In this paper we reconsider a SIS epidemic model with maturation delay developed in [
Here 
is a birth rate function. The size of mature population 
at time 
is divided into susceptible 
and infective 
classes, so that 
.
, 
are the death rates of immature and mature population, respectively. The delay 
is considered as the maturation time. The parameter
, 
and 
are respectively the constant contact rate, the disease induced death rate constant and the recovery constant rate, respectively.
In this paper we consider a special class of Equation (1) by assuming that the death rate in each stage prior to the adult stage and the disease induced death are negligible, i.e.
. Using the Rick function as the birth rate function
, Equation (1) becomes
Note that Equation (2.c) is decoupled and is called the Nicholson’s blowflies equation which proposed by [
In [
It has also been shown in [
determines the existence of equilibria as well as their stability properties. They have shown numerically that if the delay 
is sufficiently large then the positive solutions of system (2) oscillate about the positive equilibrium. The existence and stability of Hopf bifurcation were established by Wei and Zou [
For practical purposes, we need to do numerical simulations and therefore we have to transform the continuous model (2) into a discrete system. We expect that the dynamical properties of the discrete system are in accordance with its continuous counterpart (2). Kunnawuttipreechachan [
To overcome the dependence of stability condition on the time-step size, we will apply a nonstandard finite difference (NSFD) scheme. This method, which is developed by Mickens [9,10], has been applied to various problems; see e.g. [11-16], in which the numerical solutions preserve dynamical properties of the continuous model. We will show that the discrete SIS epidemic model with a delay obtained by the NSFD method maintains the stability properties of equilibria irrespective of the size of numerical time-step. Besides the stability conditions, the existence of bifurcation of the discrete model will also be investigated.
, we will only consider the last two equations of system (2). Using a transformation 
and 
with
, Equation (2.b) and (2.c) can be written as
where k is a positive integer. Then, applying the forward difference scheme for the derivative and a nonlocal approximation for the right hand sides of system (4) yields a system of difference equations
and 
are numerical approximation of 
and 
, respectively. The numerical scheme (5) is a nonstandard because it uses a nonlocal approximation; see Mickens [9,10]. Here we consider initial conditions
and
,(6)
, for
. It is easy to show that the implicit scheme (5) can be arranged to get its explicit version, i.e.
if 
. This equilibrium is called the disease free equilibrium (DFE). On the other hand, if 
and 
then, in addition to the DFE, there also exists an endemic equilibrium (EE): 
where
and
.
is a bounded, closed and connected set;
is continuous in 
and 
is a parameter. Then as 
varies, the sum of the order of the zeros of 
out of the unit circle:
by taking 
and
. The linearized system around the DFE is given by
we can rewrite system (8) in the form
and the constant matrix A is defined by
, 
and 
. The characteristic equation of matrix A is
.
and 
then there exists a 
such that all roots of Equation (10) have modulus less than one for
.
.
then 
for all
. Other roots of Equation (10) are determined by
, Equation (11) becomes
.
, has a root 
of multiplicity k and a simple root
. Consider the root 
such that
. This root depends continuously on
. Based on Equation (11) we have
. Consequently all roots of Equation (10) lie in 
for all sufficiently small
, and the existence of the maximal 
follows.
then Equation (11) has no root with modulus one for all
.
with 
be a root of Equation (11) when
. Substituting this root to Equation (11) yields
.
then
, which yields a contradiction. This completes the proof.
, then the roots 
of Equation (11) with modulus one satisfy
then the roots of Equation (11) satisfy
and 
satisfy Equation (15).
then 
and therefore
.
and 
then the DFE is asymptotically stable for all
.
and 
then there exists an infinite sequence of time delay parameter 
such that the DFE is asymptotically stable when 
and unstable when
. Map (7) has a Neimark-Sacker bifurcation at the DFE when
, 
where 
satisfy Equation (15).
and 
then it is known from Lemmas 1 and 2 that the characteristic Equation (10) has no root with modulus one for all
. Applying Theorem 1, all roots of Equation (10) have modulus less than one for all
. Thus, conclusion (a) follows.
and
. It is known from Lemma 1 that if 
then one of roots of characteristics Equation (10) has modulus less than one for all
. Other roots of Equation (10) satisfy Equation (11). From Lemmas 1 and 3 we know that all roots of Equation (11) have modulus less than one when
, and Equation (11) has at least a couple of roots with modulus greater than one when
. Hence conclusion (b) follows.
is performed by substituting 
and 
into system (7) to get
,
,
,
.
, system (16) can be written as
. The constant matrix B in Equation (17) is
.
Clearly this characteristic equation has a trivial root
. Direct calculations show that if 
then 
for all
. Since 
and
, 
is exactly the same as in Equation (11) and therefore the distribution of its roots is exactly the same as described by Lemmas 1-3. Hence we directly have the following results.
and 
then there exists a 
such that all roots of Equation (18) have modulus less than one for
.
and 
then Equation (18) has no root with modulus one for all
.
and 
then the modulus of trivial root of Equation (18), i.e.
, is less than one for all 
and other roots satisfy
and 
satisfy Equation (15).
and 
then the EE is asymptotically stable for all
.
and 
then there exists an infinite sequence of time delay parameter 
such that the EE is asymptotically stable when 
and unstable when
. System (7) has a Neimark-Sacker bifurcation at the EE when
, 
where 
satisfy Equation (15).
.
, 
and
. For the simulations of DFE and EE scenarios we use 
and
, respectively.
and vary the value of p. Using this constant contact rate we have that
. First we choose 
and therefore the conditions for the DFE to be stable are satisfied, i.e.
. 
and
. It is indicated that the numerical solutions are convergent to the DFE (0.0, 0.9523) for any
. On the other hand, if we take 
(i.e.,
) and other parameters are the same as used in 
. Such behavior is shown in 
then the numerical solution converges to the DFE (0.0, 1.3064). On the contrary, the numerical solution, especially N(t), changes its stability from asymptotic stable to periodic behavior when
.
such that R0 = 2.3810 > 1. Figures 3(a) and (b) show the numerical solutions for 
(i.e
), using 
and
, respectively. These results indicate that if 
then the EE (0.5524, 0.9523) is asymptotically stable irrespective of
.
, i.e.
. Theorem 3 predicts that a Neimark-Sacker bifurcation will occur. From Equation (15) we find that the critical delay (bifurcation point) is
. This prediction is confirmed by our numerical solutions; see 
to τ = 1.70 >
, then the behavior of numerical solutions changes from a solution converging to the EE (0.7577, 1.3064) to a solution which oscillates periodically about the EE.
then the delay does not affect the stability of the equilibria. However, in the case of
, the stability of equilibria changes when the delay passes a critical value. Here the discrete SIS model with a delay has periodic solutions when the stability is lost. In other words, a Neimark-Sacker bifurcation occurs. It is also shown that our numerical simulations have confirmed our analytical findings.