Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 293293, 12 pages
http://dx.doi.org/10.1155/2013/293293
Research Article
Stability Analysis of a Vector-Borne Disease with
Variable Human Population
Muhammad Ozair,1 Abid Ali Lashari,1 Il Hyo Jung,2 Young Il Seo,3 and Byul Nim Kim2
1
Centre for Advanced Mathematics and Physics, National University of Sciences and Technology,
H-12, Islamabad 44000, Pakistan
2
Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
3
National Fisheries Research and Development Institute, Busan 619-705, Republic of Korea
Correspondence should be addressed to Il Hyo Jung; ilhjung@pusan.ac.kr
Received 25 November 2012; Revised 1 March 2013; Accepted 2 March 2013
Academic Editor: Ferenc Hartung
Copyright © 2013 Muhammad Ozair 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.
A mathematical model of a vector-borne disease involving variable human population is analyzed. The varying population size
includes a term for disease-related deaths. Equilibria and stability are determined for the system of ordinary differential equations.
If 𝑅0 ≤ 1, the disease-“free” equilibrium is globally asymptotically stable and the disease always dies out. If 𝑅0 > 1, a unique
“endemic” equilibrium is globally asymptotically stable in the interior of feasible region and the disease persists at the “endemic”
level. Our theoretical results are sustained by numerical simulations.
1. Introduction
Vector-borne diseases such as malaria, dengue fever, plague,
and West Nile fever are infectious diseases caused by the
influx of viruses, bacteria, protozoa, or rickettsia which
are primarily transmitted by disease-transmitting biological
agents, called vectors. A vector-borne disease is transmitted
by a pathogenic microorganism from an infected host to
another organism results to form from an infection by bloodfeeding arthropods [1].
Vector-borne diseases, in particular, mosquito-borne disease, are transmitted to humans by blood-sucker mosquitoes,
which have been a big problem for the public health in the
world. The literature dealing with the mathematical theory on
vector-borne diseases is quite extensive. Many mathematical
models concerning the emergence and reemergence of the
vector-host infectious disease have been proposed and analyzed in the literature [2, 3].
By direct transmission models, we mean that the infection
moves from person to person directly, with no environmental source, intermediate vector, or host. In a vector-host
model, direct transmission may take place by transfusionrelated transmission, transplantation-related transmission,
and needle-stick-related transmission [4]. Some models have
been developed to study the dynamics of a vector-borne
disease that considers a direct mode of transmission in
human host population [5–7].
Mathematical modeling has proven to play an important
role in gaining some insights into the transmission dynamics
of infectious diseases and suggest control strategies. Appropriate mathematical models can provide a qualitative assessment for the problem. Some mathematical models discussed
in [8–10] provide, best understanding about the dynamics
and control of infectious diseases. Immense literature on the
use of mathematical models for communicable diseases is
available [11, 12]. The assumption of constant population size
in epidemiological models is usually valid when we study the
diseases of short duration with limited effects on mortality. It
may not be valid when dealing with endemic diseases such as
malaria, which has a high mortality rate. Ngwa and Shu [13]
assumed density-dependent death rates in both vector and
human populations, so that the total populations are varying
with time that includes disease-related deaths. Esteva and
Vargas [14] analyzed the effect of variable host population size
and disease-induced death rate. Recently Ozair et. al analyzed
vector-host disease model with nonlinear incidence [15].
2
Abstract and Applied Analysis
In this paper, based on the ideas posed in [6, 14], we
develop and analyze a vector-host disease model considering
a direct mode of transmission as well as a variable human
population. The aim of this paper is to establish stability
properties of equilibria and the threshold parameter 𝑅0 that
completely determines the existence of endemic or diseasefree equilibrium. If 𝑅0 ⩽ 1, the disease-free equilibrium is
globally asymptotically stable. If 𝑅0 > 1, a unique endemic
equilibrium exists and is globally asymptotically stable under
parametric restrictions. However, in numerical simulations it
is shown that the disease still can be “endemic” even if the
conditions are violated.
The rest of the paper is organized as follows. In Section 2,
we present a formulation of the extended mathematical
model. The dimensionless formulation of proposed model
is carried out in Section 3. Section 4 devotes existence
and uniqueness of “endemic” equilibria. In Section 5, we
use Lyapunov function theory to show global stability of
disease-“free” equilibrium and geometric approach to prove
global stability of “endemic” equilibrium. Discussions and
simulations are done in Section 6.
𝑑𝑁ℎ
= 𝑏1 𝑁ℎ − 𝜇ℎ 𝑁ℎ − 𝛿ℎ 𝐼ℎ .
𝑑𝑡
(2)
3. Dimensionless Formulation
Denote 𝑠ℎ = 𝑆ℎ /𝑁ℎ , 𝑖ℎ = 𝐼ℎ /𝑁ℎ , 𝑟ℎ = 𝑅ℎ /𝑁ℎ , 𝑠V = 𝑆V /𝑁V , and
𝑖V = 𝐼V /𝑁V . It is easy to verify that 𝑠ℎ , 𝑖ℎ , 𝑟ℎ , 𝑠V , and 𝑖V satisfy
the following system (see the Appendix details for):
𝑑𝑠ℎ (𝑡)
= 𝑏1 (1 − 𝑠ℎ ) − 𝛽1 𝑠ℎ 𝑖ℎ − 𝛽2 𝑠ℎ 𝑖V + 𝛿ℎ 𝑠ℎ 𝑖ℎ ,
𝑑𝑡
2. Model Formulation
The human population is partitioned into subclasses of
individuals who are susceptible, infectious, and recovered,
with sizes denoted by 𝑆ℎ (𝑡), 𝐼ℎ (𝑡), and 𝑅ℎ (𝑡), respectively.
The vector population is subdivided into susceptible and
infectious vectors, with sizes denoted by 𝑆V (𝑡) and 𝐼V (𝑡),
respectively. The mosquito population does not have an
immune class, since their infective period ends with their
death. Thus, 𝑁ℎ (𝑡) = 𝑆ℎ (𝑡) + 𝐼ℎ (𝑡) + 𝑅ℎ (𝑡) and 𝑁V (𝑡) =
𝑆V (𝑡) + 𝐼V (𝑡) are, respectively, the total human and vector
populations at time 𝑡. The model is given by the following
system of differential equations:
𝛽𝑆 𝐼
𝛽𝑆 𝐼
𝑑𝑆ℎ (𝑡)
= 𝑏1 𝑁ℎ − 1 ℎ ℎ − 2 ℎ V − 𝜇ℎ 𝑆ℎ ,
𝑑𝑡
𝑁ℎ
𝑁V
𝑑𝐼ℎ (𝑡) 𝛽1 𝑆ℎ 𝐼ℎ 𝛽2 𝑆ℎ 𝐼V
+
− 𝜇ℎ 𝐼ℎ − 𝛾ℎ 𝐼ℎ − 𝛿ℎ 𝐼ℎ ,
=
𝑑𝑡
𝑁ℎ
𝑁V
𝑑𝑅ℎ (𝑡)
= 𝛾ℎ 𝐼ℎ − 𝜇ℎ 𝑅ℎ ,
𝑑𝑡
and indirect route of transmission is given by the standard
incidence form 𝛽1 (𝑆ℎ 𝐼ℎ /𝑁ℎ ) and 𝛽2 (𝑆ℎ 𝐼V /𝑁ℎ ), respectively.
The term 𝜇ℎ is the natural mortality rate of humans. We
assume that infectious individuals acquire permanent immunity by the rate 𝛾ℎ . The infectious humans suffer from diseaseinduced death at a rate 𝛿ℎ . The recruitment and natural
death rate of vector population is assumed to be 𝜇V . The
susceptible vectors become infectious as a result of biting
effect of infectious humans at a rate 𝛽3 , so that the incidence
of newly infected vectors is again given by standard incidence
form 𝛽3 (𝑆V 𝐼ℎ /𝑁ℎ ). The total human population is governed by
the following equation:
𝑑𝑖ℎ (𝑡)
= 𝛽1 𝑠ℎ 𝑖ℎ + 𝛽2 𝑠ℎ 𝑖V − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) 𝑖ℎ + 𝛿ℎ 𝑖ℎ2 ,
𝑑𝑡
𝑑𝑟ℎ (𝑡)
= 𝛾ℎ 𝑖ℎ − 𝑏1 𝑟ℎ + 𝛿ℎ 𝑖ℎ 𝑟ℎ ,
𝑑𝑡
(3)
𝑑𝑠V (𝑡)
= 𝜇V (1 − 𝑠V ) − 𝛽3 𝑠V 𝑖ℎ ,
𝑑𝑡
𝑑𝑖V (𝑡)
= 𝛽3 𝑠V 𝑖ℎ − 𝜇V 𝑖V ,
𝑑𝑡
where solutions are restricted to 𝑠ℎ +𝑖ℎ +𝑟ℎ = 1 and 𝑠V +𝑖V = 1.
Before analyzing the unnormalized model (1) and (2), we
consider the normalized model (3) by scaling, and so we
can study the following reduced system that describes the
dynamics of the proportion of individuals in each class
𝑑𝑠ℎ (𝑡)
= 𝑏1 (1 − 𝑠ℎ ) − 𝛽1 𝑠ℎ 𝑖ℎ − 𝛽2 𝑠ℎ 𝑖V + 𝛿ℎ 𝑠ℎ 𝑖ℎ ,
𝑑𝑡
(1)
𝛽𝑆𝐼
𝑑𝑆V (𝑡)
= 𝜇V 𝑁V − 3 V ℎ − 𝜇V 𝑆V ,
𝑑𝑡
𝑁ℎ
𝑑𝐼V (𝑡) 𝛽3 𝑆V 𝐼ℎ
− 𝜇V 𝐼V .
=
𝑑𝑡
𝑁ℎ
In model (1), 𝑏1 is the recruitment rate of humans into the
population which is assumed to be susceptible. Susceptible
hosts get infected via two routes of transmission, through
a contact with an infected individual and through being
bitten by an infectious vector. We denote the infection rate of
susceptible individuals which results from effective contact
with infectious individuals by 𝛽1 and 𝛽2 is the infection
rate of susceptible humans resulting due to the biting of
infected vectors. The incidence of new infections via direct
𝑑𝑖ℎ (𝑡)
= 𝛽1 𝑠ℎ 𝑖ℎ + 𝛽2 𝑠ℎ 𝑖V − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) 𝑖ℎ + 𝛿ℎ 𝑖ℎ2 ,
𝑑𝑡
(4)
𝑑𝑖V (𝑡)
= 𝛽3 (1 − 𝑖V ) 𝑖ℎ − 𝜇V 𝑖V ,
𝑑𝑡
determining 𝑟ℎ from
𝑑𝑟ℎ (𝑡)
= 𝛾ℎ 𝑖ℎ − 𝑏1 𝑟ℎ + 𝛿ℎ 𝑖ℎ 𝑟ℎ ,
𝑑𝑡
(5)
or from 𝑟ℎ = 1−𝑠ℎ −𝑖ℎ and 𝑠V from 𝑠V = 1−𝑖V , respectively. The
correlation between normalized and unnormalized models is
explained in the Appendix. Throughout this work, we study
the reduced system (4) in the closed, positively invariant set
Γ = {(𝑠ℎ , 𝑖ℎ , 𝑖V ) ∈ 𝑅+3 , 0 ≤ 𝑠ℎ + 𝑖ℎ ≤ 1, 0 ≤ 𝑖V ≤ 1},
where 𝑅+3 denotes the nonnegative cone of 𝑅3 with its lower
dimensional faces.
Abstract and Applied Analysis
3
In order to find the “endemic” equilibrium of (4), we set
the right hand side of (4) equal to zero and get
1 or 𝑏1 /𝛿ℎ
𝑖V∗ =
𝑠ℎ∗
0
𝛽3 𝑖ℎ∗
,
𝜇V + 𝛽3 𝑖ℎ∗
𝑏1 (𝜇V + 𝛽3 𝑖ℎ∗ )
=
,
∗
(𝜇V + 𝛽3 𝑖ℎ ) (𝑏1 + (𝛽1 − 𝛿ℎ ) 𝑖ℎ∗ ) + 𝛽2 𝛽3 𝑖ℎ∗
(9)
where 𝑖ℎ∗ is a positive solution of the equation
3
2
𝑓 (𝑖ℎ∗ ) = 𝐴 3 𝑖ℎ∗ + 𝐴 2 𝑖ℎ∗ + 𝐴 1 𝑖ℎ∗ + 𝐴 0 = 0,
(10)
where
0
1
𝐴 3 = 𝛽3 𝛿ℎ (𝛽1 − 𝛿ℎ ) ,
Figure 1: 𝛽1 > 𝛿ℎ .
𝐴 2 = 𝛽2 𝛽3 𝛿ℎ + (𝜇V 𝛿ℎ − 𝛽3 (𝑏1 + 𝛾ℎ + 𝛿ℎ )) (𝛽1 − 𝛿ℎ )
+ 𝑏1 𝛽3 𝛿ℎ ,
4. Existence of Equilibria
We seek the conditions for the existence and stability of
the disease-“free” equilibrium (DFE) 𝐸0 (𝑠ℎ0 , 0, 0) and the
“endemic” proportion equilibrium 𝐸∗ (𝑠ℎ∗ , 𝑖ℎ∗ , 𝑖V∗ ). Obviously,
𝐸0 (1, 0, 0) ∈ Γ is the DFE of (4), which exists for all positive
parameters. The Jacobian matrix of (4) at an arbitrary point
𝐸(𝑠ℎ , 𝑖ℎ , 𝑖V ) takes the following form:
𝐽 (𝐸)
− (𝛽1 −𝛿ℎ ) 𝑠ℎ
−𝛽2 𝑠ℎ
−𝑏1 −𝛽1 𝑖ℎ −𝛽2 𝑖V +𝛿ℎ 𝑖ℎ
𝛽1 𝑖ℎ +𝛽2 𝑖V
𝛽1 𝑠ℎ −(𝑏1 +𝛾ℎ +𝛿ℎ ) + 2𝛿ℎ 𝑖ℎ
𝛽2 𝑠ℎ ) .
=(
0
𝛽3 (1−𝑖V )
−𝛽3 𝑖ℎ − 𝜇V
(6)
To analyze the stability of DFE, we calculate the characteristic
equation of 𝐽(𝐸) at 𝐸 = 𝐸0 as follows:
(𝜆 + 𝑏1 ) (𝜆2 + 𝜆 (𝜇V + 𝑏1 + 𝛾ℎ + 𝛿ℎ − 𝛽1 )
+𝜇V (𝑏1 + 𝛾ℎ + 𝛿ℎ ) (1 − 𝑅0 ) ) ,
(7)
where
𝑅0 =
𝛽1
𝛽2 𝛽3
+
.
𝑏1 + 𝛾ℎ + 𝛿ℎ 𝜇V (𝑏1 + 𝛾ℎ + 𝛿ℎ )
(11)
𝐴 1 = 𝑏1 (𝜇V 𝛿ℎ + 𝛽1 𝛽3 )
(8)
By Routh Hurwitz criteria [16], all roots of (7) have negative
real parts if and only if 𝑅0 < 1. So, 𝐸0 is locally asymptotically
stable for 𝑅0 < 1. If 𝑅0 > 1, the characteristic equation (7) has
positive eigenvalue, and 𝐸0 is thus unstable. We established
the following theorem.
Theorem 1. The disease-free equilibrium is locally asymptotically stable whenever 𝑅0 < 1 and unstable for 𝑅0 > 1.
− (𝑏1 + 𝛾ℎ + 𝛿ℎ ) (𝛽2 𝛽3 + 𝑏1 𝛽3 + 𝜇V (𝛽1 − 𝛿ℎ )) ,
𝐴 0 = (𝑏1 + 𝛾ℎ + 𝛿ℎ ) 𝑏1 𝜇V (𝑅0 − 1) .
From right hand side of (5), we have 𝛾ℎ 𝑖ℎ∗ = (𝑏1 − 𝛿ℎ 𝑖ℎ∗ )𝑟ℎ∗ > 0
and second equation of (9) 𝛽1 𝜇V +𝛽2 𝛽3 −𝜇V 𝛿ℎ > 𝛽3 𝛿ℎ 𝑖ℎ∗ , which
means that
0 < 𝑖ℎ∗ < min {1,
𝜇
𝑏1 𝛽1 𝜇V + 𝛽2 𝛽3
,(
− 1) V } .
𝛿ℎ
𝜇V 𝛿ℎ
𝛽3
(12)
If (𝛽1 𝜇V +𝛽2 𝛽3 )/𝜇V 𝛿ℎ ≤ 1, there is no positive 𝑖ℎ∗ , and therefore
the only equilibrium point in Γ is 𝐸0 . Note that this is a special
case of 𝑅0 < 1.
Assume that 𝑅0 > 1.
(1) If 𝛽1 > 𝛿ℎ , then 𝐴 3 > 0, we have 𝑓(−∞) < 0, 𝑓(∞) >
0 and 𝑓(0) = 𝐴 0 > 0. Further, 𝑓(1) < 0 (if 𝑏1 /𝛿ℎ ≥ 1)
and 𝑓(𝑏1 /𝛿ℎ ) < 0. Thus, there exists unique 𝑖ℎ∗ such
that 𝑓(𝑖ℎ∗ ) = 0 (see Figure 1).
2
(2) If 𝛽1 = 𝛿ℎ , then 𝐴 3 = 0 and 𝑓(𝑖ℎ∗ ) = 𝐴 2 𝑖ℎ∗ + 𝐴 1 𝑖ℎ∗ +
𝐴 0 , where 𝐴 2 = 𝛽2 𝛽3 𝛿ℎ + 𝑏1 𝛽3 𝛿ℎ > 0. We observe
that 𝑓(−∞) > 0, 𝑓(∞) > 0 and 𝑓(0) = 𝐴 0 > 0.
Moreover, 𝑓(1) < 0 (if 𝑏1 /𝛿ℎ ≥ 1) and 𝑓(𝑏1 /𝛿ℎ ) < 0.
Therefore, there exists unique 𝑖ℎ∗ such that 𝑓(𝑖ℎ∗ ) = 0
(see Figure 2).
(3) If 𝛽1 < 𝛿ℎ , then 𝐴 3 < 0, we have 𝑓(−∞) > 0, 𝑓(∞) <
0 and still 𝑓(0) = 𝐴 0 > 0, 𝑓(1) < 0 (if 𝑏1 /𝛿ℎ ≥ 1),
𝑓(𝑏1 /𝛿ℎ ) < 0. In this case, we can say that there is only
one root or three roots in the interval (0, 1) if 𝑏1 /𝛿ℎ ≥
1 or (0, 𝑏1 /𝛿ℎ ) if 𝑏1 /𝛿ℎ < 1.
We know that 𝑓(𝑖ℎ∗ ) = 0 has three real roots if and
only if
𝑞2 𝑝3
+
≤ 0,
4 27
(13)
4
Abstract and Applied Analysis
1
0.9
1 or 𝑏1 /𝛿ℎ
Proportional populations
0.8
0
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
0
50
Figure 2: 𝛽1 = 𝛿ℎ .
150
200
𝑠ℎ
𝑖ℎ
𝑖𝑣
where
Figure 3: 𝑏1 = 1; 𝛽1 = 0.02; 𝛽2 = 0.4; 𝛽3 = 0.6; 𝛾ℎ = 0.3; 𝜇V = 0.2;
𝛽1 = 𝛿ℎ = 0.02; 𝑅0 = 0.92.
2
𝑝=
(𝐴 2 )
𝐴1
−
,
𝐴 3 3(𝐴 3 )2
(14)
3
2(𝐴 2 )
𝐴
𝐴 𝐴
𝑞 = 0 − 1 22 +
,
3
𝐴 3 3(𝐴 3 )
27(𝐴 3 )
1
0.9
̂1 =
𝑅
3
3
2
2
18𝐴 0 𝐴 1 𝐴 2 𝐴 3 −4𝐴 0 (𝐴 2 ) −4(𝐴 1 ) 𝐴 3 +(𝐴 1 ) (𝐴 2 )
2
2
27(𝐴 0 ) (𝐴 3 )
≥ 1.
(15)
̂ 1 < 1, there is unique 𝑖∗ such that 𝑓(𝑖∗ ) = 0 in the
If 𝑅
ℎ
ℎ
feasible interval.
̂ 1 > 1, there are three different real roots for 𝑓(𝑖∗ ) = 0
If 𝑅
ℎ
∗ ∗ ∗
∗
∗
∗
< 𝑖ℎ2
< 𝑖ℎ3
). Note that, differentiating with
say 𝑖ℎ1 , 𝑖ℎ2 , 𝑖ℎ3 (𝑖ℎ1
respect to 𝑖ℎ∗ , we obtain
𝑓
󸀠
(𝑖ℎ∗ )
=
2
3𝐴 3 𝑖ℎ∗
+
2𝐴 2 𝑖ℎ∗
+ 𝐴 1.
(16)
The three different real roots for 𝑓(𝑖ℎ∗ ) = 0 are in the
feasible interval if and only if the following inequalities are
satisfied:
0<
−𝐴 2
< 1,
3𝐴 3
𝑓󸀠 (0) = 𝐴 1 < 0,
𝑓󸀠 (1) = 3𝐴 3 + 2𝐴 2 + 𝐴 1 < 0 (if
𝑏1
≥ 1) ,
𝛿ℎ
𝑏
𝑏 2
𝑏
𝑏
𝑓 ( 1 ) = 3𝐴 3 ( 1 ) + 2𝐴 2 ( 1 ) + 𝐴 1 < 0 (if 1 < 1) .
𝛿ℎ
𝛿ℎ
𝛿ℎ
𝛿ℎ
(17)
Proportional populations
0.8
or
󸀠
100
Time (day)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
Time (day)
150
200
𝑠ℎ
𝑖ℎ
𝑖𝑣
Figure 4: 𝑏1 = 1; 𝛽1 = 0.02; 𝛽2 = 0.4; 𝛽3 = 0.6; 𝛾ℎ = 0.3; 𝜇V = 0.2;
𝛽1 > 𝛿ℎ = 0.01; 𝑅0 = 0.93.
̂ 1 = 1, there are three real roots for 𝑓(𝑖∗ ) = 0, in which
If 𝑅
ℎ
at least two are identical. Similarly, if inequalities (17) are
satisfied, then there are three real roots for 𝑓(𝑖ℎ∗ ) = 0 in the
∗ ∗ ∗
∗
∗
, 𝑖ℎ2 , 𝑖ℎ3 (𝑖ℎ1
= 𝑖ℎ2
).
feasible interval, say 𝑖ℎ1
Assume that 𝑅0 = 1.
(1) If 𝛽1 = 𝛿ℎ , then 𝐴 3 = 0 and (10) reduces to
𝑖ℎ∗ (𝐴 2 𝑖ℎ∗ + 𝐴 1 ) = 0, which implies that 𝑖ℎ∗ = 0 or
𝑖ℎ∗ = −𝐴 1 /𝐴 2 , which is positive but it lies outside the
interval (0, 1) if 𝑏1 /𝛿ℎ ≥ 1 or (0, 𝑏1 /𝛿ℎ ) if 𝑏1 /𝛿ℎ < 1.
Abstract and Applied Analysis
5
In summary, regarding the existence and the number of
the “endemic” equilibria, we have the following.
1
0.9
Theorem 2. Suppose that 𝛽1 ≥ 𝛿ℎ . There is always a disease“free” equilibrium for system (4); if 𝑅0 > 1, then there is a
unique “endemic” equilibrium 𝐸∗ (𝑠ℎ∗ , 𝑖ℎ∗ , 𝑖V∗ ) with coordinates
satisfying (9) and (10) besides the disease-“free” equilibrium.
Proportional populations
0.8
0.7
0.6
0.5
0.4
5. Global Dynamics
0.3
5.1. Global Stability of the Disease-“Free” Equilibrium. In this
subsection, we analyze the global behavior of the equilibria
for system (4). The following theorem provides the global
property of the disease-free equilibrium 𝐸0 of the system.
0.2
0.1
0
0
50
100
Time (day)
150
200
𝑠ℎ
𝑖ℎ
𝑖𝑣
Proof. To establish the global stability of the disease-free
equilibrium, we construct the following Lyapunov function:
Figure 5: 𝑏1 = 2; 𝛽1 = 0.025; 𝛽2 = 0.65; 𝛽3 = 0.75; 𝛾ℎ = 0.000045;
𝜇V = 0.2; 𝛽1 > 𝛿ℎ = 0.000025; 𝑅0 = 1.23.
1
𝐿 (𝑡) = 𝜇V 𝑖ℎ (𝑡) + 𝛽2 𝑖V (𝑡) .
Calculating the time derivative of 𝐿 along (4), we obtain
0.8
= 𝜇V [𝛽1 𝑠ℎ 𝑖ℎ + 𝛽2 𝑠ℎ 𝑖V − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) 𝑖ℎ
0.7
+𝛿ℎ 𝑖ℎ2 ] + 𝛽2 (𝛽3 𝑠V 𝑖ℎ − 𝜇V 𝑖V )
0.6
= 𝜇V [𝛽1 (1 − 𝑖ℎ ) 𝑖ℎ + 𝛽2 (1 − 𝑖ℎ ) 𝑖V − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) 𝑖ℎ
0.5
+𝛿ℎ 𝑖ℎ2 ] + 𝛽2 [𝛽3 (1 − 𝑖V ) 𝑖ℎ − 𝜇V 𝑖V ]
0.4
0.3
= 𝜇V [𝛽1 𝑖ℎ − 𝛽1 𝑖ℎ2 + 𝛽2 𝑖V − 𝛽2 𝑖ℎ 𝑖V − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) 𝑖ℎ
0.2
+𝛿ℎ 𝑖ℎ2 ] + 𝛽2 (𝛽3 𝑖ℎ − 𝛽3 𝑖V 𝑖ℎ − 𝜇V 𝑖V )
0.1
0
(19)
𝐿󸀠 (𝑡) = 𝜇V 𝑖ℎ󸀠 (𝑡) + 𝛽2 𝑖V󸀠 (𝑡)
0.9
Proportional populations
Theorem 3. If 𝑅0 ≤ 1, then the infection-free equilibrium 𝐸0
is globally asymptotically stable in the interior of Γ.
0
50
100
Time (day)
150
200
= 𝜇V 𝛽1 𝑖ℎ − 𝜇V 𝛽1 𝑖ℎ2 + 𝜇V 𝛽2 𝑖V − 𝜇V 𝛽2 𝑖ℎ 𝑖V
− 𝜇V (𝑏1 + 𝛾ℎ + 𝛿ℎ ) 𝑖ℎ + 𝜇V 𝛿ℎ 𝑖ℎ2 + 𝛽2 𝛽3 𝑖ℎ − 𝛽2 𝛽3 𝑖V 𝑖ℎ
𝑠ℎ
𝑖ℎ
𝑖𝑣
− 𝛽2 𝜇V 𝑖V
Figure 6: 𝑏1 = 2; 𝛽1 = 0.0025; 𝛽2 = 0.65; 𝛽3 = 0.75; 𝛾ℎ = 0.000045;
𝜇V = 0.2; 𝛽1 = 𝛿ℎ = 0.0025; 𝑅0 = 1.22.
= −𝜇V (𝑏1 + 𝛾ℎ + 𝛿ℎ ) (1 − 𝑅0 ) 𝑖ℎ − 𝜇V (𝛽1 − 𝛿ℎ ) 𝑖ℎ2
− 𝜇V 𝛽2 𝑖ℎ 𝑖V − 𝛽2 𝛽3 𝑖V 𝑖ℎ .
(20)
(2)
2
If 𝛽1 > 𝛿ℎ , then 𝐴 3 > 0, we have 𝑖ℎ∗ (𝐴 3 𝑖ℎ∗ + 𝐴 2 𝑖ℎ∗ +
∗
∗
𝐴 1 ) = 0, which implies that 𝑖ℎ = 0 or 𝑖ℎ is the solution
of the equation
2
𝑔 (𝑖ℎ∗ ) = 𝐴 3 𝑖ℎ∗ + 𝐴 2 𝑖ℎ∗ + 𝐴 1 = 0,
(18)
where 𝑔(−∞) > 0, 𝑔(∞) > 0, and 𝑔(0) = 𝐴 1 < 0.
Moreover, 𝑔(1) < 0 (if 𝑏1 /𝛿ℎ ≥ 1) and 𝑔(𝑏1 /𝛿ℎ ) <
0 if 𝑏1 /𝛿ℎ < 1. Therefore, there exists no 𝑖ℎ∗ such
that 𝑔(𝑖ℎ∗ ) = 0 in the interval (0, 1) if 𝑏1 /𝛿ℎ ≥ 1 or
(0, 𝑏1 /𝛿ℎ ) if 𝑏1 /𝛿ℎ < 1.
Thus, 𝐿󸀠 (𝑡) is negative if 𝑅0 ≤ 1 and 𝐿󸀠 = 0 if and only if
𝑖ℎ = 0. Consequently, the largest compact invariant set in
{(𝑆ℎ , 𝐼ℎ , 𝐼V ) ∈ Γ, 𝐿󸀠 = 0}, when 𝑅0 ≤ 1, is the singelton
{𝐸0 }. Hence, LaSalle’s invariance principle [16] implies that
“𝐸0 ” is globally asymptotically stable in Γ. This completes the
proof.
5.2. Global Stability of “Endemic” Equilibrium. Here, we use
the geometrical approach of Li and Muldowney to investigate
the global stability of the endemic equilibrium 𝐸∗ in the
Abstract and Applied Analysis
1
1
0.9
0.9
0.8
0.8
0.7
0.7
Infectious vectors
Infectious individuals
6
0.6
0.5
0.4
0.3
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
0
20
40
60
Time (day)
80
0
100
Figure 7: 𝑏1 = 2; 𝛽1 = 0.4; 𝛽2 = 0.85; 𝛽3 = 0.75; 𝛾ℎ = 0.85; 𝜇V = 0.2;
𝛿ℎ = 0.0000001 < 𝛽1 < 𝑏1 /2 = 1, 𝛾ℎ /2 = 0.425; 𝑅0 = 1.26.
0
20
40
60
Time (day)
80
100
Figure 8: 𝑏1 = 2; 𝛽1 = 0.4; 𝛽2 = 0.85; 𝛽3 = 0.75; 𝛾ℎ = 0.85; 𝜇V = 0.2;
𝛿ℎ = 0.0000001 < 𝛽1 < 𝑏1 /2 = 1, 𝛾ℎ /2 = 0.425; 𝑅0 = 1.26.
1
󸀠
𝑥 = 𝑓 (𝑥) .
(21)
is uniquely determined by the initial value 𝑥(0, 𝑥0 ). We have
the following assumptions:
0.9
0.8
Infectious individuals
feasible region Γ. We have omitted the detailed introduction
of this approach, and we refer the interested readers to see
[17]. We summarize this approach below.
Consider a 𝐶1 map 𝑓 : 𝑥 󳨃→ 𝑓(𝑥) from an open set 𝐷 ⊂
𝑛
𝑅 to 𝑅𝑛 such that each solution 𝑥(𝑡, 𝑥0 ) to the differential
equation
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(𝐻1 ) 𝐷 is simply connected;
(𝐻2 ) there exists a compact absorbing set 𝐾 ⊂ 𝐷;
(𝐻3 ) (21) has unique equilibrium 𝑥 in 𝐷.
0
0
20
40
60
Time (day)
80
100
Let 𝑃 : 𝑥 󳨃→ 𝑃(𝑥) be a nonsingular ( 𝑛2 ) × ( 𝑛2 ) matrixvalued function which is 𝐶1 in 𝐷 and a vector norm | ⋅ | on
𝑅𝑁, where 𝑁 = ( 𝑛2 ).
Let 𝜇 be the Lozinskiı̆ measure with respect to the | ⋅ |.
Define a quantity 𝑞2 as
Figure 9: 𝑏1 = 2; 𝛽1 = 0.04; 𝛽2 = 0.85; 𝛽3 = 0.65; 𝛾ℎ = 0.85; 𝜇V = 0.1;
0.04 = 𝛿ℎ = 𝛽1 < 𝑏1 /2 = 1, 𝛾ℎ /2 = 0.425; 𝑅0 = 1.92.
1 𝑡
𝑞2 = lim sup sup ∫ 𝜇 (𝐵 (𝑥 (𝑠, 𝑥0 ))) 𝑑𝑠,
𝑡 → ∞ 𝑥0 ∈𝐾 𝑡 0
the above theorem for global stability of endemic equilibrium
𝐸∗ , we first prove the uniform persistence of (4) when the
threshold parameter 𝑅0 > 1, by applying the acyclicity
Theorem (see [18]).
(22)
where 𝐵 = 𝑃𝑓 𝑃−1 + 𝑃𝐽[2] 𝑃−1 , the matrix 𝑃𝑓 is obtained by
replacing each entry 𝑝 of 𝑃 by its derivative in the direction
of 𝑓, (𝑝𝑖𝑗 )𝑓 , and 𝐽[2] is the second additive compound matrix
of the Jacobian matrix 𝐽 of (21). The following result has been
established in Li and Muldowney [17].
Theorem 4. Suppose that (𝐻1 ), (𝐻2 ), and (𝐻3 ) hold, the
unique endemic equilibrium 𝐸∗ is globally stable in Γ if 𝑞2 < 0.
Obviously Γ is simply connected and 𝐸∗ is a unique
endemic equilibrium for 𝑅0 > 1 in Γ. To apply the result of
Definition 5 (see [19]). The system (4) is uniformly persistent, that is, there exists 𝑐 > 0 (independent of initial
conditions), such that lim inf 𝑡 → ∞ 𝑠ℎ ≥ 𝑐, lim inf 𝑡 → ∞ 𝑖ℎ ≥
𝑐, lim inf 𝑡 → ∞ 𝑖V ≥ 𝑐.
Let 𝑋 be a locally compact metric space with metric 𝑑
and let Γ be a closed nonempty subset of 𝑋 with boundary Γ
and interior Γ∘ . Clearly, Γ∘ is a closed subset of Γ. Let Φ𝑡 be
a dynamical system defined on Γ. A set 𝐵 in 𝑋 is said to be
invariant if Φ(𝐵, 𝑡) = 𝐵. Define 𝑀𝜕 := {𝑥 ∈ Γ : Φ(𝑡, 𝑥) ∈
Γ, for all 𝑡 ≥ 0}.
7
1
1
0.9
0.9
0.8
0.8
0.7
0.7
Infectious vectors
Infectious vectors
Abstract and Applied Analysis
0.6
0.5
0.4
0.3
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
0
20
40
60
Time (day)
80
100
Figure 10: 𝑏1 = 2; 𝛽1 = 0.04; 𝛽2 = 0.85; 𝛽3 = 0.65; 𝛾ℎ = 0.85;
𝜇V = 0.1; 0.04 = 𝛿ℎ = 𝛽1 < 𝑏1 /2 = 1, 𝛾ℎ /2 = 0.425; 𝑅0 = 1.92.
1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0
20
40
60
Time (day)
80
100
Figure 12: 𝑏1 = 1; 𝛽1 = 0.01; 𝛽2 = 0.85; 𝛽3 = 0.95; 𝛾ℎ = 0.015;
𝜇V = 0.25; 0.009 = 𝛿ℎ , 𝛾ℎ /2 = 0.0075 < 𝛽1 < 𝑏1 /2 = 0.5; 𝑅0 = 3.16.
𝑀𝜕 = 𝜕Γ. Since Γ is bounded and positively invariant, so
there exists a compact set 𝑀 in which all solutions of system
(4) initiated in Γ ultimately enter and remain forever. On
𝑠ℎ -axis we have 𝑠ℎ󸀠 = 𝑏1 (1 − 𝑠ℎ ) which means 𝑠ℎ → 1 as
𝑡 → ∞. Thus, 𝐸0 is the only omega limit point on 𝜕Γ, that
is, 𝜔(𝑥) = 𝐸0 for all 𝑥 ∈ 𝑀𝜕 . Furthermore, 𝑀 = 𝐸0 is a
covering of Ω = ⋃𝑥∈𝑀𝜕 𝜔(𝑥), because all solutions initiated
on the 𝑠ℎ -axis converge to 𝐸0 . Also 𝐸0 is isolated and acyclic.
This verifies that hypothesis (1) and (2) hold. When 𝑅0 > 1,
the disease-“free” equilibrium (DFE) 𝐸0 is unstable from
Theorem 1 and also 𝑊𝑠 (𝑀) = 𝜕Γ. Hypothesis (3) and (4)
hold. Therefore, there always exists a global attractor due to
ultimate boundedness of solutions.
0.9
Infectious individuals
0.6
0
20
40
60
Time (day)
80
100
Figure 11: 𝑏1 = 1; 𝛽1 = 0.01; 𝛽2 = 0.85; 𝛽3 = 0.95; 𝛾ℎ = 0.015;
𝜇V = 0.25; 0.009 = 𝛿ℎ , 𝛾ℎ /2 = 0.0075 < 𝛽1 < 𝑏1 /2 = 0.5; 𝑅0 = 3.16.
The boundedness of Γ and the above lemma imply that
(4) has a compact absorbing set 𝐾 ⊂ Γ [19]. Now we shall
prove that the quantity 𝑞2 < 0. We choose a suitable vector
norm | ⋅ | in 𝑅3 and a 3 × 3 matrix-valued function
1 0 0
𝑖
0 ℎ 0
).
𝑃 (𝑥) = ( 𝑖V
𝑖ℎ
0 0
𝑖V
Lemma 6 (see [18]). Assume that
(𝑎) Φ𝑡 has a global attractor;
(𝑏) there exists 𝑀 = {𝑀1 , . . . , 𝑀𝑘 } of pair-wise disjoint,
compact and isolated invariant set on 𝜕Γ such that
⋃𝑘𝑗=1
𝑀𝑗 ;
(1) ⋃𝑥∈𝑀𝜕 𝜔(𝑥) ⊆
(2) no subsets of 𝑀 form a cycle on 𝜕Γ;
(3) each 𝑀𝑗 is also isolated in Γ;
(4) 𝑊𝑠 (𝑀𝑗 ) ⋂ Γ∘ = 𝜙 for each 1 ≤ 𝑗 ≤ 𝑘, where
𝑊𝑠 (𝑀𝑗 ) is stable manifold of 𝑀𝑗 . Then Φ𝑡 is
uniformly persistent with respect to Γ.
Proof. We have Γ = {(𝑠ℎ , 𝑖ℎ , 𝑖V ) ∈ 𝑅+3 , 0 ≤ 𝑠ℎ + 𝑖ℎ ≤ 1, 0 ≤ 𝑖V ≤
1}, Γ∘ = {(𝑠ℎ , 𝑖ℎ , 𝑖V ) ∈ 𝑅+3 𝑠ℎ , 𝑖ℎ > 0}, 𝜕Γ = Γ/Γ∘ . Obviously
(23)
Obviously, 𝑃 is 𝐶1 and nonsingular in the interior of Ω.
Linearizing system (4) about an endemic equilibrium 𝐸∗
gives the following Jacobian matrix:
𝐽 (𝐸∗ )
𝑏1
− (𝛽1 − 𝛿ℎ ) 𝑠ℎ
−𝛽2 𝑠ℎ
𝑠ℎ
).
=(
𝛽1 𝑖ℎ + 𝛽2 𝑖V 𝛽1 𝑠ℎ − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) + 2𝛿ℎ 𝑖ℎ
𝛽2 𝑠ℎ
0
𝛽3 (1 − 𝑖V )
−𝛽3 𝑖ℎ − 𝜇V
(24)
−
8
Abstract and Applied Analysis
with
1
𝐵11 = −
0.9
Infectious individuals
0.8
0.7
0.6
0.5
𝑏1
+ 𝛽1 𝑠ℎ − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) + 2𝛿ℎ 𝑖ℎ ,
𝑠ℎ
𝑖
𝑖
𝐵12 = (𝛽2 𝑠ℎ V , 𝛽2 𝑠ℎ V ) ,
𝑖ℎ
𝑖ℎ
𝐵21
0.4
0.3
0.1
where
0
20
40
60
Time (day)
80
100
𝑄11 = −
Figure 13: 𝑏1 = 1; 𝛽1 = 0.01; 𝛽2 = 0.85; 𝛽3 = 0.95; 𝛾ℎ = 0.015;
𝜇V = 0.25; 𝛾ℎ /2 = 0.0075 < 0.01 = 𝛿ℎ = 𝛽1 < 𝑏1 /2 = 0.5; 𝑅0 = 3.16.
𝑏1
− 𝛽3 𝑖ℎ − 𝜇V ,
𝑠ℎ
𝑄12 = − (𝛽1 − 𝛿ℎ ) 𝑠ℎ ,
𝑄21 = 𝛽1 𝑖ℎ + 𝛽2 𝑖V ,
1
(29)
𝑄22 = 𝛽1 𝑠ℎ − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) + 2𝛿ℎ 𝑖ℎ − 𝛽3 𝑖ℎ − 𝜇V ,
0.9
𝑖󸀠 𝑖󸀠
𝑖V 𝑖ℎ
( ) = ℎ − V.
𝑖ℎ 𝑖V 𝑓 𝑖ℎ 𝑖V
0.8
Infectious vectors
(28)
𝑄 𝑄
𝐵22 = ( 11 12 ) ,
𝑄21 𝑄22
0.2
0
𝑖
( ℎ ) 𝛽3 (1 − 𝑖V )
= ( 𝑖V
),
0
0.7
0.6
0.2
Consider the norm in 𝑅3 as |(𝑢, V, 𝑤)| = max(|𝑢|, |V| +
|𝑤|) where (𝑢, V, 𝑤) denotes the vector in 𝑅3 . The Lozinskiı̆,
measure with respect to this norm is defined as 𝜇(𝐵) ≤
sup(𝑔1 , 𝑔2 ), where
󵄨 󵄨
󵄨 󵄨
𝑔2 = 𝜇1 (𝐵22 ) + 󵄨󵄨󵄨𝐵21 󵄨󵄨󵄨 .
𝑔1 = 𝜇1 (𝐵11 ) + 󵄨󵄨󵄨𝐵12 󵄨󵄨󵄨 ,
(30)
0.1
From system (4), we can write
0.5
0.4
0.3
0
0
20
40
60
Time (day)
80
𝑖ℎ󸀠
𝑖
= 𝛽1 𝑠ℎ + 𝛽2 𝑠ℎ V − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) + 𝛿ℎ 𝑖ℎ ,
𝑖ℎ
𝑖ℎ
100
Figure 14: 𝑏1 = 1; 𝛽1 = 0.01; 𝛽2 = 0.85; 𝛽3 = 0.95; 𝛾ℎ = 0.015;
𝜇V = 0.25; 𝛾ℎ /2 = 0.0075 < 0.01 = 𝛿ℎ = 𝛽1 < 𝑏1 /2 = 0.5; 𝑅0 = 3.16.
The second additive compound matrix of 𝐽(𝐸∗ ) is given by
𝐽[2]
𝑀11
𝛽2 𝑠ℎ
𝛽2 𝑠ℎ
𝑀22
− (𝛽1 − 𝛿ℎ ) 𝑠ℎ ) ,
= (𝛽3 (1 − 𝑖V )
0
𝛽1 𝑖ℎ + 𝛽2 𝑖V
𝑀33
𝑖V󸀠
𝑖
= 𝛽3 (1 − 𝑖V ) ℎ − 𝜇V .
𝑖V
𝑖V
Since 𝐵11 is a scalar, its Lozinskiı̆ measure with respect to any
vector norm in 𝑅1 will be equal to 𝐵11 . Thus
(25)
𝐵11 = −
where
𝑀11 = −
𝑀22 = −
𝑏1
− 𝛽3 𝑖ℎ − 𝜇V ,
𝑠ℎ
(32)
ℎ
(26)
and 𝑔1 will become
𝑔1 = −
𝑀33 = 𝛽1 𝑠ℎ − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) + 2𝛿ℎ 𝑖ℎ − 𝛽3 𝑖ℎ − 𝜇V .
The matrix 𝐵 = 𝑃𝑓 𝑃−1 +𝑃𝐽[2] 𝑃−1 can be written in block form
as
𝐵 𝐵
𝐵 = ( 11 12 ) ,
𝐵21 𝐵22
𝑏1
+ 𝛽1 𝑠ℎ − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) + 2𝛿ℎ 𝑖ℎ ,
𝑠ℎ
𝑖
󵄨󵄨 󵄨󵄨
󵄨󵄨𝐵12 󵄨󵄨 = 𝛽2 𝑠ℎ V ,
𝑖
𝑏1
+ 𝛽1 𝑠ℎ − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) + 2𝛿ℎ 𝑖ℎ ,
𝑠ℎ
(31)
(27)
𝑖
𝑏1
+ 𝛽1 𝑠ℎ − (𝑏1 + 𝛾ℎ + 𝛿ℎ ) + 2𝛿ℎ 𝑖ℎ + 𝛽2 𝑠ℎ V
𝑠ℎ
𝑖ℎ
=
𝑖ℎ󸀠 𝑏1
− + 𝛿ℎ 𝑖ℎ
𝑖ℎ 𝑠ℎ
≤
𝑖ℎ󸀠
− 𝑏1 + 𝛿ℎ 𝑖ℎ .
𝑖ℎ
(33)
Abstract and Applied Analysis
9
𝑖 𝑖
𝑏
𝜇 (𝐵22 ) = Sup { V ( ℎ ) − 1 − 𝛽3 𝑖ℎ − 𝜇V + 𝛽1 𝑖ℎ
𝑖ℎ 𝑖V 𝑓 𝑠ℎ
𝑖 𝑖
+ 𝛽2 𝑖V , V ( ℎ ) + (𝛽1 − 𝛿ℎ ) 𝑠ℎ + 𝛽1 𝑠ℎ
𝑖ℎ 𝑖V 𝑓
1
0.9
0.8
Infectious individuals
Also |𝐵21 | = (𝑖ℎ /𝑖V )𝛽3 (1 − 𝑖V ), |𝐵12 | and |𝐵21 | are the operator
norms of 𝐵12 and 𝐵21 which are mapping from 𝑅2 , to 𝑅 and
from 𝑅 to 𝑅2 respectively, and 𝑅2 is endowed with the 𝑙1 norm.
𝜇1 (𝐵22 ) is the Lozinskiı̆ measure of 2 × 2 matrix 𝐵22 with
respect to 𝑙1 norm in 𝑅2 ;
≤
𝑖ℎ
−
𝑖V󸀠
𝑖V
− 𝑏1 + 𝛿ℎ 𝑖ℎ − 𝛽3 𝑖ℎ − 𝜇V ,
(34)
0.6
0.5
0.4
0.3
0.2
0.1
− (𝑏1 + 𝛾ℎ + 𝛿ℎ ) + 2𝛿ℎ − 𝛽3 𝑖ℎ − 𝜇V 𝑖ℎ }
𝑖ℎ󸀠
0.7
0
0
20
40
60
Time (day)
80
100
Figure 15: 𝑏1 = 0.78; 𝛽1 = 0.4; 𝛽2 = 0.65; 𝛽3 = 0.55; 𝛾ℎ = 0.8;
𝜇V = 0.15; 0.35 = 𝛿ℎ , 𝑏1 /2 = 0.39 < 𝛽1 < 𝛾ℎ /2 = 0.0075; 𝑅0 = 1.44.
if 𝛽1 ≤ 𝛾ℎ /2. Hence
𝑖󸀠 𝑖󸀠
𝑖
𝑔2 ≤ ℎ − V − 𝑏1 + 𝛿ℎ 𝑖ℎ − 𝛽3 𝑖ℎ − 𝜇V + ( ℎ ) 𝛽3 (1 − 𝑖V )
𝑖ℎ 𝑖V
𝑖V
𝑖ℎ
0.9
− 𝑏1 + 𝛿ℎ 𝑖ℎ − 𝛽3 𝑖ℎ .
Thus,
𝜇 (𝐵) = Sup {𝑔1 , 𝑔2 }
𝑖󸀠
≤ ℎ − 𝑏1 + 𝛿ℎ
𝑖ℎ
≤
𝑖ℎ󸀠
𝑖ℎ
0.7
0.6
0.5
0.4
0.3
0.2
(36)
0.1
0
− 𝛽1 ,
where 𝛽1 = min(𝛾ℎ /2, 𝑏1 /2). Since (4) is uniformly persistent
when 𝑅0 > 1, so for 𝑇 > 0 such that 𝑡 > 𝑇 implies
𝑖ℎ (𝑡) ≥ 𝑐, 𝑖V (𝑡) ≥ 𝑐 and (1/𝑡) log 𝑖ℎ (𝑡) < (𝛽1 /2) for all
(𝑠ℎ (0), 𝑖ℎ (0), 𝑖V (0)) ∈ 𝐾. Thus,
𝛽
log 𝑖ℎ (𝑡)
1 𝑡
− 𝛽1 < − 1 ,
∫ 𝜇 (𝐵) 𝑑𝑡 <
𝑡 0
𝑡
2
0.8
(35)
Infectious vectors
=
𝑖ℎ󸀠
1
(37)
for all (𝑠ℎ (0), 𝑖ℎ (0), 𝑖V (0)) ∈ 𝐾, which further implies that 𝑞2 <
0. Therefore, all the conditions of Theorem 4 are satisfied.
Hence, unique endemic equilibrium 𝐸∗ is globally stable in
Γ.
6. Discussions and Simulations
This paper deals with a vector-host disease model which
allows a direct mode of transmission and varying human
population. It concerns diseases with long duration and
substantial mortality rate (e.g., malaria). Our main results
are concerned with the global dynamics of transformed proportionate system. We have constructed Lyapunov function
0
20
40
60
Time (day)
80
100
Figure 16: 𝑏1 = 0.78; 𝛽1 = 0.4; 𝛽2 = 0.65; 𝛽3 = 0.55; 𝛾ℎ = 0.8;
𝜇V = 0.15; 0.35 = 𝛿ℎ , 𝑏1 /2 = 0.39 < 𝛽1 < 𝛾ℎ /2 = 0.0075; 𝑅0 = 1.44.
to show the global stability of disease-“free” equilibrium and
the geometric approach is used to prove the global stability
of “endemic” equilibrium. The epidemiological correlations
between the two systems (normalized and unnormalized)
have also been discussed. The dynamical behavior of the
proportionate model is determined by the basic reproduction
number of the disease. The model has a globally asymptotically stable disease-“free” equilibrium whenever 𝑅0 ≤ 1
(Figures 3 and 4). When 𝑅0 > 1, the disease persists at an
“endemic” level (Figures 5 and 6) if 𝛽1 < min(𝑏1 /2, 𝛾ℎ /2).
Figures 7, 8, 9, and 10 describe numerically “endemic” level
of infectious individuals and infectious vectors under the
condition 𝛽1 < min(𝑏1 /2, 𝛾ℎ /2). We here question that what
are the dynamics of the proportionate system (4) even if
the condition 𝛽1 < min(𝑏1 /2, 𝛾ℎ /2) is not satisfied? We see
numerically that if 𝛿ℎ , 𝛾ℎ /2 < 𝛽1 < 𝑏1 /2 or 𝛾ℎ /2 < 𝛽1 = 𝛿ℎ <
𝑏1 /2 then infectious individuals and infectious vectors will
10
Abstract and Applied Analysis
the dimensionless form (3). If 𝛿ℎ = 0 and 𝑏1 = 𝜇ℎ , then
𝑁ℎ (𝑡)󸀠 = 0 and so 𝑁ℎ (𝑡) remains fixed at its initial value
𝑁ℎ0 . In this case, the system (1) becomes the model with
constant population whose dynamics are the same as the
proportionate system (3). Hence, the solutions with initial
conditions 𝑆ℎ0 + 𝐼ℎ0 + 𝑅ℎ0 = 𝑁ℎ0 tend to (𝑁ℎ0 , 0, 0) if 𝑅0 ≤ 1
and to 𝑁ℎ0 (𝑠ℎ∗ , 𝑖ℎ∗ , 𝑟ℎ∗ ) if 𝑅0 > 1. In the rest of this section,
we suppose that 𝛿ℎ > 0. From system (1) and (2), the trivial
equilibrium 𝐸 = (0, 0, 0, 0, 0) can be easily obtained. Assume
that 𝐸∗ = (𝑁ℎ∗ , 𝑆ℎ∗ , 𝐼ℎ∗ , 𝑅ℎ∗ , 𝐼V∗ ) is the endemic equilibrium of
system (1) and (2), where 𝑁ℎ∗ = 𝑆ℎ∗ + 𝐼ℎ∗ + 𝑅ℎ∗ . This equilibrium
exists if and only if the following equations are satisfied
1
0.9
Infectious individuals
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
Time (day)
80
100
Figure 17: 𝑏1 = 0.78; 𝛽1 = 0.4; 𝛽2 = 0.65; 𝛽3 = 0.55; 𝛾ℎ = 0.8;
𝜇V = 0.15; 𝑏1 /2 = 0.39 < 0.4 = 𝛿ℎ = 𝛽1 < 𝛾ℎ /2 = 0.0075; 𝑅0 = 1.40.
0.9
(A.1)
𝐼V∗
𝛽3 𝛼ℎ 𝑁V
=
,
∗
𝑁ℎ
(𝛽3 𝛼ℎ + 𝜇V ) 𝑁ℎ∗
0.8
Infectious vectors
𝐼ℎ∗
𝛼
= ℎ,
∗
𝑁ℎ
𝛿ℎ
𝑅ℎ∗
𝛾𝛼
= ℎ ℎ,
∗
𝑁ℎ
𝜇ℎ 𝛿ℎ
1
0.7
where 𝛼ℎ = 𝑏1 − 𝜇ℎ and 𝑄 = 𝜇ℎ + 𝛾ℎ + 𝛿ℎ . We introduce the
parameters
0.6
0.5
𝑏1
{
if 𝑅0 ≤ 1
{
{ 𝜇ℎ ,
𝑅1 = {
𝑏1
{
{
, if 𝑅0 > 1,
∗
{ 𝜇ℎ + 𝛿ℎ 𝑖ℎ
0.4
0.3
0.2
0.1
0
𝑆ℎ∗
𝑄 (𝛽3 𝛼ℎ + 𝜇V 𝛿ℎ )
=
,
∗
𝑁ℎ
𝛽1 (𝛽3 𝛼ℎ + 𝜇V 𝛿ℎ ) + 𝛽2 𝛽3 𝛿ℎ
0
20
40
60
Time (day)
80
100
Figure 18: 𝑏1 = 0.78; 𝛽1 = 0.4; 𝛽2 = 0.65; 𝛽3 = 0.55; 𝛾ℎ = 0.8;
𝜇V = 0.15; 𝑏1 /2 = 0.39 < 0.4 = 𝛿ℎ = 𝛽1 < 𝛾ℎ /2 = 0.0075; 𝑅0 = 1.40.
𝛽2 𝛽3
𝛽1
{
+
, if 𝑅0 ≤ 1
{
{
{ 𝜇ℎ + 𝛾ℎ + 𝛿ℎ 𝜇V (𝜇ℎ + 𝛾ℎ + 𝛿ℎ )
𝑅2 = {
{
𝛽1 𝑠ℎ∗
𝛽 𝛽 𝑠∗ (1 − 𝑖V∗ )
{
{
+ 2 3 ℎ
, if 𝑅0 > 1.
{ 𝜇ℎ + 𝛾ℎ + 𝛿ℎ 𝜇V (𝜇ℎ + 𝛾ℎ + 𝛿ℎ )
(A.2)
From (2) we have for 𝑡 → ∞
also approach to endemic level for different initial conditions
(Figures 11, 12, 13, and 14). It is also numerically shown that the
same is true for the case 𝛿ℎ , 𝑏1 /2 < 𝛽1 < 𝛾ℎ /2 or 𝑏1 /2 < 𝛽1 =
𝛿ℎ < 𝛾ℎ /2 (Figures 15, 16, 17, and 18). This implies that the
condition 𝛽1 < min(𝑏1 /2, 𝛾ℎ /2) is weak for the global stability
of unique “endemic” equilibrium.
𝑑𝑁ℎ
= 𝑁ℎ (𝑏1 − 𝜇ℎ − 𝛿ℎ 𝑖ℎ )
𝑑𝑡
𝑁 (𝑏 − 𝜇ℎ ) ,
󳨀→ { ℎ 1
𝑁ℎ (𝑏1 − 𝜇ℎ − 𝛿ℎ 𝑖ℎ ∗ ) ,
if 𝑅0 ≤ 1
if 𝑅0 > 1.
(A.3)
By the definition of 𝑅1 , we have following threshold result.
Appendix
Using the transformation 𝑠ℎ = 𝑆ℎ /𝑁ℎ , 𝑖ℎ = 𝐼ℎ /𝑁ℎ , 𝑟ℎ =
𝑅ℎ /𝑁ℎ , 𝑠V = 𝑆V /𝑁V , and 𝑖V = 𝐼V /𝑁V for scaling, their
differentials: 𝑑𝑠ℎ (𝑡)/𝑑𝑡 = (1/𝑁ℎ )(𝑑𝑆ℎ /𝑑𝑡)−(𝑆ℎ /𝑁ℎ2 )(𝑑𝑁ℎ /𝑑𝑡),
𝑑𝑖ℎ (𝑡)/𝑑𝑡 = (1/𝑁ℎ )(𝑑𝐼ℎ /𝑑𝑡) − (𝐼ℎ /𝑁ℎ2 )(𝑑𝑁ℎ /𝑑𝑡), 𝑑𝑟ℎ (𝑡)/𝑑𝑡 =
(1/𝑁ℎ )(𝑑𝑅ℎ /𝑑𝑡) − (𝑅ℎ /𝑁ℎ2 )(𝑑𝑁ℎ /𝑑𝑡), 𝑑𝑠V (𝑡)/𝑑𝑡 = (1/𝑁V )
(𝑑𝑆V /𝑑𝑡) − (𝑆V /𝑁V2 )(𝑑𝑁V /𝑑𝑡), and 𝑑𝑖V (𝑡)/𝑑𝑡 = (1/𝑁V )(𝑑𝑖V /
𝑑𝑡) − (𝑖V /𝑁V2 )(𝑑𝑁V /𝑑𝑡), and the system (1) and (2), we obtain
Theorem A.1. The total population 𝑁ℎ (𝑡) for the system (1)
decreases to zero if 𝑅1 < 1 and increases to ∞ if 𝑅1 > 1 as
𝑡 → ∞. The asymptotic rate of decrease is 𝜇ℎ (𝑅1 −1) if 𝑅0 ≤ 1,
and the asymptotic rate of increase is (𝜇ℎ + 𝛿ℎ 𝑖ℎ ∗ )(𝑅1 − 1) when
𝑅0 > 1.
Theorem A.2. Suppose 𝑅1 > 1, for 𝑡 → ∞, (𝑆ℎ (𝑡), 𝐼ℎ (𝑡),
𝑅ℎ (𝑡)) tend to (∞, 0, 0) if 𝑅2 < 1 and tend to (∞, ∞, ∞) if
𝑅2 > 1.
Abstract and Applied Analysis
11
Table 1: Asymptotic behavior with threshold criteria.
𝑅0
≤1
>1
≤1
>1
≤1
≤1
<1
>1
>1
𝑅1
=1, 𝛿ℎ = 0
=1, 𝛿ℎ = 0
<1
<1
>1
>1
=1
>1
=1
𝑅2
≤1
=1
<1
<1
<1
>1
<1
>1
=1
𝑁ℎ
𝑁ℎ = 𝑁ℎ0
𝑁ℎ = 𝑁ℎ0
𝑁ℎ → 0
𝑁ℎ → 0
𝑁ℎ → ∞
𝑁ℎ → ∞
𝑁ℎ → 𝑁ℎ∗
𝑁ℎ → ∞
𝑁ℎ → 𝑁ℎ∗
Proof. Since 𝑖V󸀠 → 0 as 𝑡 → ∞, so in the limiting case
the proportion of infectious mosquitoes is related to the
proportion of infectious humans as
𝑖V =
𝛽3 (1 − 𝑖V ) 𝑖ℎ
,
𝜇V
(A.4)
thus, the equation for 𝐼ℎ (𝑡) has limiting form
(𝑠ℎ , 𝑖ℎ , 𝑟ℎ , 𝑖V ) →
(1, 0, 0, 0)
(𝑠ℎ∗ , 𝑖ℎ∗ , 𝑟ℎ∗ , 𝑖V∗ )
(1, 0, 0, 0)
(𝑠ℎ∗ , 𝑖ℎ∗ , 𝑟ℎ∗ , 𝑖V∗ )
(1, 0, 0, 0)
(1, 0, 0, 0)
(1, 0, 0, 0)
(𝑠ℎ∗ , 𝑖ℎ∗ , 𝑟ℎ∗ , 𝑖V∗ )
(𝑠ℎ∗ , 𝑖ℎ∗ , 𝑟ℎ∗ , 𝑖V∗ )
(𝑆ℎ , 𝐼ℎ , 𝑅ℎ ) →
(𝑁ℎ0 , 0, 0)
𝑁ℎ0 (𝑠ℎ∗ , 𝑖ℎ∗ , 𝑟ℎ∗ )
(0, 0, 0)
(0, 0, 0)
(∞, 0, 0)
(∞, ∞, ∞)
(𝑁ℎ∗ , 0, 0)
(∞, ∞, ∞)
(𝑠ℎ∗ , 𝑖ℎ∗ , 𝑟ℎ∗ )
Acknowledgment
This work was supported by Basic Science Research Program
through the National Research Foundation of Korea (NRF)
funded by the Ministry of Education, Science and Technology
(MEST) (2012-000599).
References
(A.5)
[1] H. Yang, H. Wei, and X. Li, “Global stability of an epidemic
model for vector-borne disease,” Journal of Systems Science &
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which shows that 𝐼ℎ (𝑡) decreases exponentially if 𝑅2 < 1 and
increases exponentially if 𝑅2 > 1.
The solution 𝑅ℎ (𝑡) is given by
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𝑑𝐼ℎ (𝑡)
= (𝜇ℎ + 𝛾ℎ + 𝛿ℎ ) (𝑅2 − 1) 𝐼ℎ ,
𝑑𝑡
𝑡
𝑅ℎ (𝑡) = 𝑅ℎ0 𝑒−𝜇ℎ 𝑡 + 𝛾ℎ 𝑒−𝜇ℎ 𝑡 ∫ 𝐼ℎ (𝑠) 𝑒𝜇ℎ 𝑠 𝑑𝑠.
0
(A.6)
From the exponential nature of 𝐼ℎ (𝑡), it follows that 𝐼ℎ (𝑡)
declines exponentially if 𝑅2 < 1 and grows exponentially if
𝑅2 > 1.
Suppose 𝑅1 = 1, then 𝑏1 = 𝜇ℎ corresponding to 𝑅0 < 1
and the differential equation for 𝑁ℎ (𝑡) will have the form
𝑑𝑁ℎ
= −𝛿ℎ 𝐼ℎ ,
𝑑𝑡
(A.7)
which means that 𝑁ℎ (𝑡) is bounded for all 𝑡 > 0, the
equilibria (𝑁ℎ∗ , 0, 0, 0) have one eigenvalue zero, and the other
eigenvalues have negative real parts. Therefore, each orbit
approaches an equilibrium point.
If 𝑅0 > 1, the disease becomes “endemic.” From the global
stability of 𝐸∗ and the equation
𝑏 − 𝜇ℎ ∗
𝑑𝑁ℎ
− 𝑖ℎ ) − (𝑖ℎ − 𝑖ℎ∗ )] 𝑁ℎ ,
= 𝛿ℎ [( 1
𝑑𝑡
𝛿ℎ
(A.8)
we observe that (𝑁ℎ , 𝑆ℎ , 𝐼ℎ , 𝑅ℎ , 𝐼V ) approaches to (0, 0, 0, 0, 0)
or (∞, ∞, ∞, ∞, ∞) if 𝑅1 < 1 or 𝑅1 > 1. From the global
stability of 𝑖ℎ ∗ , we have 𝑁ℎ (𝑡) converges to some 𝑁ℎ ∗ as 𝑡
approaches to ∞. Since 𝑠ℎ = 𝑆ℎ /𝑁ℎ , 𝑖ℎ = 𝐼ℎ /𝑁ℎ , 𝑟ℎ = 𝑅ℎ /𝑁ℎ ,
so we have 𝑆ℎ ∗ = 𝑠ℎ ∗ 𝑁ℎ ∗ , 𝐼ℎ ∗ = 𝑖ℎ ∗ 𝑁ℎ ∗ , 𝑅ℎ ∗ = 𝑟ℎ ∗ 𝑁ℎ ∗ . All
the above discussion is summarized in Table 1.
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Volume 2014