Global Stability for a Class of Discrete Sir Epidemic Models

In this paper, we propose a class of discrete SIR epidemic models which are derived from SIR epidemic models with distributed delays by using a variation of the backward Euler method. Applying a Lyapunov functional technique, it is shown that the global dynamics of each discrete SIR epidemic model are fully determined by a single threshold parameter and the effect of discrete time delays are harmless for the global stability of the endemic equilibrium of the model.

Cooke [5] formulated an SIR epidemic model with a bilinear incidence rate and a discrete time delay which takes the form βSI to investigate the spread of an infectious disease transmitted by a vector (e.g.mosquitoes, rats, etc.) after an incubation time denoting the time during which the infectious agents develop in the vector.This is called the phenomena of time delay effect which now has important biological meanings in epidemic models (see [1,5]).
Based on their ideas, Ma et al. [13] considered a continuous-time SIR model with a discrete delay and investigated the global asymptotic stability of the equilibrium, while a continuous SIR model with distributed delays has been also studied in [1,2,12,14,17].The distributed delays is more appropriate form than the discrete one because it is considered more realistic to assume that the time delay is not a fixed time but a distributed parameter which is upper bounded by some positive finite time.Beretta and Takeuchi [1] have studied the following continuous SIR model with distributed delays: where s(t), i(t) and r(t) denote the proportions of the population susceptible to the disease, of infective members and of members who have been removed from the possibility of infection at time t, respectively.It is assumed that all newborns are susceptible and all recruitment is into the susceptible class at a constant rate b > 0. The positive constants µ 1 , µ 2 and µ 3 represent the death rates of susceptible, infectious and recovered classes, respectively.Infectiousness is assumed to vary over time from the initial time of infection until a duration h has passed and the function f (τ ) denotes the fraction of vector population in which the time taken to become infectious is τ .The mass action coefficient is β and f (τ ) is chosen so that it is nonnegative and continuous on [0, h] and assume that, for simplicity, ∫ h 0 f (τ )dτ = 1.System (1.1) always has a disease-free equilibrium E 0 = (b/µ 1 , 0, 0).Furthermore, if R 0 > 1, then system (1.1) has a unique endemic equilibrium E * = (s * , i * , r * ), where and For the case R 0 ≤ 1, it is known that disease dies out for (1.1) (see also [1,2,17]).For the case R 0 > 1, it is shown that system (1.1) is permanent by Ma et al. [12].Later, by using a Lyapunov functional for R 0 > 1, complete analysis of system (1.1) has been established by McCluskey [14].Thereafter, using the similar techniques, McCluskey [15] also established a complete analysis of an SIR epidemic models with a saturated incidence rate and a discrete delay.Summarizing the above discussion, the following result holds (see [1,2,12,17] and [14,Theorem 4.1]).
Theorem A. The following statement holds.
(i) Let us assume R 0 ≤ 1.Then the disease-free equilibrium E 0 of (1.1) is globally attractive, and locally asymptotically stable if the inequality is strict.
(ii) Let us assume R 0 > 1.Then the endemic equilibrium E * of (1.1) is globally asymptotically stable.
On the other hand, 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.Furthermore, this allows better use of statistical data for numerical simulations due to the reason that the infection data are compiled at discrete given time intervals.For a discrete epidemic model with immigration of infectives, Jang and Elaydi [10] showed the global asymptotic stability of the disease-free equilibrium, the local asymptotic stability of the endemic equilibrium and the strong persistence of susceptible class by means of the nonstandard discretization method.Recently, using a discretization called "mixed type" formula in Izzo and Vecchio [8] and Izzo et al. [9], Sekiguchi [16] obtained the permanence of a class of SIR discrete epidemic models with one delay and SEIRS discrete epidemic models with two delays if an endemic equilibrium of each model exists.For the detailed property for a class of discrete epidemic models, we refer to [3, 4, 8-11, 16, 18].
However, in those cases, how to choose the discrete schemes which guarantee the global asymptotic stability for the endemic equilibrium of the models, was still an open problem.
In this paper, motivated by the above facts, we propose the following discrete SIR epidemic model which is derived from system (1.1) by applying a variation of backward Euler method.
The initial conditions of system (1.4) is (1.5) Remark 1.1.To prove the positivity of s(p), i(p) and r(p) for any p ≥ 0, we need to use the backward Euler discretization instead of the forward Euler discretization (see, e.g., [8,9]).Moreover, to apply a discrete time analogue of the Lyapunov function proposed by McCluskey [14,15], we adopt a variation of the backward Euler method which is different from that of Sekiguchi [16].
Using the same threshold R 0 = bβ µ1(µ2+γ) and applying the both techniques in Izzo and Vecchio [8] and Izzo et al. [9] for the case R 0 ≤ 1 and McCluskey [14,Theorem 4.1] for the case R 0 > 1 to system (1.4), we establish a complete analysis of the global asymptotic stability for system (1.4).In particular, by applying Lemma 4.1, we offer a simplified proof for the permanence of system (1.4) for R 0 > 1 (cf.Sekiguchi [16]).
Our main result in this paper is as follows.
Remark 1.2.Theorem 1.1 for system (1.4) is just a discrete analogue of Theorem A for system (1.1).
The organization of this paper is as follows.In Section 2, we offer some basic results for system (1.4) and some asymptotic properties are investigated.The first part of Theorem 1.1 concerning the global asymptotic stability for R 0 ≤ 1 is given in Section 3, and the second part of Theorem 1.1 concerning the permanence and the global stability for R 0 > 1 is given in Section 4. Finally, we offer concluding remarks in Section 5.

Basic properties
For system (1.4), since the variable r does not appear in the first and the second equations, it is sufficient to consider the following 2-dimensional system: with the initial conditions 2) The following results in Sections 2 and 3 are obtained by applying techniques in Izzo and Vecchio [8] and Izzo et al. [9] to system (2.1).
( Proof.Let V (p) = s(p) + i(p).From system (2.1), we have that (2.4) Furthermore, we put then we easily obtain the following lemma: Lemma 2.4.P (s) is strictly monotone increasing function on [0, +∞) satisfying ) If R 0 > 1, then there exists a unique solution s = s * of P (s) = 1 such that Proof.By the definition of P (s),it is clear that P (s) is strictly monotone increasing on [0, +∞).By (2.4), we have (2.6) and (2.7).If R 0 > 1, then ).Hence, there exists a unique solution 0 In this section, we prove the first part of Theorem 1.1 for system (1.4).By applying techniques in Izzo et al. [9], we obtain the global asymptotic stability of the disease-free equilibrium E 0 of system (2.1) for R 0 ≤ 1. and E 0 = ( b µ1 , 0) is globally asymptotically stable.
Finally, we will prove that if R 0 ≤ 1, then E 0 = ( b µ1 , 0) is uniformly stable.First, consider the case that there exists a nonnegative integer p 1 such that s(p + 1) > s = b µ1 for any p ≥ p 1 .From the first equation of (2.1), we have that for any p > 0, Then, we obtain that b µ1 ≤ s(p+1) ≤ s(p) ≤ s(p 1 ) for any p ≥ p 1 .Next, consider the case that there exists a nonnegative integer p 2 such that s(p 2 ) ≤ s = b µ1 .Then, by the first equation of (2.1), we have that Thus, we obtain that s(p) ≤ s = b µ1 , for any p ≥ p 2 .Then, by the second equation of (2.1) and bβ µ1 ≤ µ 2 + λ, we have that for any p ≥ p 2 , Moreover, by the first equation of (2.1), we have that and hence, for any p ≥ p 2 , we obtain that Thus, by the above discussion, we can conclude that Then, by Lemma 2.4, there exists a positive constant solution (s(p), i(p)) = (s * , i * ) of system (2.1) defined by (1.2).This contradicts the fact that (3.1) holds.Hence, the proof is complete.□ Proof.By the second equation of (2.1), we have that

Proof of the first part of Theorem
Therefore, if i(p + 1) < min 0≤j≤m i(p − j), then by i(p) − i(p + 1) > 0 and ∑ m j=0 f (j)i(p − j) > i(p + 1), we have i(p + 1) > s(p+1) s * i(p + 1), from which we obtain s(p + 1) < s * .The remaining part of this lemma is evident.□ Lemma 4.2.If R 0 > 1, then for any solution of system (2.1) with the initial conditions (1.5), where for Proof.By the first equation of (2.3), it is straightforward to obtain (4.1).We now show that (4.2) holds.For any 0 < q < 1, by (1.2), one can see that We first prove the claim that any solution (s(p), i(p)) of system (2.1) does not have the following property: there exists a nonnegative integer p 0 such that i(p) ≤ qi * for all p ≥ p 0 .Suppose on the contrary that there exist a solution (s(p), i(p)) of system (2.1) and a nonnegative integer p 0 such that i(p) ≤ qi * for all p ≥ p 0 .From system (2.1), one can obtain that , for any p ≥ p 0 + m.
Therefore, we have By the second part of Lemma 4.1, we obtain Thus, there exists a positive constant î such that i(p) ≥ î for any p ≥ p 0 + m + l 0 − 1.Moreover, for the sequence {w(p)} +∞ p=p0 defined by (3.2), we have this yields lim p→+∞ w(p) = +∞.However, from Lemma 2.2, it holds that lim sup p→+∞ w(p) ≤ w, which leads to a contradiction.Hence, the claim is proved.By the claim, we are left to consider two possibilities.First, i(p) ≥ qi * for all p sufficiently large.Second, we consider the case that i(p) oscillates about qi * for all p sufficiently large.We now show that i(p) ≥ qv 2 for all p sufficiently large.
If the first case holds, then we get the conclusion of the proof.If the second case holds, let p 1 < p 2 be sufficiently large such that i(p 1 ), i(p 2 ) > qi * , and i(p) ≤ qi * for any p 1 < p < p 2 .

Discussions
Recently, in order to investigate the transmission dynamics of infectious diseases, there have been many papers focusing on the analysis of the global stability of the disease-free equilibrium and the endemic equilibrium of a class of discrete and continuous SIR epidemic models.Motivated by the fact that the discrete models are more appropriate forms than the continuous ones in order to directly fit the statistical data concerning infectious diseases with a latent period such as malaria, in this paper, we show that the disease-free equilibrium E 0 is globally asymptotically stable if R 0 ≤ 1, and the unique endemic equilibrium E * exists and is globally asymptotically stable if R 0 > 1 for the discrete SIR epidemic model (1.4) by applying a variation of the backward Euler discretization (cf.[8,9]) and Lyapunov functional techniques in McCluskey [14,15].We note that its stability conditions no longer need any restriction of the size of time delays for the model (1.4) which is discretized by a variation of the backward Euler method.From a biological viewpoint, it is noteworthy that the global dynamics is determined by a single threshold parameter R 0 without imposing any restriction on the length of an incubation period of the diseases not only for the continuous SIR model in [1,2,12,14,17] but also for this discrete SIR model.Applying these techniques to the other types of discrete epidemic models is our future work.