A Class of Nonautonomous Schistosomiasis Transmission Model with Incubation Period

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ISSN Online: 2152-7393 ISSN Print: 2152-7385

DOI: 10.4236/am.2019.103013 Mar. 29, 2019 159 Applied Mathematics

A Class of Nonautonomous Schistosomiasis

Transmission Model with Incubation Period

Yang Liu

1

, Yuying He

2

, Shuixian Yan

1*

, Shujing Gao

1

1_{College of Mathematics and Computer Science, Gannan Normal University, Ganzhou, China} 2_{Yuhua Elite School, Kaifeng, China}

Abstract

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

Keywords

Schistosomiasis, Nonautonomous Model, Permanence, Extinction

1. Introduction

Schistosomiasis (also known as bilharzia) is a disease caused by parasitic worms of the Schistosoma type [1]. Schistosomiasis affects almost 210 million people worldwide [2], and an estimated 12,000 to 200,000 people die from it each year

[3][4]. The disease is most commonly found in Africa, Asia and South America

[5]. Around 700 million people, in more than 70 countries, live in areas where the disease is common [4] [6]. Schistosomiasis is second only to malaria, as a parasitic disease with the greatest economic impact [7].

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 ([8]-[13], etc.). These models provide a detailed exposition on how to describe, analyze, and predict epidemics of schistosomiasis for the ultimate purposes of developing control strategies and tactics for schis-tosomiasis transmission.

Many diseases incubate inside the hosts for a period of time before the hosts

How to cite this paper: Liu, Y., He, Y.Y., Yan, S.X. and Gao, S.J. (2019) A Class of Nonautonomous Schistosomiasis Transmis-sion Model with Incubation Period. Applied Mathematics, 10, 159-172.

https://doi.org/10.4236/am.2019.103013

Received: December 21, 2018 Accepted: March 26, 2019 Published: March 29, 2019

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DOI: 10.4236/am.2019.103013 160 Applied Mathematics become infectious. Using a compartmental approach, one may assume that a susceptible individual first goes through an incubation period (and is said to be-come 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 con-stants, which are approximated by average values. However, we note that eco-systems in the real world often appear the nonautonomous phenomenon. Re-cently many nonautonomous epidemic systems have been studied ([14]-[20], etc.). In fact, natural factors, such as seasonal changes in moisture and tempera-ture, affect the abundance and activity of the intermediate snail host, Oncomela-nia hupensis, and the transmission dynamics of schistosomiasis are in a constant state of flux [21]. Moreover, there are many social factors related to human be-haviors accounting for the change of schistosomiasis incidence, such as marked changes of contact rates caused by daily production activities [22]. This illu-strates that the transmission of schistosomiasis shows seasonal behavior. In or-der to describe this kind of phenomenon, in the model, the parameters of the system should be functions of time. As far as we know, the research work on the nonautonomous schistosomiasis models is very few. Therefore, it is necessary to study nonautonomous schistosomiasis models.

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

( )

_{( )}

( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( )

( )

(

( ) ( ) ( )

)

( )

_{( ) ( )}

₍

_{( ) ( ) ( )}

₎

_{( )}

d

,

d 1

d

,

d 1

d

. d

S t t S t I t

t t S t t I t t E t

t S t

E t t S t I t

t t t E t

t S t

I t

t E t t t d t I t t

β

µ δ γ

α β

µ ε γ

α

ε µ δ



= Λ − − + +

 ₊



 ₌ ₋ ₊ ₊

 ₊

 

 ₌ ₋ ₊ ₊



(1.1)

with initial value

( )

0 0,

( )

0 0, 0

( )

0. S > E > I >

(1.2) Here S t

( )

, E t

( )

and I t

( )

denote the size of susceptible, exposed, infec-tious population at time t, respectively. Λ

( )

t is the growth rate of population,

( )

t

µ _{is the natural death rate of the population,} β

_{( )}

t _{is the rate of the} effi-cient 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.

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simula-DOI: 10.4236/am.2019.103013 161 Applied Mathematics tions in Section 5.

2. Preliminaries

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

(H1): 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_.

(H2): There exist positive constants ωi >0

(

i=1, 2,3

)

such that

( )

2

liminf _tt d 0,

t s s

ω

β

+

→+∞

∫

>

( )

2

liminf _tt d 0,

t s s

ω

µ

+

→+∞

∫

>

( )

2

liminf t d 0.

t

t s s

ω +

→+∞

∫

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

( )

( ) ( )

( )

( ) ( )

( )

(

( )

)

( )

d d N t

t t N t t t N t d t I t

t

t t d t N t

µ µ

µ

Λ − ≥ = Λ − −

≥ Λ − +

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

d N t

t t N t

t = Λ −µ (2.1)

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

( )

the solution of

( )

_{( )}

₍

_{( )}

₎

_{( )}

*

* d

d N t

t t d t N t

t = Λ − µ + (2.2)

with initial value (1.2). Then

( )

( ) ( )

*

( )

* t t t t .

N ≤S +E +I ≤N t

By [22], we have the following result:

Lemma 2.1. Suppose that assumptions (H1) and (H2) hold. Then:

(i) there exist positive constants m>0_and M >0_{, such that}

( )

* *

0 liminf limsup

liminf limsup .

t _t

t

m N N

M t

N t N t

→+∞ _→+∞

< ≤ ≤

≤ ≤ ≤ < +∞ (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

( )

, :

( ) ( )

_{( )}

( )

( ) ( )

1 1

( )

1

p S

G p t d

S

t t

t t t t

t p

β

δ γ ε

α

 

= + + − − +_ _

+ _ _

and

( )

, :

( ) ( )

.

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DOI: 10.4236/am.2019.103013 162 Applied Mathematics investigate the longtime behavior of system (1.1).

Lemma 2.2. If there exist positive constants p>0_and T₁>0_{such that}

( )

, 0

G p t < _{for all}t T≥ ₁_{, then there exists}T₂≥T₁_{such that}W p t

( )

, >0_or

( )

, 0

W p t ≤ _{for all} t T≥ ₂_.

Proof: Suppose that there does not exist T2≥T1 such that W p t

( )

, >0 or

( )

, 0

W p t ≤ _{for all} t T≥ ₂_{. So we have}

( )

pE s =I s

(2.5) and

( )

( ) ( ) ( )

( )

(

( ) ( ) ( )

)

( )

( ) ( )

(

( ) ( )

( )

)

( )

{

}

( )

( ) ( )

_{( )}

(

( ) ( )

( )

)

( )

(

( ) ( ) ( )

)

( )

d ,

d 1

1

0.

W p s s S s I s

p s s s E s

t S s

s E s s s d s I s

p s S s

I s s s d s

S s

s

pE s s s s

p

β

µ ε γ

α

ε µ δ

β

µ δ

α

ε

µ ε γ

 

 

= _ − + + _

+

 

 

− − + +

 

 

= _ + + + _

+

 

 

 

 

− _ + + + _

 

 

>

(2.6)

Substituting (2.5) into (2.6), we have

( )

( ) ( )

_{( )}

( )

( ) ( )

( )

( ) (

)

1

0 1

1 , . p s S s

pE s s d s s s

S s p

pE s G p s

β

δ γ ε

α

   

 

<  ₊ + + − − +  

   

 

=

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.

3. Extinction of Infectious Population

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 (H1) and (H2) hold. If there exist

0

λ> , p>0 and T1>0 such that

(

)

( )

_{( )}

*

( )

(

( ) ( ) ( )

)

1 , : limsup ₁ * d 0,

t t t

s

R p p N s s s s s

N s

λ β

λ µ ε γ

α

+

→+∞

 

 

=  − + +  <

+

 

 

∫

(3.1)

(

)

( )

(

( ) ( ) ( )

)

*

1 , : limsup 1 d 0

t t t

R p s s s d s s

p

λ

λ + ε µ δ

→+∞

 

=  − + +  <

 

∫

(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 0.

t→+∞I t =

Proof: From Lemma 2.2, we consider the following two cases: (i) pE t

( )

>I

( )

t for all t T≥ 2;

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DOI: 10.4236/am.2019.103013 163 Applied Mathematics First, we consider the cases (i). From the second equation of system (1.1), we have

( )

( ) ( )

( )

( ) ( )

(

)

(

( )

( ) ( )

)

( ) ( ) ( )

(

)

( )

( ) ( )

( )

( ) ( )

(

)

(

( )

( ) ( )

)

( ) ( ) ( )

(

)

( )

( ) ( )

( )

(

( ) ( ) ( )

)

( )

(

( ) ( ) ( )

)

* *

* * d

d 1

1

. 1

E t t I t

N t E t I t

t N t E t I t

t t t E t

p t E t

N t E t I t

t t t E t

p t E t

N t t t t E t

N t p t

E t N t t t t

N t

β α

µ ε γ

β α

µ ε γ

β

µ ε γ

α β

µ ε γ

α

= − −

+ − −

− + +

< − −

+ − −

− + +

< − + +

+

 

 

= _ − + + _

+

 

 

So we have

( )

(

( ) ( ) ( )

)

2

*

2 exp ₁ * d

t T

s

E t E T p N s s s s s

N s

β

µ

ε

γ

α

 _ _ 

< _ _ − + + _ _

+

 

 



∫

 (3.3)

for all t T≥ 2. From (3.1), we see that there exist constants δ >1 0 and T T3> 2 such that

( )

*

( )

(

( ) ( ) ( )

)

1

* d

1

t t

s

p N s s s s s

N s

λ β

µ ε γ δ

α

+  

− + + < −

 ₊ 

 

 

∫

(3.4)

for all t T> 3. From (3.3) and (3.4), we obtain limt→+∞E t

( )

=0. Therefore, it follows from pE t

( )

>I t

( )

_{, that} limt→+∞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 1 _.

d I t

I t t t t d t

t ε p µ δ

 

≤  − + + 

 

Then the following expression

( )

( ) ( )

( )

2

2 exp d

t T

s

I t I T s s d s s

p

ε

µ δ

   

≤ _  − − −  _

 

 



∫

 (3.5)

for all t T> 2 hold. Hence, by (3.2), there exist δ >2 0 and T4>T2 such that

( )

1

(

( ) ( ) ( )

)

d 2,

t

t+λ ε s _p µ s δ s d s s δ

 

− + + < −

 

 

∫

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

( )

lim 0.

t→+∞I t =

4. Permanence of Infectious Population

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DOI: 10.4236/am.2019.103013 164 Applied Mathematics Theorem 4.1. Suppose that assumptions (H1) and (H2) hold. If there are

0

λ> _, p>0_and T₁>0_{such that}

(

)

( )

_{( ) ( )}

(

( ) ( ) ( )

)

2 *

*

, : liminf d 0,

1

t t t

s

R p p N s s s s s

N s

λ β

λ µ ε γ

α

+

→+∞

 

 

=  ₊ − + +  >

 

 

∫

(4.1)

(

)

( )

(

( ) ( ) ( )

)

*

2 , : liminf 1 d 0

t t t

R p s s s d s s

p

λ

λ + ε µ δ

→+∞

 

= _ − + + _ >

 

∫

(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 T₁>0_{such that} (4.1), (4.2) and G p t

( )

, >0 hold for all t T≥ 1. Then there exists T2>T1 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

( )

I t E t

p

> _{for all}

2

t T≥ _{. From the third equation of system (1.1), we have}

( )

_{( )}

₍

_{( ) ( ) ( )}

₎

_{( )}

( ) ( )

(

( ) ( ) ( )

)

d 1

d

1 _.

I t

t I t t t d t I t

t p

I t t t t d t

p

ε µ δ

> − + +

 

= _ − + + _

 

So we obtain

( )

( ) ( )

( )

2

2 exp d

t T

s

I t I T s s d s s

p

ε

µ δ

   

> _  − − −  _

 

 



∫

 (4.3)

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

( )

1

(

( ) ( ) ( )

)

d

t

t+λ ε s _p µ s δ s d s s η

 

− + + >

 

 

∫

(4.4)

for all t T≥ _{. So the inequality (4.3) holds for all} t≥max

_{

T T₂,

_}

_{. Then}

( )

limt→+∞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_:= −m , M_:=M+, where >0 is a con-stant. In fact, Lemma 2.1 implies that for any sufficiently small >0, there ex-ists T >0 such that

( )

*

( )

*

m_ <N t ≤N t <M_ (4.5) for all t T≥ _{. The inequality (4.1) implies that for any sufficiently small}η>0_, there exists T T1> such that

( ) ( )

( )

(

( ) ( ) ( )

)

*

d 1

t t

p s N s

s s s s

N s

λ β

µ ε γ η

α

+  

− + + >

 

+

 

 

∫

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DOI: 10.4236/am.2019.103013 165 Applied Mathematics for all t T≥ 1. We define

( )

0 0 0

: sup , : sup , : sup ,

t t t t t t

β+ β µ+ µ δ+ δ

≥ ≥ ≥

= = =

( )

0 0 0

: sup , : sup , : sup .

t t t

d+ d t ε+ ε t γ+ γ t

≥ ≥ ≥

= = =

Thus, by (4.5) and (4.6), for any sufficiently small η η1< and T2>T1, there exist very small i, i∈

{

1, 2,3

}

such that

( )

(

*

( )

1 2

)

(

( ) ( ) ( )

)

1

*

d ,

1

t t

s

N s k p s s s s

N s

λ β

µ ε γ η

α

+  

− − − + + >

 ₊ 

 

 

∫

  (4.7)

( )

* 1 2

N t − −_ k_ >m__(4.8)

for all t T≥ 2, where k: 1 ₁ _M M 2

β _ω

α

+

= +

+ 



. Lemma 2.1 implies that for any

suf-ficiently small 2 >0, we have

( )

₍

_{( ) ( ) ( )}

₎

1 2

1

2

1 d 1

1 t t

s M

s s s s

M

ω β

µ ε γ η

α

+  

− + + < −

 ₊ 

 

 

∫



 

 (4.9)

for all t T≥ 2. First, we prove

( )

2

limsup .

t→+∞ I t >

In fact, if it is not true, there exists T T3≥ 2 such that

( )

2

I t ≤ _(4.10)

for all t T≥ 3. If E t

( )

≥1 for all t T≥ 3, then from (4.5) and (4.6), we have

( )

( ) ( )

_{( ) ( )}

(

( ) ( )

)

(

( ) ( ) ( )

)

( )

(

( ) ( ) ( )

)

3

3 2 1

d 1

t T

E t

s I s

E T N s E s I s s s s E s s

S s s

E T M s s s s

M

β

µ ε γ

α β

µ ε γ

α

 

 

= + _ − − − + + _

+

 

 

 

 

≤ +  ₊ − + + 

 

 

∫



 

for all t T≥ 3. It follows from inequality (4.9) that limt→+∞E t

( )

= −∞. This con-tradicts with the boundedness of solution. Hence, there exists an s T1≥ 3 such that E s

( )

1 <1. In the following we prove

( )

1 ₁ 2 2,

E t M

M

β _ω

α

+

≤ +

+ 



  (4.11)

for all t s≥ 1. If it is not true, there exists an s2>s1 such that

( )

2 1 ₁ 2 2.

E s M

M

β _ω

α

+

> +

+ 



 

Hence, there necessarily exists an s3∈

(

s s1, 2

)

such that E s

( )

3 =1 and

( )

1

E t > for all t∈

(

s s3, 2

)

. Let n≥0 be an integer such that

(

)

2 3 2, 3 1 2

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DOI: 10.4236/am.2019.103013 166 Applied Mathematics

( )

( ) ( )

_{( )}

(

( )

( ) ( )

)

(

( ) ( ) ( )

)

( )

{

}

( )

₍

_{( ) ( ) ( )}

₎

( )

3 2

3 2 2

3 3 2

2 3 2 2 * 3 2 1 1 1 2

1 2 2

d 1 d 1 d 1 , 1 s s

s n s

s s n

E s

s I s

E s N s E s I s s s s E s s

S s s M

s s s s

M s M s M M M ω ω ω β

µ ε γ

α β

µ ε γ

α β α β _ω α + + + +     = +  − − − + +  +        

< + + _ − + + _

+     < + + < + +

∫

            

This contradicts with E s

( )

2 1 ₁ _M M 2 2

β _ω α + > + +  

  . Hence, (4.11) is valid. By

Lemma 4.1, there exists T4≥s1 such that W p t

( )

, =pE t

( ) ( )

−I t ≤0 for all 4

t T≥ _{. Therefore, by (4.10) and (4.11), we have}E t

_{( ) ( )}

+I t ≤ +₁ k₂ for all

4

t T≥ _{, then}

( )

( ) ( ) ( )

(

( ) ( )

)

( )

( ) ( )

(

)

(

( ) ( ) ( )

)

( )

( ) ( )

(

( )

( ) ( )

)

( )

( ) ( )

(

)

(

( ) ( ) ( )

)

( )

( ) ( )

(

( )

( ) ( )

)

( )

(

( ) ( ) ( )

)

( )

(

( )

( ) ( )

_{( )}

)

(

( ) ( ) ( )

)

( )

(

( )

_{( )}

)

(

( ) ( ) ( )

)

* * * * * *

* 1 2

* d d 1 1 1 1 . 1

t I t N t E t I t E t

t t t E t

t N t E t I t

t I t N t E t I t

t t t E t

N t E t I t t E t N t E t I t p

t t t E t

N t

t N t E t I t p

E t t t t

N t

t N t k p

E t t t t

N t

β

µ ε γ

α β

µ ε γ

α β

µ ε γ

α β

µ ε γ

α β

µ ε γ

α − − = − + + + − − − − ≥ − + + + − − − − ≥ − + + +  − −    ≥  − + +  +      − −    ≥  − + +  +       We obtain

( )

₄

( )

(

*

( )

_{( )}

1 2

)

(

( ) ( ) ( )

)

4

*

exp d .

1 t

T

s N s k p

E t E T s s s s

N s

β

µ

ε

γ

α

 _ − − _    ≥ _  ₊ − + +  _     

∫

  

By (4.7) we obtain limt→+∞E t

( )

= +∞. This contradicts with Lemma 2.1 (E t

( )

is uniformly bounded). Hence, limsupt→+∞I t

( )

>2 is true.

Next, we prove

( )

1

liminf ,

t→+∞ I t ≥I

where I1>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

(

≥T2

)

, λ >2 0, η >2 0 such that λ3≥λ2 and t T≥ 3, we obtain

( )

(

( )

)

(

( ) ( ) ( )

)

3

* 1 2 2

* d , 1 t t s

N s k p s s s s

N s

λ β

µ ε γ η

α

+  

− − − + + >

 ₊ 

 

 

∫

  (4.12)

( )

₍

_{( ) ( ) ( )}

₎

1 3

1

2

1 d ,

1 t t

s M

s s s s M

M

λ β

µ ε γ

α

+  

− + + < −

 ₊     

∫

   

(9)

( )

1 3

1 d 2.

t

t s s

λ

β η

+

>

∫

(4.14)

Let C>0_{be a constant satisfying}

(

)

*

2

2 _{2 2} _{2 2}

1 e e

1 1

n

m M

M M

η

µ ε γ λ ν η β ω

α α

+ + + +

− + +

> +

+ +

 



(4.15)

where

(

)

22

2 2e

d

µ δ λ

ν

= − ++ ++ + , *

2

C n ≤_λ _.

Because we have proved limsupt→+∞I t

( )

>2, there are only two possibilities as follows:

(i) There exists T4≥T3, then as t T≥ 4, we obtain I t

( )

≥2; (ii) I t

( )

oscillates about 2 for all large t.

In case (i), we have liminft→+∞I t

( )

≥2=:I1. In case (ii), there necessarily ex-ist t t1 2, ≥T t3

(

2≥t1

)

such that

( )

1 2 2 2

(

1 2

)

,

, , .

I t I t

I t t t t

= =

 

< ∈



 

Suppose that t t C2− ≤ +1 2λ2. Then

( )

₍

₎

_{( )}

d

, d

I t

d I t

t µ δ

+ + +

≥ − + + _(4.16)

which implies

( )

(

)

(

)

( )

1

2

1

2

2 1

exp d

e : .

t t

d C

I t I t d s

I

µ δ λ

µ δ

+ + +

− + + +

≥ − + +

≥ =

∫



for all t∈

(

t t1 2,

)

. Suppose that t t C2− > +1 2λ2. Then

( )

(

)

( 2 2)

2e 1

d C I t ≥ −µ++δ++ + + λ =I

for all t∈

(

t t C1 1, + +2λ2

)

. Now we only prove I t

( )

≥I1 for all

[

1 2 ,2 2

)

t∈ t C+ + λ t . If E t

( )

≥1 for all t∈

[

t t1 1, +λ2

]

. By the second equation of system (1.1) and inequality (4.13), we have

(

)

( )

1 2

( )

(

( ) ( ) ( )

)

1

1 2 1 ₁ 2 1 d

0, t t

s M

E t E t s s s s

M

M M

λ β

λ µ ε γ

α

+  

+ ≤ + _ + + + _

+

 

 

< − =

∫



 

which is contradiction. Hence, there exists an s4∈

[

t t1 1, +λ2

]

such that E s

( )

4 <1. We obtain that for t s≥ 4,

( )

1 ₁ 2 2.

E t M

M

β _ω

α

+

≤ +

+ 



 

(4.17) By inequality (4.16), then for t∈

[

t t1 1, +2λ2

]

( )

(

)

22

2 2e .

d

(10)

( )

( ) ( ) ( )

(

( ) ( )

)

( )

( ) ( )

(

)

(

( ) ( ) ( )

)

( )

( ) ( )

₍

_{( )}

_{( ) ( )}

₎

₍

_{( ) ( ) ( )}

₎

_{( )}

( )

₍

₎

_{( )}

* d d 1 1 . 1

t I t N t E t I t E t

t t t E t

t N t E t I t

t I t

N t E t I t t t t E t

M t

m E t

M

β

µ ε γ

α β

µ ε γ

α β

µ ε γ

α + + + − − = − + + + − − ≥ − − − + + + ≥ − + + +   

for all t∈

[

t1+λ2 1,t +2λ2

]

. By (4.14), we have

( )

(

)

( )

(

)

(

)

( )

(

)

(

)

( )

(

)

(

)

1 2 1 2

1 2 1 2 1 2 1 2 1 2 2 2 1 2 2 2 2 2 2 2 2 e e e d 1

e e d

1 1 e . 1 t t t s t t t s t

E t E t

s m s M s m s M m M

µ ε γ λ µ ε γ λ

λ µ ε γ

λ

µ ε γ λ µ ε γ

λ

µ ε γ λ

λ β ν α β ν α ν η α + + + + + + + + + + + + + + + + + + − + + + + + + + + + + + − + + + + + + − + +  ≥  +   + ₊   ≥ + ≥ +

∫

      (4.19)

Now, we suppose there exists t0 >0 such that t0∈

(

t C1+ +2 ,λ2 t2

)

, then

( )

0 1

I t =I and I t

( )

≥I1 for all t∈

[

t t1 0,

]

. By Lemma 4.1, we assume that t1 is so large that W p t

( )

, = pE t

( ) ( )

−I t ≤0_{for all}t t≥ +₁ 2λ₂_{. Hence, by (4.8),}

we further have

( )

( ) ( ) ( )

(

( ) ( )

)

( )

( ) ( )

(

)

(

( ) ( ) ( )

)

( )

(

*

( )

_{( )}

1 2

)

(

( ) ( ) ( )

)

* d

d 1

1

t I t N t E t I t E t

t t t E t

t N t E t I t

t N t k p

E t t t t

N t

β

µ ε γ

α β

µ ε γ

α − − = − + + + − −  − −    ≥ _ − + + _ +      

for all t∈

(

t1+2 ,λ2 2t

)

. By (4.12) and (4.19), we have

( )

(

)

( )

(

( )

_{( )}

)

(

( ) ( ) ( )

)

(

)

{

}

( )

(

( )

)

( )

(

( ) ( ) ( )

)

(

)

0 1 2

1 2 2 1 2 2

1 2 1 2 2

0 *

1 2 2

2

* 1 2

1 2 ₂

*

2 2 2

1 2 ₂ ₂

* 1 2

2 ( 1)

*

2 exp d

1 2 exp d 1 1 e t t t t t t t t n E t

t N t k p

E t t t t s

N t

E t

t N t k p

t t t s

N t λ

λ λ λ λ

λ λ λ

λ λ

µ ε γ λ

β

λ µ ε γ

α λ

β

µ ε γ

α + + + + + + + + + + + + + − − + +  _ − − _    = + _  − + +  _ +        = +  + +    − −   _{ } +  ₊ − + + _{ }      ≥

∫

     * 2

2 2e . 1

n

m

M ν η η

α

+ 



Thus, by (4.17), we have

(

)

*

2 2

2 2 2 2

1 ₁ e ₁ e ,

n

M m

M M

η µ ε γ λ

β ω ν η

ε α α + + + + ₋ ₊ ₊ + ≥ +  _ + _ 

(11)

DOI: 10.4236/am.2019.103013 169 Applied Mathematics (1.1) is permanent.

5. Numerical Simulations

Numerical verification of the results is necessary for completeness of the analyt-ical 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_,

( )

t 0.3 0.1cost

β = + , α=0.3, µ=0.1, δ =0.1, γ =0.8, =0.2,

d

=

0.5

. Then, from system (1.1), we have _lim *

( )

₁

t→+∞N t = . We easily verify that as-sumptions (H1) and (H2) hold.We choose λ=1 and p=2. Then we have

(

)

(

)

(

)

(

)

1

1 ₀

1 *

1 ₀

0.3 0.1cos

, : 2 0.1 0.2 0.8 d 0.84 0,

1 0.3 1

, : 0.2 0.1 0.1 0.5 d 0.7 0.

2

t

R p t

λ λ

+

 

=  ₊ − + +  ≈ − <

 

 

=  − + +  = − <

 

∫

and

( )

1

0

0.3 0.1cos 1

, : 2 0.1 0.5 0.8 1 0.2 d

1 0.3 2

0.1385 0. t

G p t = _ + + + − − +_ _ _ t

+  

 

≈ − <

∫

for all t>0_{. From Theorem 3.1, we see that the infectious population of system} (1.1) is extinct, see Figure 1.

Fix Λ =1_, β

_{( )}

t =0.6 0.1cos+ t _, α=0.6_, µ=0.1_, δ =0.02 _, γ =0.1_, 0.5

=

 ,

d

=

0.02

. We choose λ=1 and p=2. Then we have

(

)

(

)

(

)

(

)

1

2 ₀

1 *

2 ₀

0.6 0.1cos

, : 2 0.1 0.5 0.1 d 0.05 0,

1 0.6 1

, : 0.5 0.1 0.02 0.02 d 0.11 0.

2

t

R p t

λ λ

+

 

= _ − + + _ ≈ >

+

 

 

=  − + +  = >

 

∫

and

( )

1

0

0.6 0.1cos 1

, : 2 0.02 0.02 0.1 1 0.5 d

1 0.6 2

0.06 0.

t

G p t =  +₊ + + − − +_ _  t

 

≈ − <

∫

for all t T> 1. From Theorem 4.1, we see that the infectious population of system (1.1) is permanent, see Figure 2.

6. Conclusions

(12)

[image:12.595.124.473.64.352.2]

Figure 1. The trajectories of deterministic system (1.1) with R1

(

λ,p

)

<0,

(

)

*

1 , 0

R λ p < , G p t

( )

, <0. (a) Time se-ries diagram of susceptible population, (b) Time sese-ries diagram of exposed population, (c) Time sese-ries diagram of infectious population, (d) phase diagram of three populations (susceptible, exposed, infectious), respectively.

Figure 2. The trajectories of deterministic system (1.1) with R1

(

λ,p

)

>0,

(

)

*

1 , 0

[image:12.595.123.474.409.691.2]

(13)

DOI: 10.4236/am.2019.103013 171 Applied Mathematics 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 per-manence 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.

Acknowledgements

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

Conflicts of Interest

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

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