I arg(kj > ap/2, I arg(k2)| > ap/2) [11,12,14-16]. The stability region of the fractional-order system with order a is illustrated in Fig. 1 (in which r, x refer to the real and imaginary parts of the eigenvalues, respectively, and j = V—1). From Fig. 1, it is easy to show that the stability region of the fractional-order case is greater than the stability region of the integer order case.

3. Fractional-order SIS model

Let S(t) be the number of individuals in the susceptible class at time t, I(t) be the number of individuals who are infectious at time t.

The fractional-order SIS model is given by

Da1 S(t) — K — (SI — iS + uI, Da1 I(t)—(SI —(u + 1 + a) I,

where 0 < a1 6 1 and the parameters are positive constants. The constant K is the recruitment rate of susceptible corresponding to births and immigration, i is the per capita natural mortality rate. We assume that a disease may be fatal to some infectious, so deaths due to disease can be included in a model using the disease-related death rate from infectious class, a. Let U be the rate at which individuals infectious and return to susceptible class. This together with N = S + I, implies

Da1 N — K — iN — a.I.

Figure 1 Stability region of the fractional-order system.

Thus, the total population size N may vary in time [2]. To evaluate the equilibrium points, let

Da1 S — 0,

Da11 — 0,

then (Seq, Ieq)— , 0^, (S», I»), are the equilibrium points where,

1 K i(u +1 +a) S»—j,(u + 1 + a), I»—7—■—t--^—■—r—.

( (l + a) ((i + a)

For (Seq, Ieq) = , o) we find that

—1 0

— (u +1-

and its eigenvalues are

k1 — —i < 0,

(K , x (K ,

k2 —--(u + 1 + a) < 0 if — < (u + 1 + a).

Hence the equilibrium point (Seq, Ieq)— (jj, 0^ is locally asymptotically stable if

< (u + 1 + «)•

For (Seq, Ieq) = (S», I») we find that

bK i(u+i+«) (1+a) + (l+a)

bK _ l(u+l+a)

(l+a) (l+a)

— 1 —(i + a) 0

and its eigenvalues are 1

2(i + a) 1

2(i + a)

(bK — 1u)+ V (bK — 1U)2 — 4(l + a)2[bK — i(u + 1 + a)] — (bK — 1u) — V (bK — 1u)2 — 4(1 + a)2[bK — 1(u + 1 + a)]

Figure 2 a1 = 1.0.

0 10 20 30 40 50 60 70 80 90 100 t

0 10 20 30 40 50 60 70 80 90 100

Figure 4 a1 = 0.!

A sufficient condition for the local asymptotic stability of the equilibrium point (Seq, Ieq) = (S* I*) is

| arg(k:)| > ain/2, | arg(k2)| > ain/2. (7)

4. Existence of uniformly stable solution

x1(t) = S(t), x2(t)=I(t),

fi(xl(t), x2(t)) = K - bx1(t)x2(t)- 1x1 (t) + uxi(t), and

/2 (x1 (t), x2 ( t)) = bx1(t)x2(t)-(u + 1 + a)x2(t). Let

D = fx1,x2 2 R : |x,(t)| 6 a, t 2 [0, T],i = 1,2g, then on D we have

9x1 dx1

■/1(x1, x2)

■/2(x1, X2)

■/l(x1, X2)

6 k3 and

-fi(x1, X2)

where kj, k2, k3 and k4 are positive constants.

This implies that each of the two functions fx, /2 satisfies the Lipschitz condition with respect to the two arguments x1 and x2, then each of the two functions /1, /2 is absolutely continuous with respect to the two arguments x1 and x2.

I(t)3 2 1 0

Figure 5

0 10 20 30 40 50 60 70 80 90 100

Figure 6 a1 = 1.0.

0 10 20 30 40 50 60 70 80 90 100

Figure 3 a1 = 0.9.

Consider the following initial value problem which represents the fractional-order SIS model (8) and (9)

Da1 xl{t)^fl{xl{t), x2(t)), t > 0 and x1(0) = x„1, (8) Da1 x2(t) ~f2(x\(t); x2(t)), t > 0 and x2(0) = x„2- (9)

Definition 4.1. By a solution of the fractional-order SIS model (8) and (9), we mean a column vector (x^t) x2(t))s, xi and x2 2 C[0, T], T < i where C[0, T] is the class of continuous functions defined on the interval [0, T] and s denote the transpose of the matrix.

—S —I

F(X(t)) = (fi(x1(t), x2(i)) /2(xi(i), x2(t)))s.

Now applying Theorem 2.1 [17], we deduce that the fractional-order SIS model (8) and (9) has a unique solution. Also by Theorem 3.2 [17] this solution is uniformly Lyapunov stable. □

5. Numerical methods and results

0 10 20 30 40 50 60 70 80 90 100 t

Figure 7 * = 0.9.

—S — I

0 10 20 30 40 50 60 70 80 90 100 t

Figure 8 a1 = 0.8.

Figure 9

Now we have the following theorem

Theorem 4.1. The fractional-order SIS model (8) and (9) has a unique uniformly Lyapunov stable solution.

Proof. Write the model (8) and (9) in the matrix form

Da1 X(t) — F(X(t)), t > 0 and X(0) — X, (10)

X(t) — (x1(t) x2(t))s,

An Adams-type predictor-corrector method has been introduced and investigated further in [18-20]. In this paper, we use an Adams-type predictor-corrector method for the numerical solution of fractional integral equation.

The key to the derivation of the method is to replace the original problem (5) by an equivalent fractional integral equations

S(i) = S(0) I(t) = I(0) 4

h I*1 [K - ßSI- iS + uI], I"1 [ßSI -(u + l + a) I],

and then apply the PECE (Predict, Evaluate, Correct, Evaluate) method.

The approximate solutions are displayed in Figs. 2-9 for S(0) = 20.0, I(0) = 1.0 and different 0 < a1 6 1. In Figs. 2-5 we take K = 0.1, ( = 0.1, a = 0.2, u = 0.3, a = 0.1 and found

that the equilibrium point , 0^ — (0.5,0) is locally asymptotically stable where the condition (6) — 0.05 <

(u + 1 + a) — 0.6^ is satisfied. In Fig. 5 we found that in the

fractional-order case, the peak of the infection is reduced. But the disease takes a longer time to be eradicated.

In Figs. 6-9 we take K = 0.5, ( = 0.5, 1 = 0.3, u = 0.1, a = 0.1 and found that the equilibrium point

, 0^ — (1.66667,0) is unstable where the condition (6) is

not satisfied — 0.833333 > (u + 1 + a) — 0.5^ and the

equilibrium point (S», I») is locally asymptotically stable where the condition (7) is satisfied where the equilibrium point and the eigenvalues are given as:

(S*, I») = (1.0, 0.5), k12 = —0.275 ± 0.156125i.

The equilibrium point (S», I») = (1.0, 0.5), is locally asymptotically stable where I arg (k2,3)I = 2.62524 > a1p/2. In Fig. 9 we found that in the fractional-order case, the peak of the infection is reduced. But the disease takes a longer time to be eradicated.

6. Conclusions

In this paper we study the fractional-order SIS model. The stability of equilibrium points is studied. Numerical solutions of this model are given.

The reason for considering a fractional-order system instead of its integer order counterpart is that fractional-order differential equations are generalizations of integer order differential equations. Also using fractional-order differential equations can help us to reduce the errors arising from the neglected parameters in modelling real life phenomena.

We like to argue that fractional-order equations are more suitable than integer order ones in modelling biological, economic and social systems (generally complex adaptive systems) where memory effects are important.

The stability of equilibrium points is studied. Numerical solutions of these models are given. Numerical simulations have been used to verify the theoretical analysis.

Acknowledgment

I wish to thank professor Ahmed M.A. El-Sayed (Mathematics Department, Faculty of Science, Alexandria University, Alexandria, Egypt) for his support and encouragement.

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