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A nonautonomous schistosomiasis model with latent period and saturated incidence is investigated. Further, we study the long-time behavior of the epidemic model. The weaker sufficient conditions for the permanence and extinction of infectious population of the model are obtained by constructing some auxiliary functions. Numerical simulations show agreement with the theoretical results.

Schistosomiasis (also known as bilharzia) is a disease caused by parasitic worms of the Schistosoma type [

Mathematical modeling has become an important tool in analyzing the spread and control of infectious diseases. In recent years, many schostosomiasis models have been proposed and studied ( [

Many diseases incubate inside the hosts for a period of time before the hosts become infectious. Using a compartmental approach, one may assume that a susceptible individual first goes through an incubation period (and is said to become exposed or in the class E) after infection, before becoming infectious. The resulting models are of SEIR or SEIRS types, respectively, depending on whether the acquired immunity is permanent or otherwise.

In the aforementioned framework, their coefficients are considered as constants, which are approximated by average values. However, we note that ecosystems in the real world often appear the nonautonomous phenomenon. Recently many nonautonomous epidemic systems have been studied ( [

In this paper, we assume large intermediate host population and thus ignore snail dynamics. Motivated by the above description, we develop a class of nonautonomous schistosomiasis transmission model with incubation period:

{ d S ( t ) d t = Λ ( t ) − β ( t ) S ( t ) I ( t ) 1 + α S ( t ) − μ ( t ) S ( t ) + δ ( t ) I ( t ) + γ ( t ) E ( t ) , d E ( t ) d t = β ( t ) S ( t ) I ( t ) 1 + α S ( t ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) E ( t ) , d I ( t ) d t = ε ( t ) E ( t ) − ( μ ( t ) + δ ( t ) + d ( t ) ) I ( t ) . (1.1)

with initial value

S ( 0 ) > 0 , E ( 0 ) > 0 , I ( 0 ) > 0. (1.2)

Here S ( t ) , E ( t ) and I ( t ) denote the size of susceptible, exposed, infectious population at time t, respectively. Λ ( t ) is the growth rate of population, μ ( t ) is the natural death rate of the population, β ( t ) is the rate of the efficient contact, δ ( t ) and γ ( t ) are the recovery rates of infectious population and exposed population, respectively, d ( t ) is the disease-related death rate and ε ( t ) is the rate of developing infectivity at time t.

The organization of this paper is as follows. In the next section, we present preliminaries setting and propositions, which we use to analyze the long-time behavior of system (1.1) in the following sections. In Section 3, we establish the extinction of the disease of system (1.1). In Section 4, we will discuss the permanence of the infectious population. Our results are verified by numerical simulations in Section 5.

In this section, system (1.1) satisfies the following assumptions:

(H_{1}): The functions Λ ( t ) , δ ( t ) , μ ( t ) , δ ( t ) , γ ( t ) , ε ( t ) , d ( t ) are nonnegative, bounded and continuous on [ 0, + ∞ ) and β ( 0 ) > 0 .

(H_{2}): There exist positive constants ω i > 0 ( i = 1 , 2 , 3 ) such that

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

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

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

Adding all the equations of model (1.1), then we have

Λ ( t ) − μ ( t ) N ( t ) ≥ d N ( t ) d t = Λ ( t ) − μ ( t ) N ( t ) − d ( t ) I ( t ) ≥ Λ ( t ) − ( μ ( t ) + d ( t ) ) N (t)

Let N ( t ) = S ( t ) + E ( t ) + I ( t ) be the total population in system (1.1) with the initial value N ( 0 ) = S ( 0 ) + E ( 0 ) + I ( 0 ) . We denote by N * ( t ) the solution of

d N * ( t ) d t = Λ ( t ) − μ ( t ) N * ( t ) (2.1)

with initial value (1.2), and denote by N ∗ ( t ) the solution of

d N * ( t ) d t = Λ ( t ) − ( μ ( t ) + d ( t ) ) N * ( t ) (2.2)

with initial value (1.2). Then

N * ( t ) ≤ S ( t ) + E ( t ) + I ( t ) ≤ N * ( t ) .

By [

Lemma 2.1. Suppose that assumptions (H_{1}) and (H_{2}) hold. Then:

(i) there exist positive constants m > 0 and M > 0 , such that

0 < m ≤ lim inf t → + ∞ N * ( t ) ≤ lim sup t → + ∞ N * ( t ) ≤ lim inf t → + ∞ N * ( t ) ≤ lim sup t → + ∞ N * ( t ) ≤ M < + ∞ . (2.3)

(ii) the solution ( S ( t ) , E ( t ) , I ( t ) ) of system (1.1) with the initial value (1.2) exists, is uniformly bounded and S ( t ) > 0 , E ( t ) > 0 , I ( t ) > 0 for all t > 0 . For the solution ( S ( t ) , E ( t ) , I ( t ) ) of system (1.1), we define

G ( p , t ) : = p β ( t ) S ( t ) 1 + α S ( t ) + δ ( t ) + d ( t ) − γ ( t ) − ( 1 + 1 p ) ε (t)

and

W ( p , t ) : = p E ( t ) − I ( t ) . (2.4)

for p > 0 , t > 0 . In Sections 3 and 4 we use the following lemma in order to investigate the longtime behavior of system (1.1).

Lemma 2.2. If there exist positive constants 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 W ( p , t ) > 0 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 or W ( p , t ) ≤ 0 for all t ≥ T 2 . So we have

p E ( s ) = I ( s ) (2.5)

and

d W ( p , s ) d t = p { β ( s ) S ( s ) I ( s ) 1 + α S ( s ) − ( μ ( s ) + ε ( s ) + γ ( s ) ) E ( s ) } − { ε ( s ) E ( s ) − ( μ ( s ) + δ ( s ) + d ( s ) ) I ( s ) } = I ( s ) { p β ( s ) S ( s ) 1 + α S ( s ) + ( μ ( s ) + δ ( s ) + d ( s ) ) } − p E ( s ) { ( μ ( s ) + ε ( s ) + γ ( s ) ) + ε ( s ) p } > 0. (2.6)

Substituting (2.5) into (2.6), we have

0 < p E ( s ) { p β ( s ) S ( s ) 1 + α S ( s ) + δ ( s ) + d ( s ) − γ ( s ) − ( 1 + 1 p ) ε ( s ) } = p E ( s ) G ( p , s ) .

From Lemma 2.1, we have p > 0 and E ( s ) > 0 , so G ( p , s ) > 0 , which is a contradiction with G ( p , t ) < 0 for all t ≥ T 1 . The proof is completed.

In this section, we obtain conditions for focus on the extinction of the infectious population of system (1.1).

Theorem 3.1 Suppose that assumptions (H_{1}) and (H_{2}) hold. If there exist λ > 0 , p > 0 and T 1 > 0 such that

R 1 ( λ , p ) : = lim sup t → + ∞ ∫ t t + λ { p β ( s ) 1 + α N * ( s ) N * ( s ) − ( μ ( s ) + ε ( s ) + γ ( s ) ) } d s < 0 , (3.1)

R 1 * ( λ , p ) : = lim sup t → + ∞ ∫ t t + λ { ε ( s ) 1 p − ( μ ( s ) + δ ( s ) + d ( s ) ) } d s < 0 (3.2)

and G ( p , t ) < 0 for all t ≥ T 1 , then infectious population I ( t ) in system (1.1) is extinct. i.e.

lim t → + ∞ I ( t ) = 0.

Proof: From Lemma 2.2, we consider the following two cases:

(i) p E ( t ) > I ( t ) for all t ≥ T 2 ;

(ii) p E ( t ) ≤ I ( t ) for all t ≥ T 2 .

First, we consider the cases (i). From the second equation of system (1.1), we have

d E ( t ) d t = β ( t ) I ( t ) 1 + α ( N ( t ) − E ( t ) − I ( t ) ) ( N ( t ) − E ( t ) − I ( t ) ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) E ( t ) < p β ( t ) E ( t ) 1 + α ( N * ( t ) − E ( t ) − I ( t ) ) ( N * ( t ) − E ( t ) − I ( t ) ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) E ( t ) < p β ( t ) E ( t ) 1 + α N * ( t ) N * ( t ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) E ( t ) = E ( t ) { p β ( t ) 1 + α N * ( t ) N * ( t ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) } .

So we have

E ( t ) < E ( T 2 ) exp ( ∫ T 2 t { p β ( s ) 1 + α N * ( s ) N * ( s ) − ( μ ( s ) + ε ( s ) + γ ( 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 { p β ( s ) 1 + α N * ( s ) N * ( s ) − ( μ ( s ) + ε ( s ) + γ ( s ) ) } d s < − δ 1 (3.4)

for all t > T 3 . From (3.3) and (3.4), we obtain lim t → + ∞ E ( t ) = 0 . Therefore, it follows from p E ( t ) > I ( t ) , that lim t → + ∞ I ( t ) = 0 . Now we consider the case (ii). From E ( t ) ≤ I ( t ) p for all t ≥ T 2 and the third equation of (1.1), we have

d I ( t ) d t ≤ I ( t ) { ε ( t ) 1 p − ( μ ( t ) + δ ( t ) + d ( t ) ) } .

Then the following expression

I ( t ) ≤ I ( T 2 ) exp ( ∫ T 2 t { ε ( s ) p − μ ( s ) − δ ( s ) − d ( s ) } d s ) (3.5)

for all t > T 2 hold. Hence, by (3.2), there exist δ 2 > 0 and T 4 > T 2 such that

∫ t + λ t { ε ( s ) 1 p − ( μ ( s ) + δ ( s ) + d ( s ) ) } d s < − δ 2 , (3.6)

for t ≥ T 4 . From (3.5) and (3.6), we have

lim t → + ∞ I ( t ) = 0.

In this section, we obtain the sufficient conditions for the permanence of infectious population.

Theorem 4.1. Suppose that assumptions (H_{1}) and (H_{2}) hold. If there are λ > 0 , p > 0 and T 1 > 0 such that

R 2 ( λ , p ) : = lim inf t → + ∞ ∫ t t + λ { p β ( s ) 1 + α N * ( s ) N * ( s ) − ( μ ( s ) + ε ( s ) + γ ( s ) ) } d s > 0 , (4.1)

R 2 * ( λ , p ) : = lim inf t → + ∞ ∫ t t + λ { ε ( s ) 1 p − ( μ ( s ) + δ ( s ) + d ( s ) ) } d s > 0 (4.2)

and G ( p , t ) < 0 for all t ≥ T 1 , then I ( t ) in system (1.1) is permanent.

Before we give the proof of Theorem 4.1, we first prove the following lemma.

Lemma 4.1. If there exist 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 there exists T 2 > T 1 so that W ( p , t ) ≤ 0 for all t ≥ T 2 .

Proof: From Lemma 2.2, we consider the following two cases:

(i) W ( p , t ) > 0 for all t ≥ T 2 ;

(ii) W ( p , t ) ≤ 0 for all t ≥ T 2 .

Suppose W ( p , t ) > 0 for all t ≥ T 2 , then we have E ( t ) > I ( t ) p for all t ≥ T 2 . From the third equation of system (1.1), we have

d I ( t ) d t > ε ( t ) 1 p I ( t ) − ( μ ( t ) + δ ( t ) + d ( t ) ) I ( t ) = I ( t ) { ε ( t ) 1 p − ( μ ( t ) + δ ( t ) + d ( t ) ) } .

So we obtain

I ( t ) > I ( T 2 ) exp ( ∫ T 2 t { ε ( s ) p − μ ( s ) − δ ( s ) − d ( s ) } d s ) (4.3)

for all t ≥ T 2 . From the inequality (4.2), there exist positive constants η > 0 and T > 0 such that

∫ t + λ t { ε ( s ) 1 p − ( μ ( s ) + δ ( s ) + d ( s ) ) } d s > η (4.4)

for all t ≥ T . So the inequality (4.3) holds for all t ≥ max { T 2 , T } . Then lim t → + ∞ I ( t ) = + ∞ , which contradicts with the boundedness of I ( t ) in Lemma 2.1. Now, we prove Theorem 4.1 by using Lemma 4.1.

Proof: For simplicity, let m ϵ : = m − ϵ , M ϵ : = M + ϵ , where ϵ > 0 is a constant. In fact, Lemma 2.1 implies that for any sufficiently small ϵ > 0 , there exists T > 0 such that

m ϵ < N * ( t ) ≤ N * ( t ) < M ϵ (4.5)

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

∫ t t + λ { p β ( s ) N * ( s ) 1 + α N * ( s ) − ( μ ( s ) + ε ( s ) + γ ( s ) ) } d s > η (4.6)

for all t ≥ T 1 . We define

β + : = sup t ≥ 0 β ( t ) , μ + : = sup t ≥ 0 μ ( t ) , δ + : = sup t ≥ 0 δ ( t ) ,

d + : = sup t ≥ 0 d ( t ) , ε + : = sup t ≥ 0 ε ( t ) , γ + : = sup t ≥ 0 γ ( t ) .

Thus, by (4.5) and (4.6), for any sufficiently small η 1 < η and T 2 > T 1 , there exist very small ϵ i , i ∈ { 1,2,3 } such that

∫ t t + λ { β ( s ) 1 + α N * ( s ) ( N * ( s ) − ϵ 1 − k ϵ 2 ) p − ( μ ( s ) + ε ( s ) + γ ( s ) ) } d s > η 1 , (4.7)

N * ( t ) − ϵ 1 − k ϵ 2 > m ϵ (4.8)

for all t ≥ T 2 , where k : = 1 + β + 1 + α M ϵ M ϵ ω 2 . Lemma 2.1 implies that for any sufficiently small ϵ 2 > 0 , we have

∫ t 1 t 1 + ω 2 { β ( s ) M ϵ ϵ 2 1 + α M ϵ − ( μ ( s ) + ε ( s ) + γ ( s ) ) ϵ 1 } d s < − η 1 (4.9)

for all t ≥ T 2 . First, we prove

lim sup t → + ∞ I ( t ) > ϵ 2 .

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

I ( t ) ≤ ϵ 2 (4.10)

for all t ≥ T 3 . If E ( t ) ≥ ϵ 1 for all t ≥ T 3 , then from (4.5) and (4.6), we have

E ( t ) = E ( T 3 ) + ∫ T 3 t { β ( s ) I ( s ) 1 + α S ( s ) ( N ( s ) − E ( s ) − I ( s ) ) − ( μ ( s ) + ε ( s ) + γ ( s ) ) E ( s ) } d s ≤ E ( T 3 ) + ∫ T 3 t { β ( s ) 1 + α M ϵ M ϵ ϵ 2 − ( μ ( s ) + ε ( s ) + γ ( s ) ) ϵ 1 } d s

for all t ≥ T 3 . It follows from inequality (4.9) that lim t → + ∞ E ( t ) = − ∞ . This contradicts with the boundedness of solution. Hence, there exists an s 1 ≥ T 3 such that E ( s 1 ) < ϵ 1 . In the following we prove

E ( t ) ≤ ϵ 1 + β + 1 + α M ϵ M ϵ ω 2 ϵ 2 , (4.11)

for all t ≥ s 1 . If it is not true, there exists an s 2 > s 1 such that

E ( s 2 ) > ϵ 1 + β + 1 + α M ϵ M ϵ ω 2 ϵ 2 .

Hence, there necessarily exists an s 3 ∈ ( s 1 , s 2 ) such that E ( s 3 ) = ϵ 1 and E ( t ) > ϵ 1 for all t ∈ ( s 3 , s 2 ) . Let n ≥ 0 be an integer such that s 2 ∈ [ s 3 + n ω 2 , s 3 + ( n + 1 ) ω 2 ] . By (4.9), we obtain

E ( s 2 ) = E ( s 3 ) + ∫ s 2 s 3 { β ( s ) I ( s ) 1 + α S ( s ) ( N * ( s ) − E ( s ) − I ( s ) ) − ( μ ( s ) + ε ( s ) + γ ( s ) ) E ( s ) } d s < ϵ 1 + { ∫ s 3 s 3 + n ω 2 + ∫ s 3 + n ω 2 s 2 } { β ( s ) M ϵ ϵ 2 1 + α M ϵ − ( μ ( s ) + ε ( s ) + γ ( s ) ) ϵ 1 } d s < ϵ 1 + ∫ s 3 + n ω 2 s 2 β ( s ) 1 + α M ϵ M ϵ ϵ 2 d s < ϵ 1 + β + 1 + α M ϵ M ϵ ω 2 ϵ 2 ,

This contradicts with E ( s 2 ) > ϵ 1 + β + 1 + α M ϵ M ϵ ω 2 ϵ 2 . Hence, (4.11) is valid. By Lemma 4.1, there exists T 4 ≥ s 1 such that W ( p , t ) = p E ( t ) − I ( t ) ≤ 0 for all t ≥ T 4 . Therefore, by (4.10) and (4.11), we have E ( t ) + I ( t ) ≤ ϵ 1 + k ϵ 2 for all t ≥ T 4 , then

d E ( t ) d t = β ( t ) I ( t ) ( N ( t ) − E ( t ) − I ( t ) ) 1 + α ( N ( t ) − E ( t ) − I ( t ) ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) E ( t ) ≥ β ( t ) I ( t ) ( N * ( t ) − E ( t ) − I ( t ) ) 1 + α ( N * ( t ) − E ( t ) − I ( t ) ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) E ( t ) ≥ β ( t ) E ( t ) ( N * ( t ) − E ( t ) − I ( t ) ) p 1 + α N * ( t ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) E ( t ) ≥ E ( t ) { β ( t ) ( N * ( t ) − E ( t ) − I ( t ) ) p 1 + α N * ( t ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) } ≥ E ( t ) { β ( t ) ( N * ( t ) − ϵ 1 − k ϵ 2 ) p 1 + α N * ( t ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) } .

We obtain

E ( t ) ≥ E ( T 4 ) exp ( ∫ T 4 t { β ( s ) ( N * ( s ) − ϵ 1 − k ϵ 2 ) p 1 + α N * ( s ) − ( μ ( s ) + ε ( s ) + γ ( s ) ) } d s ) .

By (4.7) we obtain lim t → + ∞ E ( t ) = + ∞ . This contradicts with Lemma 2.1 ( E ( t ) is uniformly bounded). Hence, lim 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. By inequality (4.7), (4.8), (4.9) and Lemma 2.1, there exist T ˜ 3 ( ≥ T 2 ) , λ 2 > 0 , η 2 > 0 such that λ 3 ≥ λ 2 and t ≥ T ˜ 3 , we obtain

∫ t t + λ 3 { β ( s ) 1 + α N * ( s ) ( N * ( s ) − ϵ 1 − k ϵ 2 ) p − ( μ ( s ) + ε ( s ) + γ ( s ) ) } d s > η 2 , (4.12)

∫ t 1 t 1 + λ 3 { β ( s ) M ϵ ϵ 2 1 + α M ϵ − ( μ ( s ) + ε ( s ) + γ ( s ) ) ϵ 1 } d s < − M ϵ , (4.13)

∫ t 1 t 1 + λ 3 β ( s ) d s > η 2 . (4.14)

Let C > 0 be a constant satisfying

e − ( μ + + ε + + γ + ) λ 2 m ϵ ν 2 η 2 e n * η 2 1 + α M ϵ > ϵ 1 + β + M ϵ ω 2 ϵ 2 1 + α M ϵ (4.15)

where ν 2 = ϵ 2 e − ( μ + + δ + + d + ) 2 λ 2 , n * ≤ C λ 2 .

Because we have proved lim sup t → + ∞ I ( t ) > ϵ 2 , there are only two possibilities as follows:

(i) There exists T ˜ 4 ≥ T ˜ 3 , then as t ≥ T ˜ 4 , we obtain I ( t ) ≥ ϵ 2 ;

(ii) I ( t ) oscillates about ϵ 2 for all large t.

In case (i), we have lim inf t → + ∞ I ( t ) ≥ ϵ 2 = : I 1 . In case (ii), there necessarily exist t 1 , t 2 ≥ T ˜ 3 ( t 2 ≥ t 1 ) such that

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

Suppose that t 2 − t 1 ≤ C + 2 λ 2 . Then

d I ( t ) d t ≥ − ( μ + + δ + + d + ) I ( t ) , (4.16)

which implies

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

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

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

for all t ∈ ( t 1 , t 1 + C + 2 λ 2 ) . Now we only prove I ( t ) ≥ I 1 for all t ∈ [ t 1 + C + 2 λ 2 , t 2 ) . If E ( t ) ≥ ϵ 1 for all t ∈ [ t 1 , t 1 + λ 2 ] . By the second equation of system (1.1) and inequality (4.13), we have

E ( t 1 + λ 2 ) ≤ E ( t 1 ) + ∫ t 1 t 1 + λ 2 { β ( s ) M ϵ 1 + α M ϵ ϵ 2 + ( μ ( s ) + ε ( s ) + γ ( s ) ) ϵ 1 } d s < M ϵ − M ϵ = 0 ,

which is contradiction. Hence, there exists an s 4 ∈ [ t 1 , t 1 + λ 2 ] such that E ( s 4 ) < ϵ 1 . We obtain that for t ≥ s 4 ,

E ( t ) ≤ ϵ 1 + β + 1 + α M ϵ M ϵ ω 2 ϵ 2 . (4.17)

By inequality (4.16), then for t ∈ [ t 1 , t 1 + 2 λ 2 ]

I ( t ) ≥ ν 2 = ϵ 2 e − ( μ + + δ + + d + ) 2 λ 2 . (4.18)

Therefore, by the second equation of system (1.1) and inequalities (4.8), (4.17), (4.18), we obtain that

d E ( t ) d t = β ( t ) I ( t ) ( N ( t ) − E ( t ) − I ( t ) ) 1 + α ( N ( t ) − E ( t ) − I ( t ) ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) E ( t ) ≥ β ( t ) I ( t ) 1 + α M ϵ ( N * ( t ) − E ( t ) − I ( t ) ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) E ( t ) ≥ β ( t ) 1 + α M ϵ m ϵ − ( μ + + ε + + γ + ) E ( t ) .

for all t ∈ [ t 1 + λ 2 , t 1 + 2 λ 2 ] . By (4.14), we have

E ( t ) ≥ e − ( μ + + ε + + γ + ) ( t 1 + 2 λ 2 ) { E ( t 1 + λ 2 ) e ( μ + + ε + + γ + ) ( t 1 + λ 2 ) + ∫ t 1 + λ 2 t 1 + 2 λ 2 β ( s ) 1 + α M ϵ m ϵ ν 2 e ( μ + + ε + + γ + ) s d s } ≥ e − ( μ + + ε + + γ + ) ( t 1 + 2 λ 2 ) ∫ t 1 + λ 2 t 1 + 2 λ 2 β ( s ) 1 + α M ϵ m ϵ ν 2 e ( μ + + ε + + γ + ) s d s ≥ e − ( μ + + ε + + γ + ) λ 2 1 1 + α M ϵ m ϵ ν 2 η 2 . (4.19)

Now, we suppose there exists t 0 > 0 such that t 0 ∈ ( t 1 + C + 2 λ 2 , t 2 ) , then I ( t 0 ) = I 1 and I ( t ) ≥ I 1 for all t ∈ [ t 1 , t 0 ] . By Lemma 4.1, we assume that t 1 is so large that W ( p , t ) = p E ( t ) − I ( t ) ≤ 0 for all t ≥ t 1 + 2 λ 2 . Hence, by (4.8), we further have

d E ( t ) d t = β ( t ) I ( t ) ( N ( t ) − E ( t ) − I ( t ) ) 1 + α ( N ( t ) − E ( t ) − I ( t ) ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) E ( t ) ≥ E ( t ) { β ( t ) ( N * ( t ) − ϵ 1 − k ϵ 2 ) p 1 + α N * ( t ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) }

for all t ∈ ( t 1 + 2 λ 2 , t 2 ) . By (4.12) and (4.19), we have

E ( t ) = E ( t 1 + 2 λ 2 ) exp ( ∫ t 1 + 2 λ 2 t 0 { β ( t ) ( N * ( t ) − ϵ 1 − k ϵ 2 ) p 1 + α N * ( t ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) } d s ) = E ( t 1 + 2 λ 2 ) exp ( { ∫ t 1 + 2 λ 2 t 1 + 2 λ 2 + λ 2 + ∫ t 1 + 2 λ 2 + λ 2 t 1 + 2 λ 2 + 2 λ 2 + ⋯ + ∫ t 1 + 2 λ 2 + ( n * − 1 ) λ 2 t 0 } { β ( t ) ( N * ( t ) − ϵ 1 − k ϵ 2 ) p 1 + α N * ( t ) − ( μ ( t ) + ε ( t ) + γ ( t ) ) } d s ) ≥ e − ( μ + + ε + + γ + ) λ 2 1 1 + α M ϵ m ϵ ν 2 η 2 e n * η 2 .

Thus, by (4.17), we have

ε 1 + β + M ϵ ω 2 ϵ 2 1 + α M ϵ ≥ e − ( μ + + ε + + γ + ) λ 2 m ϵ ν 2 η 2 e n * η 2 1 + α M ϵ ,

This contradicts with (4.15). Hence, I ( t ) ≥ I 1 for all t ∈ [ t 1 + C + 2 λ 2 , t 2 ] , which implies lim inf t → + ∞ I ( t ) ≥ I 1 . Thus, the infectious population of system (1.1) is permanent.

Numerical verification of the results is necessary for completeness of the analytical study. In Sections 3 and 4, we focused our attention on the dynamic analysis of system (1.1). In the present section, numerical simulations are carried out to illustrate the analytical results of system (1.1) by means of the software Matlab.

In order to testify the validity of our results, in system (1.1), fix Λ = 1 , β ( t ) = 0.3 + 0.1 cos t , α = 0.3 , μ = 0.1 , δ = 0.1 , γ = 0.8 , ϵ = 0.2 , d = 0.5 . Then, from system (1.1), we have lim t → + ∞ N * ( t ) = 1 . We easily verify that assumptions (H1) and (H2) hold.We choose λ = 1 and p = 2 . Then we have

R 1 ( λ , p ) : = ∫ 0 1 { 2 0.3 + 0.1 cos t 1 + 0.3 − ( 0.1 + 0.2 + 0.8 ) } d t ≈ − 0.84 < 0 , R 1 * ( λ , p ) : = ∫ 0 1 { 0.2 1 2 − ( 0.1 + 0.1 + 0.5 ) } d t = − 0.7 < 0.

and

G ( p , t ) : = ∫ 0 1 { 2 0.3 + 0.1 cos t 1 + 0.3 + 0.1 + 0.5 − 0.8 − ( 1 + 1 2 ) 0.2 } d t ≈ − 0.1385 < 0.

for all t > 0 . From Theorem 3.1, we see that the infectious population of system (1.1) is extinct, see

Fix Λ = 1 , β ( t ) = 0.6 + 0.1 cos t , α = 0.6 , μ = 0.1 , δ = 0.02 , γ = 0.1 , ϵ = 0.5 , d = 0.02 . We choose λ = 1 and p = 2 . Then we have

R 2 ( λ , p ) : = ∫ 0 1 { 2 0.6 + 0.1 cos t 1 + 0.6 − ( 0.1 + 0.5 + 0.1 ) } d t ≈ 0.05 > 0 , R 2 * ( λ , p ) : = ∫ 0 1 { 0.5 1 2 − ( 0.1 + 0.02 + 0.02 ) } d t = 0.11 > 0.

and

G ( p , t ) : = ∫ 0 1 { 2 0.6 + 0.1 cos t 1 + 0.6 + 0.02 + 0.02 − 0.1 − ( 1 + 1 2 ) 0.5 } d t ≈ − 0.06 < 0.

for all t > T 1 . From Theorem 4.1, we see that the infectious population of system (1.1) is permanent, see

In this paper we obtain new sufficient conditions for the permanence and extinction of system (1.1). We prove that our conditions give the threshold-type result by the basic reproduction number given as in (3.1) when every parameter is given as a constant parameter. Thus our result is an extension result of the threshold-type result in the autonomous system. Our results may contribute to predicting the disease dynamics, such as permanence and extinction of the infectious population, when the phenomena are modeled as a nonautonomous system.

In Section 5, we provide numerical examples to illustrate the validity of our results. In those examples we show that conditions in Theorems 4.1 for the permanence and extinction of infectious population of system (1.1) are not satisfied. One may argue that our conditions for the permanence and extinction may not sharp.

It is still an open problem that if the basic reproduction number for (1.1) works as a threshold parameter to determine the permanence and extinction of infectious population like in the autonomous system.

The research has been partially supported by the Natural Science Foundation of China (No. 11561004).

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

Liu, Y., He, Y.Y., Yan, S.X. and Gao, S.J. (2019) A Class of Nonautonomous Schistosomiasis Transmission Model with Incubation Period. Applied Mathematics, 10, 159-172. https://doi.org/10.4236/am.2019.103013