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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 350892, 10 pages
doi:10.1155/2011/350892
Research Article
Permanence for a Delayed Nonautonomous
SIR Epidemic Model with Density-Dependent
Birth Rate
Li Yingke,1 Chen Liang,2 and Wang Kai3
1
College of Mathematics and Physics, Xinjiang Agricultural University, Urumqi 830052, China
Department of Mathematics, Changji College, Changji 831100, China
3
Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830054, China
2
Correspondence should be addressed to Li Yingke, lykhefeng@yahoo.com.cn
Received 21 February 2011; Accepted 12 May 2011
Academic Editor: Ibrahim Yalcinkaya
Copyright q 2011 Li Yingke et al. 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.
Based on some well-known SIR models, a revised nonautonomous SIR epidemic model with
distributed delay and density-dependent birth rate was considered. Applying some classical
analysis techniques for ordinary differential equations and the method proposed by Wang 2002,
the threshold value for the permanence and extinction of the model was obtained.
1. Introduction
For understanding the spread of infectious diseases in population, mathematical models that
use the theories of ordinary differential equations in epidemiology have been developed
rapidly. Epidemic models with delay, including either the autonomous continuous systems
or the discrete ones, were discussed by many authors 1–13. The important subjects for this
models are looking for the threshold value that determines whether the infectious disease
will be permanent or extinct. Hence, permanence of disease plays an important role in
epidemiology. Furthermore, it is well known that models with distributed delay are more
appropriate than the discrete ones because it is considered more realistic to assume the
infectivity to be a function of the duration since infection and up to some maximum duration.
Recently, Song and Ma in 4 and Song et al. in 5 discussed the permanence of disease
in a generalized autonomous SIR epidemic model with density dependent birth rate. On the
other hand, for the nonautonomous systems 1 and the reference therein, the literature
is still very inadequate. Motivated by the works in 1, 2, 4, 5, 9, 10, in this paper, we will
2
Discrete Dynamics in Nature and Society
consider the following nonautonomous delayed systems with density-dependent birth rate:
Ṡt −βtSt
h
0
İt βtSt
h
Nt
,
It − sdηs − μ1 tSt bt 1 − β1 t
1 Nt
1.1
It − sdηs − μ2 tIt − λtIt,
0
Ṙt λtIt − μ3 tRt,
where Nt St It Rt denote the total number of a population at time t, S is the
susceptible population, I is the infective population and R is the removed population. It is
assumed that all newborns are susceptible. Functions μ1 t, μ2 t and μ3 t are instantaneous
per capita mortality rates of susceptible, infective, and recovered population, respectively;
functions bt and λt represent the birth rate of the population and the recovery rate of
infectives, respectively; function β1 t0 ≤ β1 t < 1 reflects the relation between the birth
rate and the density of population. The nonnegative constant h is the time delay. The function
ηs : 0, h → 0, ∞ is nondecreasing and has hounded variation such that
h
1.2
dηs ηh − η0 1.
0
2. Preliminaries
Firstly, we give some notations for convenience: if ft is a continuous bounded function
defined on 0, ∞, then we set
f L inf ft,
t≥0
f M sup ft.
2.1
t≥0
Secondly, for system 1.1, we introduce the following assumption: functions bt, βt,
β1 t, λt, μ1 t, μ2 t, μ3 t are nonnegative continuous bounded functions and have positive
lower bounds. It is biologically natural to assume that μ2 t, μ3 t ≥ μ1 t i.e., epidemics will
increase the death rates of the infective and removed.
The initial condition of 1.1 is given as
Sθ ϕ1 θ,
Iθ ϕ2 θ,
Rθ ϕ3 θ,
−∞ < θ ≤ 0,
2.2
where φ ϕ1 , ϕ2 , ϕ3 T ∈ C such that ϕi θ ≥ 0 i 1, 2, 3 for all −∞ < θ ≤ 0, and
C denotes the Banach space C−∞, 0, R of bounded continuous functions mapping the
interval −∞, 0 into R with the toplogy of uniform convergence, that is, for ϕ ∈ C, we
designate the norm of an element φ ∈ C by φ sup−∞<θ≤0 {|ϕ1 θ|, |ϕ2 θ|, |ϕ3 θ|}. By a
biological meaning, we further assume that ϕi θ > 0 for i 1, 2, 3.
Now, a very impotent lemma for investigating the dynamics of the above system will
be given.
Discrete Dynamics in Nature and Society
3
Lemma 2.1. Any solution St, It, Rt of system 1.1 with 2.2 is positive for all t ≥ 0.
Proof. Because the right-hand side of system 1.1 is completely continuous and locally
Lipschitzian on C, the solution St, It, Rt of system 1.1 with initial condition 2.2
exists and is unique on 0, α, where 0 < α ≤ ∞. Firstly, we show that St > 0 for all
t ∈ 0, α. Otherwise, there is a t1 ∈ 0, α such that St1 0, Ṡt1 ≤ 0, and St > 0 for all
t ∈ 0, t1 . Hence, we have It > 0 for all t ∈ 0, t1 . If the statement is not true, then there is
a t2 ∈ 0, t1 such that It2 0, İt2 ≤ 0, and It > 0 on t ∈ 0, t2 . Integrating the second
equation of 1.1 from 0 to t2 , we get
It2 I0e
−
t2
0
μ2 sγsds
t2 h
0
u μ sγsds
βuSuI u − ρ e t2 2
dρ du > 0,
2.3
0
which is a contradiction. This shows that It > 0 for all t ∈ 0, t1 . From the third equation of
1.1, we also get that
t
2
μ3 s ds > 0
Rt ≥ R0 exp −
2.4
0
on 0, t1 . Therefore, from the first equation of system 1.1
Ṡt1 bt1 1 − β1 t1 bt1 β1 t1 > 0,
1 It1 Rt1 2.5
which also is a contradiction. This shows that St > 0 on 0, α. By the above calculation, it is
obvious that It > 0, Rt > 0 for all t ∈ 0, α.
3. Permanence and Extinction
In this section, we investigate the permanence of system 1.1 and demonstrate how the
disease will be extinct under some conditions.
Theorem 3.1. System 1.1 is permanent if
S0 M
bM 1 − β1
− μL1 2
M
bM 1 − β1
− μL1 4μL1 bM
2μL1
>
μ2 λ
β
M
.
3.1
Proof. We will give the following several propositions to complete the proof of this theorem.
Proposition 3.2. Assume that 3.1 holds. Then
lim sup Nt ≤ S0 .
t → ∞
3.2
4
Discrete Dynamics in Nature and Society
Proof. From system 1.1, we have that
btβ1 t
Ṅt −μ1 tSt − μ2 tIt − μ3 tRt bt 1 − β1 t 1 Nt
1
≤
−μ1 tNt1 Nt bt 1 − β1 t 1 Nt btβ1 t
1 Nt
M
1
−μL1 N 2 t − μL1 Nt bM 1 − β1 Nt bM
≤
1 Nt
1
Nt − S0 Nt − S0∗ ,
−
1 Nt
3.3
where
S0∗ 2
M
M
L
M
bM 1 − β1
b 1 − β1
− μ1 −
− μL1 4μL1 bM
2μL1
<0
.
3.4
Now, we consider only the following three cases.
i Nt ≤ S0 for all sufficiently large t. Obviously, for the case 3.2 is true.
ii Nt ≥ S0 for all sufficiently large t. In this case, Ṅt ≤ 0 for all sufficiently large t
such that
from 3.2. Hence, there is a constant N
≥ S0 .
lim Nt N
t → ∞
3.5
> S0 , for all sufficiently small ε : 0 < ε < N
− S0 , there
S0 . If N
In fact, we can obtain that N
∗
∗
is a sufficiently large t such that for any t > t ,
− ε ≤ Nt ≤ N
ε.
0<N
3.6
From 3.3, we have that for all t ≥ t∗ ,
Ṅt ≤ −
1
− S0 − ε N
− S0∗ − ε < 0,
N
ε
1N
3.7
Hence,
which implies that limt → ∞ Nt −∞. This is a contradiction to limt → ∞ Nt N.
0
S .
N
iii There is a time sequence {tn } : tn → ∞ as n → ∞ such that Ntn > S0 and
Ṅtn 0. Then, for t tn ,
Ṅt −
1
Nt − S0 Nt − S0∗ < 0,
1 Nt
which is a contradiction to Ṅtn 0. This completes the proof.
3.8
Discrete Dynamics in Nature and Society
5
Proposition 3.3. Assume that 3.1 holds. Then any positive solution St, It, Rt of system
1.1 with 2.2 satisfies
L
b 1 − β1
≡ ν1 .
lim inf St ≥
t → ∞
βS0 μ1
3.9
Proof. For any sufficiently small ε > 0, from Proposition 3.2, there is a large t1 > 0 such that
for all t > t1 > 0, It ≤ S0 ε. Hence, for t > t1 h,
h
It − sdηs − μ1 tSt bt 1 − β1 t
0
≥ bt 1 − β1 t − μ1 t βt S0 ε St,
Ṡt ≥ −βtSt
3.10
which clearly implies that
lim inf St ≥
t → ∞
L
b 1 − β1
≡ ν1 .
βS0 ε μ1
3.11
Noting that ε can be arbitrarily small, the conclusion is valid. This completes the proof of
Proposition 3.3.
Proposition 3.4. Assume that 3.1 holds. Then for any positive solution St, It, Rt of system
1.1 with 2.2, one has
lim inf It ≥ e−μ1 λ
t → ∞
M
d
≡ ν2 ,
3.12
where > 0, ε0 > 0 and d > 0 satisfy
L μ2 λ M
bβ1
1
>
,
b
1
−
β
p 1
M
β
1 S0 ε0
μ1 βM
μ2 λ M
SΔ p 1 − exp − μL1 βM d >
.
β
3.13
Proof. By 3.3 and 3.1 it is obvious that
M bβ1
M
μ2 λ M
1 >
.
S L b 1 − β1
β
1 S0
μ1
0
3.14
Then there exist two positive constants > 0, ε0 > 0 and d > 0 such that p > μ2 λ/βM
and SΔ > μ2 λ/βM .
6
Discrete Dynamics in Nature and Society
Let us consider the following differential function V t:
V t It h t
0
βu sSu sIudu dηs.
3.15
t−s
Then, the derivative of V t along with the solution of 1.1 with 2.2 is
V̇ t İt h
h
βt sSt sItdηs −
0
h
βtStIt − sdηs
0
βt sSt sdηs − μ2 t λt It
0
≥
β
L
h
St sdηs − μ2 λ
M
3.16
It.
0
We claim that it is impossible that It ≤ for all t ≥ t1 t1 is any nonnegative constant. In
fact, supposing the contrary, it follows from the first equation of 1.1 and 3.1 that for any
t ≥ t1 h,
h
btβ1 t
It − sdηs − μ1 tSt bt 1 − β1 t 1 Nt
0
btβ1 t
− μ1 t βM St.
≥ b1 t 1 − β1 t 1 S0 ε0
Ṡt −βtSt
3.17
Integrating the above inequality from t1 h to t, we obtain
St ≥ St1 h exp
t h
1
μ1 s β
M
ds
t
s btβ1 t
M
μ
dτ ds
bt 1 − β1 t exp
β
τ
1
1 S0 ε0
t1 h
t
t s bt 1−β1 t btβ1 t/ 1S0 ε0 μ1 τβM exp 0 μ1 τ βM dτ ds
t1 h
≥
t
μ1 τ βM exp 0 μ1 τ βM dτ
t
1
≥
μ1 βM
M
b 1 − β1 bβ1
1 S0 ε0
L 1 − exp − μL1 βM t − t1 − h .
3.18
Discrete Dynamics in Nature and Society
7
Thus, for any t ≥ t1 h d,
L bβ1
1
St ≥ 1 − exp − μL1 βM d SΔ .
b
1
−
β
1
0
M
1 S ε0
μ1 βM
3.19
Then, by the above inequality, from 3.16 and 3.19, we get, for any t ≥ t1 h d,
M V̇ t ≥ βL SΔ − μ2 λ
It.
3.20
i min It1 d 2h θ.
3.21
Set
θ∈−h,0
Next, we will prove that It ≥ i for all t ≥ t1 d h. Suppose that this is not true. Then
there is a T ≥ 0 such that It ≥ i for all t ∈ t1 d h, t1 d 2h T , It1 d 2h T i
and İt1 d 2h T ≤ 0. On the other hand, by the second equation of 1.1 and 3.16, for
t t1 d 2h T ,
İt βtSt
h
It − sdηs − μ2 tIt − λtIt
0
M i > 0.
≥ β L SΔ − μ2 λ
3.22
This is a contradiction to İt1 d 2h T ≤ 0. Hence, It ≥ i for all t ≥ t1 d h.
Consequently, for all t ≥ t1 d h, we obtain that
M V̇ t ≥ βL SΔ − μ2 λ
i > 0,
3.23
which implies that V t → ∞ as t → ∞. In fact, from Proposition 3.2, Vt is bounded,
which is a contradiction. Consequently, the claim is proved. From the claim, we will discuss
the following possible cases:
i It ≥ for all large t;
ii It oscillates about for all large t.
M
We show that It ≥ e−μ2 λ dh for all large t. Evidently, we only need to consider case ii.
Let t1 and t2 be sufficiently large such that
It1 It2 ,
It < as t ∈ t1 , t2 .
3.24
3.25
8
Discrete Dynamics in Nature and Society
If t2 − t1 ≤ d h and from the second equation of 1.1 we have that İt ≥ −μ2 λM It,
M
it is obvious that It ≥ e−μ2 λ dh for t ∈ t1 , t2 ; if t2 − t1 > d h, and then it is clear
M
that It ≥ e−μ2 λ dh for t ∈ t1 d h, t2 ; in fact, if not, there exists a T ∗ ≥ 0, such that
M
M
It ≥ e−μ2 λ dh for t ∈ t1 , t1 d h T ∗ ⊆ t1 , t2 , It1 d h T ∗ e−μ2 λ dh and
İt1 d h T ∗ ≤ 0. Using the second equation of system 1.1, as t t1 d h T ∗ ,
İt βtSt
h
It − sdηs − μ2 tIt − λtIt
0
M −μ2 λM dh
e
≥ β L SΔ − μ2 λ
> 0.
M
This is a contradiction to İt1 d hT ∗ ≤ 0. So, It ≥ e−μ2 λ
Since this kind of interval t1 , t2 is chosen as an arbitrary way,
M
lim inf It ≥ e−μ2 λ
dh
t → ∞
dh
3.26
is valid for all t ∈ t1 , t2 .
ν2 > 0.
3.27
The proof of Proposition 3.4 is completed.
Form the third equation of system 1.1 and Proposition 3.4, we easily obtain
lim inf Rt ≥ ν2
t → ∞
λ
μ2
L
ν3 > 0.
3.28
Thus, the system 1.1 is permanent by Propositions 3.2–3.4. The proof of Theorem 3.1 is
completed.
Next, we give a useful lemma discussing the extinction of the disease.
Lemma 3.5 see 12. Consider an autonomous system of delayed differential equation
ẋt a1
h
xt − hdηs − a2 xt,
3.29
0
where a1 , a2 are two constants. If 0 ≤ a1 < a2 , then for any solutions xt with initial condition
φθ ≥ 0, θ ∈ −h, 0, one has
lim xt 0.
t → ∞
3.30
Discrete Dynamics in Nature and Society
9
Theorem 3.6. If
0
S <
μ2 λ
β
L
,
3.31
then limt → ∞ xt 0; that is, the disease in system 1.1 will be extinct.
Proof. From Lemma 3.5, for any sufficiently small ε > 0, there exists a T > 0, for t > T , such
that
St < S0 ε.
3.32
Therefore, from the second equation of 1.1, we have
İt βtSt
≤β
M
h
It − sdηs − μ2 t λt It
0
S ε
0
h
L
It − sdηs − μ2 λ It.
3.33
0
Using the comparison theorem of functional differential equations and Lemma 3.5, we can
easily get limt → ∞ It 0. The proof is completed.
4. Discussion
In this paper, we discussed a revised nonautonomous SIR epidemic model with distributed
time delay and density-dependent birth rate. Theorems 3.1 and 3.6 show that the disease
will be permanent when S0 > μ2 λ/βM , and the disease will be extinct when S0 <
μ2 λ/βL . From the point of view of biology, Theorems 3.1 and 3.6 suggest that the
inequality S0 > μ2 λ/βM may be the threshold value for an epidemic disease to occur.
The technique for obtaining the sufficient condition on the permanence of the disease in this
paper is the same as that in 4, 5 if all coefficients in system 1.1 are constants and the delay
is discrete.
Acknowledgment
The authors are grateful to the anonymous referee for his/her many valuable comments and
suggestions.
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Discrete Dynamics in Nature and Society
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