Research Article
Bifurcation Analysis of an SIR Epidemic Model with the Contact
Transmission Function
Guihua Li1
and Gaofeng Li2
1
School of Science, North University of China, Taiyuan, Shanxi 030051, China
2
Xinjiang Agriculture Second Division Korla Hospital, Korla, Xinjiang 841000, China
Correspondence should be addressed to Guihua Li; ttl1013@163.com
Received 8 December 2013; Accepted 23 December 2013; Published 21 January 2014
Academic Editor: Kaifa Wang
Copyright © 2014 G. Li and G. Li. 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.
We consider an SIR endemic model in which the contact transmission function is related to the number of infected population. By
theoretical analysis, it is shown that the model exhibits the bistability and undergoes saddle-node bifurcation, the Hopf bifurcation,
and the Bogdanov-Takens bifurcation. Furthermore, we find that the threshold value of disease spreading will be increased, when
the half-saturation coefficient is more than zero, which means that it is an effective intervention policy adopted for disease spreading.
However, when the endemic equilibria exist, we find that the disease can be controlled as long as we let the initial values lie in the
certain range by intervention policy. This will provide a theoretical basis for the prevention and control of disease.
1. Introduction
The classical SIR model for disease transmission has been
widely studied. It is one of the most important issues that the
dynamical behaviors are changed by the different incidence
rate in epidemic system. For the incidence rate, we divided
into two categories: one is that Capasso and Serio [1] proposed
the infection force which is a saturated curve, described
“crowding effect” or “protection measures;” the other is the
infection force that describes the effect of “intervention
policy,” for example, closing schools and restaurants and
postponing conferences (see Figure 1). For the model with
the saturated infection force, 𝑎𝐼2
/(𝑏 + 𝐼2
), which is one of
the typical infection forces, the rich dynamical behaviors were
founded by Ruan and Wang [2] and Tang et al. [3]. The model
with the incidence rate can be suited for many infectious
diseases, including measles, mumps, rubella, chickenpox,
and influenza. For more research literatures about nonlinear
infection rate see [4–9]. However, for some parasite-host
models, by observing macro- and microparasitic infections,
one finds that the infection rate is an increasing function of
the parasite dose, usually sigmoidal in shape [10, 11]. So we
will build a model with sigmoidal incidence rate which is
taken into account “crowding effect” and “saturated effect.”
According to the parasite-host model which is proposed by
Anderson and May (1979) [12, 13], the model is as follows:
𝑑𝑆
𝑑𝑡
= 𝐴 − 𝑑𝑆 − 𝛽 (𝐼) 𝑆,
𝑑𝐼
𝑑𝑡
= 𝛽 (𝐼) 𝑆 − (𝑑 + 𝛾 + 𝜖) 𝐼,
𝑑𝑅
𝑑𝑡
= 𝛾𝐼 − 𝑑𝑅,
(1)
where 𝑆, 𝐼, 𝑅 are susceptible hosts, infected hosts, and
removed hosts, respectively. 𝐴 is the birth rate of susceptible
host, 𝑑 is the natural death rate of a population, 𝛾 is the
removal rate, and 𝜖 is the per capita infection-related death
rate. If we denote infection force 𝛽(𝐼) = 𝑔(𝐼)𝐼, 𝑔(𝐼) can be
explained as the rate of valid contact. At the beginning of
disease, most people have poor awareness of prevention, then
the rate of valid contact 𝑔(𝐼) can be first increasing then tends
to a certain value. As the time flies, people are gradually aware
of the seriousness and take measures to prevent and control
development of the disease and will reduce to be contact with
infected, so the rate of contact 𝑔(𝐼) is first increasing then
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014,Article ID 930541, 7 pages
http://dx.doi.org/10.1155/2014/930541
2 Abstract and Applied Analysis
𝛽(I)
𝛽
O I
(a) Case 0 ≤ 𝑎
𝛽(I)
𝛽
O I1/a
(b) Case −2√ 𝑏 < 𝑎 < 0
Figure 1: The plotting for the contact transmission function 𝛽(𝐼).
decreasing. In short time, it does not tend to zero, but tends
to a nonzero constant. To simplify the study, we take
𝑔 (𝐼) =
𝛽𝐼2
𝑏 + 𝑎𝐼 + 𝐼2
, (2)
where 𝑏 > 0 and −2√ 𝑏 < 𝑎. If 𝑎 ≥ 0, 𝑔(𝐼) is increasing
monotonically and tends to 𝛽. If −2√ 𝑏 < 𝑎 < 0, 𝑔(𝐼) is first
increasing then decreasing and tends to 𝛽 (see Figure 1).
Then model (1) becomes
𝑑𝑆
𝑑𝑡
= 𝐴 − 𝑑𝑆 −
𝛽𝑆𝐼3
𝑏 + 𝑎𝐼 + 𝐼2
,
𝑑𝐼
𝑑𝑡
=
𝛽𝑆𝐼3
𝑏 + 𝑎𝐼 + 𝐼2
− (𝑑 + 𝛾 + 𝜖) 𝐼,
𝑑𝑅
𝑑𝑡
= 𝛾𝐼 − 𝑑𝑅,
(3)
where 𝛽 is the valid contact coefficient.
When 𝑎 = 0 and 𝑏 = 0, model (3) becomes
𝑑𝑆
𝑑𝑡
= 𝐴 − 𝑑𝑆 − 𝛽𝑆𝐼,
𝑑𝐼
𝑑𝑡
= 𝛽𝑆𝐼 − 𝜇𝐼,
𝑑𝑅
𝑑𝑡
= 𝛾𝐼 − 𝑑𝑅,
(4)
where 𝜇 = 𝑑 + 𝛾 + 𝜖.
We know that 𝑅0 = 𝛽𝐴/𝑑𝜇 is the basic reproduction
number of (4). It is easy to see that there is a unique positive
equilibrium 𝐼∗
in system (4) when 𝑅0 > 1 and there is no
positive equilibrium when 𝑅0 ≤ 1. In the next sections, we
will study that parameters 𝑎 and 𝑏 would have any effect on
the dynamic behaviors of model (3).
The organization of this paper is as follows. In the next
section, we analyze the existence and stability of the endemic
equilibria for model (3). Then we discuss conditions for
the Hopf bifurcation and the Bogdanov-Takens bifurcation
in Sections 3. Section 4 presents numerical simulations to
indicate dynamical behaviors and bifurcation structures, and
gives with a brief discussion.
2. Existence and Stability of Equilibria
We consider the positive equilibria of (3). Setting the right
hand sides of system (3) to zero, we find that the first and
second equations of system (3) do not include 𝑅, so we only
consider
𝐴 − 𝑑𝑆 −
𝛽𝑆𝐼3
𝑏 + 𝑎𝐼 + 𝐼2
= 0,
𝛽𝑆𝐼3
𝑏 + 𝑎𝐼 + 𝐼2
− 𝜇𝐼 = 0.
(5)
From the above two equations, except for the disease-free
equilibrium (DFE) at (𝐴/𝑑, 0), any endemic equilibrium
(EE), if exists, is the intersection of the following two curves
in the positive quadrant
𝑆 =
𝐴 − 𝜇𝐼
𝑑
,
𝑆 =
𝜇 (𝑏 + 𝑎𝐼 + 𝐼2
)
𝛽𝐼2
.
(6)
From (6), 𝐼 must satisfy the following equation:
𝐻 (𝐼) := 𝛽𝐼3
+ 𝑑 (1 − 𝑅0) 𝐼2
+ 𝑑𝑎𝐼 + 𝑑𝑏 = 0. (7)
Thus the intersection of two curves (6) is transformed into
the positive root of (7).
The derivative of 𝐻󸀠
(𝐼) is
𝐻󸀠
(𝐼) := 3𝛽𝐼2
+ 2𝑑 (1 − 𝑅0) 𝐼 + 𝑑𝑎. (8)
In the following, we consider three cases according to
the sign of 𝑎. By calculation, we have the following three
theorems.
Set
𝑎1 = 𝑑2
(1 − 𝑅0)
2
− 3𝛽𝑎𝑑,
𝑎2 = 27𝛽2
𝑑𝑏 − 𝑑𝑏 (1 − 𝑅0) (3𝛽𝑎𝑑 − 2𝑎1) .
(9)
Theorem 1. Suppose 𝑎 > 0. Then we have the following.
(a) If 𝑅0 ≤ 1 + √3𝛽𝑎/𝑑, then system (3) has no endemic
equilibrium.
Abstract and Applied Analysis 3
(b) If 𝑅0 > 1 + √3𝛽𝑎/𝑑, then we have the following.
(i) When 2𝑎3/2
1 < 𝑎2, system (3) has no endemic
equilibrium.
(ii) When 2𝑎3/2
1 = 𝑎2, system (3) has a unique endemic
equilibrium.
(iii) When 2𝑎3/2
1 > 𝑎2, system (3) has two endemic
equilibria 𝐸1(𝑆1, 𝐼1), 𝐸2(𝑆2, 𝐼2).
Theorem 2. Suppose 𝑎 = 0. Then we have the following.
(a) If 𝑅0 < 1+ 3
√27𝛽2 𝑏/4𝑑2, then system (3) has no endemic
equilibrium.
(b) If 𝑅0 = 1 + 3
√27𝛽2 𝑏/4𝑑2, then system (3) has a unique
endemic equilibrium.
(c) If 𝑅0 > 1 + 3
√27𝛽2 𝑏/4𝑑2, then system (3) has two
endemic equilibria 𝐸1(𝑆1, 𝐼1), 𝐸2(𝑆2, 𝐼2).
Theorem 3. Suppose −2√ 𝑏 < 𝑎 < 0. Then we have the
following.
(a) If 2𝑎3/2
1 < 𝑎2, then system (3) has no endemic equi-
librium.
(b) If 2𝑎3/2
1 = 𝑎2, then system (3) has a unique endemic
equilibrium 𝐸∗
(𝑆∗
, 𝐼∗
).
(c) If 2𝑎3/2
1 > 𝑎2, then system (3) has two endemic
equilibria 𝐸1(𝑆1, 𝐼1), 𝐸2(𝑆2, 𝐼2), where 𝐼1 < 𝐼∗
< 𝐼2.
Remark 4. From Theorems 1 and 2, we can find that the basic
reproduction number for the model (3) is less than that of the
standard model. It means that the disease will spread more
easily. For Theorem 3, it is obvious that the disease can exist
if 𝑅0 < 1.
For disease-free equilibrium (DFE), it is easy to calculate
that the Jacobian matrix of system (3) at DFE has eigenvalues
𝜆1 = −𝑑 and 𝜆2 = −𝜇. Hence, DFE is always stable.
In the following, the stability of the endemic equilibrium
in system (3) will be studied. Firstly, evaluating the Jacobian
matrix of system (3) at 𝐸(𝑆, 𝐼) gives
𝐽 = (
𝑗11 𝑗12
𝑗21 𝑗22
)
󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(𝑆,𝐼)
, (10)
where
𝑗11 = −𝑑 −
𝛽𝐼3
𝑏 + 𝑎𝐼 + 𝐼2
, 𝑗12 = −𝜇 −
𝜇 (2𝑏 + 𝑎𝐼)
𝑏 + 𝑎𝐼 + 𝐼2
,
𝑗21 =
𝛽𝐼3
𝑏 + 𝑎𝐼 + 𝐼2
, 𝑗22 =
𝜇 (2𝑏 + 𝑎𝐼)
𝑏 + 𝑎𝐼 + 𝐼2
.
(11)
Its characteristic equation is
𝑃 (𝜆) = 𝜆2
− tr (𝐽) 𝜆 + det (𝐽) = 0, (12)
where
det (𝐽) =
𝛽𝜇𝐼3
− 𝑑𝑎𝜇𝐼 − 2𝑑𝑏𝜇
𝑏 + 𝑎𝐼 + 𝐼2
, (13)
tr (𝐽) =
−𝛽𝐴𝐼2
+ 𝑎𝜇2
𝐼 + 2𝑏𝜇2
𝜇 (𝑏 + 𝑎𝐼 + 𝐼2)
. (14)
It is easy to calculate
𝛽𝜇𝐼∗3
− 𝑑𝑎𝜇𝐼∗
− 2𝑑𝑏𝜇 = 𝜇 [𝐼∗
𝐻󸀠
(𝐼∗
) − 2𝐻 (𝐼∗
)] = 0;
(15)
that is, det(𝐽)| 𝐼=𝐼∗ = 0.
Now suppose that the model has two endemic equilibria
𝐸1(𝑆1, 𝐼1), 𝐸2(𝑆2, 𝐼2), with 𝐼1 < 𝐼∗
< 𝐼2 < 𝐴/𝜇; that is, in
Theorem 1, the item (b) (iii) holds or Theorems 2 and 3, the
item 𝑐 holds. If 𝐽𝑖 (𝑖 = 1, 2) is the Jacobian matrix at (𝑆𝑖, 𝐼𝑖),
then (13) gives
det (𝐽𝑖) =
𝛽𝜇𝐼3
− 𝑑𝑎𝜇𝐼 − 2𝑑𝑏𝜇
𝑏 + 𝑎𝐼 + 𝐼2
󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝐼=𝐼𝑖
. (16)
Thus, it is easily obtained that det(𝐽1) is negative and det(𝐽2)
is positive. We can immediately conclude that the endemic
equilibrium 𝐸1 with low number of infected individuals is
always a saddle, and that the endemic equilibrium 𝐸2 with
high number of infected individuals is a node or focus but
the stability of 𝐸2 is determined by tr(𝐽2). From (14), we notice
that the sign of the trace of 𝐽2 is determined by
tr 1 := −𝛽𝐴𝐼2
+ 𝑎𝜇2
𝐼 + 2𝑏𝜇2
. (17)
Set
𝑏0 :=
𝑎𝜇2
+ 𝜇√ 𝑎2 𝜇2 + 8𝛽𝐴𝑏
2𝛽𝐴
,
𝑏1 :=
−𝑑 (1 − 𝑅0) + √ 𝑎1
3𝛽𝜇
,
𝑟0 := 𝛽𝐴 (𝑎𝑏0 (𝐴 + 1) + 𝐴𝑏) ,
𝑟1 := 𝑑 (𝑅0 − 1) (2 + 𝐴) (𝑎 + 2𝑏0 (1 − 𝑅0)) .
(18)
Theorem 5. Assume that (3) has two endemic equilibria. Then
𝐸2 is asymptotically stable if one of the following is satisfied.
(a) 𝑏0 < 𝑏1;
(b) 𝑏0 > 𝑏1 and 𝑟0 > 𝑟1.
Further, 𝐸2 is unstable if 𝑏0 > 𝑏1 and 𝑟0 < 𝑟1.
Proof. If 𝑏0 < 𝑏1, then −𝛽𝐴𝐼∗2
+ 𝑎𝜇2
𝐼∗
+ 2𝑏𝜇2
< 0. It follows
from 𝐼2 > 𝐼∗
that tr 1 < 0. Hence, 𝐸2 is asymptotically stable
in this case. If 𝑏0 > 𝑏1, we have 𝑏0 > 𝐼∗
. By direct calculations
we see that 𝑟0 > 𝑟1 implies 𝐼2 > 𝑏0, which leads to tr 1 < 0.
Therefore, 𝐸2 is asymptotically stable if condition (b) holds.
Similarly, if 𝑏0 < 𝑏1 and 𝑟0 < 𝑟1, we have 𝐼2 > 𝑏0, which leads
to tr 1 > 0. It follows that 𝐸2 is unstable.
4 Abstract and Applied Analysis
0.75
0.70
0.65
0.60
0.55
0.50
0.45
1.45 1.46 1.47 1.48 1.49 1.50
I
R0R1
Figure 2: The bifurcation curves the palne of (𝐼, 𝑅0).
3. Bifurcation of the System
3.1. Hopf Bifurcation. When the condition (b) (ii) in Theorem
1 and the condition (b) in Theorems 2 and 3 hold and 𝑟0 =
𝑟1, there are a pair of purely imaginary eigenvalues (Figure 2).
Thus for suitable parameter values a Hopf bifurcation may
occur, which means that there is a periodic solution around
the larger nontrivial equilibrium. In order to determine the
type of the Hopf bifurcation, we set
𝑞1 :=
𝜌𝑝3
(3𝛽𝑏𝐴 + 𝑎2
− 4𝑏)
(2𝑏𝜇𝑝 + 𝑎𝜌𝐴)
2
,
𝑞2 :=
−2𝛽𝑆2 [(𝑎2
− 𝑏) 𝑏𝐼4
2 + 4𝑎𝑏𝐼3
2 + 6𝑏2
𝐼2
2 − 𝑏3
]
(𝑏 + 𝑎𝐼2 + 𝐼2)
4
.
(19)
Then we consider the transformation 𝑋 = 𝑆 − 𝑆2, 𝑌 = 𝐼 − 𝐼2
to move (𝑆2, 𝐼2) to the origin of (𝑋, 𝑌). After some manipulations,
the model can be transformed into the following
equations:
𝑑𝑋
𝑑𝑡
= 𝑎11 𝑋 + 𝑎12 𝑌 − 𝐶 (𝑋, 𝑌) ,
𝑑𝑌
𝑑𝑡
= 𝑏11 𝑋 + 𝑏12 𝑌 + 𝐶 (𝑋, 𝑌) ,
(20)
where 𝐶(𝑋, 𝑌) represents the higher order terms and
𝑎11 = −𝑑 −
𝛽𝐼3
2
𝑏 + 𝑎𝐼2 + 𝐼2
2
, 𝑎12 = −𝜇 −
𝜇 (2𝑏 + 𝑎𝐼2)
𝑏 + 𝑎𝐼2 + 𝐼2
2
,
𝑏11 =
𝛽𝐼3
2
𝑏 + 𝑎𝐼2 + 𝐼2
2
, 𝑏12 =
𝜇 (2𝑏 + 𝑎𝐼2)
𝑏 + 𝑎𝐼2 + 𝐼2
2
.
(21)
Suppose 𝑟0 = 𝑟1. Then
tr (𝐽) = −𝑑 −
𝛽𝐼3
2
𝑏 + 𝑎𝐼2 + 𝐼2
2
+
𝜇 (2𝑏 + 𝑎𝐼2)
𝑏 + 𝑎𝐼2 + 𝐼2
2
= 0. (22)
By defining 𝜌 = 𝛽𝐼3
2 /(𝑏 + 𝑎𝐼2 + 𝐼2
2 ) and 𝑝 = 𝜇(2𝑏 + 𝑎𝐼2)/(𝑏 +
𝑎𝐼2 + 𝐼2
2 ), it can be seen that
𝑎11 = −𝑑 − 𝜌, 𝑎12 = −𝜇 − 𝑝, 𝑏11 = 𝜌,
𝑏12 = 𝑝, 𝑑 = 𝑝 − 𝜌.
(23)
Set
𝜔 = √det (𝐽2) = √ 𝜇𝜌 − 𝑑𝑝. (24)
Then the eigenvalues of 𝐽2 are 𝜆1 = 𝜔𝑖 and 𝜆2 = −𝜔𝑖.
Now, using the transformation 𝑢 = 𝑋, V = −(1/𝜔)(𝑎11 𝑋+
𝑎12 𝑌) to (20), we obtain
𝑑𝑢
𝑑𝑡
= −𝜔V + 𝐹1 (𝑢, V) ,
𝑑V
𝑑𝑡
= 𝜔𝑢 + 𝐹2 (𝑢, V) ,
(25)
where
𝐹1 (𝑢, V) = −𝐶 (𝑢,
𝜔V − 𝑝𝑢
𝜇 + 𝑝
) ,
𝐹2 (𝑢, V) =
𝜇
𝜔
𝐶 (𝑢,
𝜔V − 𝑝𝑢
𝜇 + 𝑝
) .
(26)
If
𝜎 =
1
16
[
𝜕3
𝐹1
𝜕𝑢3
+
𝜕3
𝐹2
𝜕𝑢 𝜕V2
+
𝜕3
𝐹2
𝜕𝑢2 𝜕V
+
𝜕3
𝐹2
𝜕V3
]
+
1
16𝜔
[
𝜕2
𝐹1
𝜕𝑢 𝜕V
(
𝜕2
𝐹1
𝜕𝑢2
+
𝜕2
𝐹1
𝜕V2
)−
𝜕2
𝐹2
𝜕𝑢 𝜕V
(
𝜕2
𝐹2
𝜕𝑢2
+
𝜕2
𝐹2
𝜕V2
)
−
𝜕2
𝐹1
𝜕𝑢2
𝜕2
𝐹2
𝜕𝑢2
+
𝜕2
𝐹1
𝜕V2
𝜕2
𝐹2
𝜕V2
]
󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑢=0,V=0
,
(27)
by some tedious calculations, we see that the sign of 𝜎 is
determined by 𝜉, where
𝜉 = 𝐴2
𝑞1 (2𝑑𝜌𝑞1 + 𝜇𝑝2
(2𝑑𝑝 − 𝜇𝜌 − 4𝑝2
))
+ 𝑝2
(𝑝3
(𝜇 + 𝑑) (𝜇 + 𝑝)
2
− 3𝑞2 𝜌𝐴2
𝜔2
) .
(28)
By the results in [14], the direction of the Hopf bifurcation
is determined by the sign of 𝜎. Therefore, we have the
following result.
Theorem 6. Suppose one condition of (c) in Theorem (6) holds
and 𝑟0 = 𝑟1. If 𝜉 ̸= 0, then a curve of periodic solutions
bifurcates from the endemic equilibrium 𝐸2 such that
Abstract and Applied Analysis 5
(i) for 𝜉 < 0, system (3) undergoes a supercritical Hopf
bifurcation;
(ii) for 𝜉 > 0, system (3) undergoes a subcritical Hopf
bifurcation.
Remark 7. Theorems 5 and 6 imply the occurrence of the
Allee effect because endemic equilibrium 𝐸2 and the diseasefree
equilibrium can be stable at the same time, or a stable
limit cycle and the disease-free equilibrium can be stable at
the same time.
3.2. Bogdanov-Takens Bifurcations. The purpose of this
subsection is to study the Bogdanov-Takens bifurcation of
(3) when there is a unique degenerate positive equilibrium.
Assume that
(H1)
(1) 𝑎 > 0, 𝑅0 > 1 + √3𝛽𝑎/𝑑 and 2𝑎3/2
1 = 𝑎2;
(2) 𝑎 = 0 and 𝑅0 = 1 + 3
√27𝛽2 𝑏/4𝑑2;
(3) 𝑎 < 0 and 2𝑎3/2
1 = 𝑎2.
Then system (3) admits a unique positive equilibrium
(𝑆∗
, 𝐼∗
) if one of (H1) is satisfied.
The Jacobian matrix of (3) at this point is
𝐽 = (
−𝑑 −
𝛽𝐼∗3
𝑏 + 𝑎𝐼∗ + 𝐼∗2
−𝜇 −
𝜇 (2𝑏 + 𝑎𝐼∗
)
𝑏 + 𝑎𝐼∗ + 𝐼∗2
𝛽𝐼∗3
𝑏 + 𝑎𝐼∗ + 𝐼∗2
𝜇 (2𝑏 + 𝑎𝐼∗
)
𝑏 + 𝑎𝐼∗ + 𝐼∗2
) . (29)
Since we are interested in codimension 2 bifurcations, we
assume further
(H2) 𝑟0 = 𝑟1
By (15), we have
det (𝐽) =
𝛽𝜇𝐼∗3
− 𝑑𝑎𝜇𝐼∗
− 2𝑑𝑏𝜇
𝑏 + 𝑎𝐼∗ + 𝐼∗2
= 0. (30)
Furthermore, (H2) implies that
tr (𝐽) =
−𝛽𝐴𝐼∗2
+ 𝑎𝜇2
𝐼∗
+ 2𝑏𝜇2
𝜇 (𝑏 + 𝑎𝐼∗ + 𝐼∗2)
= 0. (31)
Thus, (H1) and (H2) imply that the Jacobian matrix has a
zero eigenvalue with multiplicity 2. This suggests that (3) may
admit a Bogdanov-Takens bifurcation. The next theorem will
confirm this.
Theorem 8. Suppose that (H1) and (H2) hold. Then the
equilibrium (𝑆∗
, 𝐼∗
) of (3) is a cusp of codimension 2; that is, it
is a Bogdanov-Takens singularity.
Proof. In order to translate the interior equilibrium (𝑆∗
, 𝐼∗
)
to the origin, we set 𝑥 = 𝑆 − 𝑆∗
, 𝑦 = 𝐼 − 𝐼∗
. Expanding the
right-hand side of the system (3) in a Taylor series about the
origin, we obtain
𝑑𝑥
𝑑𝑡
= 𝑎11 𝑥 + 𝑎12 𝑦 + 𝑎21 𝑥𝑦 + 𝑎22 𝑦2
+ 𝑃1 (𝑥, 𝑦) ,
𝑑𝑦
𝑑𝑡
= −
𝑎2
11
𝑎12
𝑥 − 𝑎11 𝑦 − 𝑎21 𝑥𝑦 − 𝑎22 𝑦2
+ 𝑃2 (𝑥, 𝑦) ,
(32)
where 𝑃𝑖(𝑥, 𝑦) is a smooth function in (𝑥, 𝑦) at least of order
three and
𝑎11 = −𝑑 −
𝛽𝐼∗3
𝑏 + 𝑎𝐼∗ + 𝐼∗2
< 0,
𝑎12 = −𝜇 −
𝜇 (2𝑏 + 𝑎𝐼∗
)
𝑏 + 𝑎𝐼∗ + 𝐼∗2
2
< 0,
𝑎21 =
−𝛽𝐼∗2
(3𝑏 + 2𝑎𝐼∗
+ 𝐼∗2
)
(𝑏 + 𝑎𝐼∗ + 𝐼∗2)
2
< 0,
𝑎22 =
−2𝛽𝑆∗
𝐼∗
[3𝑏2
+ 3𝑎𝑏𝐼∗
+ (𝑎2
− 𝑏) 𝐼∗2
]
(𝑏 + 𝑎𝐼∗ + 𝐼∗2)
3
.
(33)
Set 𝑋 = 𝑥, 𝑌 = 𝑎11 𝑥 + 𝑎12 𝑦. Then (32) is transformed into
𝑑𝑋
𝑑𝑡
= 𝑌 + 𝑐1 𝑋2
+ 𝑐2 𝑋𝑌 + 𝑐3 𝑌2
+ 𝑄1 (𝑋, 𝑌) ,
𝑑𝑌
𝑑𝑡
= −𝑑1 𝑋2
+ 𝑑2 𝑋𝑌 + 𝑑3 𝑌2
+ 𝑄2 (𝑋, 𝑌) ,
(34)
where 𝑄𝑖 are smooth functions in (𝑋, 𝑌) at least of order three
and
𝑐1 =
𝑎11 (𝑎11 𝑎22 − 𝑎12 𝑎21)
𝑎2
12
,
𝑑1 =
𝑎11 (𝑎12 𝑎21 − 𝑎11 𝑎22) (𝑎11 − 𝑎12)
𝑎2
12
,
𝑐2 =
𝑎12 𝑎21 − 2𝑎11 𝑎22
𝑎2
12
,
𝑑2 =
(𝑎12 𝑎21 − 2𝑎11 𝑎22) (𝑎11 − 𝑎12)
𝑎2
12
,
𝑐3 =
𝑎22
𝑎2
12
,
𝑑3 =
𝑎22 (𝑎11 − 𝑎12)
𝑎2
12
.
(35)
Change the variables one more time by letting 𝑋 = 𝑋, 𝑃 =
𝑌 + 𝑐3 𝑌2
; we have
𝑑𝑋
𝑑𝑡
= 𝑃 + 𝑐1 𝑋2
+ 𝑐2 𝑋𝑌 + 𝑄3 (𝑋, 𝑃) ,
𝑑𝑃
𝑑𝑡
= −𝑑1 𝑋2
+ 𝑑2 𝑋𝑃 + 𝑑3 𝑃2
+ 𝑄4 (𝑋, 𝑃) .
(36)
6 Abstract and Applied Analysis
Let 𝑋 = 𝑋, 𝑍 = 𝑃 − 𝑑3 𝑋𝑃. Then system (36) becomes
𝑑𝑋
𝑑𝑡
= 𝑍 + 𝑐1 𝑋2
+ (𝑐2 + 𝑑3) 𝑋𝑍 + 𝑄5 (𝑋, 𝑍) ,
𝑑𝑍
𝑑𝑡
= −𝑑1 𝑋2
+ 𝑑2 𝑋𝑍 + 𝑄6 (𝑋, 𝑍) .
(37)
In order to obtain the canonical normal forms, we perform
the transformation of variables by
𝑢 = 𝑋 −
𝑐2 + 𝑑3
2
𝑋2
, V = 𝑍 + 𝑐1 𝑋2
. (38)
Then, we obtain
𝑑𝑢
𝑑𝑡
= V + 𝑅1 (𝑢, V) ,
𝑑V
𝑑𝑡
= −𝑑1 𝑢2
+ (𝑑2 + 2𝑐1) 𝑢V + 𝑅2 (𝑢, V) ,
(39)
where 𝑅𝑖 are smooth functions in (𝑢, V) at least of the third
order.
Note that 𝑑1 > 0 and
𝑑2 + 2𝑐1 =
−𝑎11 𝑎21 − 𝑎21 𝑎12 + 2𝑎22 𝑎11
𝑎12
. (40)
In addition, by (30) and (31), it is obtained that
𝑎11 = −
𝐴
𝑆∗
, 𝑎12 = −𝜇 −
𝐴
𝑆∗
,
𝑎21 =
𝜇𝐼∗
𝑆∗
, 𝑎22 =
𝐴
𝑆∗
,
𝐴2
𝑆∗2
=
𝜇𝐼∗
𝑆∗
(𝜇 +
𝐴
𝑆∗
) .
(41)
So
𝑑2 + 2𝑐1 =
−𝑎11 𝑎21 − 𝑎21 𝑎12 + 2𝑎22 𝑎11
𝑎12
=
1
𝑎12
𝐴 (𝜇𝐼∗
− 𝐴)
𝑆∗2
> 0.
(42)
It follows that (3) admits that a Bogdanov-Takens bifurcation
from [15, 16] or [17].
4. Simulations and Conclusions
In the following, we use numerical simulations, based upon
the MatCont package [18], to reveal how parameters 𝑎 induce
bifurcations and limit cycles in system (3). Firstly, by fixing
𝐴 = 2, 𝑑 = 0.1, 𝛽 = 0.8, 𝑏 = 2.4, 𝜖 = 0.6, 𝛾 = 0.2, we plot a
2D-plot of variable 𝐼 versus parameter 𝑎 shown in Figure 3.
We find a Hopf bifurcation at 𝑎 = −0.090429, a limit point
(fold) bifurcation at 𝑎 = 2.378982. The Lyapunov coefficient
is 1.68171 × 10−2
, which means that the periodic orbits are
unstable. Furthermore, 𝑎 is fixed −0.13; we observe the orbits
of system (3) is how to vary with 𝑡. From Figure 4, we can find
−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
a
I
H
LP
Figure 3: Bifurcation curves in (𝑎, 𝐼) plane by fixed 𝐴 = 2, 𝑑 =
0.1, 𝛽 = 0.8, 𝑏 = 2.4, 𝜖 = 0.6, 𝛾 = 0.2, where H denotes the Hopf
bifurcation, LP is the limit point (flod) bifurcation.
4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
2
0
50
S
I
t
−300
−250
−200
−150
−100
−50
Figure 4: Phase trajectory in system (3) by fixed 𝐴 = 2, 𝑑 = 0.1, 𝛽 =
0.8, 𝑏 = 2.4, 𝜖 = 0.6, 𝛾 = 0.2, 𝑎 = −0.13.
that the periodic orbits will occur, but the disease will die out
when 𝑡 → +∞, though there exist the positive equilibria
for system (3). Furthermore, we take 𝑅0 and 𝑎 as bifurcation
parameters; from Figure 5, we can show that the system has
no positive equilibrium when 𝑅0 and 𝑎 lie in the left side of
red curve and two endemic equilibria when they are in the
right side of red curve. If parameters 𝑅0 and 𝑎 are between
red and green curves, we find that system will undergo Hopf
bifurcation.
In the paper, we built a model with contact transmission
function and obtained the dynamical behaviors. From the
analysis, we find that the threshold value of disease spreading
will be larger. It means that it is an effective intervention
policy adopted for disease spreading. For the disease-free
equilibrium is always locally stable and when a positive
equilibrium exist and is stable, we can control the disease as
long as we let the initial values be in the certain range by
intervention policy. If the positive equilibrium is unstable, the
Abstract and Applied Analysis 7
0 5 10 15 20 25 30
0
2
4
6
8
10
LP
LP
H
LP
H
GH
GH
R0
−2
a
Figure 5: Bifurcation figure when 𝑅0 and 𝑎 are taken as bifurcation
parameters in system (3) by fixed 𝐴 = 2, 𝑑 = 0.1, 𝛽 = 0.8, 𝑏 = 2.4, 𝜖 =
0.6, 𝛾 = 0.2.
disease will die out. This will provide a theoretical basis for the
prevention and control of disease.
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
Acknowledgments
This work is supported by the National Science Foundation of
China (11201434, 11271369) and Shanxi Scholarship Council
of China (2013-087).
References
[1] V. Capasso and G. Serio, “A generalization of the KermackMcKendrick
deterministic epidemic model,” Mathematical Biosciences,
vol. 42, no. 1-2, pp. 43–61, 1978.
[2] S. Ruan and W. Wang, “Dynamical behavior of an epidemic
model with a nonlinear incidence rate,” Journal of Differential
Equations, vol. 188, no. 1, pp. 135–163, 2003.
[3] Y. Tang, D. Huang, S. Ruan, and W. Zhang, “Coexistence of limit
cycles and homoclinic loops in a SIRS model with a nonlinear
incidence rate,” SIAM Journal on Applied Mathematics, vol. 69,
no. 2, pp. 621–639, 2008.
[4] D. Xiao and H. Zhu, “Multiple focus and Hopf bifurcations in a
predator-prey system with nonmonotonic functional response,”
SIAM Journal on Applied Mathematics, vol. 66, no. 3, pp. 802–
819, 2006.
[5] W. Wang, “Epidemic models with nonlinear infection forces,”
Mathematical Biosciences and Engineering, vol. 3, no. 1, pp. 267–
279, 2006.
[6] D. Xiao and S. Ruan, “Global analysis of an epidemic model
with nonmonotone incidence rate,” Mathematical Biosciences,
vol. 208, no. 2, pp. 419–429, 2007.
[7] A. B. Gumel and S. M. Moghadas, “A qualitative study of a vaccination
model with non-linear incidence,” Applied Mathematics
and Computation, vol. 143, no. 2-3, pp. 409–419, 2003.
[8] W. M. Liu, H. W. Hethcote, and S. A. Levin, “Dynamical
behavior of epidemiological models with nonlinear incidence
rates,” Journal of Mathematical Biology, vol. 25, no. 4, pp. 359–
380, 1987.
[9] W. M. Liu, S. A. Levin, and Y. Iwasa, “Influence of nonlinear
incidence rates upon the behavior of SIRS epidemiological
models,” Journal of Mathematical Biology, vol. 23, no. 2, pp. 187–
204, 1986.
[10] R. R. Regoes, D. Ebert, and S. Bonhoeffer, “Dose-dependent
infection rates of parasites produce the Allee effect in epidemiology,”
Proceedings of the Royal Society B, vol. 269, no. 1488, pp.
271–279, 2002.
[11] G. Li and W. Wang, “Bifurcation analysis of an epidemic
model with nonlinear incidence,” Applied Mathematics and
Computation, vol. 214, no. 2, pp. 411–423, 2009.
[12] R. M. Anderson and R. M. May, “Population biology of
infectious diseases: Part I,” Nature, vol. 280, no. 5721, pp. 361–
367, 1979.
[13] R. M. May and R. M. Anderson, “Population biology of
infectious diseases: Part II,” Nature, vol. 280, no. 5722, pp. 455–
461, 1979.
[14] L. Perko, Differential Equations and Dynamical Systems, vol. 7,
Springer, New York, NY, USA, 2nd edition, 1996.
[15] R. Bogdanov, “Bifurcations of a limit cycle for a family of vector
fields on the plan,” Selecta Mathematica, vol. 1, pp. 373–388, 1981.
[16] R. Bogdanov, “Versal deformations of a singular point on the
plan in the case of zero eigenvalues,” Selecta Mathematica, vol.
1, pp. 389–421, 1981.
[17] F. Takens, “Forced oscillations and bifurcations,” in Applications
of Global Analysis I, pp. 1–59, Rijksuniversitat Utrecht, 1974.
[18] A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Limit cycles and
their bifucations in MatCont,” http://www.matcont.ugent.be/
Tutorial2.pdf.
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