# Permanence and Extinction for a Nonautonomous Malaria Transmission Model with Distributed Time Delay

Authors: Xiaohong Zhang; Jianwen Jia; Xinyu Song
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We study the permanence, extinction, and global asymptotic stability for a nonautonomous malaria transmission model with distributed time delay. We establish some sufficient conditions on the permanence and extinction of the disease by using inequality analytical techniques. By a Lyapunov functional method, we also obtain some sufficient conditions for global asymptotic stability of this model. A numerical analysis is given to explain the analytical findings.


---

## Body

## 1. Introduction and the Model

There have been lots of researches about SEIRS, SIRS models, in which the infectious diseases spread in a single population [1, 2]. In recent years, the study of diseases spreading among multiple populations has increased gradually, such as avian influenza and malaria. Malaria remains one of the most prevalent and lethal human infectious diseases in the world. Malaria is a protozoan infection of red blood cells caused in human by four species of the genusPlasmodium (Plasmodium falciparum, Plasmodium vivax, Plasmodium ovale, and Plasmodium malariae). The malaria parasites are generally transmitted to the human host through the bite of an infected female anopheline mosquito.There has been a great deal of work about using mathematical models to study malaria [3–5]. However, considering malaria often occurs in most tropical and some subtropical regions of the world [6], environmental and climatic factors play an important role in the geographical distribution and transmission of malaria [7]. Malaria fluctuates over time and often exhibits seasonal behaviors, especially in the northern areas. Therefore, it is meaningful and essential to take account of malaria model with periodic environment [8]. However, up to now, there have been few results about malaria model with periodic environment. In [9], a malaria transmission model with periodic birth rate and age structure for the vector population was presented by Lou and Zhao, and they further showed that ℛ
0 is the threshold value determining the extinction and the uniform persistence of the disease. Later, they used these analytic results to study the malaria transmission cases in KwaZulu-Natal Province, South Africa.Motivated by the work of [8, 10–12], studied a malaria transmission model with periodic environment. By applying the way of computing the basic reproduction number for a wide class of compartmental epidemic models in periodic environments given by Wang and Zhao [13], Lei Wang, Zhidong Teng, and Tailei Zhang calculated the basic reproduction number and indicated it was the threshold value determining the extinction and the uniform persistence of the disease. They studied the following model:
(1)
v
S
R
′
(
t
)
=
r
(
t
)
S
R
(
t
)
(
1
-
S
R
k
(
t
)
)
-
α
(
t
)
S
R
I
H
,
v
I
R
′
(
t
)
=
α
(
t
)
S
R
I
H
-
d
(
t
)
I
R
,
i
S
H
′
(
t
)
=
λ
-
β
(
t
)
S
H
I
R
-
δ
S
H
+
γ
R
H
,
v
I
H
′
(
t
)
=
β
(
t
)
S
H
I
R
-
(
δ
+
μ
+
σ
)
I
H
,
i
R
H
′
(
t
)
=
μ
I
H
-
δ
R
H
-
γ
R
H
.In [14], the researcher constructed a mathematical model to interpret the spread of wild avian influenza from the birds to the humans, after the emergence of mutant avian influenza, with nonautonomous ordinary differential equations and distributed time delay due to the intracellular delay between initial infection of a cell and the release of new virus particles. The researcher studied the following model:
(2)
X
′
(
t
)
=
c
(
t
)
-
b
(
t
)
X
(
t
)
-
ω
(
t
)
X
(
t
)
∫
0
h
‍
Y
(
t
-
s
)
d
η
(
s
)
,
Y
′
(
t
)
=
ω
(
t
)
X
(
t
)
∫
0
h
‍
Y
(
t
-
s
)
d
η
(
s
)
-
{
b
(
t
)
+
m
(
t
)
}
Y
(
t
)
,
S
′
(
t
)
=
λ
(
t
)
-
μ
(
t
)
S
(
t
)
-
β
1
(
t
)
S
(
t
)
∫
0
h
‍
Y
(
t
-
s
)
d
η
(
s
)
-
β
2
(
t
)
S
(
t
)
∫
0
h
‍
H
(
t
-
s
)
d
η
(
s
)
,
B
′
(
t
)
=
β
1
(
t
)
S
(
t
)
∫
0
h
‍
Y
(
t
-
s
)
d
η
(
s
)
-
{
μ
(
t
)
+
d
1
(
t
)
+
ξ
(
t
)
}
B
(
t
)
,
H
′
(
t
)
=
β
2
(
t
)
S
(
t
)
∫
0
h
‍
H
(
t
-
s
)
d
η
(
s
)
+
ξ
(
t
)
B
(
t
)
-
{
μ
(
t
)
+
d
2
(
t
)
+
γ
(
t
)
}
H
(
t
)
,
R
′
(
t
)
=
γ
(
t
)
H
(
t
)
-
μ
(
t
)
R
(
t
)
.Motivated by system (2), considering the intracellular delay between initial infection of a cell biting by an infected female anopheline mosquito and the release of new virus particles, we study the system (1) with distributed time delay, which can be more reasonable. We construct the following model:
(3)
S
R
′
(
t
)
=
c
(
t
)
-
α
(
t
)
S
R
∫
0
h
‍
I
H
(
t
-
s
)
d
η
(
s
)
-
b
(
t
)
S
R
,
I
R
′
(
t
)
=
α
(
t
)
S
R
∫
0
h
‍
I
H
(
t
-
s
)
d
η
(
s
)
-
d
(
t
)
I
R
-
b
(
t
)
I
R
,
S
H
′
(
t
)
=
λ
(
t
)
-
β
(
t
)
S
H
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
δ
(
t
)
S
H
+
γ
(
t
)
R
H
,
I
H
′
(
t
)
=
β
(
t
)
S
H
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
(
δ
(
t
)
+
μ
(
t
)
+
σ
(
t
)
)
I
H
,
R
H
′
(
t
)
=
μ
(
t
)
I
H
-
δ
(
t
)
R
H
-
γ
(
t
)
R
H
.
Here N
R
(
t
)
=
S
R
(
t
)
+
I
R
(
t
) and N
H
(
t
)
=
S
H
(
t
)
+
I
H
(
t
)
+
R
H
(
t
) denote the total number of mosquito and human population, respectively, at time t; S
R
(
t
), I
R
(
t
), S
H
(
t
), I
H
(
t
), and R
H
(
t
) represent the densities (or fractions) of susceptible mosquitoes, infected mosquitoes, susceptible humans, infective humans, and recovered humans, respectively, at time t. Due to the mosquito's short lifespan, it cannot recover from the infection. Consequently, we only divide the total mosquito population into two classes: the susceptible and the infected.The quantitiesc
(
t
), α
(
t
), b
(
t
), d
(
t
), λ
(
t
), β
(
t
), δ
(
t
), γ
(
t
), μ
(
t
), and σ
(
t
) are as follows:c
(
t
): the instantaneous growth rate function of the mosquito population;
b
(
t
): the instantaneous natural death rate function of the mosquito population;
d
(
t
): the instantaneous additional death rate function of the mosquito population;
α
(
t
): the transmission rate function from human to mosquito when susceptible mosquitoes contact infective humans and the rate of transmission is of the form
(4)
α
(
t
)
S
R
∫
0
h
‍
I
H
(
t
-
s
)
d
η
(
s
)
;
λ
(
t
): the instantaneous immigration rate function of the human population;
δ
(
t
): the instantaneous natural death rate function of the human population;
γ
(
t
): the instantaneous rate function of which the recovered becomes susceptible again;
μ
(
t
): the instantaneous recovered rate function of the human population;
σ
(
t
): the instantaneous disease-induced death rate function of the human population;
β
(
t
): the transmission rate function from mosquito to human when susceptible humans take contact with infected mosquitoes and the rate of transmission is of the form:
(5)
β
(
t
)
S
H
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
.The nonnegative constant h is the time delay. The function η
(
s
)
:
[
0
,
h
]
→
[
0
,
∞
) is nondecreasing and has bounded variation such that
(6)
∫
0
h
‍
d
η
(
s
)
=
η
(
h
)
-
η
(
0
)
=
1
.
The time delay is due to intracellular delay between initial infection of a cell and the release of new virions. Those infected at time t
-
s become infectious at time s
(
0
≤
s
≤
h
) later with different probabilities. Additionally, comparing with the human, the mosquito's life is much too shorter, so we suppose that
(7)
b
(
t
)
+
d
(
t
)
≫
δ
(
t
)
+
μ
(
t
)
+
σ
(
t
)
.This paper is organized as follows. In Section2, we establish some sufficient conditions on the permanence and extinction of the disease. In Section 3, we analyze global asymptotical stability of the disease. Some numerical simulations are given in Section 4. And finally, in Section 5, we come to a conclusion.


## 2. Permanence and Extinction

In this section, we first introduce the following assumptions for system (3): Functions c
(
t
), α
(
t
), b
(
t
), d
(
t
), λ
(
t
), β
(
t
), δ
(
t
), γ
(
t
), μ
(
t
), and σ
(
t
) are positive continuous bounded and have positive lower bounds.The initial conditions of (3) are given as
(8)
S
R
(
θ
)
=
φ
1
(
θ
)
,
I
R
(
θ
)
=
φ
2
(
θ
)
,
S
H
(
θ
)
=
φ
3
(
θ
)
,
I
H
(
θ
)
=
φ
4
(
θ
)
,
R
H
(
θ
)
=
φ
5
(
θ
)
,
-
h
≤
θ
≤
0
,
where φ
=
(
φ
1
,
φ
2
,
φ
3
,
φ
4
,
φ
5
)
T
∈
C such that φ
i
(
θ
)
≥
0
(
i
=
1,2
,
3,4
,
5
), ∀
θ
∈
[
-
h
,
0
]. C denotes the Banach space C
(
[
-
h
,
0
]
,
R
5
) of continuous functions mapping the interval [
-
h
,
0
] into R
5 and the norm of an element φ in C is designated by ∥
φ
∥
=
sup
⁡
-
h
≤
θ
≤
0
⁡
{
|
φ
1
(
θ
)
|
,
|
φ
2
(
θ
)
|
,
|
φ
3
(
θ
)
|
,
|
φ
4
(
θ
)
|
,
|
φ
5
(
θ
)
|
}. For a biological meaning, we further assume that φ
i
(
0
)
>
0
(
i
=
1,2
,
3,4
,
5
).Lemma 1 (see [15]).
If the functionsc
(
t
), α
(
t
), b
(
t
), d
(
t
), λ
(
t
), β
(
t
), δ
(
t
), γ
(
t
), μ
(
t
), and σ
(
t
) are continuous and bounded on [
0
,
+
∞
), then there exists a unique solution of the system (3) with initial conditions (8) defined on [
0
,
+
∞
).Firstly we discuss the permanence of the system (3).For a continuous and bounded functionf
(
t
) defined on [
0
,
+
∞
), we introduce the following signs:
(9)
f
l
=
inf
⁡
t
≥
0
f
(
t
)
,
f
u
=
sup
⁡
t
≥
0
f
(
t
)
.Definition 2 (see [16]).
The system (3) is said to be permanent, that is, the long-term survival (will not vanish in time) of all components of the system (3), if there are positive constants v
i and M
i
(
i
=
1,2
,
3,4
,
5
) such that
(10)
v
1
≤
liminf
⁡
t
→
∞
S
R
(
t
)
≤
limsup
⁡
t
→
∞
S
R
(
t
)
≤
M
1
,
v
2
≤
liminf
⁡
t
→
∞
I
R
(
t
)
≤
limsup
⁡
t
→
∞
I
R
(
t
)
≤
M
2
,
v
3
≤
liminf
⁡
t
→
∞
S
H
(
t
)
≤
limsup
⁡
t
→
∞
S
H
(
t
)
≤
M
3
,
v
4
≤
liminf
⁡
t
→
∞
I
H
(
t
)
≤
limsup
⁡
t
→
∞
I
H
(
t
)
≤
M
4
,
v
5
≤
liminf
⁡
t
→
∞
R
H
(
t
)
≤
limsup
⁡
t
→
∞
R
H
(
t
)
≤
M
5
,
hold for any solution (
S
R
(
t
)
,
I
R
(
t
)
,
S
H
(
t
)
,
I
H
(
t
)
,
R
H
(
t
)
) of (3) with initial conditions of type (8). Here v
i and M
i
(
i
=
1,2
,
3,4
,
5
) are independent of (8).Theorem 3.
Setr
*
=
(
β
l
/
(
d
+
b
)
u
)
(
λ
l
/
δ
u
), r
*
=
(
α
l
/
(
δ
+
μ
+
σ
)
u
)
(
c
l
/
b
u
), and r
0
=
(
α
l
/
(
d
+
b
)
u
)
(
c
l
/
b
u
). The system (3) with initial conditions (8) is permanent provided that r
*
≥
r
* and r
0
>
1.Proof.
We will give the following Propositions4–8 to complete the proof of this theorem.Proposition 4.
The solution(
S
R
(
t
)
,
I
R
(
t
)
,
S
H
(
t
)
,
I
H
(
t
)
,
R
H
(
t
)
) of (3) with initial conditions (8) is positive for all t
≥
0, and
(11)
limsup
⁡
t
→
+
∞
N
R
(
t
)
≤
c
u
b
l
;
limsup
⁡
t
→
+
∞
N
H
(
t
)
≤
λ
u
δ
l
.Proof.
Since the functionsc
(
t
), α
(
t
), b
(
t
), d
(
t
), λ
(
t
), β
(
t
), δ
(
t
), γ
(
t
), μ
(
t
), and σ
(
t
) are continuous and bounded on [
0
,
+
∞
), the solution of (3) with initial conditions (8) exists and is unique on [
0
,
+
∞
). Now,
(12)
S
R
(
t
)
=
S
R
(
0
)
exp
⁡
[
-
∫
0
t
‍
{
α
(
θ
)
∫
0
h
‍
I
H
(
θ
-
s
)
d
η
(
s
)
+
b
(
θ
)
}
d
θ
]
+
∫
0
t
‍
c
(
u
)
exp
⁡
[
∫
t
u
‍
{
α
(
θ
)
∫
0
h
‍
I
H
(
θ
-
s
)
d
η
(
s
)
vvvvvvvvvvvvvvvvvvv
+
b
(
θ
)
{
α
(
θ
)
∫
0
h
‍
I
H
(
θ
-
s
)
d
η
(
s
)
+
b
(
θ
)
}
}
d
θ
d
u
]
>
0
,
t
≥
0
.
Next, we show that I
R
(
t
)
>
0 for all t
≥
0. Otherwise there exists a t
1
∈
(
0
,
+
∞
) such that I
R
(
t
1
)
=
0 and I
R
(
t
)
>
0 for all t
∈
[
0
,
t
1
). We claim that I
H
(
t
)
>
0 for all t
∈
[
0
,
t
1
). If this is not true, then there exists a t
2
∈
[
0
,
t
1
) such that I
H
(
t
2
)
=
0 and I
H
(
t
)
>
0 for all t
∈
[
0
,
t
2
). From the fourth equation of system (3), we have
(13)
I
H
(
t
2
)
=
I
H
(
0
)
exp
⁡
{
-
∫
0
t
2
‍
(
δ
(
θ
)
+
μ
(
θ
)
+
σ
(
θ
)
)
d
θ
}
+
∫
0
t
2
‍
∫
0
h
‍
β
(
u
)
S
H
(
u
)
I
R
(
u
-
s
)
d
η
(
s
)
vvvvvvv
×
exp
⁡
{
∫
t
2
u
‍
(
δ
(
θ
)
+
μ
(
θ
)
+
σ
(
θ
)
)
d
θ
}
d
u
>
0
,
which contradicts with I
H
(
t
2
)
=
0. Therefore, I
H
(
t
)
>
0 for all t
∈
[
0
,
t
1
). Integrating the second equation of system (3) from 0 to t
1, we have
(14)
I
R
(
t
1
)
=
I
R
(
0
)
exp
⁡
{
-
∫
0
t
1
‍
(
d
(
θ
)
+
b
(
θ
)
)
d
θ
}
+
∫
0
t
1
‍
∫
0
h
‍
α
(
u
)
S
R
(
u
)
I
H
(
u
-
s
)
d
η
(
s
)
vvvvvvv
×
exp
⁡
{
∫
t
1
u
‍
(
d
(
θ
)
+
b
(
θ
)
)
d
θ
}
d
u
>
0
,
which contradicts with I
R
(
t
1
)
=
0. Therefore, I
R
(
t
)
>
0 for all t
≥
0. Thus I
H
(
t
)
>
0.
From the fifth equation of system (3), we have
(15)
R
H
(
t
)
=
R
H
(
0
)
exp
⁡
{
-
∫
0
t
‍
(
δ
(
s
)
+
γ
(
s
)
)
d
s
}
+
∫
0
t
‍
μ
(
u
)
I
H
(
u
)
exp
⁡
{
∫
t
u
‍
(
δ
(
s
)
+
γ
(
s
)
)
d
s
}
d
u
>
0
,
vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
t
≥
0
.
Last, from the third equation of system (3), we have
(16)
S
H
(
t
)
=
S
H
(
0
)
exp
⁡
{
-
∫
0
t
‍
(
β
(
θ
)
∫
0
h
‍
I
R
(
θ
-
s
)
d
η
(
s
)
+
δ
(
θ
)
)
d
θ
}
+
∫
0
t
‍
(
λ
(
u
)
+
γ
(
u
)
R
H
(
u
)
)
×
exp
⁡
{
∫
t
u
‍
∫
t
u
‍
(
β
(
θ
)
∫
0
h
‍
I
R
(
θ
-
s
)
d
η
(
s
)
vvvvvvvvvvvvvvvvvv
+
δ
(
θ
)

∫
0
h
‍
)
d
θ
}
d
u
>
0
,
t
≥
0
.
Therefore, S
R
(
t
)
>
0
,
I
R
(
t
)
>
0
,
S
H
(
t
)
>
0
,
I
H
(
t
)
>
0
,
R
H
(
t
)
>
0 for all t
≥
0. Thus, ∀
t
∈
[
0
,
∞
),
(17)
N
R
′
(
t
)
≤
c
(
t
)
-
b
(
t
)
N
R
(
t
)
-
d
(
t
)
I
R
(
t
)
≤
c
u
-
b
l
N
R
(
t
)
,
⟹
limsup
⁡
t
→
+
∞
N
R
(
t
)
≤
c
u
b
l
.
Similarly,
(18)
N
H
′
(
t
)
≤
λ
(
t
)
-
δ
(
t
)
N
H
(
t
)
-
σ
(
t
)
I
H
(
t
)
≤
λ
u
-
δ
l
N
H
(
t
)
,
⟹
limsup
⁡
t
→
+
∞
N
H
(
t
)
≤
λ
u
δ
l
.
This completes the proof.Proposition 5.
The solution(
S
R
(
t
)
,
I
R
(
t
)
,
S
H
(
t
)
,
I
H
(
t
)
,
R
H
(
t
)
) of (3) with initial conditions (8) satisfies
(19)
liminf
⁡
t
→
∞
S
R
(
t
)
≥
c
l
δ
l
α
u
λ
u
+
δ
l
b
u
≡
v
1
>
0
.Proof.
By Proposition4, for any ϵ
>
0 (no matter however small), there exists a t
1
>
0, such that
(20)
I
H
(
t
)
≤
λ
u
δ
l
+
ϵ
,
t
≥
t
1
.
Therefore, from the first equation of system (3), when t
≥
t
1
+
h,
(21)
S
R
′
(
t
)
≥
c
(
t
)
-
[
α
(
t
)
(
λ
u
δ
l
+
ϵ
)
+
b
(
t
)
]
S
R
(
t
)
≥
c
l
-
[
α
u
(
λ
u
δ
l
+
ϵ
)
+
b
u
]
S
R
(
t
)
,
⟹
liminf
⁡
t
→
∞
S
R
(
t
)
≥
c
l
α
u
(
λ
u
/
δ
l
+
ϵ
)
+
b
u
.
Since ϵ
>
0 can be made arbitrarily small, the result of this proposition is valid. This completes the proof.Proposition 6.
SetI
(
t
)
=
I
R
(
t
)
+
I
H
(
t
), assume that r
*
≥
r
* and r
0
>
1, and then for any solution (
S
R
(
t
)
,
I
R
(
t
)
,
S
H
(
t
)
,
I
H
(
t
)
,
R
H
(
t
)
) of (3) with initial conditions (8) we have
(22)
liminf
⁡
t
→
∞
I
(
t
)
≥
η
e
-
[
(
d
+
b
)
u
+
(
δ
+
μ
+
σ
)
u
]
(
h
+
ρ
)
≡
v
2
>
0
,
where η
>
0 and ρ
>
0 will be given in the proof.Proof.
Sincer
0
>
1 and it is obvious that c
l
/
G
→
c
l
/
b
u, as η
→
0, where G
=
b
u
+
η
α
u. Then, there exists two positive constants η and ρ such that
(23)
c
l
G
{
1
-
exp
⁡
(
-
G
ρ
)
}
α
l
(
d
+
b
)
u
>
1
.
From that condition (7), we can get (
d
+
b
)
u
>
(
δ
+
μ
+
σ
)
u. Then, we have
(24)
c
l
G
{
1
-
exp
⁡
(
-
G
ρ
)
}
α
l
(
δ
+
μ
+
σ
)
u
>
1
.
Similarly, sincer
*
≥
r
*
>
1, it is obvious that λ
l
/
H
→
λ
l
/
δ
u, as η
→
0 where H
=
δ
u
+
η
β
u. Then, for the positive constants η and ρ, we have
(25)
λ
l
H
{
1
-
exp
⁡
(
-
H
ρ
)
}
β
l
(
d
+
b
)
u
>
1
.
Firstly, from the second and the fourth equation of (3) together [17]. We get the following equivalent system:
(26)
S
R
′
(
t
)
=
c
(
t
)
-
α
(
t
)
S
R
∫
0
h
‍
I
H
(
t
-
s
)
d
η
(
s
)
-
b
(
t
)
S
R
,
I
′
(
t
)
=
α
(
t
)
S
R
∫
0
h
‍
I
H
(
t
-
s
)
d
η
(
s
)
-
(
d
(
t
)
+
b
(
t
)
)
I
R
+
β
(
t
)
S
H
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
(
δ
(
t
)
+
μ
(
t
)
+
σ
(
t
)
)
I
H
,
S
H
′
(
t
)
=
λ
(
t
)
-
β
(
t
)
S
H
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
δ
(
t
)
S
H
+
γ
(
t
)
R
H
,
R
H
′
(
t
)
=
μ
(
t
)
I
H
-
δ
(
t
)
R
H
-
γ
(
t
)
R
H
.
Let us consider the following differential function V
(
t
):
(27)
V
(
t
)
=
I
(
t
)
+
∫
0
h
‍
∫
t
-
s
t
‍
α
(
u
+
s
)
S
R
(
u
+
s
)
I
H
(
u
)
d
u
d
η
(
s
)
+
∫
0
h
‍
∫
t
-
s
t
‍
β
(
u
+
s
)
S
H
(
u
+
s
)
I
R
(
u
)
d
u
d
η
(
s
)
.
The derivative V
(
t
) along solution of (26) is
(28)
V
′
(
t
)
=
α
(
t
)
S
R
∫
0
h
‍
I
H
(
t
-
s
)
d
η
(
s
)
-
(
d
(
t
)
+
b
(
t
)
)
I
R
+
β
(
t
)
S
H
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
(
δ
(
t
)
+
μ
(
t
)
+
σ
(
t
)
)
I
H
+
∫
0
h
‍
α
(
t
+
s
)
S
R
(
t
+
s
)
I
H
(
t
)
d
η
(
s
)
-
∫
0
h
‍
α
(
t
)
S
R
(
t
)
I
H
(
t
-
s
)
d
η
(
s
)
+
∫
0
h
‍
β
(
t
+
s
)
S
H
(
t
+
s
)
I
R
(
t
)
d
η
(
s
)
-
∫
0
h
‍
β
(
t
)
S
H
(
t
)
I
R
(
t
-
s
)
d
η
(
s
)
=
[
∫
0
h
‍
α
(
t
+
s
)
S
R
(
t
+
s
)
d
η
(
s
)
-
(
δ
(
t
)
+
μ
(
t
)
+
σ
(
t
)
)
∫
0
h
‍
]
I
H
(
t
)
+
[
∫
0
h
‍
β
(
t
+
s
)
S
H
(
t
+
s
)
d
η
(
s
)
-
(
d
(
t
)
+
b
(
t
)
)
]
I
R
(
t
)
.
We claim that it is impossible thatI
(
t
)
≤
η, ∀
t
≥
t
1 (t
1 is any nonnegative constant). Suppose the contrary; then as t
≥
t
1
+
h,
(29)
S
R
′
(
t
)
≥
c
(
t
)
-
(
η
α
(
t
)
+
b
(
t
)
)
S
R
(
t
)
≥
c
l
-
G
S
R
(
t
)
.
Fort
>
t
1
+
h, integrating the above inequality from t
1
+
h to t, we obtain
(30)
S
R
(
t
)
≥
S
R
(
t
1
+
h
)
exp
⁡
(
∫
t
t
1
+
h
‍
G
d
s
)
+
∫
t
1
+
h
t
‍
c
l
exp
⁡
(
∫
t
s
‍
G
d
θ
)
d
s
≥
∫
t
1
+
h
t
‍
c
l
exp
⁡
(
∫
0
s
‍
G
d
θ
-
∫
0
t
‍
G
d
θ
)
d
s
=
∫
t
1
+
h
t
‍
c
l
exp
⁡
⁡
(
∫
0
s
‍
G
d
θ
)
d
s
exp
⁡
⁡
(
∫
0
t
‍
G
d
θ
)
=
(
c
l
exp
⁡
⁡
(
G
t
)
)
∫
t
1
+
h
t
‍
exp
⁡
(
G
s
)
d
s
.
Hence
(31)
S
R
(
t
)
≥
(
c
l
G
)
exp
⁡
(
G
t
)
-
exp
⁡
(
G
(
t
1
+
h
)
)
exp
⁡
(
G
t
)
=
(
c
l
G
)
[
1
-
exp
⁡
(
-
G
(
t
-
t
1
-
h
)
)
]
.
Therefore, S
R
(
t
)
≥
(
c
l
/
G
)
[
1
-
exp
⁡
(
-
G
ρ
)
]
≡
S
R
∆, ∀
t
≥
t
1
+
h
+
ρ
≡
t
2.
Similarly, from the third equation of system (26), we have
(32)
S
H
(
t
)
≥
(
λ
l
H
)
[
1
-
exp
⁡
(
-
H
ρ
)
]
≡
S
H
∆
,
∀
t
≥
t
2
.
Notice that r
0
>
1; from the previous discussion, we have α
l
S
R
∆
/
(
d
+
b
)
u
>
1. And from r
*
≥
r
*
>
1, we can get β
l
S
H
Δ
/
(
d
+
b
)
u
≥
α
l
S
R
Δ
/
(
δ
+
μ
+
σ
)
u
>
1. Since (
d
+
b
)
u
>
(
δ
+
μ
+
σ
)
u, we have
(33)
β
l
S
H
∆
-
(
d
+
b
)
u
≥
α
l
S
R
∆
-
(
δ
+
μ
+
σ
)
u
>
0
.
Thus
(34)
V
′
(
t
)
≥
[
α
l
S
R
∆
-
(
δ
+
μ
+
σ
)
u
]
I
H
(
t
)
+
[
β
l
S
H
∆
-
(
d
+
b
)
u
]
I
R
(
t
)
≥
[
α
l
S
R
∆
-
(
δ
+
μ
+
σ
)
u
]
[
I
H
(
t
)
+
I
R
(
t
)
]
=
(
δ
+
μ
+
σ
)
u
[
α
l
S
R
∆
(
δ
+
μ
+
σ
)
u
-
1
]
I
(
t
)
,
∀
t
≥
t
2
.
Let us takei
_
=
min
⁡
t
2
≤
t
≤
t
2
+
h
I
(
t
). Next we will prove that I
(
t
)
≥
i
_, ∀
t
≥
t
2.
Suppose that it is not true; then there existsT
>
0, such that I
(
t
)
≥
i
_, for all t
∈
[
t
2
,
t
2
+
h
+
T
], I
(
t
2
+
h
+
T
)
=
i
_, and I
′
(
t
2
+
h
+
T
)
≤
0. On the other hand, by the second equation of (26), as t
=
t
2
+
h
+
T, we have
(35)
I
′
(
t
)
≥
α
l
S
R
∆
∫
0
h
‍
I
H
(
t
-
s
)
d
η
(
s
)
-
(
d
+
b
)
u
I
R
+
β
l
S
H
∆
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
(
δ
+
μ
+
σ
)
u
I
H
=
α
l
S
R
∆
∫
0
h
‍
I
H
(
t
-
s
)
d
η
(
s
)
+
α
l
S
R
∆
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
α
l
S
R
∆
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
+
β
l
S
H
∆
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
(
d
+
b
)
u
I
R
-
(
d
+
b
)
u
I
H
+
(
d
+
b
)
u
I
H
-
(
δ
+
μ
+
σ
)
u
I
H
=
α
l
S
R
∆
∫
0
h
‍
I
(
t
-
s
)
d
η
(
s
)
-
(
d
+
b
)
u
I
(
t
)
+
(
β
l
S
H
∆
-
α
l
S
R
∆
)
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
+
[
(
d
+
b
)
u
-
(
δ
+
μ
+
σ
)
u
]
I
H
.
Since β
l
S
H
∆
-
(
d
+
b
)
u
≥
α
l
S
R
∆
-
(
δ
+
μ
+
σ
)
u
>
0, so β
l
S
H
∆
-
α
l
S
R
∆
≥
(
d
+
b
)
u
-
(
δ
+
μ
+
σ
)
u
>
0.
Thus(36)
I
′
(
t
)
≥
(
d
+
b
)
u
[
α
l
S
R
∆
(
d
+
b
)
u
-
1
]
i
_
>
0
,
since from α
l
S
R
∆
/
(
d
+
b
)
u
>
1. This is a contradiction. Hence, I
(
t
)
≥
i
_, ∀
t
≥
t
2. Consequently,
(37)
V
′
(
t
)
≥
(
δ
+
μ
+
σ
)
u
[
α
l
S
R
∆
(
δ
+
μ
+
σ
)
u
-
1
]
i
_
>
0
,
∀
t
≥
t
2
,
which implies V
(
t
)
→
∞ as t
→
∞. From Proposition 4, V
(
t
) is bounded. This is a contradiction. Hence the claim is proved. From this claim, we will discuss the following two possibilities:(1)
I
(
t
)
≥
η for all large t;
(2)
I
(
t
) oscillates about η for all large t.
Finally, we will show thatI
(
t
)
≥
η
e
-
[
(
d
+
b
)
u
+
(
δ
+
μ
+
σ
)
u
]
(
h
+
ρ
) for sufficiently large t. Evidently, we only need to consider case (2).
Lett
1 and t
2 be sufficiently large times satisfying I
(
t
1
)
=
I
(
t
2
)
=
η, I
(
t
)
<
η as t
∈
(
t
1
,
t
2
).
Ift
2
-
t
1
≤
h
+
ρ, since I
′
(
t
)
≥
-
(
d
+
u
)
u
I
R
-
(
δ
+
μ
+
σ
)
u
I
H
≥
-
[
(
d
+
u
)
u
+
(
δ
+
μ
+
σ
)
u
]
I
(
t
) and I
(
t
1
)
=
η, integrating the above inequality from t
1 to t, according to the comparison theorem, I
(
t
)
≥
η
e
-
[
(
d
+
b
)
u
+
(
δ
+
μ
+
σ
)
u
]
(
h
+
ρ
), ∀
t
∈
[
t
1
,
t
2
]. If t
2
-
t
1
>
h
+
ρ, then it is obvious that I
(
t
)
≥
η
e
-
[
(
d
+
b
)
u
+
(
δ
+
μ
+
σ
)
u
]
(
h
+
ρ
) for all t
∈
[
t
1
,
t
1
+
h
+
ρ
]. From the above discussion, we see that S
R
(
t
)
≥
S
R
∆
,
S
H
(
t
)
≥
S
H
∆, ∀
t
∈
[
t
1
+
h
+
ρ
,
t
2
]; we will show that I
(
t
)
≥
η
e
-
[
(
d
+
b
)
u
+
(
δ
+
μ
+
σ
)
u
]
(
h
+
ρ
), ∀
t
∈
[
t
1
+
h
+
ρ
,
t
2
]. If it is not true, then there exists a T
*
≥
0, such that I
(
t
)
≥
η
e
-
[
(
d
+
b
)
u
+
(
δ
+
μ
+
σ
)
u
]
(
h
+
ρ
), ∀
t
∈
[
t
1
,
t
1
+
h
+
ρ
+
T
*
], I
(
t
1
+
h
+
ρ
+
T
*
)
=
η
e
-
[
(
d
+
b
)
u
+
(
δ
+
μ
+
σ
)
u
]
(
h
+
ρ
), and I
′
(
t
1
+
h
+
ρ
+
T
*
)
≤
0.
Using the second equation of system (26), as t
=
t
1
+
h
+
ρ
+
T
*, we have
(38)
I
′
(
t
)
≥
α
l
S
R
∆
∫
0
h
‍
I
(
t
-
s
)
d
η
(
s
)
-
(
d
+
b
)
u
I
(
t
)
+
(
β
l
S
H
∆
-
α
l
S
R
∆
)
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
+
[
(
d
+
b
)
u
-
(
δ
+
μ
+
σ
)
u
]
I
H
≥
(
d
+
b
)
u
[
α
l
S
R
∆
(
d
+
b
)
u
-
1
]
η
e
-
[
(
d
+
b
)
u
+
(
δ
+
μ
+
σ
)
u
]
(
h
+
ρ
)
>
0
,
We get the last inequality by use of α
l
S
R
∆
/
(
d
+
b
)
u
>
1. This is a contradiction. Therefore, I
(
t
)
≥
η
e
-
[
(
d
+
b
)
u
+
(
δ
+
μ
+
σ
)
u
]
(
h
+
ρ
), ∀
t
∈
[
t
1
,
t
2
]. Hence
(39)
liminf
⁡
t
→
∞
I
(
t
)
≥
η
e
-
[
(
d
+
b
)
u
+
(
δ
+
μ
+
σ
)
u
]
(
h
+
ρ
)
≡
v
2
>
0
.
This completes the proof of Proposition 6.Proposition 7.
The solution(
S
R
(
t
)
,
I
R
(
t
)
,
S
H
(
t
)
,
I
H
(
t
)
,
R
H
(
t
)
) of (3) with initial conditions (8) satisfies
(40)
liminf
⁡
t
→
∞
S
H
(
t
)
≥
λ
l
b
l
β
u
c
u
+
δ
u
b
l
≡
v
3
>
0
.Proof.
From the third equation of system (26), we have
(41)
S
H
′
(
t
)
≥
λ
(
t
)
-
β
(
t
)
S
H
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
δ
(
t
)
S
H
.
From Proposition 4, for any ϵ
>
0 (no matter however small), there exists a t
1
>
0 such that
(42)
I
R
(
t
)
≤
c
u
b
l
+
ϵ
,
as t
≥
t
1. Thus when t
≥
t
1
+
h,
(43)
S
H
′
(
t
)
≥
λ
(
t
)
-
{
β
(
t
)
(
c
u
b
l
+
ϵ
)
+
δ
(
t
)
}
S
H
(
t
)
≥
λ
l
-
{
β
u
(
c
u
b
l
+
ϵ
)
+
δ
u
}
S
H
(
t
)
,
⟹
liminf
⁡
t
→
∞
S
H
(
t
)
≥
λ
l
β
u
(
c
u
/
b
l
+
ϵ
)
+
δ
u
.
Since ϵ
>
0 can be made arbitrarily small, the result of this proposition is valid. This completes the proof.Proposition 8.
Assume thatr
*
≥
r
* and r
0
>
1; then for any solution (
S
R
(
t
)
,
I
R
(
t
)
,
S
H
(
t
)
,
I
H
(
t
)
,
R
H
(
t
)
) of (3) with initial conditions (8), we have
(44)
liminf
⁡
t
→
∞
R
H
(
t
)
≥
μ
l
δ
u
+
γ
u
[
v
2
-
c
u
b
l
]
≡
v
4
>
0
.Proof.
From the fourth equation of system (26), we have
(45)
R
H
′
(
t
)
=
μ
(
t
)
I
(
t
)
-
μ
(
t
)
I
R
(
t
)
-
δ
(
t
)
R
H
-
γ
(
t
)
R
H
.
From Proposition 4, for any ϵ
>
0, there exists a t
1
>
0 such that
(46)
I
R
(
t
)
≤
c
u
b
l
+
ϵ
,
t
≥
t
1
.
From Proposition 6, when t
≥
t
1, we have
(47)
R
H
′
(
t
)
≥
μ
(
t
)
v
2
-
μ
(
t
)
(
c
u
b
l
+
ϵ
)
-
(
δ
(
t
)
+
γ
(
t
)
)
R
H
=
μ
(
t
)
[
v
2
-
(
c
u
b
l
+
ϵ
)
]
-
(
δ
(
t
)
+
γ
(
t
)
)
R
H
≥
μ
l
[
v
2
-
(
c
u
b
l
+
ϵ
)
]
-
(
δ
u
+
γ
u
)
R
H
(
t
)
.
So, when v
2
>
c
u
/
b
l, according to the comparison theorem and the arbitrariness of ϵ, we have
(48)
liminf
⁡
t
→
∞
R
H
(
t
)
≥
μ
l
δ
u
+
γ
u
[
v
2
-
c
u
b
l
]
>
0
.
This completes the proof.Thus, the system (3) with initial conditions (8) is permanent provided that r
*
≥
r
* and r
0
>
1.Remark 9.
In this paper, we only find the inferior limit ofI
(
t
) for system (26), that is, the inferior limit of I
R
(
t
)
+
I
H
(
t
). We cannot find the inferior limits of I
R
(
t
) and I
H
(
t
) for system (3), respectively. Even though we can also obtain the permanence of system (3), as liminf
⁡
t
→
∞
⁡
I
(
t
)
≥
v
2
>
0, there exist the following three possibilities:(1)
liminf
⁡
t
→
∞
⁡
I
R
(
t
)
>
0 and liminf
⁡
t
→
∞
⁡
I
H
(
t
)
>
0: in this case, it is obvious that system (3) is permanent;
(2)
liminf
⁡
t
→
∞
⁡
I
R
(
t
)
>
0 and liminf
⁡
t
→
∞
⁡
I
H
(
t
)
=
0: in this case, infected mosquitoes exist all the time; then as long as the effective infection occurs between infected mosquitoes and susceptible humans, the human population will be infected ultimately, so system (3) is permanent;
(3)
liminf
⁡
t
→
∞
⁡
I
R
(
t
)
=
0 and liminf
⁡
t
→
∞
I
H
(
t
)
>
0: in this case, infective humans exist all the time; then as long as the effective infection occurs between infective humans and susceptible mosquitoes, the mosquito population will be infected ultimately, so system (3) is permanent. In fact, we only pay attention to the human population of system (3). We do not care whether mosquito population is infected or not.Next, we will use the following lemma to discuss the extinction of the epidemic.Lemma 10 (see [14]).
Consider an autonomous delay differential equation(49)
x
′
(
t
)
=
a
1
∫
0
h
‍
x
(
t
-
s
)
d
η
(
s
)
-
a
2
x
(
t
)
,
where a
1
,
a
2 are two constants. If 0
≤
a
1
<
a
2, then for any solution x
(
t
) with initial condition φ
(
θ
)
≥
0
,
θ
∈
[
-
h
,
0
], we have
(50)
lim
⁡
t
→
∞
x
(
t
)
=
0
.Theorem 11.
SetR
*
=
(
β
u
/
(
d
+
b
)
l
)
(
λ
u
/
δ
l
)
,
R
*
=
(
α
u
/
(
δ
+
μ
+
σ
)
l
)
(
c
u
/
b
l
). If R
*
<
R
*
<
1, then lim
⁡
t
→
∞
⁡
I
(
t
)
=
0; that is, the disease in system (3) will be extinct.Proof.
Note that(51)
I
′
(
t
)
=
α
(
t
)
S
R
∫
0
h
‍
I
H
(
t
-
s
)
d
η
(
s
)
-
(
d
(
t
)
+
b
(
t
)
)
I
R
+
β
(
t
)
S
H
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
(
δ
(
t
)
+
μ
(
t
)
+
σ
(
t
)
)
I
H
≤
α
u
S
¯
R
∫
0
h
‍
I
H
(
t
-
s
)
d
η
(
s
)
-
(
d
+
b
)
l
I
R
+
β
u
S
¯
H
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
(
δ
+
μ
+
σ
)
l
I
H
=
α
u
S
¯
R
∫
0
h
‍
I
H
(
t
-
s
)
d
η
(
s
)
+
α
u
S
¯
R
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
(
δ
+
μ
+
σ
)
l
I
H
-
(
δ
+
μ
+
σ
)
l
I
R
-
α
u
S
¯
R
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
+
β
u
S
¯
H
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
(
d
+
b
)
l
I
R
+
(
δ
+
μ
+
σ
)
l
I
R
=
(
δ
+
μ
+
σ
)
l
[
α
u
S
¯
R
(
δ
+
μ
+
σ
)
l
∫
0
h
‍
I
(
t
-
s
)
d
η
(
s
)
-
I
(
t
)
]
+
[
(
d
+
b
)
l
-
(
δ
+
μ
+
σ
)
l
]
×
[
β
u
S
¯
H
-
α
u
S
¯
R
(
d
+
b
)
l
-
(
δ
+
μ
+
σ
)
l
∫
0
h
‍
I
R
(
t
-
s
)
d
η
(
s
)
-
I
R
]
,
where S
¯
H
,
S
¯
R are some upper bounds of S
H
,
S
R, respectively, and will be given later.
From Proposition4, there exists a t
1
>
0 and a sufficiently small ϵ
>
0, such that
(52)
S
H
(
t
)
≤
λ
u
δ
l
+
ϵ
≜
S
¯
H
,
S
R
(
t
)
≤
c
u
b
l
+
ϵ
≜
S
¯
R
,
t
≥
t
1
.
As R
*
<
R
*
<
1, we have
(53)
β
u
S
¯
H
(
d
+
b
)
l
<
α
u
S
¯
R
(
δ
+
μ
+
σ
)
l
<
1
.
And from condition (7), we can obtain (
d
+
b
)
l
>
(
δ
+
μ
+
σ
)
l. So from (53) we have
(54)
β
u
S
¯
H
-
(
d
+
b
)
l
<
α
u
S
¯
R
-
(
δ
+
μ
+
σ
)
l
<
0
.
Thus
(55)
β
u
S
¯
H
-
α
u
S
¯
R
<
(
d
+
b
)
l
-
(
δ
+
μ
+
σ
)
l
.
Next we will discuss the extinction in two cases.(1)
Ifβ
u
S
¯
H
-
α
u
S
¯
R
≤
0, then the second part of (51) is negative, so we have
(56)
I
′
(
t
)
≤
(
δ
+
μ
+
σ
)
l
[
α
u
S
¯
R
(
δ
+
μ
+
σ
)
l
∫
0
h
‍
I
(
t
-
s
)
d
η
(
s
)
-
I
(
t
)
]
.
Using the Lemma10 and the comparison theorem, we come toα
u
S
¯
R
/
(
δ
+
μ
+
σ
)
l
<
1; that is, R
*
<
1, lim
⁡
t
→
∞
⁡
I
(
t
)
=
0.
(2)
Ifβ
u
S
¯
H
-
α
u
S
¯
R
>
0, then (55) becomes
(57)
0
<
β
u
S
¯
H
-
α
u
S
¯
R
(
d
+
b
)
l
-
(
δ
+
μ
+
σ
)
l
<
1
.Now we denote the second part of (51) as Y
′
(
t
); that is,
(58)
Y
′
(
t
)
=
[
(
d
+
b
)
l
-
(
δ
+
μ
+
σ
)
l
]
×
[
β
u
S
¯
H
-
α
u
S
¯
R
(
d
+
b
)
l
-
(
δ
+
μ
+
σ
)
l
∫
0
h
‍
Y
(
t
-
s
)
d
η
(
s
)
-
Y
(
t
)
]
.
Using the Lemma 10, we have lim
⁡
t
→
∞
Y
(
t
)
=
0. Thus when t
→
∞, (51) becomes
(59)
I
′
(
t
)
≤
(
δ
+
μ
+
σ
)
l
[
α
u
S
¯
R
(
δ
+
μ
+
σ
)
l
∫
0
h
‍
I
(
t
-
s
)
d
η
(
s
)
-
I
(
t
)
]
.
Using the Lemma 10 and the comparison theorem again, we come toα
u
S
¯
R
/
(
δ
+
μ
+
σ
)
l
<
1; that is, R
*
<
1, lim
⁡
t
→
∞
⁡
I
(
t
)
=
0. This completes the proof.


## 3. Global Asymptotic Stability

In this section, we derive sufficient conditions for the global asymptotic stability of system (3) with initial conditions (8).Definition 12 (see [16]).
System (3) with initial conditions (8) is said to be globally asymptotically stable if
(60)
lim
⁡
t
→
∞
|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
=
0
,
lim
⁡
t
→
∞
|
I
R
1
(
t
)
-
I
R
2
(
t
)
|
=
0
,
lim
⁡
t
→
∞
|
S
H
1
(
t
)
-
S
H
2
(
t
)
|
=
0
,
lim
⁡
t
→
∞
|
I
H
1
(
t
)
-
I
H
2
(
t
)
|
=
0
,
lim
⁡
t
→
∞
|
R
H
1
(
t
)
-
R
H
2
(
t
)
|
=
0
hold for two solutions (
S
R
1
(
t
)
,
I
R
1
(
t
)
,
S
H
1
(
t
)
,
I
H
1
(
t
)
,
R
H
1
(
t
)
) and (
S
R
2
(
t
)
,
I
R
2
(
t
)
,
S
H
2
(
t
)
,
I
H
2
(
t
)
,
R
H
2
(
t
)
) of (3) with initial conditions of type (8).Assume that(
S
R
(
t
)
,
I
R
(
t
)
,
S
H
(
t
)
,
I
H
(
t
)
,
R
H
(
t
)
) is a solution of (3). By the uniform boundedness of solutions of (3), there is a L
>
0 (in fact, L
=
max
⁡
{
c
u
/
b
l
,
λ
u
/
δ
l
}
+
ϵ, where ϵ
>
0 can be made arbitrary small) independent of initial conditions (8) such that
(61)
0
≤
S
R
(
t
)
≤
L
,
0
≤
I
R
(
t
)
≤
L
,
0
≤
S
H
(
t
)
≤
L
,
0
≤
I
H
(
t
)
≤
L
,
0
≤
R
H
(
t
)
≤
L
,
for large enough t. Without loss of generality, we may assume that
(62)
0
≤
S
R
(
t
)
≤
L
,
0
≤
I
R
(
t
)
≤
L
,
0
≤
S
H
(
t
)
≤
L
,
vvvvvvvvvvvvvvv
0
≤
I
H
(
t
)
≤
L
,
0
≤
R
H
(
t
)
≤
L
,
vvvvvvvvvvvvvvivvvvvvvvvvvvvvvvvvvvvv
∀
t
≥
0
.Theorem 13.
If there existc
1
,
c
2
,
c
3
>
0, such that the functions B
i
(
t
)
(
i
=
1,2
,
3,4
,
5
) are nonnegative on [
0
,
∞
) and for any interval sequence {
[
a
i
,
b
i
]
}
1
∞, [
a
i
,
b
i
]
∩
[
a
j
,
b
j
]
=
∅, and b
i
-
a
i
=
b
j
-
a
j
>
0, for all i
,
j
=
1,2
,
3
,
…
,
i
≠
j, one has ∑
k
=
1
∞
∫
a
k
b
k
‍
B
i
(
t
)
d
t
=
∞, then system (3) with initial conditions (8) is globally asymptotically stable. Here
(63)
B
1
(
t
)
=
c
1
b
(
t
)
-
c
2
L
α
(
t
)
,
B
2
(
t
)
=
c
2
(
d
(
t
)
+
b
(
t
)
)
-
(
c
1
+
c
2
)
L
∫
0
h
‍
β
(
t
+
s
)
d
η
(
s
)
,
B
3
(
t
)
=
c
2
δ
(
t
)
-
c
1
L
β
(
t
)
,
B
4
(
t
)
=
c
1
(
δ
(
t
)
+
σ
(
t
)
)
+
(
c
1
-
c
3
)
μ
(
t
)
+
(
c
1
+
c
2
)
L
∫
0
h
‍
α
(
t
+
s
)
d
η
(
s
)
,
B
5
(
t
)
=
c
3
δ
(
t
)
+
(
c
3
-
c
2
)
γ
(
t
)
.Proof.
Assume that(
S
R
1
(
t
)
,
I
R
1
(
t
)
,
S
H
1
(
t
)
,
I
H
1
(
t
)
,
R
H
1
(
t
)
) and (
S
R
2
(
t
)
,
I
R
2
(
t
)
,
S
H
2
(
t
)
,
I
H
2
(
t
)
,
R
H
2
(
t
)
) are any two solutions of system (3) with initial conditions of type (8).
The right-upper derivatives of|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
,
|
I
R
1
(
t
)
-
I
R
2
(
t
)
|
,
|
S
H
1
(
t
)
-
S
H
2
(
t
)
|, |
I
H
1
(
t
)
-
I
H
2
(
t
)
|
,
|
R
H
1
(
t
)
-
R
H
2
(
t
)
| along the solution of system (3) and (8) are given below:
(64)
D
+
|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
=
sgn
⁡
(
S
R
1
(
t
)
-
S
R
2
(
t
)
)
×
{
-
α
(
t
)
S
R
1
(
t
)
∫
0
h
‍
I
H
1
(
t
-
s
)
d
η
(
s
)
-
b
(
t
)
S
R
1
(
t
)
+
α
(
t
)
S
R
2
(
t
)
∫
0
h
‍
I
H
2
(
t
-
s
)
d
η
(
s
)
+
b
(
t
)
S
R
2
(
t
)
∫
0
h
‍
}
=
sgn
⁡
(
S
R
1
(
t
)
-
S
R
2
(
t
)
)
×
{
-
α
(
t
)
S
R
1
(
t
)
∫
0
h
‍
I
H
1
(
t
-
s
)
d
η
(
s
)
+
α
(
t
)
S
R
2
(
t
)
∫
0
h
‍
I
H
1
(
t
-
s
)
d
η
(
s
)
-
α
(
t
)
S
R
2
(
t
)
∫
0
h
‍
I
H
1
(
t
-
s
)
d
η
(
s
)
+
α
(
t
)
S
R
2
(
t
)
∫
0
h
‍
I
H
2
(
t
-
s
)
d
η
(
s
)
-
b
(
t
)
(
S
R
1
(
t
)
-
S
R
2
(
t
)
)
∫
0
h
‍
}
=
sgn
⁡
(
S
R
1
(
t
)
-
S
R
2
(
t
)
)
×
{
-
α
(
t
)
∫
0
h
‍
I
H
1
(
t
-
s
)
d
η
(
s
)
(
S
R
1
(
t
)
-
S
R
2
(
t
)
)
+
α
(
t
)
S
R
2
(
t
)
∫
0
h
‍
(
I
H
2
(
t
-
s
)
-
I
H
1
(
t
-
s
)
)
d
η
(
s
)
-
b
(
t
)
(
S
R
1
(
t
)
-
S
R
2
(
t
)
)
∫
0
h
‍
}
,
so
(65)
D
+
|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
≤
α
(
t
)
L
∫
0
h
‍
|
I
H
1
(
t
-
s
)
-
I
H
2
(
t
-
s
)
|
d
η
(
s
)
-
b
(
t
)
|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
,
(66)
D
+
|
I
R
1
(
t
)
-
I
R
2
(
t
)
|
=
sgn
⁡
(
I
R
1
(
t
)
-
I
R
2
(
t
)
)
×
{
α
(
t
)
S
R
1
(
t
)
∫
0
h
‍
I
H
1
(
t
-
s
)
d
η
(
s
)
-
α
(
t
)
S
R
2
(
t
)
∫
0
h
‍
I
H
1
(
t
-
s
)
d
η
(
s
)
+
α
(
t
)
S
R
2
(
t
)
∫
0
h
‍
I
H
1
(
t
-
s
)
d
η
(
s
)
-
α
(
t
)
S
R
2
(
t
)
∫
0
h
‍
I
H
2
(
t
-
s
)
d
η
(
s
)
-
(
d
(
t
)
+
b
(
t
)
)
(
I
R
1
(
t
)
-
I
R
2
(
t
)
)
∫
0
h
‍
}
=
sgn
⁡
(
I
R
1
(
t
)
-
I
R
2
(
t
)
)
×
{
α
(
t
)
∫
0
h
‍
I
H
1
(
t
-
s
)
d
η
(
s
)
(
S
R
1
(
t
)
-
S
R
2
(
t
)
)
+
α
(
t
)
S
R
2
(
t
)
∫
0
h
‍
(
I
H
1
(
t
-
s
)
-
I
H
2
(
t
-
s
)
)
d
η
(
s
)
-
(
d
(
t
)
+
b
(
t
)
)
(
I
R
1
(
t
)
-
I
R
2
(
t
)
)
∫
0
h
‍
}
,
so
(67)
D
+
|
I
R
1
(
t
)
-
I
R
2
(
t
)
|
≤
α
(
t
)
L
|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
+
α
(
t
)
L
∫
0
h
‍
|
I
H
1
(
t
-
s
)
-
I
H
2
(
t
-
s
)
|
d
η
(
s
)
-
(
d
(
t
)
+
b
(
t
)
)
|
I
R
1
(
t
)
-
I
R
2
(
t
)
|
,
(68)
D
+
|
S
H
1
(
t
)
-
S
H
2
(
t
)
|
=
sgn
⁡
(
S
H
1
(
t
)
-
S
H
2
(
t
)
)
×
{
-
β
(
t
)
S
H
1
(
t
)
∫
0
h
‍
I
R
1
(
t
-
s
)
d
η
(
s
)
+
β
(
t
)
S
H
2
(
t
)
∫
0
h
‍
I
R
1
(
t
-
s
)
d
η
(
s
)
-
β
(
t
)
S
H
2
(
t
)
∫
0
h
‍
I
R
1
(
t
-
s
)
d
η
(
s
)
+
β
(
t
)
S
H
2
(
t
)
∫
0
h
‍
I
R
2
(
t
-
s
)
d
η
(
s
)
-
δ
(
t
)
(
S
H
1
-
S
H
2
)
+
γ
(
t
)
(
R
H
1
-
R
H
2
)
∫
0
h
‍
}
=
sgn
⁡
(
S
H
1
(
t
)
-
S
H
2
(
t
)
)
×
{
-
β
(
t
)
∫
0
h
‍
I
R
1
(
t
-
s
)
d
η
(
s
)
(
S
H
1
-
S
H
2
)
+
β
(
t
)
S
H
2
(
t
)
∫
0
h
‍
(
I
R
2
(
t
-
s
)
-
I
R
1
(
t
-
s
)
)
d
η
(
s
)
-
δ
(
t
)
(
S
H
1
-
S
H
2
)
+
γ
(
t
)
(
R
H
1
-
R
H
2
)
∫
0
h
‍
}
,
so
(69)
D
+
|
S
H
1
(
t
)
-
S
H
2
(
t
)
|
≤
β
(
t
)
L
∫
0
h
‍
|
I
R
1
(
t
-
s
)
-
I
R
2
(
t
-
s
)
|
d
η
(
s
)
-
δ
(
t
)
|
S
H
1
-
S
H
2
|
+
γ
(
t
)
|
R
H
1
-
R
H
2
|
,
(70)
D
+
|
I
H
1
(
t
)
-
I
H
2
(
t
)
|
=
sgn
⁡
(
I
H
1
(
t
)
-
I
H
2
(
t
)
)
×
{
β
(
t
)
S
H
1
∫
0
h
‍
I
R
1
(
t
-
s
)
d
η
(
s
)
-
β
(
t
)
S
H
2
∫
0
h
‍
I
R
1
(
t
-
s
)
d
η
(
s
)
+
β
(
t
)
S
H
2
∫
0
h
‍
I
R
1
(
t
-
s
)
d
η
(
s
)
-
β
(
t
)
S
H
2
(
t
)
∫
0
h
‍
I
R
2
(
t
-
s
)
d
η
(
s
)
-
(
δ
(
t
)
+
μ
(
t
)
+
σ
(
t
)
)
(
I
H
1
-
I
H
2
)
∫
0
h
‍
}
=
sgn
⁡
(
I
H
1
(
t
)
-
I
H
2
(
t
)
)
×
{
β
(
t
)
∫
0
h
‍
I
R
1
(
t
-
s
)
d
η
(
s
)
(
S
H
1
-
S
H
2
)
+
β
(
t
)
S
H
2
∫
0
h
‍
(
I
R
1
(
t
-
s
)
-
I
R
2
(
t
-
s
)
)
d
η
(
s
)
-
(
δ
(
t
)
+
μ
(
t
)
+
σ
(
t
)
)
(
I
H
1
-
I
H
2
)
∫
0
h
‍
}
,
so
(71)
D
+
|
I
H
1
(
t
)
-
I
H
2
(
t
)
|
≤
β
(
t
)
L
|
S
H
1
-
S
H
2
|
+
β
(
t
)
L
∫
0
h
‍
|
I
R
1
(
t
-
s
)
-
I
R
2
(
t
-
s
)
|
d
η
(
s
)
-
(
δ
(
t
)
+
μ
(
t
)
+
σ
(
t
)
)
|
I
H
1
-
I
H
2
|
,
(72)
D
+
|
R
H
1
(
t
)
-
R
H
2
(
t
)
|
=
sgn
⁡
(
R
H
1
(
t
)
-
R
H
2
(
t
)
)
×
{
μ
(
t
)
(
I
H
1
-
I
H
2
)
-
(
δ
(
t
)
+
γ
(
t
)
)
(
R
H
1
-
R
H
2
)
}
,
so
(73)
D
+
|
R
H
1
(
t
)
-
R
H
2
(
t
)
|
≤
μ
(
t
)
|
I
H
1
-
I
H
2
|
-
(
δ
(
t
)
+
γ
(
t
)
)
|
R
H
1
-
R
H
2
|
.
DefineV
1
(
t
)
=
|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
+
|
I
H
1
(
t
)
-
I
H
2
(
t
)
|, V
2
(
t
)
=
|
I
R
1
(
t
)
-
I
R
2
(
t
)
|
+
|
S
H
1
(
t
)
-
S
H
2
(
t
)
|
,
V
3
(
t
)
=
|
R
H
1
(
t
)
-
R
H
2
(
t
)
|.
Calculating the right-upper derivatives ofV
1
(
t
)
,
V
2
(
t
)
,
V
3
(
t
) along the solution of system (3) and (8), we have
(74)
D
+
V
1
(
t
)
≤
-
b
(
t
)
|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
+
β
(
t
)
L
|
S
H
1
-
S
H
2
|
-
(
δ
(
t
)
+
μ
(
t
)
+
σ
(
t
)
)
|
I
H
1
-
I
H
2
|
+
α
(
t
)
L
∫
0
h
‍
|
I
H
1
(
t
-
s
)
-
I
H
2
(
t
-
s
)
|
d
η
(
s
)
+
β
(
t
)
L
∫
0
h
‍
|
I
R
1
(
t
-
s
)
-
I
R
2
(
t
-
s
)
|
d
η
(
s
)
,
D
+
V
2
(
t
)
≤
α
(
t
)
L
|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
-
(
d
(
t
)
+
b
(
t
)
)
|
I
R
1
(
t
)
-
I
R
2
(
t
)
|
-
δ
(
t
)
|
S
H
1
-
S
H
2
|
+
γ
(
t
)
|
R
H
1
-
R
H
2
|
+
α
(
t
)
L
∫
0
h
‍
|
I
H
1
(
t
-
s
)
-
I
H
2
(
t
-
s
)
|
d
η
(
s
)
+
β
(
t
)
L
∫
0
h
‍
|
I
R
1
(
t
-
s
)
-
I
R
2
(
t
-
s
)
|
d
η
(
s
)
,
D
+
V
3
(
t
)
≤
μ
(
t
)
|
I
H
1
-
I
H
2
|
-
(
δ
(
t
)
+
γ
(
t
)
)
|
R
H
1
-
R
H
2
|
.
Define V
4
(
t
) as
(75)
V
4
(
t
)
=
∫
0
h
‍
∫
t
-
s
t
‍
α
(
u
+
s
)
L
|
I
H
1
(
u
)
-
I
H
2
(
u
)
|
d
u
d
η
(
s
)
+
∫
0
h
‍
∫
t
-
s
t
‍
β
(
u
+
s
)
L
|
I
R
1
(
u
)
-
I
R
2
(
u
)
|
d
u
d
η
(
s
)
.
The right-upper derivative ofV
4
(
t
) along the solution of system (3) and (8) is given below:
(76)
D
+
V
4
(
t
)
=
L
|
I
H
1
(
t
)
-
I
H
2
(
t
)
|
∫
0
h
‍
α
(
t
+
s
)
d
η
(
s
)
-
α
(
t
)
L
∫
0
h
‍
|
I
H
1
(
t
-
s
)
-
I
H
2
(
t
-
s
)
|
d
η
(
s
)
+
L
|
I
R
1
(
t
)
-
I
R
2
(
t
)
|
∫
0
h
‍
β
(
t
+
s
)
d
η
(
s
)
-
β
(
t
)
L
∫
0
h
‍
|
I
R
1
(
t
-
s
)
-
I
R
2
(
t
-
s
)
|
d
η
(
s
)
.
LetV
(
t
)
=
c
1
V
1
(
t
)
+
c
2
V
2
(
t
)
+
c
3
V
3
(
t
)
+
(
c
1
+
c
2
)
V
4
(
t
); then by using (74) and (76), we have
(77)
D
+
V
(
t
)
≤
-
B
1
(
t
)
|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
-
B
2
(
t
)
|
I
R
1
(
t
)
-
I
R
2
(
t
)
|
-
B
3
(
t
)
|
S
H
1
-
S
H
2
|
-
B
4
(
t
)
|
I
H
1
(
t
-
s
)
-
I
H
2
(
t
-
s
)
|
-
B
5
(
t
)
|
R
H
1
-
R
H
2
|
,
∀
t
≥
h
,
where B
i
(
t
)
(
i
=
1,2
,
3,4
,
5
) are defined in (63).
Integrating (77) from h to t, we have
(78)
∫
h
t
‍
{
∫
h
t
‍
B
1
(
t
)
|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
+
B
2
(
t
)
|
I
R
1
(
t
)
-
I
R
2
(
t
)
|
+
B
3
(
t
)
|
S
H
1
-
S
H
2
|
+
B
4
(
t
)
|
I
H
1
(
t
-
s
)
-
I
H
2
(
t
-
s
)
|
+
B
5
(
t
)
|
R
H
1
-
R
H
2
|
∫
h
t
‍
}
d
t
≤
V
(
h
)
-
V
(
t
)
.
So
(79)
∫
h
t
‍
{
∫
h
t
‍
B
1
(
t
)
|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
+
B
2
(
t
)
|
I
R
1
(
t
)
-
I
R
2
(
t
)
|
+
B
3
(
t
)
|
S
H
1
-
S
H
2
|
+
B
4
(
t
)
|
I
H
1
(
t
-
s
)
-
I
H
2
(
t
-
s
)
|
+
B
5
(
t
)
|
R
H
1
-
R
H
2
|
∫
h
t
‍
}
d
t
<
∞
.
By assumptions aboutB
i
(
t
) and the boundedness of (
S
R
1
(
t
)
,
I
R
1
(
t
)
,
S
H
1
(
t
)
,
I
H
1
(
t
)
,
R
H
1
(
t
)
) and (
S
R
2
(
t
)
,
I
R
2
(
t
)
,
S
H
2
(
t
)
,
I
H
2
(
t
)
,
R
H
2
(
t
)
) on [
0
,
∞
], we obtain from system (3) that |
S
R
1
(
t
)
-
S
R
2
(
t
)
|, |
I
R
1
(
t
)
-
I
R
2
(
t
)
|, |
S
H
1
(
t
)
-
S
H
2
(
t
)
|, |
I
H
1
(
t
)
-
I
H
2
(
t
)
|, and |
R
H
1
(
t
)
-
R
H
2
(
t
)
| are bounded and uniformly continuous on [
0
,
∞
).
It follows from (79) that
(80)
lim
⁡
t
→
∞
|
S
R
1
(
t
)
-
S
R
2
(
t
)
|
=
0
,
lim
⁡
t
→
∞
|
I
R
1
(
t
)
-
I
R
2
(
t
)
|
=
0
,
lim
⁡
t
→
∞
|
S
H
1
(
t
)
-
S
H
2
(
t
)
|
=
0
,
lim
⁡
t
→
∞
|
I
H
1
(
t
)
-
I
H
2
(
t
)
|
=
0
,
lim
⁡
t
→
∞
|
R
H
1
(
t
)
-
R
H
2
(
t
)
|
=
0
.
This shows that system (3) with initial conditions (8) is globally asymptotically stable. This completes the proof.Corollary 14.
If there existc
1
,
c
2
,
c
3
>
0 such that
(81)
liminf
⁡
t
→
∞
{
c
1
b
(
t
)
-
c
2
L
α
(
t
)
}
>
0
,
liminf
⁡
t
→
∞
{
∫
0
h
‍
c
2
(
d
(
t
)
+
b
(
t
)
)
-
(
c
1
+
c
2
)
L
∫
0
h
‍
β
(
t
+
s
)
d
η
(
s
)
}
>
0
,
liminf
⁡
t
→
∞
{
c
2
δ
(
t
)
-
c
1
L
β
(
t
)
}
>
0
,
liminf
⁡
t
→
∞
{
∫
0
h
‍
c
1
(
δ
(
t
)
+
σ
(
t
)
)
+
(
c
1
-
c
3
)
μ
(
t
)
+
(
c
1
+
c
2
)
L
∫
0
h
‍
α
(
t
+
s
)
d
η
(
s
)
}
>
0
,
liminf
⁡
t
→
∞
{
c
3
δ
(
t
)
+
(
c
3
-
c
2
)
γ
(
t
)
}
>
0
,
then system (3) with initial conditions (8) is globally asymptotically stable.Theorem 15.
Specially, if system (3) is ω-periodic and there are positive constants v
i and M
i
(
i
=
1,2
,
3,4
,
5
) such that
(82)
v
1
≤
liminf
⁡
t
→
∞
S
R
(
t
)
≤
limsup
⁡
t
→
∞
S
R
(
t
)
≤
M
1
,
v
2
≤
liminf
⁡
t
→
∞
I
R
(
t
)
≤
limsup
⁡
t
→
∞
I
R
(
t
)
≤
M
2
,
v
3
≤
liminf
⁡
t
→
∞
S
H
(
t
)
≤
limsup
⁡
t
→
∞
S
H
(
t
)
≤
M
3
,
v
4
≤
liminf
⁡
t
→
∞
I
H
(
t
)
≤
limsup
⁡
t
→
∞
I
H
(
t
)
≤
M
4
,
v
5
≤
liminf
⁡
t
→
∞
R
H
(
t
)
≤
limsup
⁡
t
→
∞
R
H
(
t
)
≤
M
5
hold for any solution (
S
R
(
t
)
,
I
R
(
t
)
,
S
H
(
t
)
,
I
H
(
t
)
,
R
H
(
t
)
) of (3) with initial conditions (8), then system (3) has positive periodic solution with period ω.Corollary 16.
If system (3) is ω-periodic and the conditions in Theorems 3 and 13 are valid, then there exists a unique positive ω-periodic solution which is globally asymptotically stable.


## 4. Numerical Simulations

To demonstrate the theoretical results obtained in this paper, we will give some numerical simulations.Firstly, for system (3) we consider the special case; that is, the parameter values are constants as λ
(
t
)
=
0.29, β
(
t
)
=
0.29, α
(
t
)
=
0.19, μ
(
t
)
=
0.022, c
(
t
)
=
0.4, d
(
t
)
=
0.22, σ
(
t
)
=
0.011, b
(
t
)
=
0.11, γ
(
t
)
=
0.01, and δ
(
t
)
=
0.011. The delay h
=
Π
/
2 and η
(
s
)
=
s
/
h. It is easy to verify r
*
≥
r
*
,
r
0
>
1; the conditions in Theorem 3 are all valid. So according to Theorem 3, we know that the system (3) is permanent. Figure 1 shows trajectories of S
R
(
t
), I
R
(
t
), S
H
(
t
), I
H
(
t
), and R
H
(
t
), respectively.Figure 1
(a)–(e) show that system (3) with initial conditions (
S
R
(
0
)
,
I
R
(
0
)
,
S
H
(
0
)
,
I
H
(
0
)
,
R
H
(
0
)
)
=
(
2,0.12,3
,
0.15,0.1
) is permanent.Now, we consider the general case; that is, we take account of malaria model with periodic environment, we chose the parameter values asλ
(
t
)
=
0.3
+
0.01
cos
⁡
t
,
β
(
t
)
=
0.3
+
0.01
sin
t, α
(
t
)
=
0.2
+
0.01
cos
⁡
t
,
μ
(
t
)
=
0.02
+
0.002
sin
t, c
(
t
)
=
0.5
+
0.1
sin
t
,
d
(
t
)
=
0.2
+
0.02
sin
t, σ
(
t
)
=
0.012
+
0.001
sin
t
,
b
(
t
)
=
0.1
+
0.01
sin
t
,
γ
(
t
)
=
0.01
+
0.0001
cos
⁡
t, δ
(
t
)
=
0.01
+
0.001
sin
t, and the conditions in Theorem 11 are all valid. Figure 2 shows trajectories of S
R
(
t
), I
R
(
t
), S
H
(
t
), I
H
(
t
), and R
H
(
t
), respectively.Figure 2
(a)–(e) show that system (3) with initial conditions (
S
R
(
0
)
,
I
R
(
0
)
,
S
H
(
0
)
,
I
H
(
0
)
,
R
H
(
0
)
)
=
(
2,0.12,3
,
0.15,0.1
) is permanent.Next, we also consider the special case firstly; we chose the parameter values asλ
(
t
)
=
0.31, β
(
t
)
=
0.031
,
α
(
t
)
=
0.011, μ
(
t
)
=
0.018, c
(
t
)
=
0.6, d
(
t
)
=
0.48, σ
(
t
)
=
0.011, b
(
t
)
=
0.09, γ
(
t
)
=
0.01, and δ
(
t
)
=
0.09. The delay h
=
Π
/
2 and η
(
s
)
=
s
/
h. It is easy to verify R
*
<
R
*
<
1; the conditions in Theorem 11 are all valid. So according to Theorem 11, we know that the disease in system (3) will be extinct. Figure 3 shows trajectories of S
R
(
t
), I
R
(
t
), S
H
(
t
), I
H
(
t
), and R
H
(
t
), respectively.Figure 3
(a)–(e) show that the disease in system (3) with initial conditions (
S
R
(
0
)
,
I
R
(
0
)
,
S
H
(
0
)
,
I
H
(
0
)
,
R
H
(
0
)
)
=
(
2.5,0.15,3.5,0.1,0.2
) will be extinct.Now, we consider the general case; we chose the parameter values asλ
(
t
)
=
0.3
+
0.01
cos
⁡
t, β
(
t
)
=
0.03
+
0.001
sin
t, α
(
t
)
=
0.01
+
0.001
cos
⁡
t, μ
(
t
)
=
0.02
+
0.002
sin
t, c
(
t
)
=
0.5
+
0.1
sin
t, d
(
t
)
=
0.5
+
0.02
sin
t, σ
(
t
)
=
0.012
+
0.001
sin
t, b
(
t
)
=
0.1
+
0.01
sin
t, γ
(
t
)
=
0.01
+
0.0001
cos
⁡
t, and δ
(
t
)
=
0.1
+
0.01
sin
t and the conditions in Theorem 11 are all valid. Figure 4 shows trajectories of S
R
(
t
), I
R
(
t
), S
H
(
t
), I
H
(
t
), and R
H
(
t
), respectively.Figure 4
(a)–(e) show that the disease in system (3) with initial conditions (
S
R
(
0
)
,
I
R
(
0
)
,
S
H
(
0
)
,
I
H
(
0
)
,
R
H
(
0
)
)
=
(
2.5,0.15,3.5,0.1,0.2
) will be extinct.


## 5. Conclusions

In this paper, we study the permanence, extinction, and global asymptotic stability for a nonautonomous malaria transmission model with distributed time delay, that is, (3). We establish some sufficient conditions on the permanence and extinction of the disease by using inequality analytical techniques. When r
*
≥
r
* and r
0
>
1, the system is permanent. When R
*
<
R
*
<
1, the disease in system will be extinct. By a Lyapunov functional method, according to the definition of global asymptotic stability, we also obtain some sufficient conditions for global asymptotic stability of this model. This method has been used in many references [11, 13, 14].



---

*Source: 139046-2014-05-08.xml*