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2017 International Conference on Mathematics, Modelling and Simulation Technologies and Applications (MMSTA 2017) ISBN: 978-1-60595-530-8

Persistence and Extinction of a Stochastic Epidemic Model with Delay

and Proportional Vaccination

Shu-qi GAN and Feng-ying WEI

*

College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, P.R. China *Corresponding author

Keywords: Stochastic epidemic model, Time delay, Extinction, Persistence in the mean, Threshold.

Abstract. A type of susceptible-infection-vaccinated epidemic model with proportional vaccination and generalized nonlinear rate is formulated and investigated in the paper. We show that the stochastic epidemic model admits a unique and global positive solution with probability one when constructing a proper 2

C -function therewith. Then the sufficient condition that guarantees the diseases vanish is derived when the indicator R01. Further, if R0 1, then we obtain that the

solution is weakly permanent with probability one. And we also derived the sufficient conditions of the persistence in the mean for the susceptible and infected under the condition R01.

Formulation

We formulation an epidemic model of having three states involved: the susceptible, the infected and the vaccinated in the paper. We always assume that, the transmission rate between the susceptible and the infected is supposed to be a nonlinear incidence rate, say f(S(t))g(I(t)). The susceptible

individuals with probability pare chosen to be the vaccinated individuals and recovered to the susceptible again with the effective survival rate e, where is the nature death rate and the

period of temporary immunity. We then reach a stochastic epidemic model if the fluctuation is taken into account

( ) [ ( ) ( ) ( ( )) ( ( )) ( ) ] ( ( )) ( ( )) ( ),

( ) [ ( ( )) ( ( )) ( ) ( )] ( ( )) ( ( )) ( ),

( ) [ ( ) ( ) ( )] ,

t t

dS t A S t pS t f S t g I t pS t e dt f S t g I t dB t

dI t f S t g I t I t dt f S t g I t dB t

dV t pS t pS t e V t dt

   

   

 

      

   

    (1)

where A denotes the recruitment rate of the population,  the death rate induced by the diseases,  is the intensity of the white noise, B t( ) is a standard Brownian motion which is defined on a

complete probability space (,{Ft}t0,P) with its filtration {Ft}t0 satisfying the usual conditions.

All the coefficients are assumed to be nonnegative throughout this paper. The authors would like to mention the related works hereby, for example, the stochastic epidemic models [1,2,3,4,5]. Besides, we also assume that (A1) and (A2) hold throughout this paper:

(A1) Function f(S) is continuously differentiable and monotonically increasing with f(0)0,

and for some constant l0 such that ( )

0

: inf f S

l S l S

m

 

  , ( )

0

: sup f S

l S

S l

M

 

  .

(A2) Function ( )g I is twice continuously differentiable, and g I( )I is monotonically decreasing on

R with g(0)0 and g'(0) 0 .

Since the first two equations do not depend on the third equation of model (1), we equivalently consider model (2) of having two compartments ( )S t and ( )I t herewith:

( ) [ ( ) ( ) ( ( )) ( ( )) ( ) ] ( ( )) ( ( )) ( ),

( ) [ ( ( )) ( ( )) ( ) ( )] ( ( )) ( ( )) ( ).

t

dS t A S t pS t f S t g I t pS t e dt f S t g I t dB t

dI t f S t g I t I t dt f S t g I t dB t

   

   

      

(2)

The initial condition of model (2) is S( )  ( ) 0 ,  [ ,0] , I(0)0.

Next, we will investigate the existence and uniqueness of the global solution to model (2) in Section 2. And the sufficient conditions guarantee the extinction of the disease will be discussed in Section 3. Thus the threshold of the persistence in the mean of the susceptible and the infected to model (2) is obtained.

Existence and Uniqueness of Global Solution

Theorem 2.1 Model (2) admits a unique solution (S(t),I(t)) on t0, and the solution will remain

in 2

R with probability one.

Proof It is easily to check that the local Lipschitz condition is valid for model (2) on the interval )

, 0

[ e , where e is the explosion time. Let m0 1 be an integer such that  ( ) ( [ ,0]) and

(0)

I lying within the closed interval [m01,m0]. For each integer mm0, we define the stopping time

1 1

inf{ [ , ) : ( ) ( , ) or ( ) ( , )}.

m t e S t m m I t m m

   Then,

m

 is an increasing function as

 

m . Denote lim m

m



 , thus  e. We claim that  . The proof goes by contradiction

from now on. If the claim is false, then there exists a constant T 0 and (0,1) satisfying

 }

{

P . That is, there exists an integer m1m0 such that  m P{mT} for mm1. Let N t( )S t( )I t( )V t( ) , we obtain that N t( ) AN t( )I t( ) , which implies that

0

lim ( )

tN tN as t[0, ]m where

1

0 max{ (0), }

NN A . We define a C2-function as follows:

( ( ), ( )) ( ) ln ( ) ( ) 1 ln ( ) t ( ) ,

t

W S t I t S t a a S t I t I t pe  S s ds

 

      

(3) where a is a positive constant that will be determined later. Then generalize Ito’s formula acts on

( ( ), ( ))

W S t I t , which gives that

1 1

d ( ( ), ( ))W S t I tLW S t I t dt( ( ), ( )) (aS ( )tI ( ))tf S t g I t dB t( ( )) ( ( )) ( ), (4) where

1 1 2 2 2 2

2

2 2 2 2

1 2

( ( ), ( )) ( ) ( ) ( )

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

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

LW S t I t A S t pS t pS t e a ap

a f S t g I t S t a f S t g I t S t

I t pS t e f S t g I t I t





  

 

    

 

 

      

 

     

0

2 2 2 2 2

1

2 ( ( ) ( )) ( ( )) ( ( ))

( (1 )) ( ) ( N '(0) ( )) ( ).

A a ap aS t I t f S t g I t

p e  S t a M g I t

   

   

 

      

     

(5)

Choosing ( )( '(0)) 1

0

 

M g

a    N such that

0

2 2

0 2 2

0

( ) '(0)

( ( ), ( )) ( ) ' (0) : ,

2

N

M g N

LW S t I t A aap      f N g K

  

        

  (6)

where K is a constant. Integrating (4) from 0 to mT and taking expectation then yields that

( ( m ), ( m )) ( (0), (0)) .

EW S  T I  TW S IKT (7)

Each component of (S(mT),I(mT)) equals either m or m 1

for all

m  

 . Consequently,

we have ( (0), (0)) min{ 1 ln , 1 1 ln }

W S I KTm m mm

        , letting m leads to the

(3)

Remark 2.1 Model (1) admits a unique solution, which will remain in 3

R with probability one.

Sufficient Condition of the Extinction of the Infected

The extinction and the persistence are two important issues when people studied epidemic models.

For the simplicity, we denote 1

0

( ) t t ( )

x t x s ds

  

.

Theorem 3.1 Let ( ( ), ( ))S t I t be any solution of model (2), if

2 0

( ) '(0)

f N g

  ,

2 2 2

0 0

0

( ) '(0) ( ) ' (0)

1

2( )

f N g f N g

R  

   

  

  , (8)

then the infected individuals will decline with the exponential rate (  )(R01) almost surely.

Proof From Theorem 2.1, for an arbitrary 0 , there exists a constant T0 such that

0

( ) ( )

S tI tN  for all tT0. The generalized Ito’s formula acts on lnI(t), we then get that

0

2 2

2

0 0

ln ( ) ln (0) ( ( ))

( ( )) ( )

( )

( ( )) ( ( ))

( ( )) ( ( )) ( ).

2 ( ) ( )

t

t t

I t I g I s

f S s ds

t t t I s

g I s g I s

f S s ds f S s dB s

t I s t I s

   

 

   

   

   

(9)

Let us define a function 1 2 2

2

( ) ( ) ( )

G x    x    x   , which is a monotonically increasing

function as [0, ( ) 2)

x     . The condition  2f(N0)g'(0) yields that there exists a constant

   

0 such that f N( 0 ) '(0) (g   ) 2

   , which implies that

0

( ( ))

( ( ) ( ( ) '(0))

( ) g I t

G f S t G f N g

I t

 

 

  . (10)

There, for tT0, we have that

0 0

0

0 0

ln ( ) ln (0) 1 ( ( )) ( ( ))

( ( )) ( ( ) '(0)) ( ( )) ( ).

( ) ( )

T t T t

I t I g I s g I s

G f S s ds G f N g f S s dB s

t t t I s t t I s

 

  

    

 

(11)

Taking superior limit on both sides of (11), together with the arbitrariness of  and , which

finally shows that ln ( )

0

lim sup I t ( )( 1) 0

t t

R

 

     . The proof is complete.

The Permanence of the Susceptible and the Infected

In the section, we will demonstrate same useful results about the weak permanence of the disease. Lemma 4.1 Let ( ( ), ( ))S t I t be the solution of model (2), then

( )

lim 0

t

S t t

  ,

( )

lim 0

t

I t t

  ,

1

lim t ( ) 0

t

t pe t S s ds

 

 



 a.s.. (12)

The proof is similar to the approach used in Liu et al.[4] and we omit the proof herewith.

Theorem 4.1 Suppose that R0 1, the solution ( ( ), ( ))S t I t of model (2) is weakly permanent with probability one if there exists a constant  0 satisfying lim sup ( )

t

I t

  .

Proof Noting that ( ( )) ( ) ( ) 0

lim g I t '(0)

I t

I t g , then there exists a constant (0, ] satisfying

( ( )) ( '(0) ) ( )

(4)

go by contradiction herewith. If the statement is false, we then set 0 { : lim sup ( ) }

t

I t

 



   , then

there exists a constant 0(0,1) satisfying P( 0) 0 As lim( ( ) ( )) 0

t S tI tN , thus for the above

0

  , there exists a T00 such that S t( )I t( )N0, for all tT0. For any 0 there exists

a T1T1( ) T0 satisfying I t( )   for all tT1. Then, for all tT1, we have

1

0 1

0 0 ( )

0 0

( ) (0)

( ( )) ( ( )) ( ) max ( ( ))( )

( ) ( ( )) ( ( )) ( ),

T

I t

t t

S t S

A f S s g I s ds f N g I t t T

t t

p

S s ds f S s g I s dB s

t t

 

 

 

 

(13)

where

1

0 0

( ) (0) 1 1

lim lim T ( ( )) ( ( )) lim t ( ( )) ( ( )) ( ) 0.

t t t

S t S

f S s g I s ds f S s g I s dB s

t t t

  

(14)

There exists a constant T2T1 such that for all tT2, then (13) gives that

2 0 0 ( )

1

lim inf t ( ) ( 2 ) max ( ( )).

T

t I t

A

S s ds f N g I t

t p p

 



      

(15) The first equation of model (2) gives that for all tT2

2

2

2 2

0 2

ln ( ) ( '(0) )

( ) ( ) ( ) ' (0)( ) ( ),

2

t T

I t g

Q t f s ds f N g t T

t t t

    

 

    

(16) where

2 2 2 2

2

0 0

ln (0) 1 ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( ))

( ) ( ).

( ) 2 ( ) ( )

T t

I f S s g I s f S s g I s f S s g I s

Q t ds dB s

t t I s I s t I s

  

 

 

 

(17) Lagrange's mean value theorem gives that f s( ) f '( )( ( ) S sN0 ) f N( 0), for s[ , ]T t2

2

2 0

0 0

0

0 0 2

0

2

0 0

0 0

0

( '(0) )

( )

1 ( '(0) )

max '( )( '(0) ) ( ) ( )( )

max '( )( '(0) )( ) max '( )( '(0) )( )

max '( )( '(0) ) (

t T

t T N

N N

N

g

f s ds t

g

f g N S s ds f N t T

t t

T

f g N f g N

t

A

f g f N

p p

 

   

 

 

 

    

       

  

 

  

     

   

       

      

  

 

0 0 ( )

2

0 0

2 ) max ( ( ))

( )( '(0) ) ( )( '(0) ) ,

I t g I t

T

f N g f N g

t

     

 

 

 

     

(5)

0

0

2

2 2

2 2

0 0

0

2

2 2

2

0 0 0

0 0

0 0 ( )

ln ( )

( ) max '( )( '(0) )( ) ( ) ' (0)

2

( )( '(0) ) ( )( '(0) ) ( ) ' (0) ( )

2

max '( )( '(0) ) ( 2 ) max ( ( ))

N

N I t

T T

I t

Q t f g N f N g

t t t

T

f N g f N g f N g

t

A

f g N f N g I t

p p

 

  

    

        

    

 

  

    

     

         

     

  .

 

  (19)

According to condition R0 1, there must exist a positive constant small enough such that

0

0 0 0 0 0 ( )

2

2 2

0

( '(0) ) ( ) max '( ) ( 2 ) max ( ( ))

( ) ' (0) ( ) 0.

2

N I t

A

g f N f N f N g I t

p p

f N g

  

     

 

 

    

  

     

 

 

 

    

(20) We therefore take the inferior limit of (19) as t tends to infinity, together with (20), and then get that the density of the infected individuals tends to infinity. This is a contradiction. The proof is complete.

We will discuss the persistence in the mean of the disease and to obtain Theorem 4.2. Theorem 4.2 If

0

2 2 2 0

0

0 0

0

max '( )

( ) '(0) ( ) ' (0)

1,

2( )

N f N

f N g f N g

R

 

     

 

   

  

(21) then the solution ( ( ), ( ))S t I t of model (2) has the property

0 0

0

0 0

( 1)( (1 ))

lim inf ( ) , liminf ( ) .

max '( ) '(0) '(0)

t t

N N

R p e A

I t S t

f g M g N p



   

 

 

  

 

  

(22) Proof Differentiating along model (2) gives

( ) ( ) t ( )

( (1 )) ( ) ( ) ( ),

t

d S t I t pe  S s ds A p e  S t I t

   

 

 

     

(23) then integrating on both sides of (23) shows that

( ) ( )

( ) ( ),

(1 )

A I t

S t t

p e 

 

 

 

 

  (24)

where

0

1

( ) ( ) ( ) ( ) (0) (0) ( ) .

( (1 ))

t t

t S t I t pe S s ds S I pe S s ds

p e t

 



 

     

 

(25)

Lemma 4.1 indicates that lim ( ) 0.

t t  The generalized Ito's formula acts on ln ( )I t , together with

(6)

0 0 2 0 2 0 0 0 0 0 0 0 0 2 2

2 2 2

2

0 0 0

ln ( )

( ) max '( )( '(0) )( ) max '( )( '(0) )( )

1

max '( )( '(0) ) ( ) ( ) ( )( '(0) )

( )( '(0) ) ( ) ' (0) ( )

2 2 N N t T N T I t

Q t f g N f g N

t t

f g S s ds S s ds f N g

t T

f N g f N g f N g

t                                                        

2 2

' (0)T ( ).

t    (26)

Substituting (24) into (26) yields that

0

2 0 0 0 2 0 0 0 2

2 2 2

0 0

0 0

2

2 2

0 0 0 0

ln ( ) ( ) ( '(0) ) max '( )( ) ( )

1

max '( ) ( ) max '( ) ( ) ( ) ' (0)

2

( ) ' (0) ( '(0) ) ( ) max '( )( )

2 N T N N N T I t

Q t g f N f N

t t

T

f S s ds f t f N g

t t

f N g g f N f N

                                                        

0 0

( '(0) ) max '( ) ( ) ( ).

(1 )

N

g f I t

p e 

                   (27)

We easily obtain that the limit of the first five terms of (27) approaches zero as time scale t tends to infinity. Expression (27) thus gives that

0

0 2

2 2

0 0 0 0

0

ln ( )

( ) ' (0) ( '(0) ) ( ) max '( )( )

2

( '(0) ) max '( ) ( ) ( ),

(1 )

N N

I t

f N g g f N f N

t

g f I t

p e                                              

  (28)

which then gives that

0 0

1

0

2

2 2

0 0 0 0

lim inf ( ) max '( )( '(0) )( ) ( (1 ))

( '(0) ) ( ) max '( )( ) ( ) ' (0) ( ) .

2

t N

N

I t f g p e

g f N f N f N g

                                               

 (29)

From the arbitrariness of  and condition R0 1, therefore the first expression of assertion (22) is

derived as letting time t approaches infinity. By assumptions (A1) and (A2), there exists a constant

3 2 0

TT  such that for all tT3,

0 0

( ( )) ( ( )) N '(0)( ) ( ).

f S t g I tM g N  S t Integrating both sides

of the first equation of model (2) gives

3

0 3

0

0 0

( ) (0) 1 [ ( ( )) ( ( )) ( ) ( )]

1

[ '(0)( ) ] ( ) ( ( )) ( ( )) ( ).

T

t t

N T

S t S

A f S s g I s p S s ds

t t

M g N p S s ds f S s g I s dB s

tt

             

(30)

The arbitrariness of  and Lemma 4.1 therefore gives (22). The proof is complete.

Acknowledgement

(7)

References

[1] D. Zhao, T. Zhang, S. Yuan. The threshold of a stochastic SIVS epidemic model with nonlinear saturated incidence. Physica A, 443 (2016) 372-379.

[2] Q. Liu, D. Jiang, N. Shi, T., Hayat, A. Alsaedi. Asymptotic behaviors of a stochastic delayed SIR epidemic model with nonlinear incidence. Commun. Nonlinear Sci. Numer. Simul., 40 (2016) 89-99. [3] Z. Teng, L. Wang. Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate. Physica A, 451 (2016) 507-518.

[4] Q. Liu, Q. Chen, D. Jiang. The threshold of a stochastic delayed SIR epidemic model with temporary immunity. Physica A, 450 (2016) 115-125.

References

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