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In this paper, we construct a backward difference scheme for a class of
*SIR* epidemic model with general incidence
*f* . The step size
*τ* used in our discretization is one. The dynamical properties are investigated (positivity and the boundedness of solution). By constructing the Lyapunov function, the general incidence function
*f* must satisfy certain assumptions, under which, we establish the global stability of endemic equilibrium when R
_{0} >1. The global stability of diseases-free equilibrium is also established when
R
_{0} ≤1. In addition we present numerical results of the continuous and discrete model of the different class according to the value of basic reproduction number
R
_{0}.

In certain epidemiological modeling, the population is generally divided into three classes which are susceptible represented by S, infected individual represented by I and recovered individual represented by R. This kind of mathematical model is noted SIR. Recently, many authors have studied the dynamical behavior of epidemic models (see [

The delay appears in the incidence term which is typically the only non-linearity, and is therefore the “cause” of all “interesting behavior”. Various forms have been used for the incidence term, both for ODEs and for delay equations. Common forms include mass action β S I [

incidence β S I 1 + c I [

[

In this paper we study the discrete mathematical model which result from the continuous-time model presented and study in [

From this we use the general incidence term β ∑ j = 0 h k ( j ) f ( S n , I n j ) , where h > 0 is a time delay. We choose the constant β so that ∑ j = 0 h k ( j ) = 1 . The discrete

model is obtained by using the backward Euler method.

The studied of discrete epidemic models is motivate by the fact that there occur situations such that constructing discrete epidemic models is more appropriate approach to understand disease transmission dynamics and to evaluate eradication policies because they permit arbitrary time-step units, preserving the basic features of corresponding continuous-time models [

This paper is organized as follows. In Section 2, we give the discrete model, the equilibrium point and the reproduction rate R 0 . In Section 3, the positivity and boundedness of the solution of system (3) are obtained. In addition we proved the existence and uniqueness of disease-free equilibrium E 0 and endemic equilibrium E * . In Section 4, we study the stability of disease-free equilibrium point for R 0 ≤ 1 . In Section 5, we study the global stability of the endemic equilibrium point for R 0 > 1 . In section 6, we give the numerical result and their comment. In the last part we give the conclusion.

In this section we describe the discrete mathematical model derived by the continuous time model study in [

{ S ˙ = B − μ S S − β ∫ 0 h k ( τ ) f ( S , I τ ) d τ , I ˙ = β ∫ 0 h k ( τ ) f ( S , I τ ) d τ − ( μ I + γ ) I , R ˙ = γ I − μ R R , (1)

where I τ = I ( t − τ ) .

1) A population is divided into susceptible, infectious and recovered classes with sizes S = S ( t ) , I = I ( t ) and R = R ( t ) respectively.

2) B is the recruitment of new individuals, it is into the susceptible class.

3) μ S , μ I and μ R denote respectively the death rates of susceptible, infectious and recovered class.

4) The total exit rate for infectious is μ I + γ , which, for biological reasons we assume is at least as large as μ S ; that is, μ I + γ ≥ μ S .

5) The incidence at time t is β ∫ 0 h k ( τ ) f ( S , I τ ) d τ where the maximum delay

h > 0 , k is a Lebesgue integral function which gives the relative infectivity of vectors of different infection ages. We choose β so that ∫ 0 h k ( τ ) d τ = 1 . It is assumed that the support of k has positive measure in any open interval having supremum h so the interval of integration is not artificially extended by concluding with an interval for which the integral is automatically zero.

The form of the function f is of fundamental importance. In this paper we use a general incidence function used in one of he’s discrete version. So we use assumption:

H1 f is a non-negative differentiable function on the non-negative quadrant. Furthermore, f is positive if and only if both arguments are positive.

H2 for all ( S , I ) ∈ ℝ + 2 f ( S , 0 ) = f ( 0 , I ) = 0 .

The partial derivative of f are denoted by f 1 and f 2 from the first and second variable.

H3 f ( S n , I n j ) ≤ f 2 ( S 0 ,0 ) I n for all n.

H4 f ( S * , I * ) f ( S n + 1 , I n + 1 j ) ≤ S * S n + 1 ≤ I * I n + 1 for all n.

Now, we use the backward Euler difference scheme to discretize the model (1). The time step size of our discretization is one. Thus, we obtain the following discrete SIR epidemic model with nonlinear general incidence given by:

{ S n + 1 − S n = B − μ S S n + 1 − β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) I n + 1 − I n = β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) − ( μ I + γ ) I n + 1 R n + 1 − R n = γ I n + 1 − μ R R n + 1 (2)

where I j = I ( t − j ) .

Since R does not appear in the first and second equations of system above, it is sufficient to analyses the behavior of solutions of the following system:

( S n + 1 − S n = B − μ S S n + 1 − β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) I n + 1 − I n = β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) − ( μ I + γ ) I n + 1 (3)

The constants B , μ S , μ I , γ and the relation between this constants are given above. In the discrete model the incidence function at time t is

β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) ,

where the maximum delay h > 0 . Let E ( S , I ) be a equilibrium point model of (3) so we have,

( B − μ S S − β f ( S , I ) = 0 β f ( S , I ) − ( μ I + γ ) I = 0 (4)

By adding the equations of system above we get

B − μ S S − ( μ I + γ ) I = 0

⇒ S = B − ( μ I + γ ) I μ S . (5)

Let E 0 and E * be respectively disease-free equilibrium and endemic equilibrium point of model (3). The disease-free equilibrium correspond to the case where the infectious class is nil ( I = 0 ) . Thus, we have E 0 = ( S 0 , 0 ) ; with

S 0 = B μ S . The endemic equilibrium E * is given by: E * = ( S * , I * ) ; with

S * = B − ( μ I + γ ) I * μ S .

Proposition 2.1. The basic reproduction number is given by R 0 = β f 2 ( E 0 ) μ I + γ .

Proof: The Jacobian matrix of system (3) at equilibrium E 0 is define by:

J E 0 = ( − μ S − β f 1 ( E 0 ) − β f 2 ( E 0 ) β f 1 ( E 0 ) β f 2 ( E 0 ) − ( μ I + γ ) ) . (6)

Let

A = β 2 ( E 0 ) − ( μ I + γ ) , (7)

M = β f 2 ( E 0 ) et D = μ I + γ . (8)

Thus, we have:

R 0 = M D − 1 (9)

R 0 = β f 2 ( E 0 ) μ I + γ . (10)

We suppose that initial condition of system (3) satisfy:

S ( 0 ) > 0 and I ( θ ) = Φ ( θ ) for all θ ∈ [ − h , 0 ] , (11)

where Φ ∈ C = C ( [ − h ,0 ] , ℝ + ) , the space of continuous functions from [ − h , 0 ] to ℝ + equipped with the sup norm: ‖ Φ ‖ = s u p θ ∈ [ − h ,0 ] Φ ( θ ) . Standard theory of functional differential equations [

Lemma 3.1. Let ( S n , I n ) be a solution of system (3) with initial condition (11), then we have S n > 0 and I n > 0 for all n.

Proof: Assume that S n > 0 and I n > 0 . From system (3) we have the following system:

( ( 1 + μ S ) S n + 1 = B + S n − β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) ( 1 + μ I + γ ) I n + 1 = I n + β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) . (12)

By using second equation of system above and the fact that I n > 0 , we have, I n + 1 > 0 . So I n > 0 , ∀ n ∈ ℕ . From the non-negativity of S n + 1 we used the assumption H4. Thus, S n , I n > 0 for all n.

Lemma 3.2. Any solution ( S n , I n ) of system (3), with initial condition (11) satisfy

lim sup n → + ∞ ( S n + I n ) ≤ B μ S .

Proof: Let N n = S n + I n and N n + 1 = S n + 1 + I n + 1 ; for biological reasons we assume is at least as large as μ S ; that is, μ I + γ ≥ μ S .

By adding the different equation of system (3) we get:

N n + 1 − N n = B − μ S S n + 1 − ( μ I + γ ) I n + 1 ≤ B − μ S S n + 1 − μ S I n + 1 ≤ B − μ S ( S n + 1 + I n + 1 ) ≤ B − μ S N n + 1

μ S N n + 1 + N n + 1 − N n ≤ B (13)

lim sup n → + ∞ μ S N n + 1 + N n + 1 − N n = lim sup n → + ∞ μ S N n + 1 (14)

⇒ lim sup n → + ∞ μ S N n + 1 ≤ B

⇒ l i m s u p n → + ∞ N n + 1 ≤ B μ S . (15)

Hence, we have

l i m s u p n → + ∞ ( S n + I n ) ≤ B μ S . (16)

Proposition 3.1.

1) When R 0 < 1 , then model (3) has only a unique disease-free equilibrium E 0 = ( S 0 , 0 ) .

2) When R 0 > 1 , then model (3) has only a unique endemic equilibrium E * = ( S * , I * ) .

Proof: Any equilibrium point E = ( S , I ) of system (3) verified the following system:

( B − μ S S − β f ( S , I ) = 0 β f ( S , I ) − ( μ I + γ ) I = 0. (17)

By using the second equation of system above we have:

β f ( S , I ) I = ( μ I + γ ) . (18)

So we can consider the function G defined by,

G ( I ) = β f ( S 0 − μ I + γ μ S I , I ) I − ( μ I + γ ) . (19)

Hence we have

lim I → 0 + G ( I ) = β ∂ f ∂ I ( S 0 , 0 ) − ( μ I + γ ) = ( μ I + γ ) ( β f 2 ( S 0 , 0 ) μ I + γ − 1 ) = ( μ I + γ ) ( R 0 − 1 ) . (20)

And also we have,

G ( I ¯ ) = − ( μ I + γ ) , where I ¯ = μ S S 0 μ I + γ . (21)

when R 0 ≤ 1 , we have lim I → 0 + G ( I ) ≤ 0 . Consequently, there is not any I * > 0

such that G ( I * ) = 0 . Therefore, model (3) has a unique disease-free equilibrium E 0 .

When, R 0 > 1 , we have lim I → 0 + G ( I ) > 0 . Therefore, there exists a unique I * ∈ ] 0 ; I ¯ [ such that G ( I * ) = 0 .

This implies that model (3) has unique endemic equilibrium E * = ( S * , I * ) .

Remark 3.1. The space K = ℝ + × C is positively invariant and attracting domain for system (3).

Now, let us analyze the behavior of system (3) when the basic reproduction rate R 0 is less than one.

In this section, we study the stability of diseases-free equilibrium E 0 = ( S 0 , 0 ) , with S 0 = B μ S .

Theorem 4.1. If R 0 ≤ 1 , then the diseases-free equilibrium E 0 of system (3) is locally asymptotically stable.

Proof: The linearization of system (3) at diseases-free equilibrium point E 0 is given by:

{ S n + 1 − S n = [ − μ S − β f 1 ( E 0 ) ] S n + 1 − β f 2 ( E 0 ) I n + 1 I n + 1 − I n = β f 1 ( E 0 ) S n + 1 + [ β f 2 ( E 0 ) − ( μ I + γ ) ] I n + 1 . (22)

Thus, we have:

( [ 1 + μ S + β f 1 ( E 0 ) ] S n + 1 + β f 2 ( E 0 ) I n + 1 = S n − β f 1 ( E 0 ) S n + 1 + [ 1 − β f 2 ( E 0 ) + ( μ I + γ ) ] I n + 1 = I n . (23)

The matrix M associate of the linearization (23) is given by:

M = ( 1 + μ S + β f 1 ( E 0 ) β f 2 ( E 0 ) − β f 1 ( E 0 ) 1 − β f 2 ( E 0 ) + ( μ I + γ ) ) , (24)

and the linearization system can be rewrite by:

X n + 1 = M − 1 X n , (25)

with X n = ( S n , I n ) t .

The model (3) is locally asymptotically stable at diseases-free equilibrium point E 0 if all eigenvalue of matrix M is greater than one.

Let P ( X ) be characteristic polynomial associate of matrix M. We have,

P ( X ) = d e t ( M − X I 2 ) = ( 1 + μ S + β f 1 ( E 0 ) − X ) ( 1 − β f 2 ( E 0 ) + ( μ I + γ ) − X ) + ( β f 1 ( E 0 ) ) ( β f 2 ( E 0 ) ) . (26)

Let λ be a eigenvalue of matrix M, thus P ( λ ) = 0 . This implies that:

( 1 + μ S + β f 1 ( E 0 ) − λ ) ( 1 − β f 2 ( E 0 ) + ( μ I + γ ) − λ ) + ( β f 1 ( E 0 ) ) ( β f 2 ( E 0 ) ) = 0. (27)

From the Equation (27) we have

( 1 + μ S + β f 1 ( E 0 ) − λ ) ( 1 − β f 2 ( E 0 ) + ( μ I + γ ) − λ ) = − ( β f 1 ( E 0 ) ) ( β f 2 ( E 0 ) ) . (28)

By using the second member of (28), the fact that R 0 < 1 and we assume that the matrix M have a eigenvalue which is less than one. So we have:

− ( β f 1 ( E 0 ) ) ( β f 2 ( E 0 ) ) = ( β f 1 ( E 0 ) ) ( − β f 2 ( E 0 ) ) ≤ ( μ S + β f 1 ( E 0 ) ) ( − β f 2 ( E 0 ) + ( μ I + γ ) ) < ( 1 + μ S + β f 1 ( E 0 ) − λ ) ( 1 − β f 2 ( E 0 ) + ( μ I + γ ) − λ ) (29)

as a result of, the Equation (28) cannot have roots which is less than one. Hence, E 0 is locally asymptotically stable according to the theorem 2 in [

Theorem 4.2. When R 0 ≤ 1 , the disease-free equilibrium E 0 of system (3) is globally asymptotically stable in K.

Proof: In this proof we used the comparison theorem [

I n = ( 1 + ( μ I + γ ) ) I n + 1 − β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) (30)

≥ ( 1 + ( μ I + γ ) − β ∑ j = 0 h k ( j ) f ( S 0 , 0 ) ) I n + 1 . (31)

Hence we have:

I n ≥ ( 1 + ( μ I + γ ) − β f 2 ( S 0 ,0 ) ) I n + 1 . (32)

Thus,

M − 1 I n ≥ I n + 1 with M = 1 + ( μ I + γ ) − β f 2 ( S 0 , 0 ) . (33)

By using the fact R 0 ≤ 1 we have μ I + γ − β f 2 ( S 0 ,0 ) ≥ 0 . So the constant M is greater than one. we conclude that the linearized Equation (32) is stable whenever R 0 ≤ 1 . By a standard comparison theorem [

In this section, we study the stability the stability of endemic equilibrium E * = ( S * , I * ) , with

S * = S 0 − μ I + γ μ S I * . (34)

Theorem 5.1. If R 0 > 1 , then the endemic equilibrium E * of system (3) is globally asymptotically stable.

Proof: From the equation of system (3), at endemic equilibrium E * , we have:

B = μ S S * + β ∑ j = 0 h k ( j ) f ( S * , I * ) (35)

and

( μ I + γ ) I * = β f ( S * , I * ) , (36)

which will be used as substitutions in the calculations below. Let g ( x ) = x − 1 − ln x and

V S ( n ) = S * g ( S n S * ) (37)

V I ( n ) = I * g ( I n I * ) (38)

V + ( n ) = ∑ j = 0 h α ( j ) g ( I n − j I * ) , (39)

where

α ( j ) = β ∑ s = j h k ( s ) f ( S * , I * ) . (40)

We will study the behavior of the Lyapunov functional

V ( n ) = V S ( n ) + V I ( n ) + V + ( n ) ; (41)

which satisfies V n ≥ 0 with equality if and only if

S n S * = I n I * = 1 and I n − j I * = 1

for all j ∈ [ 0, h ] . For clarity, the difference V S ( n + 1 ) − V S ( n ) , V I ( n + 1 ) − V I ( n ) and V + ( n + 1 ) − V + ( n ) will be calculated separately and then combined to obtain V ( n + 1 ) − V ( n ) .

Calculation of the variation V S ( n + 1 ) − V S ( n ) : in this calculation, we used the value theorem and we assume that S n + 1 > S n . Note that we have the same result when S n > S n + 1 .

V S ( n + 1 ) − V S ( n ) = S * g ( S n + 1 S * ) − S * g ( S n S * ) = ( 1 − S * ln S n + 1 − ln S n S n + 1 − S n ) ( S n + 1 − S n ) ≤ ( 1 − S * S n + 1 ) ( S n + 1 − S n ) ≤ ( 1 − S * S n + 1 ) ( B − μ S S n + 1 − β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) ) ≤ ( 1 − S * S n + 1 ) ( μ S S * + β f ( S * , I * ) ∑ j = 0 h k ( j ) − μ S S n + 1 − β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) )

≤ − μ S ( S n + 1 − S * ) 2 S n + 1 + β ∑ j = 0 h k ( j ) f ( S * , I * ) [ ( 1 − S * S n + 1 ) ( 1 − f ( S n + 1 , I n + 1 j ) f ( S * , I * ) ) ] ≤ − μ S ( S n + 1 − S * ) 2 S n + 1 + β ∑ j = 0 h k ( j ) f ( S * , I * ) ( 1 − f ( S n + 1 , I n + 1 j ) f ( S * , I * ) − S * S n + 1 + S * S n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) )

V S ( n + 1 ) − V S ( n ) = − μ S ( S n + 1 − S * ) 2 S n + 1 + β ∑ j = 0 h k ( j ) f ( S * , I * ) ( 1 − f ( S n + 1 , I n + 1 j ) f ( S * , I * ) − S * S n + 1 + S * S n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) ) (42)

Let us calculate of the variation V I ( n + 1 ) − V I ( n ) : in this calculation we used the mane value theorem and we assume that I n + 1 > I n . Note that we have the same result when I n > I n + 1 .

V I ( n + 1 ) − V I ( n ) = I * g ( I n + 1 I * ) − I * g ( I n I * )

= ( 1 − I * ln I n + 1 − ln I n I n + 1 − I n ) ( I n + 1 − I n ) = ( 1 − I * ln I n + 1 − ln I n I n + 1 − I n ) ( β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) − ( μ I + γ ) I n + 1 ) ≤ ( 1 − I * I n + 1 ) ( β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) − ( μ I + γ ) I * I n + 1 I * )

≤ ( 1 − I * I n + 1 ) ( β ∑ j = 0 h k ( j ) f ( S n + 1 , I n + 1 j ) − β f ( S * , I * ) ∑ j = 0 h k ( j ) I n + 1 I * ) ≤ ( 1 − I * I n + 1 ) ( β ∑ j = 0 h k ( j ) [ f ( S n + 1 , I n + 1 j ) − f ( S * , I * ) I n + 1 I * ] ) ≤ β ∑ j = 0 h k ( j ) f ( S * , I * ) [ ( 1 − I * I n + 1 ) ( f ( S n + 1 , I n + 1 j ) f ( S * , I * ) − I n + 1 I * ) ] ≤ β ∑ j = 0 h k ( j ) f ( S * , I * ) [ f ( S n + 1 , I n + 1 j ) f ( S * , I * ) − I n + 1 I * − I * I n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) + 1 ] ,

V I ( n + 1 ) − V I ( n ) ≤ β f ( S * , I * ) ∑ j = 0 h k ( j ) [ f ( S n + 1 , I n + 1 j ) f ( S * , I * ) − I n + 1 I * − I * I n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) + 1 ] . (43)

Let now evaluate the variation V + ( n + 1 ) − V + ( n ) :

V + ( n + 1 ) − V + ( n ) = ∑ j = 0 h α ( j ) g ( I n + 1 − j I * ) − ∑ j = 0 h α ( j ) g ( I n − j I * ) = α ( 0 ) g ( I n + 1 I * ) − α ( 0 ) g ( I n − j I * ) = ∑ j = 0 h β k ( j ) f ( S * , I * ) [ g ( I n + 1 I * ) − g ( I n − j I * ) ] = ∑ j = 0 h β k ( j ) f ( S * , I * ) [ I n + 1 I * − I n − j I * − ln I n + 1 I * + ln I n − j I * ]

V + ( n + 1 ) − V + ( n ) = β ∑ j = 0 h k ( j ) f ( S * , I * ) [ I n + 1 I * − I n − j I * − ln I n + 1 I * + ln I n − j I * ] . (44)

By adding Equations (42)-(44) we obtain

V ( n + 1 ) − V ( n ) ≤ − μ S ( S n + 1 − S * ) 2 S n + 1 + β ∑ j = 0 h k ( j ) f ( S * , I * ) ( 1 − f ( S n + 1 , I n + 1 j ) f ( S * , I * ) − S * S n + 1 + S * S n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) ) + β ∑ j = 0 h k ( j ) f ( S * , I * ) [ f ( S n + 1 , I n + 1 j ) f ( S * , I * ) − I n + 1 I * − I * I n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) + 1 ]

+ β ∑ j = 0 h k ( j ) f ( S * , I * ) [ I n + 1 I * − I n − j I * − ln I n + 1 I * + ln I n − j I * ] ≤ − μ S ( S n + 1 − S * ) 2 S n + 1 + β ∑ j = 0 h k ( j ) f ( S * , I * ) Q ( j ) ,

where

Q ( j ) = 2 − S * S n + 1 + S * S n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) − I * I n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) − I n − j I * − ln I n + 1 I * + ln I n − j I * . (45)

By adding and subtracting 1 + l n S * S n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) + l n I * I n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) to (45) we obtain:

Q ( j ) = ( − I n − j I * + 1 + ln I n − j I * ) + 1 − I * I n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) − ln I n + 1 I * − S * S n + 1 + S * S n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) = − g ( I n − j I * ) + ( − I * I n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) + 1 + ln I * I n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) ) + ( − S * S n + 1 + 1 + ln S * S n + 1 ) + ( S * S n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) − 1 − ln S * S n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) )

Q ( j ) = − g ( I n − j I * ) − g ( I * I n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) ) − g ( S * S n + 1 ) + g ( S * S n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) ) . (46)

By using the assumption H4 and the fact that the function g is increasing on ] 1, + ∞ [ , we have

g ( S * S n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) ) − g ( I * I n + 1 f ( S n + 1 , I n + 1 j ) f ( S * , I * ) ) ≤ 0

so we have Q ( j ) ≤ 0 . This implies that V ( n + 1 ) − V ( n ) ≤ 0 . Hence, by the Lyapunovs theorems on the global asymptotical stability for difference equations [

In this section, we presented a numerical result of continuous-time model (1) and the discrete one (2) study above. From this we used a particular incidence

function define by f ( S , I ) = β S I τ 1 + c I τ , which is the saturating incidence. In

addition we discuss from the different value of the basic reproduction number R 0 . We have the case R 0 ≤ 1 and R 0 > 1 . The parameters values used in the simulation are:

B = 100 ; μ S = 0.1 ; μ I = 0.02 ; μ R = 0.03 ; γ = 0.2 ; β = 0.00021 ; c = 0.2 ;

from this value we have R 0 = 0.095 < 1 .

When we change the value of β by β = 0.021 , we get R 0 = 9.54 > 1 . It’ is important to notice that the software used is Scilab and the time is in term of weeks or months. In our graphic the red curve give the evolution of the class in the discrete model and the dashed ones give the evolution of the class in continuous-time model.

For all these cuves, we can see the convergence of the red cuves (the discrete model) and the dashed ones (the continuous model)

In this paper, we have studied a discrete SIR epidemic model with general

incidence. We have proved the global stability of discrete SIR epidemic model by using the comparison theorem from the global stability of disease free equilibrium, when R 0 ≤ 1 , on the positive invariant set K and we have also proved the local stability of disease free equilibrium. The technique of Lyapunov function is used to proved the global stability of endemic equilibrium, when R 0 > 1 . We have made the numerical simulation to corroborate theoretical results. From the results obtained in this paper, we can conclude that the Euler backward difference scheme, that is, the discrete dynamical model (2), is obtained with excellent dynamical properties for the step size τ = 1 in the local and global stability of equilibra. These properties are nearly the same as the corresponding continuous-time model (1). In our future work, it shall be important for us to study the same model, but with general positive step size τ and see how bifurcation can happen.

The authors want to thank the anonymous referee for his valuable comments on the paper.

Aboudramane Guiro provide the subject, wrote the introduction and the conclusion and verified some calculation. Dramane Ouédraogo conceived the study and computed the equilibria and their local stabilities. Harouna Ouédraogo rote mathematical formula, bring up the Lyapunov functional and did all the calculus with the other authors. All the authors read and approved the final manuscript.

The authors declare that they have no competing interests.

Guiro, A., Ouedraogo, D. and Ouedraogo, H. (2018) Stability Analysis for a Discrete SIR Epidemic Model with Delay and General Nonlinear Incidence Function. Applied Mathematics, 9, 1039-1054. https://doi.org/10.4236/am.2018.99070