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In the article, we established a non-autonomous vector infectious disease model, studied the long-term dynamic behavior of the system, and obtained sufficient conditions for the extinction and persistence of infectious diseases by constructing integral functions.

In real life, we are often confused by infectious diseases. Infectious diseases include humans, animals, plant infectious diseases, especially human infectious diseases, such as tuberculosis, AIDS/HIV, malaria, which are the top three single disease killers of health in the world. According to the World Health Organization statistics, in 2002 about 70 million people are infected with AIDS, causing around 20 million deaths. In recent years, each year more than 560 million people infected with AIDS [

In recent years, some mathematical models incorporating treatment have been established and investigated by many researchers [

Some infectious diseases are transmitted by vector, such as Malaria, Dengue and West Nile virus, which spread by Mosquitoes. The maintenance and resurgence of vector-borne diseases are related to ecological changes that favor increased vector densities or vector-host interactions, among other factors [

The structure of this paper is organized as follows. Section 2 presents the vector-borne diseases model. And positivity and boundary of the model (1.1) are studied. In Section 3 and 4, we deal with the existence and permanence of model (1.1). In Section 5, we had a brief discussion.

Based on [

{ d S ( t ) d t = Λ ( t ) − β 1 ( t ) S ( t ) I ( t ) − β 2 ( t ) S ( t ) Y ( t ) − μ ( t ) S ( t ) , d I ( t ) d t = β 1 ( t ) S ( t ) I ( t ) + β 2 ( t ) S ( t ) Y ( t ) − μ ( t ) I ( t ) , d X ( t ) d t = L ( t ) − γ ( t ) I ( t ) X ( t ) − μ v ( t ) X ( t ) , d Y ( t ) d t = γ ( t ) I ( t ) X ( t ) − μ v ( t ) Y ( t ) . (2.1)

with initial value

S ( 0 ) > 0 , I ( 0 ) > 0 , X ( 0 ) > 0 , Y ( 0 ) > 0. (2.2)

where the variables S ( t ) , I ( t ) , X ( t ) and Y ( t ) represent susceptible host, infected host, susceptible vector and infection vector, respectively. Λ ( t ) represents the input rate of susceptible hosts, β i ( t ) ( i = 1 , 2 ) means effective contact rate. μ ( t ) and μ v ( t ) represents the natural mortality of the host and the vector, respectively. L ( t ) represents the birth rate of the newborn vectors, A represents the effective bite rate of the vector.

Assumption 2.1

1) Functions Λ ( t ) , β 1 ( t ) , β 2 ( t ) , μ ( t ) , γ ( t ) and μ v ( t ) are positive, bounded and continuous on [ 0, + ∞ ) .

2) There exist constants ω i > 0 ( i = 1 , 2 , 3 , 4 ) such that

∫ t t + ω 1 Λ ( s ) d s > 0 , lim inf t → + ∞ ∫ t t + ω 2 β 1 ( s ) d s > 0 ,

lim inf t → + ∞ ∫ t t + ω 3 β 2 ( s ) d s > 0 , lim inf t → + ∞ ∫ t t + ω 4 μ ( s ) d s > 0.

In what follows, we denote N ( t ) = S ( t ) + I ( t ) , N v ( t ) = X ( t ) + Y ( t )

and N ( t ) the solution of

d N ( t ) d t = Λ ( t ) − μ ( t ) N ( t ) (2.3)

N v ( t ) the solution of

d N v ( t ) d t = L ( t ) − μ v ( t ) N v ( t ) (2.4)

with initial value S ( 0 ) > 0 , S ( 0 ) > 0 , N ( 0 ) = S ( 0 ) + I ( 0 ) > 0 , N v ( 0 ) = X ( 0 ) + Y ( 0 ) > 0 .

Proposition 2.2

1) There exist constants m 1 > 0 and M 1 > 0 , which are independent from the chioce of initial value N ( 0 ) > 0 , such that

0 < m 1 ≤ lim inf t → + ∞ N ( t ) ≤ lim sup t → + ∞ N ( t ) ≤ M 1 < + ∞ . (2.5)

2) There exist constants m 2 > 0 and M 2 > 0 , which are independent from the chioce of initial value N v ( 0 ) > 0 , such that

0 < m 2 ≤ lim inf t → + ∞ N v ( t ) ≤ lim sup t → + ∞ N v ( t ) ≤ M 2 < + ∞ . (2.6)

3) The solution ( S ( t ) , I ( t ) , X ( t ) , Y ( t ) ) of system (1.1) with initial value (2.2) exists, uniformly bounded and

S ( t ) > 0 , I ( t ) > 0 , X ( t ) > 0 , Y ( t ) > 0 ,

for all t > 0 .

For p > 0 , q > 0 and t > 0 we define

G ( p , t ) = [ β 1 ( t ) + p β 2 ( t ) ] N ( t ) − μ ( t ) + μ v ( t )

and

W ( p , t ) = p I ( t ) − Y ( t ) , (2.7)

where I ( t ) and Y ( t ) are solutions of system (1.1). In Sections 3 and 4 we use the following lemma in order to investigate the longtime behavior of system (2.1).

Lemma 2.3 If there exist positive contants p > 0 and T 1 > 0 such that G ( p , t ) < 0 for all t ≥ T 1 , then there exists T 2 ≥ T 1 such that either W ( p , t ) > 0 for all t ≥ T 2 or W ( p , t ) ≤ 0 for all t ≥ T 2 .

Proof. Suppose that there does not exist T 2 ≥ T 1 such that W ( p , t ) > 0 for all t ≥ T 2 or W ( p , t ) ≤ 0 for all t ≥ T 2 hold. Then there necessarily exists s 1 ≥ T 1 such that W ( p , s 1 ) = 0 and d W ( p , t ) d t | t = s 1 > 0 . Hence we have

p I ( s 1 ) = Y ( s 1 ) (2.8)

and

p [ β 1 ( s 1 ) S ( s 1 ) I ( s 1 ) + β 2 ( s 1 ) S ( s 1 ) Y ( s 1 ) − μ ( s 1 ) I ] ( s 1 ) − γ ( s 1 ) I ( s 1 ) X ( s 1 ) + μ v ( s 1 ) Y ( s 1 ) = p I ( s 1 ) [ β 1 ( s 1 ) S ( s 1 ) − μ ( s 1 ) ] + Y ( s 1 ) [ p β 2 ( s 1 ) S ( s 1 ) + μ v ( s 1 ) ] − γ ( s 1 ) I ( s 1 ) X ( s 1 ) > 0 (2.9)

Substituting (2.8) into (2.9) we have

0 < p I ( s 1 ) [ β 1 ( s 1 ) S ( s 1 ) − μ ( s 1 ) + p β 2 ( s 1 ) S ( s 1 ) + μ v ( s 1 ) ] − γ ( s 1 ) I ( s 1 ) S v ( s 1 ) ≤ p I ( s 1 ) G ( p , s 1 ) .

From 3) of Proposition 2.2, we have G ( p , s 1 ) > 0 , which is a contradiction. □

In this section, we obtain conditions for the extinction of infectious population of system (2.1). The definition of the extinction is as follows:

Definition 3.1. We say that the infectious population of system (2.1) is extinct if

lim t → + ∞ I ( t ) = 0 , lim t → + ∞ Y ( t ) = 0.

From system (2.1), it’s easy to prove that if one of the above equalities hold, then the other one is certainly hold. We give one of the main results of this paper.

Theorem 3.2. If there exist positive constants λ > 0 , p > 0 , q > 0 and T 1 > 0 such that

R 1 ( λ , p ) ≜ lim sup t → + ∞ ∫ t t + λ { ( β 1 ( s ) + p β 1 ( s ) ) N ( s ) p − μ ( s ) } d s < 0 , (3.1)

R ′ 1 ( λ , p ) ≜ lim sup t → + ∞ ∫ t t + λ { 1 p γ ( s ) N v ( s ) − μ v ( s ) } d s < 0 , (3.2)

and G ( p , t ) < 0 for all t ≥ T 1 , then the infectious population of system (2.1) is extinct.

Proof. From Lemma 2.3, we only have to consider the following two cases.

1) W ( p , t ) > 0 for all t ≥ T 2 .

2) W ( p , t ) ≤ 0 for all t ≥ T 2 .

First we consider the case 1). From the second equation of system (2.1), we have

d I ( t ) d t = β 1 ( t ) S ( t ) I ( t ) + β 2 ( t ) S ( t ) Y ( t ) − μ ( t ) I ( t ) < { β 1 ( t ) S ( t ) + p β 2 ( t ) S ( t ) − μ ( t ) } I ( t ) < { ( β 1 ( t ) + p β 1 ( t ) ) N ( s ) p − μ ( t ) } I ( t ) .

Hence, we obtain

I ( t ) < I ( T 2 ) exp ( ∫ T 2 t { ( β 1 ( s ) + p β 1 ( s ) ) N ( s ) p − μ ( s ) } d s ) (3.3)

for all t ≥ T 2 . From (3.1) we see that there exist constants δ 1 > 0 and T 3 > T 2 such that

∫ t t + λ { ( β 1 ( s ) + p β 1 ( s ) ) N ( s ) p − μ ( s ) } d s < − δ 1 , (3.4)

for all t ≥ T 3 . From (3.3) and (3.4), we have lim t → + ∞ I ( t ) = 0 . Then it follows from p I ( t ) > Y ( t ) for all t ≥ T 2 that lim t → + ∞ Y ( t ) = 0 .

Next we consider the case 2). From the fourth equation of system (2.1), we have

d Y ( t ) d t = γ ( t ) I ( t ) X ( t ) − μ v ( t ) I v ( t ) ≤ 1 p γ ( t ) Y ( t ) X ( t ) − μ v ( t ) Y ( t ) < ( 1 p γ ( t ) N v ( t ) − μ v ( t ) ) Y ( t ) (3.5)

Hence we have

Y ( t ) < Y ( T 2 ) exp ( ∫ T 2 t { 1 p γ ( s ) N v ( t ) − μ v ( s ) } d s ) (3.6)

From (3.2) we see that there exist constants δ 2 > 0 and T 4 > T 2 such that

∫ t t + λ { 1 p γ ( s ) N v ( t ) − μ v ( s ) } d s < − δ 2 , (3.7)

for all t ≥ T 4 . From (3.6) and (3.7), we have lim t → + ∞ Y ( t ) = 0 . Then it follows from I ( t ) ≤ 1 p Y ( t ) for all t ≥ T 2 that lim t → + ∞ I ( t ) = 0 . □

In this section, we get sufficient conditions for the permanence of infectious population of system (2.1). The definition of the permanence is as follows:

Definition 4.1. We say that the infectious population of system (2.1) is permanent if there exist positive constants I 1 ≥ 0 and I 2 ≥ 0 , which are independent from the choice of initial value satisfying (2.2), such that

0 < I 1 ≤ lim inf t → + ∞ I ( t ) ≤ lim sup t → + ∞ I ( t ) ≤ I 2 < + ∞ .

We give one of the main results of this paper.

Theorem 4.2. If there exist positive constants λ > 0 , p > 0 , q > 0 and T 1 > 0 such that

R 2 ( λ , p ) ≜ lim inf t → + ∞ ∫ t t + λ { ( β 1 ( s ) + p β 1 ( s ) ) N ( s ) p − μ ( s ) } d s > 0 , (4.1)

R ′ 2 ( λ , q ) ≜ lim inf t → + ∞ ∫ t t + λ { 1 p γ ( s ) N v ( s ) − μ v ( s ) } d s > 0 , (4.2)

and G ( p , t ) < 0 for all t ≥ T 1 , then the infectious population of system (2.1) is permanent.

Before we give the Proof of Theorem 4.2, we introduce the following lemma.

Lemma 4.3. If there exist positive constants λ > 0 , p > 0 and T 1 > 0 such that (4.1), (4.2) and G ( p , t ) < 0 hold for all t ≥ T 1 , then W ( p , t ) > 0 for all t ≥ T 2 ≥ T 1 , where T 2 is given as in lemma 2.3.

Proof. From Lemma 2.3 we have only two cases to discuss, W ( p , t ) > 0 for all t ≥ T 2 or W ( p , t ) ≤ 0 for all t ≥ T 2 . Suppose that W ( p , t ) ≤ 0 for all t ≥ T 2 . Then p I ( t ) ≤ Y ( t ) for all t ≥ T 2 . It follows from the last equation of system (2.1) that

d Y ( t ) d t > 1 p γ ( t ) X ( t ) Y ( t ) − μ ( t ) Y ( t ) = ( 1 p γ ( t ) X ( t ) − μ ( t ) ) Y ( t )

for all t ≥ T 2 . Hence, we obtain

Y ( t ) > Y ( T 2 ) exp ( ∫ T 2 t { 1 p γ ( s ) N v ( s ) − μ ( s ) } d s ) (4.3)

for all t ≥ T 2 . From the equality (4.2), we see that there exist constants η 1 > 0 and T > 0 such that

∫ t t + λ { 1 p γ ( s ) N v ( s ) − μ ( s ) } d s > η 1 (4.4)

for all t > T . For convenience, we choose T 2 satisfying T 2 ≥ T . Then the inequality (4.3) holds for t ≥ T 2 , it follows from (4.4) that lim t → + ∞ I ( t ) = + ∞ . This contradicts with the boundedness of Y ( t ) , stated in 2) of Proposition 2.2. Thus we have W ( p , t ) > 0 for all t ≥ T 2 . □

Using Lemma 4.4 we prove Theorem 4.2.

Proof (Proof of Theorem 4.2). For simplicity, let m 1 ε ≜ m 1 − ε , M 1 ε ≜ M 1 + ε , m 2 ε ≜ m 2 − ε , and M 2 ε ≜ M 2 + ε , where ε > 0 is a constant. From the inequality (2.7) and (2.8), we see that for any ε > 0 , there exists T > 0 such that

m 1 ε < N ( t ) < M 1 ε , (4.5)

m 2 ε < N v ( t ) < M 2 ε , (4.6)

for all t ≥ T . The inequality (4.1) and (4.2) implies that for sufficient small η > 0 , there exists T 1 ≥ T such that

∫ t t + λ { ( β 1 ( s ) + p β 1 ( s ) ) N ( s ) p − μ ( s ) } d s > η , (4.7)

∫ t t + λ { 1 p γ ( s ) N v ( s ) − μ v ( s ) } d s > η , (4.8)

for all t ≥ T 1 . We define

β 1 + ≜ sup t ≥ 0 β 1 ( t ) , β 2 + ≜ sup t ≥ 0 β 2 ( t ) , μ + ≜ sup t ≥ 0 μ ( t ) , μ v + ≜ sup t ≥ 0 μ v ( t ) , γ + ≜ sup t ≥ 0 γ ( t ) .

From (4.6) and (4.8), we see that for positive constants η 1 < η and T 2 ≥ T 1 there exist small ε i > 0 , ( i = 1 , 2 , 3 , 4 ) such that

∫ t t + λ { 1 p γ ( s ) ( N v ( s ) − ε 1 − γ + M 2 ε ω 4 ε 2 ) − μ ( s ) } d s > η 1 , (4.9)

N v ( t ) − ε 1 − γ + M 2 ε ω 4 ε 2 > m 2 ε , (4.10)

hold for all t ≥ T 2 . From 2) of Assumption 2.1, ε 1 , ε 2 can be chosen sufficiently small satisfying

∫ t t + ω 4 ( γ ( s ) M 2 ε ε 2 − μ v ( s ) ε 1 ) d s < − η 1 , (4.11)

hold for all t ≥ T 2 .

First we claim that sup t → + ∞ I ( t ) > ε 2 .

In fact, if it is not true, then there exists T 3 ≥ T 2 such that

I ( t ) ≤ ε 2 , (4.12)

for all t ≥ T 3 . Suppose that Y ( t ) ≥ ε 1 for all t ≥ T 3 . Then, from (4.5) and (4.12) we have

Y ( t ) = Y ( T 3 ) + ∫ T 3 t { γ ( s ) I ( s ) ( N v ( s ) − Y ( s ) − Y ( s ) ) − μ v ( s ) Y ( s ) } d s ≤ Y ( T 3 ) + ∫ T 3 t ( γ ( s ) M 2 ε ε 2 − μ v ( s ) ε 1 ) d s .

for all t ≥ T 3 . Thus, from (4.11), we have lim t → + ∞ Y ( t ) = − ∞ , which contradicts with 2) of Proposition 2.2. Therefore we see that there exists s 1 ≥ T 3 such that Y ( s 1 ) < ε 1 . Suppose that there exists an s 2 ≥ s 1 such that Y ( s 2 ) > ε 1 + γ + M 2 ε ω 4 ε 2 . Then, we see that there necessarily exists an s 3 ∈ ( s 1 , s 2 ) such that Y ( s 3 ) = ε 1 and Y ( t ) > ε 1 for all t ∈ ( s 3 , s 2 ] . Let n be an integer such that s 2 ∈ [ s 3 + n ω 4 , s 3 + ( n + 1 ) ω 4 ) . Then from (4.11), we have

ε 1 + γ + M 2 ε ω 4 ε 2 < Y ( s 2 ) = Y ( s 3 ) + ∫ s 3 s 2 { γ ( s ) I ( s ) ( N v ( s ) − Y ( s ) ) − μ v ( s ) Y ( s ) } d s

< ε 1 + { ∫ s 3 s 3 + n ω 4 + ∫ s 3 + n ω 4 s 2 } { γ ( s ) M 2 ε ε 2 − μ v ( s ) ε 1 } d s < ε 1 + ∫ s 3 + n ω 4 s 2 γ ( s ) M 2 ε ε 2 d s < ε 1 + γ + M 2 ε ω 4 ε 2

which is a contradiction. Therefore, we see that

Y ( t ) ≤ ε 1 + γ + M 2 ε ω 4 ε 2 , (4.13)

for all t ≥ s 1 . Now, from lemma 4.4, there exists T 4 ≥ s 1 ˜ such that W ( p , t ) > 0 for all t ≥ T 4 . Then

d Y ( t ) d t = γ ( t ) I ( t ) ( N v ( t ) − Y ( t ) ) − μ v ( t ) Y ( t ) ≥ Y ( t ) { 1 p γ ( t ) ( N v ( t ) − I v ( t ) ) − μ v ( t ) } ≥ Y ( t ) { 1 p γ ( t ) ( N v ( t ) − ε 1 − γ + M 2 ε ω 4 ε 2 ) − μ v ( t ) }

for all t ≥ T 4 . Hence, we have

Y ( t ) ≥ Y ( T 4 ) exp ( ∫ T 4 t { 1 p γ ( s ) ( N v ( s ) − ε 1 − γ + M 2 ε ω 4 ε 2 ) − μ v ( s ) } d s )

It follows from (4.9) that lim t → + ∞ Y ( t ) = + ∞ and this contradicts with the boundedness of I v ( t ) , stated in 2) of Proposition 2.2. Thus, we see that our claim sup t → + ∞ I ( t ) > ε 2 is true.

Next, we prove

lim inf t → + ∞ I ( t ) ≥ I 1 ,

where I 1 > 0 is a constant given in the following lines. For the following convenience, we let ω be the least common multiple of ω 4 and λ . If we define

lim inf t → + ∞ { 1 p γ ( t ) ( N v ( t ) − ε 1 − γ + M 2 ε ω 4 ε 2 ) − μ ( t ) } : = m _

Then we have two cases to discuss, namely 1) m _ > 0 and 2) m _ ≤ 0 . Firstly, we discuss the case 1). We set ε > 0 such that m _ − ε > 0 , then there exist T 3 ˜ ( ≥ T 2 ) such that

1 p γ ( t ) ( N v ( t ) − ε 1 − γ + M 2 ε ω 4 ε 2 ) − μ ( t ) > m _ − ε

for all t ≥ T 3 ˜ . Then, from inequalities (4.9), (4.11)-(4.12) and 2) of Assumption 2.1, we see that there exist constants T 4 ˜ ( ≥ T 3 ˜ ) , λ 2 > 0 , which is an integral multiple of ω , and η 2 > 0 such that

∫ t t + λ 3 { γ ( s ) M 2 ε ε 2 − μ v ( s ) ε 1 } d s < − M 2 ε , (4.14)

∫ t t + λ 3 { 1 p γ ( s ) ( N v ( s ) − ε 1 − γ + M 2 ε ω 4 ε 2 ) − μ ( s ) } d s > η 2 , (4.15)

∫ t t + λ 3 γ ( s ) d s > η 2 , (4.16)

for all t ≥ T 4 ˜ and λ 3 ≥ λ 2 . Let C > 0 be an integer multiple of λ 2 satisfying

e − μ + λ 2 η 2 m 2 ε ν 2 e C λ 2 η 2 > ε 1 + γ + M 2 ε ω 4 ε 2 , (4.17)

where ν 2 ≜ ε 2 e − 2 μ x + λ 2 . Since we have proved sup t → + ∞ I ( t ) > ε 2 . There are only two possibilities as follows:

1) I ( t ) ≥ ε 2 for all t ≥ ∃ T 5 ˜ ≥ T 4 ˜ .

2) I ( t ) oscillates about ε 2 for large t ≥ T 4 ˜ . In case 1), we have lim inf t → + ∞ I ( t ) ≥ ε 2 ≜ I 1 . In case 2), there necessarily exist two constants t 1 , t 2 ≥ T 4 ˜ ( t 2 ≥ t 1 ) such that

{ I ( t 1 ) = I ( t 2 ) = ε 2 , I ( t ) < ε 2 , for all t ∈ ( t 1 , t 2 ) .

Suppose that t 2 − t 1 ≤ C + 2 λ 2 . Then, from (1.1) we have

d I ( t ) d t ≥ − μ + I ( t ) , (4.18)

Hence, we obtain

I ( t ) ≥ I ( t 1 ) exp ( ∫ t 1 t − μ + d s ) ≥ ε 2 e − μ + ( C + 2 λ 2 ) : = I 1 , (4.19)

for all t ∈ ( t 1 , t 2 ) . Suppose that t 2 − t 1 > C + 2 λ 2 . Then, from (4.18), we have

I ( t ) ≥ ε 2 e − μ + ( C + 2 λ 2 ) = I 1

for all t ∈ ( t 1 , t 1 + C + 2 λ 2 ) . Now, we are in a position to show that I ( t ) ≥ I 1 for all t ∈ [ t 1 + C + 2 λ 2 , t 2 ) . Suppose that I ( t ) ≥ ε 1 for all t ∈ [ t 1 , t 1 + 2 λ 2 ] . Then, from (4.14), we have

Y ( t 1 + λ 2 ) ≤ Y ( t 1 ) + ∫ t 1 t 1 + λ 2 { γ ( s ) M 2 ε ε 2 − μ v ( s ) ε 1 } d s < M 2 ε − M 2 ε = 0

which is a contradiction. Therefore, there exists an s 4 ∈ [ t 1 , t 1 + 2 λ 2 ] such that Y ( s 4 ) < ε 1 . Then, as is in the proof of sup t → + ∞ I ( t ) > ε 2 , we can show that

Y ( t ) ≤ ε 1 + γ + M 2 ε ω 4 ε 2 , (4.20)

for all t ≥ s 4 . From (4.18), we have

I ( t ) ≥ ν 2 = ε 2 e − 2 μ + λ 2 , (4.21)

for all t ∈ [ t 1 , t 1 + 2 λ 2 ] . Thus, from (4.10), (4.20), (4.21), we have

d Y ( t ) d t = γ ( t ) I ( t ) X ( t ) − μ v ( t ) Y ( t ) ≥ γ ( t ) m 2 ε ν 2 − μ v + Y ( t )

for all t ∈ [ t 1 + λ 2 , t 1 + 2 λ 2 ] . Hence, from (4.16), we obtain

Y ( t 1 + 2 λ 2 ) ≥ e − μ + ( t 1 + 2 λ 2 ) { Y ( t 1 + λ 2 ) e μ + ( t 1 + λ 2 ) + ∫ t 1 + λ 2 t 1 + 2 λ 2 γ ( s ) m 2 ε ν 2 e μ + s d s } ≥ e − μ + ( t 1 + 2 λ 2 ) ∫ t 1 + λ 2 t 1 + 2 λ 2 γ ( s ) m 2 ε ν 2 e μ + s d s ≥ e − μ + λ 2 η 2 m 2 ε ν 2 . (4.22)

Now we suppose that there exists a t 0 > 0 such that t 0 ∈ ( t 1 + 2 λ 2 + C , t 2 ) , I ( t 0 ) = I 1 and I ( t ) ≥ I 1 for all t ∈ [ t 1 , t 0 ] . Then there exists m ∈ N such that t 0 ∈ [ t 1 + 2 λ 2 + C + m ω , t 1 + 2 λ 2 + C + ( m + 1 ) ω ) . Note that from Lemma 4.4. without loss of generality, we can assume that t 1 is so large that W ( p , t ) = p I ( t ) − Y ( t ) > 0 for all t ≥ t 1 + 2 λ 2 . Then, from (4.20), we have

d Y ( t ) d t = γ ( t ) ( N v ( t ) − Y ( t ) ) I ( t ) − μ v ( t ) Y ( t ) ≥ Y ( t ) { 1 p γ ( t ) ( N v ( t ) − Y ) ( t ) − μ v ( t ) } ≥ Y ( t ) { 1 p γ ( t ) ( N v ( t ) − ε 1 − γ + M 2 ε ω 4 ε 2 ) − μ v ( t ) }

for all t ∈ ( t 1 + 2 λ 2 , t 2 ) . Thus, from (4.15) and (4.22), we have

Y ( t 0 ) ≥ Y ( t 1 + 2 λ 2 ) exp ( ∫ t 1 + 2 λ 2 t 0 { 1 p γ ( t ) ( N v ( s ) − ε 1 − γ + M 2 ε ω 4 ε 2 ) − μ v ( s ) } d s ) ≥ e − μ + λ 2 η 2 m 2 ε ν 2 exp ( { ∫ t 1 + 2 λ 2 t 1 + 2 λ 2 + C + ∫ t 1 + 2 λ 2 t 1 + 2 λ 2 + C + m ω + ∫ t 1 + 2 λ 2 + C + m ω t 0 } { 1 p γ ( t ) ( N v ( s ) − ε 1 − γ + M 2 ε ω 4 ε 2 ) − μ v ( s ) } d s ) ≥ e − μ + λ 2 η 2 m 2 ε ν 2 e C λ 2 η 2

Thus, from (4.20), we have

ε 1 + γ + M 2 ε ω 4 ε 2 ≥ e − μ + λ 2 η 2 m 2 ε ν 2 e C λ 2 η 2 ,

which contradicts with (4.17). Finally, if m _ ≤ 0 , we let C > 0 be the integral multiple of λ 2 satisfying

e − μ 1 + λ 2 η 2 m 2 ε ν 2 e C λ 2 η 2 + ( m _ − ε ) ω > ε 1 + γ + M 2 ε ω 4 ε 2 , (4.23)

Then, repeating the above steps, we have

Y ( t 0 ) ≥ e − μ + λ 2 η 2 m 2 ε ν 2 e C λ 2 η 2 + ( m _ − ε ) ω

Thus, from (4.20), we have

ε 1 + γ + M 2 ε ω 4 ε 2 ≥ e − μ + λ 2 η 2 m 2 ε ν 2 e C λ 2 η 2 + ( m _ − ε ) ω

which is contradictive with (4.23). Therefore, I ( t ) ≥ I 1 for all t ∈ [ t 1 + 2 λ 2 + C , t 2 ) , which implies lim inf t → + ∞ I ( t ) ≥ I 1 .

Since lim sup t → + ∞ I ( t ) ≤ lim sup t → + ∞ N ( t ) ≤ M 1 < + ∞ , the infectious population of system (1.1) is permanent.

In the paper, we have extended the epidemic models of vector-borne disease with direct mode of transmission presented in [

The work was supported by the Science Fund of Education Department of Jiangxi Province (171373, 171374, 181361).

The authors declare no conflicts of interest regarding the publication of this paper.

Ji, W.W., Zou, S.H., Liu, J.S., Sun, Q.B. and Xia, L.J. (2020) Dynamic of Non-Autonomous Vector Infectious Disease Model with Cross Infection. American Journal of Computational Mathematics, 10, 591-602. https://doi.org/10.4236/ajcm.2020.104034