Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 502635 7 pageshttpdxdoiorg1011552013502635

Research ArticleThe Limit Behavior of a Stochastic Logistic Model withIndividual Time-Dependent Rates

Yilun Shang12

1 Institute for Cyber Security University of Texas at San Antonio San Antonio TX 78249 USA2 Singapore University of Technology and Design Singapore 138682

Correspondence should be addressed to Yilun Shang shylmathhotmailcom

Received 28 January 2013 Revised 21 April 2013 Accepted 7 May 2013

Academic Editor William E Fitzgibbon

Copyright copy 2013 Yilun Shang This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate a variant of the stochastic logistic model that allows individual variation and time-dependent infection and recoveryrates The model is described as a heterogeneous density dependent Markov chain We show that the process can be approximatedby a deterministic process defined by an integral equation as the population size grows

1 Introduction

The stochastic logistic model also called the endemic SISmodel in the epidemiological context was first discussed byWeiss and Dishon [1] This model describes the evolution ofan infection in a fixed population of size 119899 by a continuous-time Markov chain for the number of infected individuals119884119899 The state space is 0 1 119899 and the transition rates are

given by

119884119899

997888rarr 119884119899

+ 1 at rate 119899minus1120582119884119899 (119899 minus 119884119899)

119884119899

997888rarr 119884119899

minus 1 at rate 120583119884119899(1)

where 120582 and 120583 are the infection rate of susceptibles and therecovery rate of infectives respectivelyThis simplemodel hasfound applications in a variety of fields including populationbiology metapopulation ecology chemistry and physicsProperties such asmean epidemic size [2 3] mean extinctiontime [1 4 5] and quasi-stationary distributions [6ndash8] havebeen extensively studied under different initial conditions

The stochastic logistic model has an interesting limitproperty that it can be approximated by deterministic dif-ferential equations In particular based on Theorem 31 of[9] the rescaled process 119899minus1119884119899 converges in probabilityuniformly on finite time intervals to the solution of anordinary differential equation The issue of differential equa-tion approximations for stochastic processes traces back

to the pioneer work of Kurtz [9 10] where deterministiclimit of pure jump density-dependent Markov processeswas add-ressed via Trotter type approximation theoremsRecently McVinish and Pollett [11] show that a deterministiclimit process can be established for a stochastic logisticmodel with individual variation where the coefficients ofthe transition rates of the Markov chain can vary with thenodes We refer the interested reader to a comprehensivesurvey [12] for numerous sufficient conditions for this type ofconvergence

In this paper along the above line of study we investigatea variant of the stochastic logistic model where both indi-vidual variation and time-dependent infection and recoveryrates are allowed Specifically we consider a continuous-timeMarkov chain 119883119899 = (119883119899

1 119883

119899

119899) on the state space 0 1119899

with transition rates given by

119883119899

997888rarr 119883119899

+ 119890119899

119894at rate 120582

119894119905(1 minus 119883

119899

119894) 119891(119899

minus1

119899

sum

119895=1

119892119895(119883119899

119895))

119883119899

997888rarr 119883119899

minus 119890119899

119894at rate 120583

119894119905119883119899

119894

(2)

where 119894 = 1 2 119899 120582119894119905 120583119894119905isin R+ and 119890119899

119894is the 119899-dimen-

sional vector whose 119894th entry is 1 and other entries are 0 Let119883119899

119894= 0 represent that individual 119894 is susceptible and 119883119899

119894= 1

2 Journal of Mathematics

represent that individual 119894 is infected Then 119884119899 = sum119899119894=1119883119899

119894is

equivalent to the stochastic logistic model by setting 120582119894119905= 120582

120583119894119905= 120583 119891(119909) = 119909 and 119892

119895(119909) = 119909 for all 119894 119895 and 119905

The above model (2) can be viewed as a generalization ofthat treated in [11] in twofolds Firstly the infection and recov-ery rates are time varying incorporating realistic scenarioswhere the infection and recovery capacities may change overtime [13] As such the transition rates of theMarkov chain areexplicitly time dependent making the previously obtainedsufficient conditions no longer applicable

Secondly the introduction of functions 119892119895(119909) accommo-

dates needed flexibility for some applications Indeed since119883119899

119895isin 0 1 the linear form 119892

119895(119909) = 120581

119895119909 used in [11] implies

that no contribution can be made by susceptible individualsThis is only a rough approximation For one thing susceptibleindividuals aware of a disease in their proximity can takemeasures (such as wearing masks avoiding public placesand frequent hand washing) to reduce their susceptibilitywhich in turn affect the epidemic dynamics dramatically [1415] For another the epidemic progression strongly dependson the contact patterns between susceptible and infectedindividuals especially in the network context Identifyingindividual role is an interesting and demanding task [16]In the present framework each node 119895 applies individualcontribution 119892

119895(0) (and 119892

119895(1)) indicating the underlying

interaction structurestrength among individualsThe rest of the paper is organized as follows We state the

main result in Section 2 and provide the proof in Section 3

2 The Result

In what follows we assume naturally that 119892119895(1) ge 119892

119895(0) ge 0

for all 119895 We will show that the stochastic logistic model (2)converges weakly to the solution of an integral equation asthe population size 119899 rarr infin

Let (119878B) be a measurable space with 119878 sube R2+and

B the Borel 120590-algebra on 119878 Denote by 119862119887(119878) the set of

bounded continuous functions on 119878 Let Ω be the set of 120590-finite measures on 119878 For 119899 isin N and ℎ isin 119862

119887(119878) we define the

measure-valued nonrandom process 120590119899119905 119905 isin R

+ and the

measure-valued Markov process 120588119899119905 119905 isin R

+ by

120590119899

119905(ℎ) = int

119878

ℎ (120583 120582) 120590119899

119905(d120583 d120582)

= 119899minus1

119899

sum

119894=1

(119892119894(1) minus 119892

119894(0)) ℎ (120583

119894119905 120582119894119905)

120588119899

119905(ℎ) = int

119878

ℎ (120583 120582) 120588119899

119905(d120583 d120582)

= 119899minus1

119899

sum

119894=1

(119892119894(1) minus 119892

119894(0))119883

119899

119894119905ℎ (120583119894119905 120582119894119905)

(3)

For some set 119860 let 119863(R+ 119860) be the set of right-continuous

functions with left-hand limits mapping R+to 119860 Thus

for 119899 isin N 120590119899 isin 119863(R+ Ω) and the sample paths of 120588119899

belong to 119863(R+ Ω) since the sample paths of 119883119899 belong

to 119863(R+ 0 1

119899

) (application of Theorem 1616 in [17 page316])

We assume that the following assumptions hold(A1) 119899minus1sum119899

119894=1119892119894(0) rarr 119892 lt infin and sup

1198991198921119899lt infin where

119892119886119899= 119899minus119886

sum119899

119894=1(119892119894(1) minus 119892

119894(0))119886 for 119886 isin N

(A2) 120590119899119905

119889

997888rarr 120590119905uniformly for 119905 isin R

+ and 120588119899

0

119889

997888rarr 1205880in

Ω as 119899 rarr infin where 120590119905and 120588

0are nonrandom

measures and 119889997888rarr means weak convergence (ielim119899rarrinfin

120590119899

119905(ℎ) = 120590

119905(ℎ) and 120588119899

0(ℎ) converges to 120588

0(ℎ)

in distribution as 119899 rarr infin for all ℎ isin 119862119887(119878) [17 page

316])(A3) 119878 is bounded in R2

+

(A4) 119891 R+rarr R+is Lipschitz continuous

(A5) 120582119894119905and 120583119894119905as functions of 119905 are piecewise constant for

119894 = 1 119899

Theorem 1 If assumptions (A1)ndash(A5) hold then the measure-valued Markov process 120588119899

119905converges weakly to a measure-

valued nonrandom process 120588119905isin 119863(R

+ Ω) that is

120588119899119889

997888997888997888rarr 120588 (4)

as 119899 rarr infin where 120588 is the unique solution of

0 = 120588119905(ℎ) minus 120588

0(ℎ) minus int

119905

0

119866120588119904(ℎ) d119904 (5)

with

120588119905(ℎ) = int

119878

ℎ (120583 120582) 120588119905(d120583 d120582)

119866120588119905(ℎ) = int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119905(d1205831015840 d1205821015840) + 119892)120590

119905(d120583 d120582)

minus int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119905(d1205831015840 d1205821015840) + 119892) 120588

119905(d120583 d120582)

minus int

119878

120583ℎ (120583 120582) 120588119905(d120583 d120582)

(6)

Let 1198711(119878) be the space of 120590119905-integrable functions on 119878 We

have the following corollary

Corollary 2 Suppose that 120588 is the unique solution to (5)Thenthere exists a unique function

120593 R+997888rarr 1198711

(119878)

119905 997891997888rarr 120593119905

(7)

with 0 le 120593119905le 1 such that

120588119905(119860) = int

119860

120593119905(120583 120582) 120590

119905(d120583 d120582) (8)

for all Borel set 119860 isin 119878 and

120593119905= 1205930+ int

119905

0

(120582119891(int

119878

120593119904(1205831015840

1205821015840

) 120590119904(d1205831015840 d1205821015840) + 119892)

times (1 minus 120593119904) minus 120583120593

119904) d119904

(9)

Journal of Mathematics 3

3 Proofs

In this section Theorem 1 will be proved through a series oflemmas by tightness and uniqueness arguments [18]The ideaof proofs is similar to that in [11] andwe include the completeproofs here not only for the convenience of the reader butalso to convince the reader that the results do hold in oursetting

Lemma 3 The sequence 120588119899 is tight in119863(R+ Ω)

Proof Recall that [17 Theorem 1627 page 324] 120588119899 is tight in119863(R+ Ω) if and only if 120588119899

119905(ℎ) is tight in 119863(R

+R) for every

ℎ isin 119862119887(119878) According to Aldousrsquos tightness criterion [18

Theorem 1610 page 178] the tightness of 120588119899119905(ℎ) holds if the

following two conditions are satisfied

(A) For each 120576 gt 0 120578 gt 0 and 119903 ge 1 there exist a 1205750gt 0

and an 1198990isin N such that if 120591

119899is a discreteF119899

119905-stopping

time satisfying 120591119899le 119903 then

sup119899ge1198990

sup120575le1205750

119875 (

10038161003816100381610038161003816120588119899

120591119899

(ℎ) minus 120588119899

120591119899+120575(ℎ)

10038161003816100381610038161003816ge 120578) le 120576 (10)

(B) For each 119903 gt 0

lim119886rarrinfin

lim sup119899rarrinfin

119875(sup119905le119903

120588119899

119905(ℎ) gt 119886) = 0 (11)

To show (A) we express the generator of119883119899 by

119866119899119897 (119909) =

119899

sum

119894=1

(119897 (119909 + 119890119899

119894) minus 119897 (119909)) 120582

119894119905(1 minus 119909

119894)

times 119891(119899minus1

119899

sum

119895=1

119892119895(119909119895))

+

119899

sum

119894=1

(119897 (119909 minus 119890119899

119894) minus 119897 (119909)) 120583

119894119905119909119894

(12)

for all real-valued functions 119897(119909) with 119909 = (1199091 119909

119899) isin

0 1119899 Hence using (3) we obtain

119866119899120588119899

119905(ℎ)

= int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840)

+ 119899minus1

119899

sum

119895=1

119892119895(0))120590

119899

119905(d120583 d120582)

minus int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840)

+ 119899minus1

119899

sum

119895=1

119892119895(0))120588

119899

119905(d120583 d120582)

minus int

119878

120583ℎ (120583 120582) 120588119899

119905(d120583 d120582)

(13)

Applying Dynkinrsquos formula to 120588119899119905(ℎ) (see eg [19 Propo-

sition 17 page 162]) we have that

119872119899

119905(ℎ) = 120588

119899

119905(ℎ) minus 120588

119899

0(ℎ) minus int

119905

0

119866119899120588119899

119904(ℎ) d119904 (14)

is a martingale with respect to the filtrationF119899119905= 120590119883

119899

119894119904 0 le

119904 le 119905 and1198721198990(ℎ) = 0 Moreover119872119899

119905(ℎ) is square integrable

The condition (A) holds if for each 120576 gt 0 120578 gt 0 and 119903 ge 1there exist a 120575

0gt 0 and an 119899

0isin N such that for any discrete

F119899119905-stopping time satisfying 120591

119899le 119903

sup119899ge1198990

sup120575le1205750

119875(

100381610038161003816100381610038161003816100381610038161003816

int

120591119899+120575

120591119899

119866119899120588119899

119904(ℎ) d119904

100381610038161003816100381610038161003816100381610038161003816

ge 120578) le 120576 (15)

sup119899ge1198990

sup120575le1205750

119875 (

10038161003816100381610038161003816119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ)

10038161003816100381610038161003816ge 120578) le 120576 (16)

To see (15) we note that1003816100381610038161003816119866119899120588119899

119905(ℎ)1003816100381610038161003816

le int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 120590119899

119905(d120583 d120582)

+ int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 120588119899

119905(d120583 d120582)

+ int

119878

120583ℎ (120583 120582) 120588119899

119905(d120583 d120582)

le int

119878

ℎ (120583 120582)(2120582119891(1198921119899+ 119899minus1

119899

sum

119895=1

119892119895(0)) + 120583)

times 120590119899

119905(d120583 d120582)

le 1198921119899

times sup(120583120582)isin119878

ℎ (120583 120582)(2120582119891(1198921119899+119899minus1

119899

sum

119895=1

119892119895(0))+120583)

(17)

by using (3) It follows fromMarkovrsquos inequality and (17) that

119875(

100381610038161003816100381610038161003816100381610038161003816

int

120591119899+120575

120591119899

119866119899120588119899

119904(ℎ) d119904

100381610038161003816100381610038161003816100381610038161003816

ge 120578)

le 120578minus1

119864(int

120591119899+120575

120591119899

1003816100381610038161003816119866119899120588119899

119904(ℎ)1003816100381610038161003816d119904)

le 120578minus1

1205751198921119899

times sup(120583120582)isin119878

ℎ (120583 120582)(2120582119891(1198921119899+ 119899minus1

119899

sum

119895=1

119892119895(0)) + 120583)

(18)

4 Journal of Mathematics

From assumptions (A1) and (A4) we obtain sup119899119891(1198921119899+

119899minus1

sum119899

119895=1119892119895(0)) lt infin Therefore we can find 120575

0and 119899

0such

that (15) is satisfied by using the assumptions (A1) (A3) (A4)and the fact that ℎ isin 119862

119887(119878)

To prove (16) we need to introduce the quadratic vari-ation process [119872119899(ℎ)] for 119872119899(ℎ) Since 119872119899

119905(ℎ) is a square

integrable martingale it follows from Proposition 61 in [19page 79] that (119872119899(ℎ))2minus[119872119899(ℎ)] is also amartingale By usingMarkovrsquos inequality and martingale properties we obtain

119875 (

10038161003816100381610038161003816119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ)

10038161003816100381610038161003816ge 120578)

le 120578minus2

119864((119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ))

2

)

= 120578minus2

119864(119864 ((119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ))

2

| F119899

120591119899

))

= 120578minus2

(119864 ((119872119899

120591119899+120575(ℎ))

2

) + 119864 ((119872119899

120591119899

(ℎ))

2

)

minus2119864 (119872119899

120591119899

(ℎ) 119864 (119872119899

120591119899+120575(ℎ) | F

119899

120591119899

)) )

= 120578minus2

(119864 ((119872119899

120591119899+120575(ℎ))

2

) minus 119864 ((119872119899

120591119899

(ℎ))

2

))

= 120578minus2

(119864 ((119872119899

120591119899+120575(ℎ))

2

minus [119872119899

(ℎ)]120591119899+120575)

+ 119864 ([119872119899

(ℎ)]120591119899+120575)

minus119864 ((119872119899

120591119899

(ℎ))

2

minus [119872119899

(ℎ)]120591119899

) minus 119864 ([119872119899

(ℎ)]120591119899

))

= 120578minus2

(119864 ([119872119899

(ℎ)]120591119899+120575

minus [119872119899

(ℎ)]120591119899

))

(19)

Applying Dynkinrsquos formula to (120588119899119905(ℎ))2 similarly as in the

derivation of (14) we have that

119899

119905(ℎ) = (120588

119899

119905(ℎ))2

minus (120588119899

0(ℎ))2

minus 2int

119905

0

120588119899

119904(ℎ) 119866119899120588119899

119904(ℎ) d119904 minus 119899minus1 int

119905

0

119866119899120588119899

119904(ℎ) d119904

(20)

is a martingale where

119866119899120588119899

119905(ℎ)

= int

119878

120582ℎ2

(120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 119899

119905(d120583 d120582)

minus int

119878

120582ℎ2

(120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 120588119899

119905(d120583 d120582)

minus int

119878

120583ℎ2

(120583 120582) 120588119899

119905(d120583 d120582)

(21)

with

int

119878

ℎ (120583 120582) 119899

119905(d120583 d120582)

= 119899minus1

119899

sum

119894=1

(119892119894(1) minus 119892

119894(0))2

ℎ (120583119894119905 120582119894119905)

int

119878

ℎ (120583 120582) 120588119899

119905(d120583 d120582)

= 119899minus1

119899

sum

119894=1

(119892119894(1) minus 119892

119894(0))2

119883119899

119894119905ℎ (120583119894119905 120582119894119905)

(22)

By using (14) and (20) we obtain for any 0 le 119904 lt 119905

(120588119899

119905(ℎ) minus 120588

119899

119904(ℎ))2

= (120588119899

119905(ℎ))2

minus (120588119899

119904(ℎ))2

minus 2120588119899

119904(ℎ) (120588

119899

119905(ℎ) minus 120588

119899

119904(ℎ))

= 119899

119905(ℎ) minus

119899

119904(ℎ) + 119899

minus1

int

119905

119904

119866119899120588119899

119903(ℎ) d119903

minus 2120588119899

119904(ℎ) (119872

119899

119905(ℎ) minus 119872

119899

119904(ℎ))

+ 2int

119905

119904

120588119899

119903(ℎ) 119866119899120588119899

119903(ℎ) d119903

minus 2120588119899

119904(ℎ) int

119905

119904

119866119899120588119899

119903(ℎ) d119903

(23)

It follows from (3) and the assumption (A5) that 120588119899119905(ℎ) has

piecewise constant sample paths and int1199050

119866119899120588119899

119904(ℎ)d119904 is a con-

tinuous finite variation process Therefore (14) (23) and thefact 119899

0(ℎ) = 0 imply that

[119872119899

(ℎ)]119905= [120588119899

(ℎ)]119905

= 119899

119905(ℎ) minus 2int

119905

0

120588119899

119904(ℎ) 119889119872

119899

119904(ℎ)

+ 119899minus1

int

119905

0

119866119899120588119899

119904(ℎ) d119904

(24)

From (19) (24) and martingale properties we obtain

119875 (

10038161003816100381610038161003816119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ)

10038161003816100381610038161003816ge 120578)

le 119899minus1

120578minus2

119864(int

120591119899+120575

120591119899

119866119899120588119899

119904(ℎ) d119904)

le 119899minus1

120578minus2

times int

120591119899+120575

120591119899

int

119878

ℎ2

(120583 120582)(2120582119891(1198921119899+ 119899minus1

119899

sum

119895=1

119892119895(0)) + 120583)

times 119899

119904(d120583 d120582) d119904

le 119899minus1

120578minus2

1205751198922119899

times sup(120583120582)isin119878

ℎ2

(120583 120582)(2120582119891(1198921119899+119899minus1

119899

sum

119895=1

119892119895(0))+120583)

(25)

Journal of Mathematics 5

The assumption (A1) implies that lim sup119894rarrinfin

(119892119894(1)minus119892

119894(0)) lt

infin and hence sup1198991198922119899lt infinTherefore as in the derivation of

(15) we can choose some 1198990and 1205750such that (16) is satisfied

Now the only thing remaining to verify is the condition(B) Since ℎ isin 119862

119887(119878) and the assumption (A3) holds there

exists a constant 119862 gt 0 such that

sup119905le119903

1003816100381610038161003816120588119899

119905(ℎ)1003816100381610038161003816le 119862 119892

1119899 (26)

for all 119899 isin N and 119903 gt 0 FromMarkovrsquos inequality and the as-sumption (A1) we have

lim sup119899rarrinfin

119875(sup119905le119903

120588119899

119905(ℎ) gt 119886) le lim sup

119899rarrinfin

119886minus1

119864(sup119905le119903

1003816100381610038161003816120588119899

119905(ℎ)1003816100381610038161003816)

le 119886minus1

119862 sup119899

1198921119899997888rarr 0

(27)

as 119886 rarr infin The proof of Lemma 3 is complete

Lemma 4 For any ℎ isin 119862119887(119878) 119872119899(ℎ) 119889997888rarr 0 in 119863(R

+R) as

119899 rarr infin

Proof By the corollary in [18 page 28] it is sufficient toshow that 1198890

infin(119872119899

(ℎ) 0)

119901

997888rarr 0 as 119899 rarr infin where119901

997888rarr meansconvergence in probability Here by definition the space119863(R+R) is the so-called Skorohod space 119863[0infin) and 1198890

infin

is the metric on it which defines the Skorohod topology (seeeg [18 page 168]) Theorem 167 in [18 page 174] furtherimplies that it is sufficient to show that 1198890

119905(119872119899

(ℎ) 0)

119901

997888rarr 0 foreach 119905 ge 0 Themetric 1198890

119905generates the topology of Skorohod

space119863[0 119905] (see [18 pages 166 and 125] for definitions)Since 1198890

119905(119872119899

(ℎ) 0) le sup119904le119905|119872119899

119904(ℎ)| by Doobrsquos martin-

gale inequality we have for any 120576 gt 0

119875 (1198890

119905(119872119899

(ℎ) 0) ge 120576) le 119875( sup119904isin[0119905]

1003816100381610038161003816119872119899

119904(ℎ)1003816100381610038161003816ge 120576)

le 120576minus2

119864 ((119872119899

119905(ℎ))2

)

= 120576minus2

119864[119872119899

(ℎ)]119905

= 120576minus2

119899minus1

119864(int

119905

0

119866119899120588119899

119904(ℎ) d119904)

(28)

where the first equality uses the fact that (119872119899119905(ℎ))2

minus[119872119899

(ℎ)]119905

is a martingale with expectation 0 and the second equalityfollows from (24) It then follows from the derivation in (25)that

lim119899rarrinfin

119875 (1198890

119905(119872119899

(ℎ) 0) ge 120576) = 0 (29)

which concludes the proof of Lemma 4

From Lemma 3 the sequence 120588119899 is tight and then thereexists a weakly convergent subsequence 120588119899119896 in119863(R

+ Ω) We

still denote it by 120588119899 in the next lemma for convenience

Lemma 5 If 120588119899 119889997888rarr 120588 as 119899 rarr infin then for any ℎ isin 119862119887(119878)

119872119899

(ℎ)

119889

997888rarr 119872(ℎ) in119863(R+R) as 119899 rarr infin where

119872119905(ℎ) = 120588

119905(ℎ) minus 120588

0(ℎ) minus int

119905

0

119866120588119904(ℎ) d119904 (30)

Proof For each 119899 isin N define a random process119873119899(ℎ) by

119873119899

119905(ℎ) = 120588

119899

119905(ℎ) minus 120588

119899

0(ℎ) minus int

119905

0

119866120588119899

119904(ℎ) d119904 (31)

From Theorem 31 in [18 page 28] and the arguments in theproof of Lemma 4 it is sufficient to show the following twoconditions

(A) sup119904le119905|119872119899

119904(ℎ)minus119873

119899

119904(ℎ)|

119901

997888rarr 0 as 119899 rarr infin for each 119905 ge 0

(B) 119873119899(ℎ) 119889997888rarr 119872(ℎ) as 119899 rarr infin

To show (A) note that

sup119904le119905

1003816100381610038161003816119872119899

119904(ℎ) minus 119873

119899

119904(ℎ)1003816100381610038161003816

= int

119905

0

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119904(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times (120590119899

119904(d120583 d120582) minus 120590

119904(d120583 d120582))

10038161003816100381610038161003816100381610038161003816100381610038161003816

d119904

(32)

From the argument following (17) there is some constant119862 gt0 satisfying

119891(int

119878

120588119899

119904(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0)) le 119862 (33)

Hence by the dominated convergence theorem

sup119904le119905

1003816100381610038161003816119872119899

119904(ℎ) minus 119873

119899

119904(ℎ)1003816100381610038161003816

le 119862int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) (120590119899

119904(d120583 d120582) minus 120590

119904(d120583 d120582))

10038161003816100381610038161003816100381610038161003816

d119904119901

997888997888997888rarr 0

(34)

as 119899 rarr infin since 120582ℎ isin 119862119887(119878) and the assumption (A2) holds

To show (B) it is sufficient to show that [18 Theorem 21page 16]

119864 (119897 (119873119899

(ℎ))) 997888rarr 119864 (119897 (119872 (ℎ))) (35)

as 119899 rarr infin for any bounded uniformly continuous function119897 119863(R

+R) rarr R By the definition of 119873119899(ℎ) and the

assumption 120588119899 119889997888rarr 120588 we only need to show that 119897(119873119899(ℎ))is a continuous function of 120588119899 mapping 119863(R

+ Ω) to R

invoking the dominated convergence theorem Furthermoreit is sufficient to show that119873119899(ℎ) is a continuous function of120588119899 mapping119863(R

+ Ω) to119863(R

+R) In the following we take

one term in119873119899(ℎ) as an example to show the continuity withrespect to 120588119899 Other terms can be shownanalogously

6 Journal of Mathematics

Define

119897119899

119905= int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119892) 120588119899

119905(d120583 d120582)

119897119905= int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119905(d1205831015840 d1205821015840) + 119892) 120588

119905(d120583 d120582)

(36)

Suppose that 120588119899 rarr 120588 in 119863(R+ Ω) holds We need to

show int1199050

119897119899

119904d119904 rarr int

119905

0

119897119904d119904 in 119863(R

+R) as 119899 rarr infin Indeed

since 120582ℎ isin 119862119887(119878) and int

119878

ℎ(120583 120582)120588119905(d120583 d120582) 119863(R

+ Ω) rarr

119863(R+R) is continuous for any ℎ isin 119862

119887(119878) we have

int119878

120582ℎ(120583 120582)120588119899

119905(d120583 d120582) rarr int

119878

120582ℎ(120583 120582)120588119905(d120583 d120582) in 119863(R

+R)

and int119878

120588119899

119905(d120583 d120582) rarr int

119878

120588119905(d120583 d120582) in 119863(R

+R) as 119899 rarr

infin Hence from the assumptions (A3) and (A4) we obtain119897119899

119905rarr 119897119905in 119863(R

+R) It follows from [18 Equation (1214)

page 124] that 119897119899119905

rarr 119897119905for all but countably many 119905

The dominated convergence theorem then yields int1199050

119897119899

119904d119904 rarr

int

119905

0

119897119904d119904 in supremum norm on finite time intervals and hence

in119863(R+R)

Lemma 6 The solution to (5) is unique in119863(R+ Ω)

Proof Suppose that 120588 and 120588 are two solutions to (5) and theyare the limits of two weakly convergent subsequences of 120588119899respectively By the assumption (A2) 120588

0= 1205880 We need to

show that 120588119905= 120588119905for all 119905 isin R

+

From (3) we have

int

119878

ℎ (120583 120582) 120588119899

119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119899

119905(d120583 d120582) (37)

for any ℎ isin 119862119887(119878) Therefore by the assumption (A2)

int

119878

ℎ (120583 120582) 120588119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119905(d120583 d120582) (38)

int

119878

ℎ (120583 120582) 120588119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119905(d120583 d120582) (39)

hold Define 119889(120588119905 120588119905) = sup

|ℎ|le1ℎisin119862119887(119878)|120588119905(ℎ) minus 120588

119905(ℎ)| In view

of (5) we obtain

1003816100381610038161003816120588119905(ℎ) minus 120588

119905(ℎ)1003816100381610038161003816

= int

119905

0

119866120588119904(ℎ) minus 119866120588

119904(ℎ) d119904

le int

119905

0

int

119878

120582ℎ (120583 120582)

10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

minus119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

times 120590119904(d120583 d120582) d119904

+ int

119905

0

int

119878

120582ℎ (120583 120582)

10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

minus119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

times 120588119904(d120583 d120582) d119904

+ int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) 120588119904(d120583 d120582) minus int

119878

120582ℎ (120583 120582) 120588119904(d120583 d120582)

10038161003816100381610038161003816100381610038161003816

times 119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892) d119904

+ int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120583ℎ (120583 120582) 120588119904(d120583 d120582)

minusint

119878

120583ℎ (120583 120582) 120588119904(d120583 d120582)

10038161003816100381610038161003816100381610038161003816

d119904

(40)

The assumption (A4) indicates that10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892) minus 119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

le 119862119891

10038161003816100381610038161003816100381610038161003816

int

119878

120588119904(d1205831015840 d1205821015840) minus int

119878

120588119904(d1205831015840 d1205821015840)

10038161003816100381610038161003816100381610038161003816

le 119862119891119889 (120588119904 120588119904)

(41)

for some constant119862119891gt 0Therefore using (38) we can derive

1003816100381610038161003816120588119905(ℎ) minus 120588

119905(ℎ)1003816100381610038161003816

le 2119862119891int

119905

0

(int

119878

120582ℎ (120583 120582) 120590119904(d120583 d120582)) 119889 (120588

119904 120588119904) d119904

+ 119862( sup(120583120582)isin119878

120582)int

119905

0

119889 (120588119904 120588119904) d119904

+ ( sup(120583120582)isin119878

120583)int

119905

0

119889 (120588119904 120588119904) d119904

(42)

where 119862 gt 0 is some constant To see this note that by theassumptions (A1) (A2) and (38) we have

int

119878

120588119904(d1205831015840 d1205821015840) + 119892 le int

119878

120590119904(d1205831015840 d1205821015840) + 119892 lt infin (43)

for any 0 le 119904 le 119905 The assumption (A4) then implies

sup119904le119905

119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892) le 119862 lt infin (44)

Again employing the assumption (A2) we have int119878

120582ℎ(120583

120582)120590119904(d120583 d120582) lt infin for all 0 le 119904 le 119905 Hence by using the

assumption (A3) the inequality (42) becomes

119889 (120588119904 120588119904) le 1198621015840

int

119905

0

119889 (120588119904 120588119904) d119904 (45)

for some constant 1198621015840 gt 0 A simple application of Gronwallrsquoslemma yields 119889(120588

119905 120588119905) = 0 for all 119905 isin R

+ which concludes the

proof

Proof of Theorem 1 By Lemma 4 and Lemma 5 the limit ofany weakly convergent subsequence of 120588119899 must satisfy (5) ByLemma 6 we find that the sequence 120588119899 must itself convergeweakly to that unique solution (see the corollary in [18 page59]) The proof of Theorem 1 is thus completed

Journal of Mathematics 7

Proof of Corollary 2 For any open set 119865 sube 119878 we obtain from(38) that

int

119878

119891119899

(120583 120582) 120588119905(d120583 d120582) le int

119878

119891119899

(120583 120582) 120590119905(d120583 d120582) (46)

where119891119899 is taken as a continuous function upwardly converg-ing to the indicator function of 119865 By using the dominatedconvergence theorem we know that 120588

119905(119865) le 120590

119905(119865) A reg-

ularity property (see eg [17 page 18 Lemma 134]) impliesthat 120588

119905(119860) le 120590

119905(119860) for all 119860 isin 119878 This means that 120588

119905is

absolutely continuous with respect to 120590119905for any 119905 ge 0 An

application of the Radon-Nykodym theorem yields the exis-tence of 120593

119905such that (8) holds with 0 le 120593

119905le 1 Now the result

follows straightforwardly fromTheorem 1

Acknowledgments

The author expresses his sincere gratitude to the anonymousreferee and the editor for careful reading of the original paperand useful comments that helped to improve presentation ofresults

References

[1] G H Weiss and M Dishon ldquoOn the asymptotic behaviorof the stochastic and deterministic models of an epidemicrdquoMathematical Biosciences vol 11 pp 261ndash265 1971

[2] D A Kessler ldquoEpidemic size in the SIS model of endemicinfectionsrdquo Journal of Applied Probability vol 45 no 3 pp 757ndash778 2008

[3] Y Shang ldquoA Lie algebra approach to susceptible-infected-susceptible epidemicsrdquo Electronic Journal of Differential Equa-tions vol 2012 no 233 pp 1ndash7 2012

[4] H Andersson and B Djehiche ldquoA threshold limit theorem forthe stochastic logistic epidemicrdquo Journal of Applied Probabilityvol 35 no 3 pp 662ndash670 1998

[5] C R Doering K V Sargsyan and L M Sander ldquoExtinc-tion times for birth-death processes exact results continuumasymptotics and the failure of the Fokker-Planck approxima-tionrdquo Multiscale Modeling amp Simulation vol 3 no 2 pp 283ndash299 2005

[6] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[7] O Ovaskainen ldquoThe quasistationary distribution of thestochastic logistic modelrdquo Journal of Applied Probability vol 38no 4 pp 898ndash907 2001

[8] D Clancy and P K Pollett ldquoA note on quasi-stationary distri-butions of birth-death processes and the SIS logistic epidemicrdquoJournal of Applied Probability vol 40 no 3 pp 821ndash825 2003

[9] TGKurtz ldquoExtensions of Trotterrsquos operator semigroup approx-imation theoremsrdquo Journal of Functional Analysis vol 3 pp354ndash375 1969

[10] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970

[11] R McVinish and P K Pollett ldquoThe deterministic limit of astochastic logistic model with individual variationrdquoMathemat-ical Biosciences vol 241 no 1 pp 109ndash114 2013

[12] R W R Darling and J R Norris ldquoDifferential equationapproximations for Markov chainsrdquo Probability Surveys vol 5pp 37ndash79 2008

[13] Y Shang ldquoOptimal control strategies for virus spreading ininhomogeneousepidemicdynamicsrdquo Canadian MathematicalBulletin In press

[14] T Gross C J D DrsquoLima and B Blasius ldquoEpidemic dynamicson an adaptivenetworkrdquo Physical Review Letters vol 96 no 20Article ID 208701 2006

[15] Y Shang ldquoDiscrete-time epidemic dynamics with awareness inrandomnetworksrdquo International Journal of Biomathematics vol6 no 2 Article ID 1350007 2013

[16] K Klemm M A Serrano V M Eguıluz and M S MiguelldquoA measure of individualrole in collective dynamicsrdquo ScientificReports vol 2 article 292 2012

[17] O Kallenberg Foundations of Modern Probability SpringerNew York NY USA 2nd edition 2002

[18] P Billingsley Convergence of Probability Measures John Wileyamp Sons New York NY USA 2nd edition 1999

[19] S N Ethier and T G KurtzMarkov Processes Characterizationand Convergence JohnWiley amp Sons Hoboken NJ USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Mathematics

represent that individual 119894 is infected Then 119884119899 = sum119899119894=1119883119899

119894is

equivalent to the stochastic logistic model by setting 120582119894119905= 120582

120583119894119905= 120583 119891(119909) = 119909 and 119892

119895(119909) = 119909 for all 119894 119895 and 119905

The above model (2) can be viewed as a generalization ofthat treated in [11] in twofolds Firstly the infection and recov-ery rates are time varying incorporating realistic scenarioswhere the infection and recovery capacities may change overtime [13] As such the transition rates of theMarkov chain areexplicitly time dependent making the previously obtainedsufficient conditions no longer applicable

Secondly the introduction of functions 119892119895(119909) accommo-

dates needed flexibility for some applications Indeed since119883119899

119895isin 0 1 the linear form 119892

119895(119909) = 120581

119895119909 used in [11] implies

that no contribution can be made by susceptible individualsThis is only a rough approximation For one thing susceptibleindividuals aware of a disease in their proximity can takemeasures (such as wearing masks avoiding public placesand frequent hand washing) to reduce their susceptibilitywhich in turn affect the epidemic dynamics dramatically [1415] For another the epidemic progression strongly dependson the contact patterns between susceptible and infectedindividuals especially in the network context Identifyingindividual role is an interesting and demanding task [16]In the present framework each node 119895 applies individualcontribution 119892

119895(0) (and 119892

119895(1)) indicating the underlying

interaction structurestrength among individualsThe rest of the paper is organized as follows We state the

main result in Section 2 and provide the proof in Section 3

2 The Result

In what follows we assume naturally that 119892119895(1) ge 119892

119895(0) ge 0

for all 119895 We will show that the stochastic logistic model (2)converges weakly to the solution of an integral equation asthe population size 119899 rarr infin

Let (119878B) be a measurable space with 119878 sube R2+and

B the Borel 120590-algebra on 119878 Denote by 119862119887(119878) the set of

bounded continuous functions on 119878 Let Ω be the set of 120590-finite measures on 119878 For 119899 isin N and ℎ isin 119862

119887(119878) we define the

measure-valued nonrandom process 120590119899119905 119905 isin R

+ and the

measure-valued Markov process 120588119899119905 119905 isin R

+ by

120590119899

119905(ℎ) = int

119878

ℎ (120583 120582) 120590119899

119905(d120583 d120582)

= 119899minus1

119899

sum

119894=1

(119892119894(1) minus 119892

119894(0)) ℎ (120583

119894119905 120582119894119905)

120588119899

119905(ℎ) = int

119878

ℎ (120583 120582) 120588119899

119905(d120583 d120582)

= 119899minus1

119899

sum

119894=1

(119892119894(1) minus 119892

119894(0))119883

119899

119894119905ℎ (120583119894119905 120582119894119905)

(3)

For some set 119860 let 119863(R+ 119860) be the set of right-continuous

functions with left-hand limits mapping R+to 119860 Thus

for 119899 isin N 120590119899 isin 119863(R+ Ω) and the sample paths of 120588119899

belong to 119863(R+ Ω) since the sample paths of 119883119899 belong

to 119863(R+ 0 1

119899

) (application of Theorem 1616 in [17 page316])

We assume that the following assumptions hold(A1) 119899minus1sum119899

119894=1119892119894(0) rarr 119892 lt infin and sup

1198991198921119899lt infin where

119892119886119899= 119899minus119886

sum119899

119894=1(119892119894(1) minus 119892

119894(0))119886 for 119886 isin N

(A2) 120590119899119905

119889

997888rarr 120590119905uniformly for 119905 isin R

+ and 120588119899

0

119889

997888rarr 1205880in

Ω as 119899 rarr infin where 120590119905and 120588

0are nonrandom

measures and 119889997888rarr means weak convergence (ielim119899rarrinfin

120590119899

119905(ℎ) = 120590

119905(ℎ) and 120588119899

0(ℎ) converges to 120588

0(ℎ)

in distribution as 119899 rarr infin for all ℎ isin 119862119887(119878) [17 page

316])(A3) 119878 is bounded in R2

+

(A4) 119891 R+rarr R+is Lipschitz continuous

(A5) 120582119894119905and 120583119894119905as functions of 119905 are piecewise constant for

119894 = 1 119899

Theorem 1 If assumptions (A1)ndash(A5) hold then the measure-valued Markov process 120588119899

119905converges weakly to a measure-

valued nonrandom process 120588119905isin 119863(R

+ Ω) that is

120588119899119889

997888997888997888rarr 120588 (4)

as 119899 rarr infin where 120588 is the unique solution of

0 = 120588119905(ℎ) minus 120588

0(ℎ) minus int

119905

0

119866120588119904(ℎ) d119904 (5)

with

120588119905(ℎ) = int

119878

ℎ (120583 120582) 120588119905(d120583 d120582)

119866120588119905(ℎ) = int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119905(d1205831015840 d1205821015840) + 119892)120590

119905(d120583 d120582)

minus int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119905(d1205831015840 d1205821015840) + 119892) 120588

119905(d120583 d120582)

minus int

119878

120583ℎ (120583 120582) 120588119905(d120583 d120582)

(6)

Let 1198711(119878) be the space of 120590119905-integrable functions on 119878 We

have the following corollary

Corollary 2 Suppose that 120588 is the unique solution to (5)Thenthere exists a unique function

120593 R+997888rarr 1198711

(119878)

119905 997891997888rarr 120593119905

(7)

with 0 le 120593119905le 1 such that

120588119905(119860) = int

119860

120593119905(120583 120582) 120590

119905(d120583 d120582) (8)

for all Borel set 119860 isin 119878 and

120593119905= 1205930+ int

119905

0

(120582119891(int

119878

120593119904(1205831015840

1205821015840

) 120590119904(d1205831015840 d1205821015840) + 119892)

times (1 minus 120593119904) minus 120583120593

119904) d119904

(9)

Journal of Mathematics 3

3 Proofs

In this section Theorem 1 will be proved through a series oflemmas by tightness and uniqueness arguments [18]The ideaof proofs is similar to that in [11] andwe include the completeproofs here not only for the convenience of the reader butalso to convince the reader that the results do hold in oursetting

Lemma 3 The sequence 120588119899 is tight in119863(R+ Ω)

Proof Recall that [17 Theorem 1627 page 324] 120588119899 is tight in119863(R+ Ω) if and only if 120588119899

119905(ℎ) is tight in 119863(R

+R) for every

ℎ isin 119862119887(119878) According to Aldousrsquos tightness criterion [18

Theorem 1610 page 178] the tightness of 120588119899119905(ℎ) holds if the

following two conditions are satisfied

(A) For each 120576 gt 0 120578 gt 0 and 119903 ge 1 there exist a 1205750gt 0

and an 1198990isin N such that if 120591

119899is a discreteF119899

119905-stopping

time satisfying 120591119899le 119903 then

sup119899ge1198990

sup120575le1205750

119875 (

10038161003816100381610038161003816120588119899

120591119899

(ℎ) minus 120588119899

120591119899+120575(ℎ)

10038161003816100381610038161003816ge 120578) le 120576 (10)

(B) For each 119903 gt 0

lim119886rarrinfin

lim sup119899rarrinfin

119875(sup119905le119903

120588119899

119905(ℎ) gt 119886) = 0 (11)

To show (A) we express the generator of119883119899 by

119866119899119897 (119909) =

119899

sum

119894=1

(119897 (119909 + 119890119899

119894) minus 119897 (119909)) 120582

119894119905(1 minus 119909

119894)

times 119891(119899minus1

119899

sum

119895=1

119892119895(119909119895))

+

119899

sum

119894=1

(119897 (119909 minus 119890119899

119894) minus 119897 (119909)) 120583

119894119905119909119894

(12)

for all real-valued functions 119897(119909) with 119909 = (1199091 119909

119899) isin

0 1119899 Hence using (3) we obtain

119866119899120588119899

119905(ℎ)

= int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840)

+ 119899minus1

119899

sum

119895=1

119892119895(0))120590

119899

119905(d120583 d120582)

minus int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840)

+ 119899minus1

119899

sum

119895=1

119892119895(0))120588

119899

119905(d120583 d120582)

minus int

119878

120583ℎ (120583 120582) 120588119899

119905(d120583 d120582)

(13)

Applying Dynkinrsquos formula to 120588119899119905(ℎ) (see eg [19 Propo-

sition 17 page 162]) we have that

119872119899

119905(ℎ) = 120588

119899

119905(ℎ) minus 120588

119899

0(ℎ) minus int

119905

0

119866119899120588119899

119904(ℎ) d119904 (14)

is a martingale with respect to the filtrationF119899119905= 120590119883

119899

119894119904 0 le

119904 le 119905 and1198721198990(ℎ) = 0 Moreover119872119899

119905(ℎ) is square integrable

The condition (A) holds if for each 120576 gt 0 120578 gt 0 and 119903 ge 1there exist a 120575

0gt 0 and an 119899

0isin N such that for any discrete

F119899119905-stopping time satisfying 120591

119899le 119903

sup119899ge1198990

sup120575le1205750

119875(

100381610038161003816100381610038161003816100381610038161003816

int

120591119899+120575

120591119899

119866119899120588119899

119904(ℎ) d119904

100381610038161003816100381610038161003816100381610038161003816

ge 120578) le 120576 (15)

sup119899ge1198990

sup120575le1205750

119875 (

10038161003816100381610038161003816119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ)

10038161003816100381610038161003816ge 120578) le 120576 (16)

To see (15) we note that1003816100381610038161003816119866119899120588119899

119905(ℎ)1003816100381610038161003816

le int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 120590119899

119905(d120583 d120582)

+ int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 120588119899

119905(d120583 d120582)

+ int

119878

120583ℎ (120583 120582) 120588119899

119905(d120583 d120582)

le int

119878

ℎ (120583 120582)(2120582119891(1198921119899+ 119899minus1

119899

sum

119895=1

119892119895(0)) + 120583)

times 120590119899

119905(d120583 d120582)

le 1198921119899

times sup(120583120582)isin119878

ℎ (120583 120582)(2120582119891(1198921119899+119899minus1

119899

sum

119895=1

119892119895(0))+120583)

(17)

by using (3) It follows fromMarkovrsquos inequality and (17) that

119875(

100381610038161003816100381610038161003816100381610038161003816

int

120591119899+120575

120591119899

119866119899120588119899

119904(ℎ) d119904

100381610038161003816100381610038161003816100381610038161003816

ge 120578)

le 120578minus1

119864(int

120591119899+120575

120591119899

1003816100381610038161003816119866119899120588119899

119904(ℎ)1003816100381610038161003816d119904)

le 120578minus1

1205751198921119899

times sup(120583120582)isin119878

ℎ (120583 120582)(2120582119891(1198921119899+ 119899minus1

119899

sum

119895=1

119892119895(0)) + 120583)

(18)

4 Journal of Mathematics

From assumptions (A1) and (A4) we obtain sup119899119891(1198921119899+

119899minus1

sum119899

119895=1119892119895(0)) lt infin Therefore we can find 120575

0and 119899

0such

that (15) is satisfied by using the assumptions (A1) (A3) (A4)and the fact that ℎ isin 119862

119887(119878)

To prove (16) we need to introduce the quadratic vari-ation process [119872119899(ℎ)] for 119872119899(ℎ) Since 119872119899

119905(ℎ) is a square

integrable martingale it follows from Proposition 61 in [19page 79] that (119872119899(ℎ))2minus[119872119899(ℎ)] is also amartingale By usingMarkovrsquos inequality and martingale properties we obtain

119875 (

10038161003816100381610038161003816119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ)

10038161003816100381610038161003816ge 120578)

le 120578minus2

119864((119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ))

2

)

= 120578minus2

119864(119864 ((119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ))

2

| F119899

120591119899

))

= 120578minus2

(119864 ((119872119899

120591119899+120575(ℎ))

2

) + 119864 ((119872119899

120591119899

(ℎ))

2

)

minus2119864 (119872119899

120591119899

(ℎ) 119864 (119872119899

120591119899+120575(ℎ) | F

119899

120591119899

)) )

= 120578minus2

(119864 ((119872119899

120591119899+120575(ℎ))

2

) minus 119864 ((119872119899

120591119899

(ℎ))

2

))

= 120578minus2

(119864 ((119872119899

120591119899+120575(ℎ))

2

minus [119872119899

(ℎ)]120591119899+120575)

+ 119864 ([119872119899

(ℎ)]120591119899+120575)

minus119864 ((119872119899

120591119899

(ℎ))

2

minus [119872119899

(ℎ)]120591119899

) minus 119864 ([119872119899

(ℎ)]120591119899

))

= 120578minus2

(119864 ([119872119899

(ℎ)]120591119899+120575

minus [119872119899

(ℎ)]120591119899

))

(19)

Applying Dynkinrsquos formula to (120588119899119905(ℎ))2 similarly as in the

derivation of (14) we have that

119899

119905(ℎ) = (120588

119899

119905(ℎ))2

minus (120588119899

0(ℎ))2

minus 2int

119905

0

120588119899

119904(ℎ) 119866119899120588119899

119904(ℎ) d119904 minus 119899minus1 int

119905

0

119866119899120588119899

119904(ℎ) d119904

(20)

is a martingale where

119866119899120588119899

119905(ℎ)

= int

119878

120582ℎ2

(120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 119899

119905(d120583 d120582)

minus int

119878

120582ℎ2

(120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 120588119899

119905(d120583 d120582)

minus int

119878

120583ℎ2

(120583 120582) 120588119899

119905(d120583 d120582)

(21)

with

int

119878

ℎ (120583 120582) 119899

119905(d120583 d120582)

= 119899minus1

119899

sum

119894=1

(119892119894(1) minus 119892

119894(0))2

ℎ (120583119894119905 120582119894119905)

int

119878

ℎ (120583 120582) 120588119899

119905(d120583 d120582)

= 119899minus1

119899

sum

119894=1

(119892119894(1) minus 119892

119894(0))2

119883119899

119894119905ℎ (120583119894119905 120582119894119905)

(22)

By using (14) and (20) we obtain for any 0 le 119904 lt 119905

(120588119899

119905(ℎ) minus 120588

119899

119904(ℎ))2

= (120588119899

119905(ℎ))2

minus (120588119899

119904(ℎ))2

minus 2120588119899

119904(ℎ) (120588

119899

119905(ℎ) minus 120588

119899

119904(ℎ))

= 119899

119905(ℎ) minus

119899

119904(ℎ) + 119899

minus1

int

119905

119904

119866119899120588119899

119903(ℎ) d119903

minus 2120588119899

119904(ℎ) (119872

119899

119905(ℎ) minus 119872

119899

119904(ℎ))

+ 2int

119905

119904

120588119899

119903(ℎ) 119866119899120588119899

119903(ℎ) d119903

minus 2120588119899

119904(ℎ) int

119905

119904

119866119899120588119899

119903(ℎ) d119903

(23)

It follows from (3) and the assumption (A5) that 120588119899119905(ℎ) has

piecewise constant sample paths and int1199050

119866119899120588119899

119904(ℎ)d119904 is a con-

tinuous finite variation process Therefore (14) (23) and thefact 119899

0(ℎ) = 0 imply that

[119872119899

(ℎ)]119905= [120588119899

(ℎ)]119905

= 119899

119905(ℎ) minus 2int

119905

0

120588119899

119904(ℎ) 119889119872

119899

119904(ℎ)

+ 119899minus1

int

119905

0

119866119899120588119899

119904(ℎ) d119904

(24)

From (19) (24) and martingale properties we obtain

119875 (

10038161003816100381610038161003816119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ)

10038161003816100381610038161003816ge 120578)

le 119899minus1

120578minus2

119864(int

120591119899+120575

120591119899

119866119899120588119899

119904(ℎ) d119904)

le 119899minus1

120578minus2

times int

120591119899+120575

120591119899

int

119878

ℎ2

(120583 120582)(2120582119891(1198921119899+ 119899minus1

119899

sum

119895=1

119892119895(0)) + 120583)

times 119899

119904(d120583 d120582) d119904

le 119899minus1

120578minus2

1205751198922119899

times sup(120583120582)isin119878

ℎ2

(120583 120582)(2120582119891(1198921119899+119899minus1

119899

sum

119895=1

119892119895(0))+120583)

(25)

Journal of Mathematics 5

The assumption (A1) implies that lim sup119894rarrinfin

(119892119894(1)minus119892

119894(0)) lt

infin and hence sup1198991198922119899lt infinTherefore as in the derivation of

(15) we can choose some 1198990and 1205750such that (16) is satisfied

Now the only thing remaining to verify is the condition(B) Since ℎ isin 119862

119887(119878) and the assumption (A3) holds there

exists a constant 119862 gt 0 such that

sup119905le119903

1003816100381610038161003816120588119899

119905(ℎ)1003816100381610038161003816le 119862 119892

1119899 (26)

for all 119899 isin N and 119903 gt 0 FromMarkovrsquos inequality and the as-sumption (A1) we have

lim sup119899rarrinfin

119875(sup119905le119903

120588119899

119905(ℎ) gt 119886) le lim sup

119899rarrinfin

119886minus1

119864(sup119905le119903

1003816100381610038161003816120588119899

119905(ℎ)1003816100381610038161003816)

le 119886minus1

119862 sup119899

1198921119899997888rarr 0

(27)

as 119886 rarr infin The proof of Lemma 3 is complete

Lemma 4 For any ℎ isin 119862119887(119878) 119872119899(ℎ) 119889997888rarr 0 in 119863(R

+R) as

119899 rarr infin

Proof By the corollary in [18 page 28] it is sufficient toshow that 1198890

infin(119872119899

(ℎ) 0)

119901

997888rarr 0 as 119899 rarr infin where119901

997888rarr meansconvergence in probability Here by definition the space119863(R+R) is the so-called Skorohod space 119863[0infin) and 1198890

infin

is the metric on it which defines the Skorohod topology (seeeg [18 page 168]) Theorem 167 in [18 page 174] furtherimplies that it is sufficient to show that 1198890

119905(119872119899

(ℎ) 0)

119901

997888rarr 0 foreach 119905 ge 0 Themetric 1198890

119905generates the topology of Skorohod

space119863[0 119905] (see [18 pages 166 and 125] for definitions)Since 1198890

119905(119872119899

(ℎ) 0) le sup119904le119905|119872119899

119904(ℎ)| by Doobrsquos martin-

gale inequality we have for any 120576 gt 0

119875 (1198890

119905(119872119899

(ℎ) 0) ge 120576) le 119875( sup119904isin[0119905]

1003816100381610038161003816119872119899

119904(ℎ)1003816100381610038161003816ge 120576)

le 120576minus2

119864 ((119872119899

119905(ℎ))2

)

= 120576minus2

119864[119872119899

(ℎ)]119905

= 120576minus2

119899minus1

119864(int

119905

0

119866119899120588119899

119904(ℎ) d119904)

(28)

where the first equality uses the fact that (119872119899119905(ℎ))2

minus[119872119899

(ℎ)]119905

is a martingale with expectation 0 and the second equalityfollows from (24) It then follows from the derivation in (25)that

lim119899rarrinfin

119875 (1198890

119905(119872119899

(ℎ) 0) ge 120576) = 0 (29)

which concludes the proof of Lemma 4

From Lemma 3 the sequence 120588119899 is tight and then thereexists a weakly convergent subsequence 120588119899119896 in119863(R

+ Ω) We

still denote it by 120588119899 in the next lemma for convenience

Lemma 5 If 120588119899 119889997888rarr 120588 as 119899 rarr infin then for any ℎ isin 119862119887(119878)

119872119899

(ℎ)

119889

997888rarr 119872(ℎ) in119863(R+R) as 119899 rarr infin where

119872119905(ℎ) = 120588

119905(ℎ) minus 120588

0(ℎ) minus int

119905

0

119866120588119904(ℎ) d119904 (30)

Proof For each 119899 isin N define a random process119873119899(ℎ) by

119873119899

119905(ℎ) = 120588

119899

119905(ℎ) minus 120588

119899

0(ℎ) minus int

119905

0

119866120588119899

119904(ℎ) d119904 (31)

From Theorem 31 in [18 page 28] and the arguments in theproof of Lemma 4 it is sufficient to show the following twoconditions

(A) sup119904le119905|119872119899

119904(ℎ)minus119873

119899

119904(ℎ)|

119901

997888rarr 0 as 119899 rarr infin for each 119905 ge 0

(B) 119873119899(ℎ) 119889997888rarr 119872(ℎ) as 119899 rarr infin

To show (A) note that

sup119904le119905

1003816100381610038161003816119872119899

119904(ℎ) minus 119873

119899

119904(ℎ)1003816100381610038161003816

= int

119905

0

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119904(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times (120590119899

119904(d120583 d120582) minus 120590

119904(d120583 d120582))

10038161003816100381610038161003816100381610038161003816100381610038161003816

d119904

(32)

From the argument following (17) there is some constant119862 gt0 satisfying

119891(int

119878

120588119899

119904(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0)) le 119862 (33)

Hence by the dominated convergence theorem

sup119904le119905

1003816100381610038161003816119872119899

119904(ℎ) minus 119873

119899

119904(ℎ)1003816100381610038161003816

le 119862int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) (120590119899

119904(d120583 d120582) minus 120590

119904(d120583 d120582))

10038161003816100381610038161003816100381610038161003816

d119904119901

997888997888997888rarr 0

(34)

as 119899 rarr infin since 120582ℎ isin 119862119887(119878) and the assumption (A2) holds

To show (B) it is sufficient to show that [18 Theorem 21page 16]

119864 (119897 (119873119899

(ℎ))) 997888rarr 119864 (119897 (119872 (ℎ))) (35)

as 119899 rarr infin for any bounded uniformly continuous function119897 119863(R

+R) rarr R By the definition of 119873119899(ℎ) and the

assumption 120588119899 119889997888rarr 120588 we only need to show that 119897(119873119899(ℎ))is a continuous function of 120588119899 mapping 119863(R

+ Ω) to R

invoking the dominated convergence theorem Furthermoreit is sufficient to show that119873119899(ℎ) is a continuous function of120588119899 mapping119863(R

+ Ω) to119863(R

+R) In the following we take

one term in119873119899(ℎ) as an example to show the continuity withrespect to 120588119899 Other terms can be shownanalogously

6 Journal of Mathematics

Define

119897119899

119905= int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119892) 120588119899

119905(d120583 d120582)

119897119905= int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119905(d1205831015840 d1205821015840) + 119892) 120588

119905(d120583 d120582)

(36)

Suppose that 120588119899 rarr 120588 in 119863(R+ Ω) holds We need to

show int1199050

119897119899

119904d119904 rarr int

119905

0

119897119904d119904 in 119863(R

+R) as 119899 rarr infin Indeed

since 120582ℎ isin 119862119887(119878) and int

119878

ℎ(120583 120582)120588119905(d120583 d120582) 119863(R

+ Ω) rarr

119863(R+R) is continuous for any ℎ isin 119862

119887(119878) we have

int119878

120582ℎ(120583 120582)120588119899

119905(d120583 d120582) rarr int

119878

120582ℎ(120583 120582)120588119905(d120583 d120582) in 119863(R

+R)

and int119878

120588119899

119905(d120583 d120582) rarr int

119878

120588119905(d120583 d120582) in 119863(R

+R) as 119899 rarr

infin Hence from the assumptions (A3) and (A4) we obtain119897119899

119905rarr 119897119905in 119863(R

+R) It follows from [18 Equation (1214)

page 124] that 119897119899119905

rarr 119897119905for all but countably many 119905

The dominated convergence theorem then yields int1199050

119897119899

119904d119904 rarr

int

119905

0

119897119904d119904 in supremum norm on finite time intervals and hence

in119863(R+R)

Lemma 6 The solution to (5) is unique in119863(R+ Ω)

Proof Suppose that 120588 and 120588 are two solutions to (5) and theyare the limits of two weakly convergent subsequences of 120588119899respectively By the assumption (A2) 120588

0= 1205880 We need to

show that 120588119905= 120588119905for all 119905 isin R

+

From (3) we have

int

119878

ℎ (120583 120582) 120588119899

119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119899

119905(d120583 d120582) (37)

for any ℎ isin 119862119887(119878) Therefore by the assumption (A2)

int

119878

ℎ (120583 120582) 120588119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119905(d120583 d120582) (38)

int

119878

ℎ (120583 120582) 120588119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119905(d120583 d120582) (39)

hold Define 119889(120588119905 120588119905) = sup

|ℎ|le1ℎisin119862119887(119878)|120588119905(ℎ) minus 120588

119905(ℎ)| In view

of (5) we obtain

1003816100381610038161003816120588119905(ℎ) minus 120588

119905(ℎ)1003816100381610038161003816

= int

119905

0

119866120588119904(ℎ) minus 119866120588

119904(ℎ) d119904

le int

119905

0

int

119878

120582ℎ (120583 120582)

10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

minus119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

times 120590119904(d120583 d120582) d119904

+ int

119905

0

int

119878

120582ℎ (120583 120582)

10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

minus119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

times 120588119904(d120583 d120582) d119904

+ int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) 120588119904(d120583 d120582) minus int

119878

120582ℎ (120583 120582) 120588119904(d120583 d120582)

10038161003816100381610038161003816100381610038161003816

times 119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892) d119904

+ int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120583ℎ (120583 120582) 120588119904(d120583 d120582)

minusint

119878

120583ℎ (120583 120582) 120588119904(d120583 d120582)

10038161003816100381610038161003816100381610038161003816

d119904

(40)

The assumption (A4) indicates that10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892) minus 119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

le 119862119891

10038161003816100381610038161003816100381610038161003816

int

119878

120588119904(d1205831015840 d1205821015840) minus int

119878

120588119904(d1205831015840 d1205821015840)

10038161003816100381610038161003816100381610038161003816

le 119862119891119889 (120588119904 120588119904)

(41)

for some constant119862119891gt 0Therefore using (38) we can derive

1003816100381610038161003816120588119905(ℎ) minus 120588

119905(ℎ)1003816100381610038161003816

le 2119862119891int

119905

0

(int

119878

120582ℎ (120583 120582) 120590119904(d120583 d120582)) 119889 (120588

119904 120588119904) d119904

+ 119862( sup(120583120582)isin119878

120582)int

119905

0

119889 (120588119904 120588119904) d119904

+ ( sup(120583120582)isin119878

120583)int

119905

0

119889 (120588119904 120588119904) d119904

(42)

where 119862 gt 0 is some constant To see this note that by theassumptions (A1) (A2) and (38) we have

int

119878

120588119904(d1205831015840 d1205821015840) + 119892 le int

119878

120590119904(d1205831015840 d1205821015840) + 119892 lt infin (43)

for any 0 le 119904 le 119905 The assumption (A4) then implies

sup119904le119905

119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892) le 119862 lt infin (44)

Again employing the assumption (A2) we have int119878

120582ℎ(120583

120582)120590119904(d120583 d120582) lt infin for all 0 le 119904 le 119905 Hence by using the

assumption (A3) the inequality (42) becomes

119889 (120588119904 120588119904) le 1198621015840

int

119905

0

119889 (120588119904 120588119904) d119904 (45)

for some constant 1198621015840 gt 0 A simple application of Gronwallrsquoslemma yields 119889(120588

119905 120588119905) = 0 for all 119905 isin R

+ which concludes the

proof

Proof of Theorem 1 By Lemma 4 and Lemma 5 the limit ofany weakly convergent subsequence of 120588119899 must satisfy (5) ByLemma 6 we find that the sequence 120588119899 must itself convergeweakly to that unique solution (see the corollary in [18 page59]) The proof of Theorem 1 is thus completed

Journal of Mathematics 7

Proof of Corollary 2 For any open set 119865 sube 119878 we obtain from(38) that

int

119878

119891119899

(120583 120582) 120588119905(d120583 d120582) le int

119878

119891119899

(120583 120582) 120590119905(d120583 d120582) (46)

where119891119899 is taken as a continuous function upwardly converg-ing to the indicator function of 119865 By using the dominatedconvergence theorem we know that 120588

119905(119865) le 120590

119905(119865) A reg-

ularity property (see eg [17 page 18 Lemma 134]) impliesthat 120588

119905(119860) le 120590

119905(119860) for all 119860 isin 119878 This means that 120588

119905is

absolutely continuous with respect to 120590119905for any 119905 ge 0 An

application of the Radon-Nykodym theorem yields the exis-tence of 120593

119905such that (8) holds with 0 le 120593

119905le 1 Now the result

follows straightforwardly fromTheorem 1

Acknowledgments

The author expresses his sincere gratitude to the anonymousreferee and the editor for careful reading of the original paperand useful comments that helped to improve presentation ofresults

References

[1] G H Weiss and M Dishon ldquoOn the asymptotic behaviorof the stochastic and deterministic models of an epidemicrdquoMathematical Biosciences vol 11 pp 261ndash265 1971

[2] D A Kessler ldquoEpidemic size in the SIS model of endemicinfectionsrdquo Journal of Applied Probability vol 45 no 3 pp 757ndash778 2008

[3] Y Shang ldquoA Lie algebra approach to susceptible-infected-susceptible epidemicsrdquo Electronic Journal of Differential Equa-tions vol 2012 no 233 pp 1ndash7 2012

[4] H Andersson and B Djehiche ldquoA threshold limit theorem forthe stochastic logistic epidemicrdquo Journal of Applied Probabilityvol 35 no 3 pp 662ndash670 1998

[5] C R Doering K V Sargsyan and L M Sander ldquoExtinc-tion times for birth-death processes exact results continuumasymptotics and the failure of the Fokker-Planck approxima-tionrdquo Multiscale Modeling amp Simulation vol 3 no 2 pp 283ndash299 2005

[6] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[7] O Ovaskainen ldquoThe quasistationary distribution of thestochastic logistic modelrdquo Journal of Applied Probability vol 38no 4 pp 898ndash907 2001

[8] D Clancy and P K Pollett ldquoA note on quasi-stationary distri-butions of birth-death processes and the SIS logistic epidemicrdquoJournal of Applied Probability vol 40 no 3 pp 821ndash825 2003

[9] TGKurtz ldquoExtensions of Trotterrsquos operator semigroup approx-imation theoremsrdquo Journal of Functional Analysis vol 3 pp354ndash375 1969

[10] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970

[11] R McVinish and P K Pollett ldquoThe deterministic limit of astochastic logistic model with individual variationrdquoMathemat-ical Biosciences vol 241 no 1 pp 109ndash114 2013

[12] R W R Darling and J R Norris ldquoDifferential equationapproximations for Markov chainsrdquo Probability Surveys vol 5pp 37ndash79 2008

[13] Y Shang ldquoOptimal control strategies for virus spreading ininhomogeneousepidemicdynamicsrdquo Canadian MathematicalBulletin In press

[14] T Gross C J D DrsquoLima and B Blasius ldquoEpidemic dynamicson an adaptivenetworkrdquo Physical Review Letters vol 96 no 20Article ID 208701 2006

[15] Y Shang ldquoDiscrete-time epidemic dynamics with awareness inrandomnetworksrdquo International Journal of Biomathematics vol6 no 2 Article ID 1350007 2013

[16] K Klemm M A Serrano V M Eguıluz and M S MiguelldquoA measure of individualrole in collective dynamicsrdquo ScientificReports vol 2 article 292 2012

[17] O Kallenberg Foundations of Modern Probability SpringerNew York NY USA 2nd edition 2002

[18] P Billingsley Convergence of Probability Measures John Wileyamp Sons New York NY USA 2nd edition 1999

[19] S N Ethier and T G KurtzMarkov Processes Characterizationand Convergence JohnWiley amp Sons Hoboken NJ USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 3

3 Proofs

In this section Theorem 1 will be proved through a series oflemmas by tightness and uniqueness arguments [18]The ideaof proofs is similar to that in [11] andwe include the completeproofs here not only for the convenience of the reader butalso to convince the reader that the results do hold in oursetting

Lemma 3 The sequence 120588119899 is tight in119863(R+ Ω)

Proof Recall that [17 Theorem 1627 page 324] 120588119899 is tight in119863(R+ Ω) if and only if 120588119899

119905(ℎ) is tight in 119863(R

+R) for every

ℎ isin 119862119887(119878) According to Aldousrsquos tightness criterion [18

Theorem 1610 page 178] the tightness of 120588119899119905(ℎ) holds if the

following two conditions are satisfied

(A) For each 120576 gt 0 120578 gt 0 and 119903 ge 1 there exist a 1205750gt 0

and an 1198990isin N such that if 120591

119899is a discreteF119899

119905-stopping

time satisfying 120591119899le 119903 then

sup119899ge1198990

sup120575le1205750

119875 (

10038161003816100381610038161003816120588119899

120591119899

(ℎ) minus 120588119899

120591119899+120575(ℎ)

10038161003816100381610038161003816ge 120578) le 120576 (10)

(B) For each 119903 gt 0

lim119886rarrinfin

lim sup119899rarrinfin

119875(sup119905le119903

120588119899

119905(ℎ) gt 119886) = 0 (11)

To show (A) we express the generator of119883119899 by

119866119899119897 (119909) =

119899

sum

119894=1

(119897 (119909 + 119890119899

119894) minus 119897 (119909)) 120582

119894119905(1 minus 119909

119894)

times 119891(119899minus1

119899

sum

119895=1

119892119895(119909119895))

+

119899

sum

119894=1

(119897 (119909 minus 119890119899

119894) minus 119897 (119909)) 120583

119894119905119909119894

(12)

for all real-valued functions 119897(119909) with 119909 = (1199091 119909

119899) isin

0 1119899 Hence using (3) we obtain

119866119899120588119899

119905(ℎ)

= int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840)

+ 119899minus1

119899

sum

119895=1

119892119895(0))120590

119899

119905(d120583 d120582)

minus int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840)

+ 119899minus1

119899

sum

119895=1

119892119895(0))120588

119899

119905(d120583 d120582)

minus int

119878

120583ℎ (120583 120582) 120588119899

119905(d120583 d120582)

(13)

Applying Dynkinrsquos formula to 120588119899119905(ℎ) (see eg [19 Propo-

sition 17 page 162]) we have that

119872119899

119905(ℎ) = 120588

119899

119905(ℎ) minus 120588

119899

0(ℎ) minus int

119905

0

119866119899120588119899

119904(ℎ) d119904 (14)

is a martingale with respect to the filtrationF119899119905= 120590119883

119899

119894119904 0 le

119904 le 119905 and1198721198990(ℎ) = 0 Moreover119872119899

119905(ℎ) is square integrable

The condition (A) holds if for each 120576 gt 0 120578 gt 0 and 119903 ge 1there exist a 120575

0gt 0 and an 119899

0isin N such that for any discrete

F119899119905-stopping time satisfying 120591

119899le 119903

sup119899ge1198990

sup120575le1205750

119875(

100381610038161003816100381610038161003816100381610038161003816

int

120591119899+120575

120591119899

119866119899120588119899

119904(ℎ) d119904

100381610038161003816100381610038161003816100381610038161003816

ge 120578) le 120576 (15)

sup119899ge1198990

sup120575le1205750

119875 (

10038161003816100381610038161003816119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ)

10038161003816100381610038161003816ge 120578) le 120576 (16)

To see (15) we note that1003816100381610038161003816119866119899120588119899

119905(ℎ)1003816100381610038161003816

le int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 120590119899

119905(d120583 d120582)

+ int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 120588119899

119905(d120583 d120582)

+ int

119878

120583ℎ (120583 120582) 120588119899

119905(d120583 d120582)

le int

119878

ℎ (120583 120582)(2120582119891(1198921119899+ 119899minus1

119899

sum

119895=1

119892119895(0)) + 120583)

times 120590119899

119905(d120583 d120582)

le 1198921119899

times sup(120583120582)isin119878

ℎ (120583 120582)(2120582119891(1198921119899+119899minus1

119899

sum

119895=1

119892119895(0))+120583)

(17)

by using (3) It follows fromMarkovrsquos inequality and (17) that

119875(

100381610038161003816100381610038161003816100381610038161003816

int

120591119899+120575

120591119899

119866119899120588119899

119904(ℎ) d119904

100381610038161003816100381610038161003816100381610038161003816

ge 120578)

le 120578minus1

119864(int

120591119899+120575

120591119899

1003816100381610038161003816119866119899120588119899

119904(ℎ)1003816100381610038161003816d119904)

le 120578minus1

1205751198921119899

times sup(120583120582)isin119878

ℎ (120583 120582)(2120582119891(1198921119899+ 119899minus1

119899

sum

119895=1

119892119895(0)) + 120583)

(18)

4 Journal of Mathematics

From assumptions (A1) and (A4) we obtain sup119899119891(1198921119899+

119899minus1

sum119899

119895=1119892119895(0)) lt infin Therefore we can find 120575

0and 119899

0such

that (15) is satisfied by using the assumptions (A1) (A3) (A4)and the fact that ℎ isin 119862

119887(119878)

To prove (16) we need to introduce the quadratic vari-ation process [119872119899(ℎ)] for 119872119899(ℎ) Since 119872119899

119905(ℎ) is a square

integrable martingale it follows from Proposition 61 in [19page 79] that (119872119899(ℎ))2minus[119872119899(ℎ)] is also amartingale By usingMarkovrsquos inequality and martingale properties we obtain

119875 (

10038161003816100381610038161003816119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ)

10038161003816100381610038161003816ge 120578)

le 120578minus2

119864((119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ))

2

)

= 120578minus2

119864(119864 ((119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ))

2

| F119899

120591119899

))

= 120578minus2

(119864 ((119872119899

120591119899+120575(ℎ))

2

) + 119864 ((119872119899

120591119899

(ℎ))

2

)

minus2119864 (119872119899

120591119899

(ℎ) 119864 (119872119899

120591119899+120575(ℎ) | F

119899

120591119899

)) )

= 120578minus2

(119864 ((119872119899

120591119899+120575(ℎ))

2

) minus 119864 ((119872119899

120591119899

(ℎ))

2

))

= 120578minus2

(119864 ((119872119899

120591119899+120575(ℎ))

2

minus [119872119899

(ℎ)]120591119899+120575)

+ 119864 ([119872119899

(ℎ)]120591119899+120575)

minus119864 ((119872119899

120591119899

(ℎ))

2

minus [119872119899

(ℎ)]120591119899

) minus 119864 ([119872119899

(ℎ)]120591119899

))

= 120578minus2

(119864 ([119872119899

(ℎ)]120591119899+120575

minus [119872119899

(ℎ)]120591119899

))

(19)

Applying Dynkinrsquos formula to (120588119899119905(ℎ))2 similarly as in the

derivation of (14) we have that

119899

119905(ℎ) = (120588

119899

119905(ℎ))2

minus (120588119899

0(ℎ))2

minus 2int

119905

0

120588119899

119904(ℎ) 119866119899120588119899

119904(ℎ) d119904 minus 119899minus1 int

119905

0

119866119899120588119899

119904(ℎ) d119904

(20)

is a martingale where

119866119899120588119899

119905(ℎ)

= int

119878

120582ℎ2

(120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 119899

119905(d120583 d120582)

minus int

119878

120582ℎ2

(120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 120588119899

119905(d120583 d120582)

minus int

119878

120583ℎ2

(120583 120582) 120588119899

119905(d120583 d120582)

(21)

with

int

119878

ℎ (120583 120582) 119899

119905(d120583 d120582)

= 119899minus1

119899

sum

119894=1

(119892119894(1) minus 119892

119894(0))2

ℎ (120583119894119905 120582119894119905)

int

119878

ℎ (120583 120582) 120588119899

119905(d120583 d120582)

= 119899minus1

119899

sum

119894=1

(119892119894(1) minus 119892

119894(0))2

119883119899

119894119905ℎ (120583119894119905 120582119894119905)

(22)

By using (14) and (20) we obtain for any 0 le 119904 lt 119905

(120588119899

119905(ℎ) minus 120588

119899

119904(ℎ))2

= (120588119899

119905(ℎ))2

minus (120588119899

119904(ℎ))2

minus 2120588119899

119904(ℎ) (120588

119899

119905(ℎ) minus 120588

119899

119904(ℎ))

= 119899

119905(ℎ) minus

119899

119904(ℎ) + 119899

minus1

int

119905

119904

119866119899120588119899

119903(ℎ) d119903

minus 2120588119899

119904(ℎ) (119872

119899

119905(ℎ) minus 119872

119899

119904(ℎ))

+ 2int

119905

119904

120588119899

119903(ℎ) 119866119899120588119899

119903(ℎ) d119903

minus 2120588119899

119904(ℎ) int

119905

119904

119866119899120588119899

119903(ℎ) d119903

(23)

It follows from (3) and the assumption (A5) that 120588119899119905(ℎ) has

piecewise constant sample paths and int1199050

119866119899120588119899

119904(ℎ)d119904 is a con-

tinuous finite variation process Therefore (14) (23) and thefact 119899

0(ℎ) = 0 imply that

[119872119899

(ℎ)]119905= [120588119899

(ℎ)]119905

= 119899

119905(ℎ) minus 2int

119905

0

120588119899

119904(ℎ) 119889119872

119899

119904(ℎ)

+ 119899minus1

int

119905

0

119866119899120588119899

119904(ℎ) d119904

(24)

From (19) (24) and martingale properties we obtain

119875 (

10038161003816100381610038161003816119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ)

10038161003816100381610038161003816ge 120578)

le 119899minus1

120578minus2

119864(int

120591119899+120575

120591119899

119866119899120588119899

119904(ℎ) d119904)

le 119899minus1

120578minus2

times int

120591119899+120575

120591119899

int

119878

ℎ2

(120583 120582)(2120582119891(1198921119899+ 119899minus1

119899

sum

119895=1

119892119895(0)) + 120583)

times 119899

119904(d120583 d120582) d119904

le 119899minus1

120578minus2

1205751198922119899

times sup(120583120582)isin119878

ℎ2

(120583 120582)(2120582119891(1198921119899+119899minus1

119899

sum

119895=1

119892119895(0))+120583)

(25)

Journal of Mathematics 5

The assumption (A1) implies that lim sup119894rarrinfin

(119892119894(1)minus119892

119894(0)) lt

infin and hence sup1198991198922119899lt infinTherefore as in the derivation of

(15) we can choose some 1198990and 1205750such that (16) is satisfied

Now the only thing remaining to verify is the condition(B) Since ℎ isin 119862

119887(119878) and the assumption (A3) holds there

exists a constant 119862 gt 0 such that

sup119905le119903

1003816100381610038161003816120588119899

119905(ℎ)1003816100381610038161003816le 119862 119892

1119899 (26)

for all 119899 isin N and 119903 gt 0 FromMarkovrsquos inequality and the as-sumption (A1) we have

lim sup119899rarrinfin

119875(sup119905le119903

120588119899

119905(ℎ) gt 119886) le lim sup

119899rarrinfin

119886minus1

119864(sup119905le119903

1003816100381610038161003816120588119899

119905(ℎ)1003816100381610038161003816)

le 119886minus1

119862 sup119899

1198921119899997888rarr 0

(27)

as 119886 rarr infin The proof of Lemma 3 is complete

Lemma 4 For any ℎ isin 119862119887(119878) 119872119899(ℎ) 119889997888rarr 0 in 119863(R

+R) as

119899 rarr infin

Proof By the corollary in [18 page 28] it is sufficient toshow that 1198890

infin(119872119899

(ℎ) 0)

119901

997888rarr 0 as 119899 rarr infin where119901

997888rarr meansconvergence in probability Here by definition the space119863(R+R) is the so-called Skorohod space 119863[0infin) and 1198890

infin

is the metric on it which defines the Skorohod topology (seeeg [18 page 168]) Theorem 167 in [18 page 174] furtherimplies that it is sufficient to show that 1198890

119905(119872119899

(ℎ) 0)

119901

997888rarr 0 foreach 119905 ge 0 Themetric 1198890

119905generates the topology of Skorohod

space119863[0 119905] (see [18 pages 166 and 125] for definitions)Since 1198890

119905(119872119899

(ℎ) 0) le sup119904le119905|119872119899

119904(ℎ)| by Doobrsquos martin-

gale inequality we have for any 120576 gt 0

119875 (1198890

119905(119872119899

(ℎ) 0) ge 120576) le 119875( sup119904isin[0119905]

1003816100381610038161003816119872119899

119904(ℎ)1003816100381610038161003816ge 120576)

le 120576minus2

119864 ((119872119899

119905(ℎ))2

)

= 120576minus2

119864[119872119899

(ℎ)]119905

= 120576minus2

119899minus1

119864(int

119905

0

119866119899120588119899

119904(ℎ) d119904)

(28)

where the first equality uses the fact that (119872119899119905(ℎ))2

minus[119872119899

(ℎ)]119905

is a martingale with expectation 0 and the second equalityfollows from (24) It then follows from the derivation in (25)that

lim119899rarrinfin

119875 (1198890

119905(119872119899

(ℎ) 0) ge 120576) = 0 (29)

which concludes the proof of Lemma 4

From Lemma 3 the sequence 120588119899 is tight and then thereexists a weakly convergent subsequence 120588119899119896 in119863(R

+ Ω) We

still denote it by 120588119899 in the next lemma for convenience

Lemma 5 If 120588119899 119889997888rarr 120588 as 119899 rarr infin then for any ℎ isin 119862119887(119878)

119872119899

(ℎ)

119889

997888rarr 119872(ℎ) in119863(R+R) as 119899 rarr infin where

119872119905(ℎ) = 120588

119905(ℎ) minus 120588

0(ℎ) minus int

119905

0

119866120588119904(ℎ) d119904 (30)

Proof For each 119899 isin N define a random process119873119899(ℎ) by

119873119899

119905(ℎ) = 120588

119899

119905(ℎ) minus 120588

119899

0(ℎ) minus int

119905

0

119866120588119899

119904(ℎ) d119904 (31)

From Theorem 31 in [18 page 28] and the arguments in theproof of Lemma 4 it is sufficient to show the following twoconditions

(A) sup119904le119905|119872119899

119904(ℎ)minus119873

119899

119904(ℎ)|

119901

997888rarr 0 as 119899 rarr infin for each 119905 ge 0

(B) 119873119899(ℎ) 119889997888rarr 119872(ℎ) as 119899 rarr infin

To show (A) note that

sup119904le119905

1003816100381610038161003816119872119899

119904(ℎ) minus 119873

119899

119904(ℎ)1003816100381610038161003816

= int

119905

0

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119904(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times (120590119899

119904(d120583 d120582) minus 120590

119904(d120583 d120582))

10038161003816100381610038161003816100381610038161003816100381610038161003816

d119904

(32)

From the argument following (17) there is some constant119862 gt0 satisfying

119891(int

119878

120588119899

119904(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0)) le 119862 (33)

Hence by the dominated convergence theorem

sup119904le119905

1003816100381610038161003816119872119899

119904(ℎ) minus 119873

119899

119904(ℎ)1003816100381610038161003816

le 119862int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) (120590119899

119904(d120583 d120582) minus 120590

119904(d120583 d120582))

10038161003816100381610038161003816100381610038161003816

d119904119901

997888997888997888rarr 0

(34)

as 119899 rarr infin since 120582ℎ isin 119862119887(119878) and the assumption (A2) holds

To show (B) it is sufficient to show that [18 Theorem 21page 16]

119864 (119897 (119873119899

(ℎ))) 997888rarr 119864 (119897 (119872 (ℎ))) (35)

as 119899 rarr infin for any bounded uniformly continuous function119897 119863(R

+R) rarr R By the definition of 119873119899(ℎ) and the

assumption 120588119899 119889997888rarr 120588 we only need to show that 119897(119873119899(ℎ))is a continuous function of 120588119899 mapping 119863(R

+ Ω) to R

invoking the dominated convergence theorem Furthermoreit is sufficient to show that119873119899(ℎ) is a continuous function of120588119899 mapping119863(R

+ Ω) to119863(R

+R) In the following we take

one term in119873119899(ℎ) as an example to show the continuity withrespect to 120588119899 Other terms can be shownanalogously

6 Journal of Mathematics

Define

119897119899

119905= int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119892) 120588119899

119905(d120583 d120582)

119897119905= int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119905(d1205831015840 d1205821015840) + 119892) 120588

119905(d120583 d120582)

(36)

Suppose that 120588119899 rarr 120588 in 119863(R+ Ω) holds We need to

show int1199050

119897119899

119904d119904 rarr int

119905

0

119897119904d119904 in 119863(R

+R) as 119899 rarr infin Indeed

since 120582ℎ isin 119862119887(119878) and int

119878

ℎ(120583 120582)120588119905(d120583 d120582) 119863(R

+ Ω) rarr

119863(R+R) is continuous for any ℎ isin 119862

119887(119878) we have

int119878

120582ℎ(120583 120582)120588119899

119905(d120583 d120582) rarr int

119878

120582ℎ(120583 120582)120588119905(d120583 d120582) in 119863(R

+R)

and int119878

120588119899

119905(d120583 d120582) rarr int

119878

120588119905(d120583 d120582) in 119863(R

+R) as 119899 rarr

infin Hence from the assumptions (A3) and (A4) we obtain119897119899

119905rarr 119897119905in 119863(R

+R) It follows from [18 Equation (1214)

page 124] that 119897119899119905

rarr 119897119905for all but countably many 119905

The dominated convergence theorem then yields int1199050

119897119899

119904d119904 rarr

int

119905

0

119897119904d119904 in supremum norm on finite time intervals and hence

in119863(R+R)

Lemma 6 The solution to (5) is unique in119863(R+ Ω)

Proof Suppose that 120588 and 120588 are two solutions to (5) and theyare the limits of two weakly convergent subsequences of 120588119899respectively By the assumption (A2) 120588

0= 1205880 We need to

show that 120588119905= 120588119905for all 119905 isin R

+

From (3) we have

int

119878

ℎ (120583 120582) 120588119899

119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119899

119905(d120583 d120582) (37)

for any ℎ isin 119862119887(119878) Therefore by the assumption (A2)

int

119878

ℎ (120583 120582) 120588119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119905(d120583 d120582) (38)

int

119878

ℎ (120583 120582) 120588119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119905(d120583 d120582) (39)

hold Define 119889(120588119905 120588119905) = sup

|ℎ|le1ℎisin119862119887(119878)|120588119905(ℎ) minus 120588

119905(ℎ)| In view

of (5) we obtain

1003816100381610038161003816120588119905(ℎ) minus 120588

119905(ℎ)1003816100381610038161003816

= int

119905

0

119866120588119904(ℎ) minus 119866120588

119904(ℎ) d119904

le int

119905

0

int

119878

120582ℎ (120583 120582)

10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

minus119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

times 120590119904(d120583 d120582) d119904

+ int

119905

0

int

119878

120582ℎ (120583 120582)

10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

minus119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

times 120588119904(d120583 d120582) d119904

+ int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) 120588119904(d120583 d120582) minus int

119878

120582ℎ (120583 120582) 120588119904(d120583 d120582)

10038161003816100381610038161003816100381610038161003816

times 119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892) d119904

+ int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120583ℎ (120583 120582) 120588119904(d120583 d120582)

minusint

119878

120583ℎ (120583 120582) 120588119904(d120583 d120582)

10038161003816100381610038161003816100381610038161003816

d119904

(40)

The assumption (A4) indicates that10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892) minus 119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

le 119862119891

10038161003816100381610038161003816100381610038161003816

int

119878

120588119904(d1205831015840 d1205821015840) minus int

119878

120588119904(d1205831015840 d1205821015840)

10038161003816100381610038161003816100381610038161003816

le 119862119891119889 (120588119904 120588119904)

(41)

for some constant119862119891gt 0Therefore using (38) we can derive

1003816100381610038161003816120588119905(ℎ) minus 120588

119905(ℎ)1003816100381610038161003816

le 2119862119891int

119905

0

(int

119878

120582ℎ (120583 120582) 120590119904(d120583 d120582)) 119889 (120588

119904 120588119904) d119904

+ 119862( sup(120583120582)isin119878

120582)int

119905

0

119889 (120588119904 120588119904) d119904

+ ( sup(120583120582)isin119878

120583)int

119905

0

119889 (120588119904 120588119904) d119904

(42)

where 119862 gt 0 is some constant To see this note that by theassumptions (A1) (A2) and (38) we have

int

119878

120588119904(d1205831015840 d1205821015840) + 119892 le int

119878

120590119904(d1205831015840 d1205821015840) + 119892 lt infin (43)

for any 0 le 119904 le 119905 The assumption (A4) then implies

sup119904le119905

119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892) le 119862 lt infin (44)

Again employing the assumption (A2) we have int119878

120582ℎ(120583

120582)120590119904(d120583 d120582) lt infin for all 0 le 119904 le 119905 Hence by using the

assumption (A3) the inequality (42) becomes

119889 (120588119904 120588119904) le 1198621015840

int

119905

0

119889 (120588119904 120588119904) d119904 (45)

for some constant 1198621015840 gt 0 A simple application of Gronwallrsquoslemma yields 119889(120588

119905 120588119905) = 0 for all 119905 isin R

+ which concludes the

proof

Proof of Theorem 1 By Lemma 4 and Lemma 5 the limit ofany weakly convergent subsequence of 120588119899 must satisfy (5) ByLemma 6 we find that the sequence 120588119899 must itself convergeweakly to that unique solution (see the corollary in [18 page59]) The proof of Theorem 1 is thus completed

Journal of Mathematics 7

Proof of Corollary 2 For any open set 119865 sube 119878 we obtain from(38) that

int

119878

119891119899

(120583 120582) 120588119905(d120583 d120582) le int

119878

119891119899

(120583 120582) 120590119905(d120583 d120582) (46)

where119891119899 is taken as a continuous function upwardly converg-ing to the indicator function of 119865 By using the dominatedconvergence theorem we know that 120588

119905(119865) le 120590

119905(119865) A reg-

ularity property (see eg [17 page 18 Lemma 134]) impliesthat 120588

119905(119860) le 120590

119905(119860) for all 119860 isin 119878 This means that 120588

119905is

absolutely continuous with respect to 120590119905for any 119905 ge 0 An

application of the Radon-Nykodym theorem yields the exis-tence of 120593

119905such that (8) holds with 0 le 120593

119905le 1 Now the result

follows straightforwardly fromTheorem 1

Acknowledgments

The author expresses his sincere gratitude to the anonymousreferee and the editor for careful reading of the original paperand useful comments that helped to improve presentation ofresults

References

[1] G H Weiss and M Dishon ldquoOn the asymptotic behaviorof the stochastic and deterministic models of an epidemicrdquoMathematical Biosciences vol 11 pp 261ndash265 1971

[2] D A Kessler ldquoEpidemic size in the SIS model of endemicinfectionsrdquo Journal of Applied Probability vol 45 no 3 pp 757ndash778 2008

[3] Y Shang ldquoA Lie algebra approach to susceptible-infected-susceptible epidemicsrdquo Electronic Journal of Differential Equa-tions vol 2012 no 233 pp 1ndash7 2012

[4] H Andersson and B Djehiche ldquoA threshold limit theorem forthe stochastic logistic epidemicrdquo Journal of Applied Probabilityvol 35 no 3 pp 662ndash670 1998

[5] C R Doering K V Sargsyan and L M Sander ldquoExtinc-tion times for birth-death processes exact results continuumasymptotics and the failure of the Fokker-Planck approxima-tionrdquo Multiscale Modeling amp Simulation vol 3 no 2 pp 283ndash299 2005

[6] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[7] O Ovaskainen ldquoThe quasistationary distribution of thestochastic logistic modelrdquo Journal of Applied Probability vol 38no 4 pp 898ndash907 2001

[8] D Clancy and P K Pollett ldquoA note on quasi-stationary distri-butions of birth-death processes and the SIS logistic epidemicrdquoJournal of Applied Probability vol 40 no 3 pp 821ndash825 2003

[9] TGKurtz ldquoExtensions of Trotterrsquos operator semigroup approx-imation theoremsrdquo Journal of Functional Analysis vol 3 pp354ndash375 1969

[10] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970

[11] R McVinish and P K Pollett ldquoThe deterministic limit of astochastic logistic model with individual variationrdquoMathemat-ical Biosciences vol 241 no 1 pp 109ndash114 2013

[12] R W R Darling and J R Norris ldquoDifferential equationapproximations for Markov chainsrdquo Probability Surveys vol 5pp 37ndash79 2008

[13] Y Shang ldquoOptimal control strategies for virus spreading ininhomogeneousepidemicdynamicsrdquo Canadian MathematicalBulletin In press

[14] T Gross C J D DrsquoLima and B Blasius ldquoEpidemic dynamicson an adaptivenetworkrdquo Physical Review Letters vol 96 no 20Article ID 208701 2006

[15] Y Shang ldquoDiscrete-time epidemic dynamics with awareness inrandomnetworksrdquo International Journal of Biomathematics vol6 no 2 Article ID 1350007 2013

[16] K Klemm M A Serrano V M Eguıluz and M S MiguelldquoA measure of individualrole in collective dynamicsrdquo ScientificReports vol 2 article 292 2012

[17] O Kallenberg Foundations of Modern Probability SpringerNew York NY USA 2nd edition 2002

[18] P Billingsley Convergence of Probability Measures John Wileyamp Sons New York NY USA 2nd edition 1999

[19] S N Ethier and T G KurtzMarkov Processes Characterizationand Convergence JohnWiley amp Sons Hoboken NJ USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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4 Journal of Mathematics

From assumptions (A1) and (A4) we obtain sup119899119891(1198921119899+

119899minus1

sum119899

119895=1119892119895(0)) lt infin Therefore we can find 120575

0and 119899

0such

that (15) is satisfied by using the assumptions (A1) (A3) (A4)and the fact that ℎ isin 119862

119887(119878)

To prove (16) we need to introduce the quadratic vari-ation process [119872119899(ℎ)] for 119872119899(ℎ) Since 119872119899

119905(ℎ) is a square

integrable martingale it follows from Proposition 61 in [19page 79] that (119872119899(ℎ))2minus[119872119899(ℎ)] is also amartingale By usingMarkovrsquos inequality and martingale properties we obtain

119875 (

10038161003816100381610038161003816119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ)

10038161003816100381610038161003816ge 120578)

le 120578minus2

119864((119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ))

2

)

= 120578minus2

119864(119864 ((119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ))

2

| F119899

120591119899

))

= 120578minus2

(119864 ((119872119899

120591119899+120575(ℎ))

2

) + 119864 ((119872119899

120591119899

(ℎ))

2

)

minus2119864 (119872119899

120591119899

(ℎ) 119864 (119872119899

120591119899+120575(ℎ) | F

119899

120591119899

)) )

= 120578minus2

(119864 ((119872119899

120591119899+120575(ℎ))

2

) minus 119864 ((119872119899

120591119899

(ℎ))

2

))

= 120578minus2

(119864 ((119872119899

120591119899+120575(ℎ))

2

minus [119872119899

(ℎ)]120591119899+120575)

+ 119864 ([119872119899

(ℎ)]120591119899+120575)

minus119864 ((119872119899

120591119899

(ℎ))

2

minus [119872119899

(ℎ)]120591119899

) minus 119864 ([119872119899

(ℎ)]120591119899

))

= 120578minus2

(119864 ([119872119899

(ℎ)]120591119899+120575

minus [119872119899

(ℎ)]120591119899

))

(19)

Applying Dynkinrsquos formula to (120588119899119905(ℎ))2 similarly as in the

derivation of (14) we have that

119899

119905(ℎ) = (120588

119899

119905(ℎ))2

minus (120588119899

0(ℎ))2

minus 2int

119905

0

120588119899

119904(ℎ) 119866119899120588119899

119904(ℎ) d119904 minus 119899minus1 int

119905

0

119866119899120588119899

119904(ℎ) d119904

(20)

is a martingale where

119866119899120588119899

119905(ℎ)

= int

119878

120582ℎ2

(120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 119899

119905(d120583 d120582)

minus int

119878

120582ℎ2

(120583 120582) 119891(int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times 120588119899

119905(d120583 d120582)

minus int

119878

120583ℎ2

(120583 120582) 120588119899

119905(d120583 d120582)

(21)

with

int

119878

ℎ (120583 120582) 119899

119905(d120583 d120582)

= 119899minus1

119899

sum

119894=1

(119892119894(1) minus 119892

119894(0))2

ℎ (120583119894119905 120582119894119905)

int

119878

ℎ (120583 120582) 120588119899

119905(d120583 d120582)

= 119899minus1

119899

sum

119894=1

(119892119894(1) minus 119892

119894(0))2

119883119899

119894119905ℎ (120583119894119905 120582119894119905)

(22)

By using (14) and (20) we obtain for any 0 le 119904 lt 119905

(120588119899

119905(ℎ) minus 120588

119899

119904(ℎ))2

= (120588119899

119905(ℎ))2

minus (120588119899

119904(ℎ))2

minus 2120588119899

119904(ℎ) (120588

119899

119905(ℎ) minus 120588

119899

119904(ℎ))

= 119899

119905(ℎ) minus

119899

119904(ℎ) + 119899

minus1

int

119905

119904

119866119899120588119899

119903(ℎ) d119903

minus 2120588119899

119904(ℎ) (119872

119899

119905(ℎ) minus 119872

119899

119904(ℎ))

+ 2int

119905

119904

120588119899

119903(ℎ) 119866119899120588119899

119903(ℎ) d119903

minus 2120588119899

119904(ℎ) int

119905

119904

119866119899120588119899

119903(ℎ) d119903

(23)

It follows from (3) and the assumption (A5) that 120588119899119905(ℎ) has

piecewise constant sample paths and int1199050

119866119899120588119899

119904(ℎ)d119904 is a con-

tinuous finite variation process Therefore (14) (23) and thefact 119899

0(ℎ) = 0 imply that

[119872119899

(ℎ)]119905= [120588119899

(ℎ)]119905

= 119899

119905(ℎ) minus 2int

119905

0

120588119899

119904(ℎ) 119889119872

119899

119904(ℎ)

+ 119899minus1

int

119905

0

119866119899120588119899

119904(ℎ) d119904

(24)

From (19) (24) and martingale properties we obtain

119875 (

10038161003816100381610038161003816119872119899

120591119899+120575(ℎ) minus 119872

119899

120591119899

(ℎ)

10038161003816100381610038161003816ge 120578)

le 119899minus1

120578minus2

119864(int

120591119899+120575

120591119899

119866119899120588119899

119904(ℎ) d119904)

le 119899minus1

120578minus2

times int

120591119899+120575

120591119899

int

119878

ℎ2

(120583 120582)(2120582119891(1198921119899+ 119899minus1

119899

sum

119895=1

119892119895(0)) + 120583)

times 119899

119904(d120583 d120582) d119904

le 119899minus1

120578minus2

1205751198922119899

times sup(120583120582)isin119878

ℎ2

(120583 120582)(2120582119891(1198921119899+119899minus1

119899

sum

119895=1

119892119895(0))+120583)

(25)

Journal of Mathematics 5

The assumption (A1) implies that lim sup119894rarrinfin

(119892119894(1)minus119892

119894(0)) lt

infin and hence sup1198991198922119899lt infinTherefore as in the derivation of

(15) we can choose some 1198990and 1205750such that (16) is satisfied

Now the only thing remaining to verify is the condition(B) Since ℎ isin 119862

119887(119878) and the assumption (A3) holds there

exists a constant 119862 gt 0 such that

sup119905le119903

1003816100381610038161003816120588119899

119905(ℎ)1003816100381610038161003816le 119862 119892

1119899 (26)

for all 119899 isin N and 119903 gt 0 FromMarkovrsquos inequality and the as-sumption (A1) we have

lim sup119899rarrinfin

119875(sup119905le119903

120588119899

119905(ℎ) gt 119886) le lim sup

119899rarrinfin

119886minus1

119864(sup119905le119903

1003816100381610038161003816120588119899

119905(ℎ)1003816100381610038161003816)

le 119886minus1

119862 sup119899

1198921119899997888rarr 0

(27)

as 119886 rarr infin The proof of Lemma 3 is complete

Lemma 4 For any ℎ isin 119862119887(119878) 119872119899(ℎ) 119889997888rarr 0 in 119863(R

+R) as

119899 rarr infin

Proof By the corollary in [18 page 28] it is sufficient toshow that 1198890

infin(119872119899

(ℎ) 0)

119901

997888rarr 0 as 119899 rarr infin where119901

997888rarr meansconvergence in probability Here by definition the space119863(R+R) is the so-called Skorohod space 119863[0infin) and 1198890

infin

is the metric on it which defines the Skorohod topology (seeeg [18 page 168]) Theorem 167 in [18 page 174] furtherimplies that it is sufficient to show that 1198890

119905(119872119899

(ℎ) 0)

119901

997888rarr 0 foreach 119905 ge 0 Themetric 1198890

119905generates the topology of Skorohod

space119863[0 119905] (see [18 pages 166 and 125] for definitions)Since 1198890

119905(119872119899

(ℎ) 0) le sup119904le119905|119872119899

119904(ℎ)| by Doobrsquos martin-

gale inequality we have for any 120576 gt 0

119875 (1198890

119905(119872119899

(ℎ) 0) ge 120576) le 119875( sup119904isin[0119905]

1003816100381610038161003816119872119899

119904(ℎ)1003816100381610038161003816ge 120576)

le 120576minus2

119864 ((119872119899

119905(ℎ))2

)

= 120576minus2

119864[119872119899

(ℎ)]119905

= 120576minus2

119899minus1

119864(int

119905

0

119866119899120588119899

119904(ℎ) d119904)

(28)

where the first equality uses the fact that (119872119899119905(ℎ))2

minus[119872119899

(ℎ)]119905

is a martingale with expectation 0 and the second equalityfollows from (24) It then follows from the derivation in (25)that

lim119899rarrinfin

119875 (1198890

119905(119872119899

(ℎ) 0) ge 120576) = 0 (29)

which concludes the proof of Lemma 4

From Lemma 3 the sequence 120588119899 is tight and then thereexists a weakly convergent subsequence 120588119899119896 in119863(R

+ Ω) We

still denote it by 120588119899 in the next lemma for convenience

Lemma 5 If 120588119899 119889997888rarr 120588 as 119899 rarr infin then for any ℎ isin 119862119887(119878)

119872119899

(ℎ)

119889

997888rarr 119872(ℎ) in119863(R+R) as 119899 rarr infin where

119872119905(ℎ) = 120588

119905(ℎ) minus 120588

0(ℎ) minus int

119905

0

119866120588119904(ℎ) d119904 (30)

Proof For each 119899 isin N define a random process119873119899(ℎ) by

119873119899

119905(ℎ) = 120588

119899

119905(ℎ) minus 120588

119899

0(ℎ) minus int

119905

0

119866120588119899

119904(ℎ) d119904 (31)

From Theorem 31 in [18 page 28] and the arguments in theproof of Lemma 4 it is sufficient to show the following twoconditions

(A) sup119904le119905|119872119899

119904(ℎ)minus119873

119899

119904(ℎ)|

119901

997888rarr 0 as 119899 rarr infin for each 119905 ge 0

(B) 119873119899(ℎ) 119889997888rarr 119872(ℎ) as 119899 rarr infin

To show (A) note that

sup119904le119905

1003816100381610038161003816119872119899

119904(ℎ) minus 119873

119899

119904(ℎ)1003816100381610038161003816

= int

119905

0

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119904(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times (120590119899

119904(d120583 d120582) minus 120590

119904(d120583 d120582))

10038161003816100381610038161003816100381610038161003816100381610038161003816

d119904

(32)

From the argument following (17) there is some constant119862 gt0 satisfying

119891(int

119878

120588119899

119904(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0)) le 119862 (33)

Hence by the dominated convergence theorem

sup119904le119905

1003816100381610038161003816119872119899

119904(ℎ) minus 119873

119899

119904(ℎ)1003816100381610038161003816

le 119862int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) (120590119899

119904(d120583 d120582) minus 120590

119904(d120583 d120582))

10038161003816100381610038161003816100381610038161003816

d119904119901

997888997888997888rarr 0

(34)

as 119899 rarr infin since 120582ℎ isin 119862119887(119878) and the assumption (A2) holds

To show (B) it is sufficient to show that [18 Theorem 21page 16]

119864 (119897 (119873119899

(ℎ))) 997888rarr 119864 (119897 (119872 (ℎ))) (35)

as 119899 rarr infin for any bounded uniformly continuous function119897 119863(R

+R) rarr R By the definition of 119873119899(ℎ) and the

assumption 120588119899 119889997888rarr 120588 we only need to show that 119897(119873119899(ℎ))is a continuous function of 120588119899 mapping 119863(R

+ Ω) to R

invoking the dominated convergence theorem Furthermoreit is sufficient to show that119873119899(ℎ) is a continuous function of120588119899 mapping119863(R

+ Ω) to119863(R

+R) In the following we take

one term in119873119899(ℎ) as an example to show the continuity withrespect to 120588119899 Other terms can be shownanalogously

6 Journal of Mathematics

Define

119897119899

119905= int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119892) 120588119899

119905(d120583 d120582)

119897119905= int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119905(d1205831015840 d1205821015840) + 119892) 120588

119905(d120583 d120582)

(36)

Suppose that 120588119899 rarr 120588 in 119863(R+ Ω) holds We need to

show int1199050

119897119899

119904d119904 rarr int

119905

0

119897119904d119904 in 119863(R

+R) as 119899 rarr infin Indeed

since 120582ℎ isin 119862119887(119878) and int

119878

ℎ(120583 120582)120588119905(d120583 d120582) 119863(R

+ Ω) rarr

119863(R+R) is continuous for any ℎ isin 119862

119887(119878) we have

int119878

120582ℎ(120583 120582)120588119899

119905(d120583 d120582) rarr int

119878

120582ℎ(120583 120582)120588119905(d120583 d120582) in 119863(R

+R)

and int119878

120588119899

119905(d120583 d120582) rarr int

119878

120588119905(d120583 d120582) in 119863(R

+R) as 119899 rarr

infin Hence from the assumptions (A3) and (A4) we obtain119897119899

119905rarr 119897119905in 119863(R

+R) It follows from [18 Equation (1214)

page 124] that 119897119899119905

rarr 119897119905for all but countably many 119905

The dominated convergence theorem then yields int1199050

119897119899

119904d119904 rarr

int

119905

0

119897119904d119904 in supremum norm on finite time intervals and hence

in119863(R+R)

Lemma 6 The solution to (5) is unique in119863(R+ Ω)

Proof Suppose that 120588 and 120588 are two solutions to (5) and theyare the limits of two weakly convergent subsequences of 120588119899respectively By the assumption (A2) 120588

0= 1205880 We need to

show that 120588119905= 120588119905for all 119905 isin R

+

From (3) we have

int

119878

ℎ (120583 120582) 120588119899

119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119899

119905(d120583 d120582) (37)

for any ℎ isin 119862119887(119878) Therefore by the assumption (A2)

int

119878

ℎ (120583 120582) 120588119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119905(d120583 d120582) (38)

int

119878

ℎ (120583 120582) 120588119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119905(d120583 d120582) (39)

hold Define 119889(120588119905 120588119905) = sup

|ℎ|le1ℎisin119862119887(119878)|120588119905(ℎ) minus 120588

119905(ℎ)| In view

of (5) we obtain

1003816100381610038161003816120588119905(ℎ) minus 120588

119905(ℎ)1003816100381610038161003816

= int

119905

0

119866120588119904(ℎ) minus 119866120588

119904(ℎ) d119904

le int

119905

0

int

119878

120582ℎ (120583 120582)

10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

minus119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

times 120590119904(d120583 d120582) d119904

+ int

119905

0

int

119878

120582ℎ (120583 120582)

10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

minus119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

times 120588119904(d120583 d120582) d119904

+ int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) 120588119904(d120583 d120582) minus int

119878

120582ℎ (120583 120582) 120588119904(d120583 d120582)

10038161003816100381610038161003816100381610038161003816

times 119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892) d119904

+ int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120583ℎ (120583 120582) 120588119904(d120583 d120582)

minusint

119878

120583ℎ (120583 120582) 120588119904(d120583 d120582)

10038161003816100381610038161003816100381610038161003816

d119904

(40)

The assumption (A4) indicates that10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892) minus 119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

le 119862119891

10038161003816100381610038161003816100381610038161003816

int

119878

120588119904(d1205831015840 d1205821015840) minus int

119878

120588119904(d1205831015840 d1205821015840)

10038161003816100381610038161003816100381610038161003816

le 119862119891119889 (120588119904 120588119904)

(41)

for some constant119862119891gt 0Therefore using (38) we can derive

1003816100381610038161003816120588119905(ℎ) minus 120588

119905(ℎ)1003816100381610038161003816

le 2119862119891int

119905

0

(int

119878

120582ℎ (120583 120582) 120590119904(d120583 d120582)) 119889 (120588

119904 120588119904) d119904

+ 119862( sup(120583120582)isin119878

120582)int

119905

0

119889 (120588119904 120588119904) d119904

+ ( sup(120583120582)isin119878

120583)int

119905

0

119889 (120588119904 120588119904) d119904

(42)

where 119862 gt 0 is some constant To see this note that by theassumptions (A1) (A2) and (38) we have

int

119878

120588119904(d1205831015840 d1205821015840) + 119892 le int

119878

120590119904(d1205831015840 d1205821015840) + 119892 lt infin (43)

for any 0 le 119904 le 119905 The assumption (A4) then implies

sup119904le119905

119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892) le 119862 lt infin (44)

Again employing the assumption (A2) we have int119878

120582ℎ(120583

120582)120590119904(d120583 d120582) lt infin for all 0 le 119904 le 119905 Hence by using the

assumption (A3) the inequality (42) becomes

119889 (120588119904 120588119904) le 1198621015840

int

119905

0

119889 (120588119904 120588119904) d119904 (45)

for some constant 1198621015840 gt 0 A simple application of Gronwallrsquoslemma yields 119889(120588

119905 120588119905) = 0 for all 119905 isin R

+ which concludes the

proof

Proof of Theorem 1 By Lemma 4 and Lemma 5 the limit ofany weakly convergent subsequence of 120588119899 must satisfy (5) ByLemma 6 we find that the sequence 120588119899 must itself convergeweakly to that unique solution (see the corollary in [18 page59]) The proof of Theorem 1 is thus completed

Journal of Mathematics 7

Proof of Corollary 2 For any open set 119865 sube 119878 we obtain from(38) that

int

119878

119891119899

(120583 120582) 120588119905(d120583 d120582) le int

119878

119891119899

(120583 120582) 120590119905(d120583 d120582) (46)

where119891119899 is taken as a continuous function upwardly converg-ing to the indicator function of 119865 By using the dominatedconvergence theorem we know that 120588

119905(119865) le 120590

119905(119865) A reg-

ularity property (see eg [17 page 18 Lemma 134]) impliesthat 120588

119905(119860) le 120590

119905(119860) for all 119860 isin 119878 This means that 120588

119905is

absolutely continuous with respect to 120590119905for any 119905 ge 0 An

application of the Radon-Nykodym theorem yields the exis-tence of 120593

119905such that (8) holds with 0 le 120593

119905le 1 Now the result

follows straightforwardly fromTheorem 1

Acknowledgments

The author expresses his sincere gratitude to the anonymousreferee and the editor for careful reading of the original paperand useful comments that helped to improve presentation ofresults

References

[1] G H Weiss and M Dishon ldquoOn the asymptotic behaviorof the stochastic and deterministic models of an epidemicrdquoMathematical Biosciences vol 11 pp 261ndash265 1971

[2] D A Kessler ldquoEpidemic size in the SIS model of endemicinfectionsrdquo Journal of Applied Probability vol 45 no 3 pp 757ndash778 2008

[3] Y Shang ldquoA Lie algebra approach to susceptible-infected-susceptible epidemicsrdquo Electronic Journal of Differential Equa-tions vol 2012 no 233 pp 1ndash7 2012

[4] H Andersson and B Djehiche ldquoA threshold limit theorem forthe stochastic logistic epidemicrdquo Journal of Applied Probabilityvol 35 no 3 pp 662ndash670 1998

[5] C R Doering K V Sargsyan and L M Sander ldquoExtinc-tion times for birth-death processes exact results continuumasymptotics and the failure of the Fokker-Planck approxima-tionrdquo Multiscale Modeling amp Simulation vol 3 no 2 pp 283ndash299 2005

[6] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[7] O Ovaskainen ldquoThe quasistationary distribution of thestochastic logistic modelrdquo Journal of Applied Probability vol 38no 4 pp 898ndash907 2001

[8] D Clancy and P K Pollett ldquoA note on quasi-stationary distri-butions of birth-death processes and the SIS logistic epidemicrdquoJournal of Applied Probability vol 40 no 3 pp 821ndash825 2003

[9] TGKurtz ldquoExtensions of Trotterrsquos operator semigroup approx-imation theoremsrdquo Journal of Functional Analysis vol 3 pp354ndash375 1969

[10] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970

[11] R McVinish and P K Pollett ldquoThe deterministic limit of astochastic logistic model with individual variationrdquoMathemat-ical Biosciences vol 241 no 1 pp 109ndash114 2013

[12] R W R Darling and J R Norris ldquoDifferential equationapproximations for Markov chainsrdquo Probability Surveys vol 5pp 37ndash79 2008

[13] Y Shang ldquoOptimal control strategies for virus spreading ininhomogeneousepidemicdynamicsrdquo Canadian MathematicalBulletin In press

[14] T Gross C J D DrsquoLima and B Blasius ldquoEpidemic dynamicson an adaptivenetworkrdquo Physical Review Letters vol 96 no 20Article ID 208701 2006

[15] Y Shang ldquoDiscrete-time epidemic dynamics with awareness inrandomnetworksrdquo International Journal of Biomathematics vol6 no 2 Article ID 1350007 2013

[16] K Klemm M A Serrano V M Eguıluz and M S MiguelldquoA measure of individualrole in collective dynamicsrdquo ScientificReports vol 2 article 292 2012

[17] O Kallenberg Foundations of Modern Probability SpringerNew York NY USA 2nd edition 2002

[18] P Billingsley Convergence of Probability Measures John Wileyamp Sons New York NY USA 2nd edition 1999

[19] S N Ethier and T G KurtzMarkov Processes Characterizationand Convergence JohnWiley amp Sons Hoboken NJ USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 5

The assumption (A1) implies that lim sup119894rarrinfin

(119892119894(1)minus119892

119894(0)) lt

infin and hence sup1198991198922119899lt infinTherefore as in the derivation of

(15) we can choose some 1198990and 1205750such that (16) is satisfied

Now the only thing remaining to verify is the condition(B) Since ℎ isin 119862

119887(119878) and the assumption (A3) holds there

exists a constant 119862 gt 0 such that

sup119905le119903

1003816100381610038161003816120588119899

119905(ℎ)1003816100381610038161003816le 119862 119892

1119899 (26)

for all 119899 isin N and 119903 gt 0 FromMarkovrsquos inequality and the as-sumption (A1) we have

lim sup119899rarrinfin

119875(sup119905le119903

120588119899

119905(ℎ) gt 119886) le lim sup

119899rarrinfin

119886minus1

119864(sup119905le119903

1003816100381610038161003816120588119899

119905(ℎ)1003816100381610038161003816)

le 119886minus1

119862 sup119899

1198921119899997888rarr 0

(27)

as 119886 rarr infin The proof of Lemma 3 is complete

Lemma 4 For any ℎ isin 119862119887(119878) 119872119899(ℎ) 119889997888rarr 0 in 119863(R

+R) as

119899 rarr infin

Proof By the corollary in [18 page 28] it is sufficient toshow that 1198890

infin(119872119899

(ℎ) 0)

119901

997888rarr 0 as 119899 rarr infin where119901

997888rarr meansconvergence in probability Here by definition the space119863(R+R) is the so-called Skorohod space 119863[0infin) and 1198890

infin

is the metric on it which defines the Skorohod topology (seeeg [18 page 168]) Theorem 167 in [18 page 174] furtherimplies that it is sufficient to show that 1198890

119905(119872119899

(ℎ) 0)

119901

997888rarr 0 foreach 119905 ge 0 Themetric 1198890

119905generates the topology of Skorohod

space119863[0 119905] (see [18 pages 166 and 125] for definitions)Since 1198890

119905(119872119899

(ℎ) 0) le sup119904le119905|119872119899

119904(ℎ)| by Doobrsquos martin-

gale inequality we have for any 120576 gt 0

119875 (1198890

119905(119872119899

(ℎ) 0) ge 120576) le 119875( sup119904isin[0119905]

1003816100381610038161003816119872119899

119904(ℎ)1003816100381610038161003816ge 120576)

le 120576minus2

119864 ((119872119899

119905(ℎ))2

)

= 120576minus2

119864[119872119899

(ℎ)]119905

= 120576minus2

119899minus1

119864(int

119905

0

119866119899120588119899

119904(ℎ) d119904)

(28)

where the first equality uses the fact that (119872119899119905(ℎ))2

minus[119872119899

(ℎ)]119905

is a martingale with expectation 0 and the second equalityfollows from (24) It then follows from the derivation in (25)that

lim119899rarrinfin

119875 (1198890

119905(119872119899

(ℎ) 0) ge 120576) = 0 (29)

which concludes the proof of Lemma 4

From Lemma 3 the sequence 120588119899 is tight and then thereexists a weakly convergent subsequence 120588119899119896 in119863(R

+ Ω) We

still denote it by 120588119899 in the next lemma for convenience

Lemma 5 If 120588119899 119889997888rarr 120588 as 119899 rarr infin then for any ℎ isin 119862119887(119878)

119872119899

(ℎ)

119889

997888rarr 119872(ℎ) in119863(R+R) as 119899 rarr infin where

119872119905(ℎ) = 120588

119905(ℎ) minus 120588

0(ℎ) minus int

119905

0

119866120588119904(ℎ) d119904 (30)

Proof For each 119899 isin N define a random process119873119899(ℎ) by

119873119899

119905(ℎ) = 120588

119899

119905(ℎ) minus 120588

119899

0(ℎ) minus int

119905

0

119866120588119899

119904(ℎ) d119904 (31)

From Theorem 31 in [18 page 28] and the arguments in theproof of Lemma 4 it is sufficient to show the following twoconditions

(A) sup119904le119905|119872119899

119904(ℎ)minus119873

119899

119904(ℎ)|

119901

997888rarr 0 as 119899 rarr infin for each 119905 ge 0

(B) 119873119899(ℎ) 119889997888rarr 119872(ℎ) as 119899 rarr infin

To show (A) note that

sup119904le119905

1003816100381610038161003816119872119899

119904(ℎ) minus 119873

119899

119904(ℎ)1003816100381610038161003816

= int

119905

0

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) 119891(int

119878

120588119899

119904(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0))

times (120590119899

119904(d120583 d120582) minus 120590

119904(d120583 d120582))

10038161003816100381610038161003816100381610038161003816100381610038161003816

d119904

(32)

From the argument following (17) there is some constant119862 gt0 satisfying

119891(int

119878

120588119899

119904(d1205831015840 d1205821015840) + 119899minus1

119899

sum

119895=1

119892119895(0)) le 119862 (33)

Hence by the dominated convergence theorem

sup119904le119905

1003816100381610038161003816119872119899

119904(ℎ) minus 119873

119899

119904(ℎ)1003816100381610038161003816

le 119862int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) (120590119899

119904(d120583 d120582) minus 120590

119904(d120583 d120582))

10038161003816100381610038161003816100381610038161003816

d119904119901

997888997888997888rarr 0

(34)

as 119899 rarr infin since 120582ℎ isin 119862119887(119878) and the assumption (A2) holds

To show (B) it is sufficient to show that [18 Theorem 21page 16]

119864 (119897 (119873119899

(ℎ))) 997888rarr 119864 (119897 (119872 (ℎ))) (35)

as 119899 rarr infin for any bounded uniformly continuous function119897 119863(R

+R) rarr R By the definition of 119873119899(ℎ) and the

assumption 120588119899 119889997888rarr 120588 we only need to show that 119897(119873119899(ℎ))is a continuous function of 120588119899 mapping 119863(R

+ Ω) to R

invoking the dominated convergence theorem Furthermoreit is sufficient to show that119873119899(ℎ) is a continuous function of120588119899 mapping119863(R

+ Ω) to119863(R

+R) In the following we take

one term in119873119899(ℎ) as an example to show the continuity withrespect to 120588119899 Other terms can be shownanalogously

6 Journal of Mathematics

Define

119897119899

119905= int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119892) 120588119899

119905(d120583 d120582)

119897119905= int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119905(d1205831015840 d1205821015840) + 119892) 120588

119905(d120583 d120582)

(36)

Suppose that 120588119899 rarr 120588 in 119863(R+ Ω) holds We need to

show int1199050

119897119899

119904d119904 rarr int

119905

0

119897119904d119904 in 119863(R

+R) as 119899 rarr infin Indeed

since 120582ℎ isin 119862119887(119878) and int

119878

ℎ(120583 120582)120588119905(d120583 d120582) 119863(R

+ Ω) rarr

119863(R+R) is continuous for any ℎ isin 119862

119887(119878) we have

int119878

120582ℎ(120583 120582)120588119899

119905(d120583 d120582) rarr int

119878

120582ℎ(120583 120582)120588119905(d120583 d120582) in 119863(R

+R)

and int119878

120588119899

119905(d120583 d120582) rarr int

119878

120588119905(d120583 d120582) in 119863(R

+R) as 119899 rarr

infin Hence from the assumptions (A3) and (A4) we obtain119897119899

119905rarr 119897119905in 119863(R

+R) It follows from [18 Equation (1214)

page 124] that 119897119899119905

rarr 119897119905for all but countably many 119905

The dominated convergence theorem then yields int1199050

119897119899

119904d119904 rarr

int

119905

0

119897119904d119904 in supremum norm on finite time intervals and hence

in119863(R+R)

Lemma 6 The solution to (5) is unique in119863(R+ Ω)

Proof Suppose that 120588 and 120588 are two solutions to (5) and theyare the limits of two weakly convergent subsequences of 120588119899respectively By the assumption (A2) 120588

0= 1205880 We need to

show that 120588119905= 120588119905for all 119905 isin R

+

From (3) we have

int

119878

ℎ (120583 120582) 120588119899

119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119899

119905(d120583 d120582) (37)

for any ℎ isin 119862119887(119878) Therefore by the assumption (A2)

int

119878

ℎ (120583 120582) 120588119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119905(d120583 d120582) (38)

int

119878

ℎ (120583 120582) 120588119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119905(d120583 d120582) (39)

hold Define 119889(120588119905 120588119905) = sup

|ℎ|le1ℎisin119862119887(119878)|120588119905(ℎ) minus 120588

119905(ℎ)| In view

of (5) we obtain

1003816100381610038161003816120588119905(ℎ) minus 120588

119905(ℎ)1003816100381610038161003816

= int

119905

0

119866120588119904(ℎ) minus 119866120588

119904(ℎ) d119904

le int

119905

0

int

119878

120582ℎ (120583 120582)

10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

minus119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

times 120590119904(d120583 d120582) d119904

+ int

119905

0

int

119878

120582ℎ (120583 120582)

10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

minus119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

times 120588119904(d120583 d120582) d119904

+ int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) 120588119904(d120583 d120582) minus int

119878

120582ℎ (120583 120582) 120588119904(d120583 d120582)

10038161003816100381610038161003816100381610038161003816

times 119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892) d119904

+ int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120583ℎ (120583 120582) 120588119904(d120583 d120582)

minusint

119878

120583ℎ (120583 120582) 120588119904(d120583 d120582)

10038161003816100381610038161003816100381610038161003816

d119904

(40)

The assumption (A4) indicates that10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892) minus 119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

le 119862119891

10038161003816100381610038161003816100381610038161003816

int

119878

120588119904(d1205831015840 d1205821015840) minus int

119878

120588119904(d1205831015840 d1205821015840)

10038161003816100381610038161003816100381610038161003816

le 119862119891119889 (120588119904 120588119904)

(41)

for some constant119862119891gt 0Therefore using (38) we can derive

1003816100381610038161003816120588119905(ℎ) minus 120588

119905(ℎ)1003816100381610038161003816

le 2119862119891int

119905

0

(int

119878

120582ℎ (120583 120582) 120590119904(d120583 d120582)) 119889 (120588

119904 120588119904) d119904

+ 119862( sup(120583120582)isin119878

120582)int

119905

0

119889 (120588119904 120588119904) d119904

+ ( sup(120583120582)isin119878

120583)int

119905

0

119889 (120588119904 120588119904) d119904

(42)

where 119862 gt 0 is some constant To see this note that by theassumptions (A1) (A2) and (38) we have

int

119878

120588119904(d1205831015840 d1205821015840) + 119892 le int

119878

120590119904(d1205831015840 d1205821015840) + 119892 lt infin (43)

for any 0 le 119904 le 119905 The assumption (A4) then implies

sup119904le119905

119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892) le 119862 lt infin (44)

Again employing the assumption (A2) we have int119878

120582ℎ(120583

120582)120590119904(d120583 d120582) lt infin for all 0 le 119904 le 119905 Hence by using the

assumption (A3) the inequality (42) becomes

119889 (120588119904 120588119904) le 1198621015840

int

119905

0

119889 (120588119904 120588119904) d119904 (45)

for some constant 1198621015840 gt 0 A simple application of Gronwallrsquoslemma yields 119889(120588

119905 120588119905) = 0 for all 119905 isin R

+ which concludes the

proof

Proof of Theorem 1 By Lemma 4 and Lemma 5 the limit ofany weakly convergent subsequence of 120588119899 must satisfy (5) ByLemma 6 we find that the sequence 120588119899 must itself convergeweakly to that unique solution (see the corollary in [18 page59]) The proof of Theorem 1 is thus completed

Journal of Mathematics 7

Proof of Corollary 2 For any open set 119865 sube 119878 we obtain from(38) that

int

119878

119891119899

(120583 120582) 120588119905(d120583 d120582) le int

119878

119891119899

(120583 120582) 120590119905(d120583 d120582) (46)

where119891119899 is taken as a continuous function upwardly converg-ing to the indicator function of 119865 By using the dominatedconvergence theorem we know that 120588

119905(119865) le 120590

119905(119865) A reg-

ularity property (see eg [17 page 18 Lemma 134]) impliesthat 120588

119905(119860) le 120590

119905(119860) for all 119860 isin 119878 This means that 120588

119905is

absolutely continuous with respect to 120590119905for any 119905 ge 0 An

application of the Radon-Nykodym theorem yields the exis-tence of 120593

119905such that (8) holds with 0 le 120593

119905le 1 Now the result

follows straightforwardly fromTheorem 1

Acknowledgments

The author expresses his sincere gratitude to the anonymousreferee and the editor for careful reading of the original paperand useful comments that helped to improve presentation ofresults

References

[1] G H Weiss and M Dishon ldquoOn the asymptotic behaviorof the stochastic and deterministic models of an epidemicrdquoMathematical Biosciences vol 11 pp 261ndash265 1971

[2] D A Kessler ldquoEpidemic size in the SIS model of endemicinfectionsrdquo Journal of Applied Probability vol 45 no 3 pp 757ndash778 2008

[3] Y Shang ldquoA Lie algebra approach to susceptible-infected-susceptible epidemicsrdquo Electronic Journal of Differential Equa-tions vol 2012 no 233 pp 1ndash7 2012

[4] H Andersson and B Djehiche ldquoA threshold limit theorem forthe stochastic logistic epidemicrdquo Journal of Applied Probabilityvol 35 no 3 pp 662ndash670 1998

[5] C R Doering K V Sargsyan and L M Sander ldquoExtinc-tion times for birth-death processes exact results continuumasymptotics and the failure of the Fokker-Planck approxima-tionrdquo Multiscale Modeling amp Simulation vol 3 no 2 pp 283ndash299 2005

[6] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[7] O Ovaskainen ldquoThe quasistationary distribution of thestochastic logistic modelrdquo Journal of Applied Probability vol 38no 4 pp 898ndash907 2001

[8] D Clancy and P K Pollett ldquoA note on quasi-stationary distri-butions of birth-death processes and the SIS logistic epidemicrdquoJournal of Applied Probability vol 40 no 3 pp 821ndash825 2003

[9] TGKurtz ldquoExtensions of Trotterrsquos operator semigroup approx-imation theoremsrdquo Journal of Functional Analysis vol 3 pp354ndash375 1969

[10] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970

[11] R McVinish and P K Pollett ldquoThe deterministic limit of astochastic logistic model with individual variationrdquoMathemat-ical Biosciences vol 241 no 1 pp 109ndash114 2013

[12] R W R Darling and J R Norris ldquoDifferential equationapproximations for Markov chainsrdquo Probability Surveys vol 5pp 37ndash79 2008

[13] Y Shang ldquoOptimal control strategies for virus spreading ininhomogeneousepidemicdynamicsrdquo Canadian MathematicalBulletin In press

[14] T Gross C J D DrsquoLima and B Blasius ldquoEpidemic dynamicson an adaptivenetworkrdquo Physical Review Letters vol 96 no 20Article ID 208701 2006

[15] Y Shang ldquoDiscrete-time epidemic dynamics with awareness inrandomnetworksrdquo International Journal of Biomathematics vol6 no 2 Article ID 1350007 2013

[16] K Klemm M A Serrano V M Eguıluz and M S MiguelldquoA measure of individualrole in collective dynamicsrdquo ScientificReports vol 2 article 292 2012

[17] O Kallenberg Foundations of Modern Probability SpringerNew York NY USA 2nd edition 2002

[18] P Billingsley Convergence of Probability Measures John Wileyamp Sons New York NY USA 2nd edition 1999

[19] S N Ethier and T G KurtzMarkov Processes Characterizationand Convergence JohnWiley amp Sons Hoboken NJ USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Mathematics

Define

119897119899

119905= int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119899

119905(d1205831015840 d1205821015840) + 119892) 120588119899

119905(d120583 d120582)

119897119905= int

119878

120582ℎ (120583 120582) 119891 (int

119878

120588119905(d1205831015840 d1205821015840) + 119892) 120588

119905(d120583 d120582)

(36)

Suppose that 120588119899 rarr 120588 in 119863(R+ Ω) holds We need to

show int1199050

119897119899

119904d119904 rarr int

119905

0

119897119904d119904 in 119863(R

+R) as 119899 rarr infin Indeed

since 120582ℎ isin 119862119887(119878) and int

119878

ℎ(120583 120582)120588119905(d120583 d120582) 119863(R

+ Ω) rarr

119863(R+R) is continuous for any ℎ isin 119862

119887(119878) we have

int119878

120582ℎ(120583 120582)120588119899

119905(d120583 d120582) rarr int

119878

120582ℎ(120583 120582)120588119905(d120583 d120582) in 119863(R

+R)

and int119878

120588119899

119905(d120583 d120582) rarr int

119878

120588119905(d120583 d120582) in 119863(R

+R) as 119899 rarr

infin Hence from the assumptions (A3) and (A4) we obtain119897119899

119905rarr 119897119905in 119863(R

+R) It follows from [18 Equation (1214)

page 124] that 119897119899119905

rarr 119897119905for all but countably many 119905

The dominated convergence theorem then yields int1199050

119897119899

119904d119904 rarr

int

119905

0

119897119904d119904 in supremum norm on finite time intervals and hence

in119863(R+R)

Lemma 6 The solution to (5) is unique in119863(R+ Ω)

Proof Suppose that 120588 and 120588 are two solutions to (5) and theyare the limits of two weakly convergent subsequences of 120588119899respectively By the assumption (A2) 120588

0= 1205880 We need to

show that 120588119905= 120588119905for all 119905 isin R

+

From (3) we have

int

119878

ℎ (120583 120582) 120588119899

119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119899

119905(d120583 d120582) (37)

for any ℎ isin 119862119887(119878) Therefore by the assumption (A2)

int

119878

ℎ (120583 120582) 120588119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119905(d120583 d120582) (38)

int

119878

ℎ (120583 120582) 120588119905(d120583 d120582) le int

119878

ℎ (120583 120582) 120590119905(d120583 d120582) (39)

hold Define 119889(120588119905 120588119905) = sup

|ℎ|le1ℎisin119862119887(119878)|120588119905(ℎ) minus 120588

119905(ℎ)| In view

of (5) we obtain

1003816100381610038161003816120588119905(ℎ) minus 120588

119905(ℎ)1003816100381610038161003816

= int

119905

0

119866120588119904(ℎ) minus 119866120588

119904(ℎ) d119904

le int

119905

0

int

119878

120582ℎ (120583 120582)

10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

minus119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

times 120590119904(d120583 d120582) d119904

+ int

119905

0

int

119878

120582ℎ (120583 120582)

10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

minus119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

times 120588119904(d120583 d120582) d119904

+ int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120582ℎ (120583 120582) 120588119904(d120583 d120582) minus int

119878

120582ℎ (120583 120582) 120588119904(d120583 d120582)

10038161003816100381610038161003816100381610038161003816

times 119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892) d119904

+ int

119905

0

10038161003816100381610038161003816100381610038161003816

int

119878

120583ℎ (120583 120582) 120588119904(d120583 d120582)

minusint

119878

120583ℎ (120583 120582) 120588119904(d120583 d120582)

10038161003816100381610038161003816100381610038161003816

d119904

(40)

The assumption (A4) indicates that10038161003816100381610038161003816100381610038161003816

119891 (int

119878

120588119904(d1205831015840 d1205821015840) + 119892) minus 119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892)

10038161003816100381610038161003816100381610038161003816

le 119862119891

10038161003816100381610038161003816100381610038161003816

int

119878

120588119904(d1205831015840 d1205821015840) minus int

119878

120588119904(d1205831015840 d1205821015840)

10038161003816100381610038161003816100381610038161003816

le 119862119891119889 (120588119904 120588119904)

(41)

for some constant119862119891gt 0Therefore using (38) we can derive

1003816100381610038161003816120588119905(ℎ) minus 120588

119905(ℎ)1003816100381610038161003816

le 2119862119891int

119905

0

(int

119878

120582ℎ (120583 120582) 120590119904(d120583 d120582)) 119889 (120588

119904 120588119904) d119904

+ 119862( sup(120583120582)isin119878

120582)int

119905

0

119889 (120588119904 120588119904) d119904

+ ( sup(120583120582)isin119878

120583)int

119905

0

119889 (120588119904 120588119904) d119904

(42)

where 119862 gt 0 is some constant To see this note that by theassumptions (A1) (A2) and (38) we have

int

119878

120588119904(d1205831015840 d1205821015840) + 119892 le int

119878

120590119904(d1205831015840 d1205821015840) + 119892 lt infin (43)

for any 0 le 119904 le 119905 The assumption (A4) then implies

sup119904le119905

119891(int

119878

120588119904(d1205831015840 d1205821015840) + 119892) le 119862 lt infin (44)

Again employing the assumption (A2) we have int119878

120582ℎ(120583

120582)120590119904(d120583 d120582) lt infin for all 0 le 119904 le 119905 Hence by using the

assumption (A3) the inequality (42) becomes

119889 (120588119904 120588119904) le 1198621015840

int

119905

0

119889 (120588119904 120588119904) d119904 (45)

for some constant 1198621015840 gt 0 A simple application of Gronwallrsquoslemma yields 119889(120588

119905 120588119905) = 0 for all 119905 isin R

+ which concludes the

proof

Proof of Theorem 1 By Lemma 4 and Lemma 5 the limit ofany weakly convergent subsequence of 120588119899 must satisfy (5) ByLemma 6 we find that the sequence 120588119899 must itself convergeweakly to that unique solution (see the corollary in [18 page59]) The proof of Theorem 1 is thus completed

Journal of Mathematics 7

Proof of Corollary 2 For any open set 119865 sube 119878 we obtain from(38) that

int

119878

119891119899

(120583 120582) 120588119905(d120583 d120582) le int

119878

119891119899

(120583 120582) 120590119905(d120583 d120582) (46)

where119891119899 is taken as a continuous function upwardly converg-ing to the indicator function of 119865 By using the dominatedconvergence theorem we know that 120588

119905(119865) le 120590

119905(119865) A reg-

ularity property (see eg [17 page 18 Lemma 134]) impliesthat 120588

119905(119860) le 120590

119905(119860) for all 119860 isin 119878 This means that 120588

119905is

absolutely continuous with respect to 120590119905for any 119905 ge 0 An

application of the Radon-Nykodym theorem yields the exis-tence of 120593

119905such that (8) holds with 0 le 120593

119905le 1 Now the result

follows straightforwardly fromTheorem 1

Acknowledgments

The author expresses his sincere gratitude to the anonymousreferee and the editor for careful reading of the original paperand useful comments that helped to improve presentation ofresults

References

[1] G H Weiss and M Dishon ldquoOn the asymptotic behaviorof the stochastic and deterministic models of an epidemicrdquoMathematical Biosciences vol 11 pp 261ndash265 1971

[2] D A Kessler ldquoEpidemic size in the SIS model of endemicinfectionsrdquo Journal of Applied Probability vol 45 no 3 pp 757ndash778 2008

[3] Y Shang ldquoA Lie algebra approach to susceptible-infected-susceptible epidemicsrdquo Electronic Journal of Differential Equa-tions vol 2012 no 233 pp 1ndash7 2012

[4] H Andersson and B Djehiche ldquoA threshold limit theorem forthe stochastic logistic epidemicrdquo Journal of Applied Probabilityvol 35 no 3 pp 662ndash670 1998

[5] C R Doering K V Sargsyan and L M Sander ldquoExtinc-tion times for birth-death processes exact results continuumasymptotics and the failure of the Fokker-Planck approxima-tionrdquo Multiscale Modeling amp Simulation vol 3 no 2 pp 283ndash299 2005

[6] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[7] O Ovaskainen ldquoThe quasistationary distribution of thestochastic logistic modelrdquo Journal of Applied Probability vol 38no 4 pp 898ndash907 2001

[8] D Clancy and P K Pollett ldquoA note on quasi-stationary distri-butions of birth-death processes and the SIS logistic epidemicrdquoJournal of Applied Probability vol 40 no 3 pp 821ndash825 2003

[9] TGKurtz ldquoExtensions of Trotterrsquos operator semigroup approx-imation theoremsrdquo Journal of Functional Analysis vol 3 pp354ndash375 1969

[10] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970

[11] R McVinish and P K Pollett ldquoThe deterministic limit of astochastic logistic model with individual variationrdquoMathemat-ical Biosciences vol 241 no 1 pp 109ndash114 2013

[12] R W R Darling and J R Norris ldquoDifferential equationapproximations for Markov chainsrdquo Probability Surveys vol 5pp 37ndash79 2008

[13] Y Shang ldquoOptimal control strategies for virus spreading ininhomogeneousepidemicdynamicsrdquo Canadian MathematicalBulletin In press

[14] T Gross C J D DrsquoLima and B Blasius ldquoEpidemic dynamicson an adaptivenetworkrdquo Physical Review Letters vol 96 no 20Article ID 208701 2006

[15] Y Shang ldquoDiscrete-time epidemic dynamics with awareness inrandomnetworksrdquo International Journal of Biomathematics vol6 no 2 Article ID 1350007 2013

[16] K Klemm M A Serrano V M Eguıluz and M S MiguelldquoA measure of individualrole in collective dynamicsrdquo ScientificReports vol 2 article 292 2012

[17] O Kallenberg Foundations of Modern Probability SpringerNew York NY USA 2nd edition 2002

[18] P Billingsley Convergence of Probability Measures John Wileyamp Sons New York NY USA 2nd edition 1999

[19] S N Ethier and T G KurtzMarkov Processes Characterizationand Convergence JohnWiley amp Sons Hoboken NJ USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 7

Proof of Corollary 2 For any open set 119865 sube 119878 we obtain from(38) that

int

119878

119891119899

(120583 120582) 120588119905(d120583 d120582) le int

119878

119891119899

(120583 120582) 120590119905(d120583 d120582) (46)

where119891119899 is taken as a continuous function upwardly converg-ing to the indicator function of 119865 By using the dominatedconvergence theorem we know that 120588

119905(119865) le 120590

119905(119865) A reg-

ularity property (see eg [17 page 18 Lemma 134]) impliesthat 120588

119905(119860) le 120590

119905(119860) for all 119860 isin 119878 This means that 120588

119905is

absolutely continuous with respect to 120590119905for any 119905 ge 0 An

application of the Radon-Nykodym theorem yields the exis-tence of 120593

119905such that (8) holds with 0 le 120593

119905le 1 Now the result

follows straightforwardly fromTheorem 1

Acknowledgments

The author expresses his sincere gratitude to the anonymousreferee and the editor for careful reading of the original paperand useful comments that helped to improve presentation ofresults

References

[1] G H Weiss and M Dishon ldquoOn the asymptotic behaviorof the stochastic and deterministic models of an epidemicrdquoMathematical Biosciences vol 11 pp 261ndash265 1971

[2] D A Kessler ldquoEpidemic size in the SIS model of endemicinfectionsrdquo Journal of Applied Probability vol 45 no 3 pp 757ndash778 2008

[3] Y Shang ldquoA Lie algebra approach to susceptible-infected-susceptible epidemicsrdquo Electronic Journal of Differential Equa-tions vol 2012 no 233 pp 1ndash7 2012

[4] H Andersson and B Djehiche ldquoA threshold limit theorem forthe stochastic logistic epidemicrdquo Journal of Applied Probabilityvol 35 no 3 pp 662ndash670 1998

[5] C R Doering K V Sargsyan and L M Sander ldquoExtinc-tion times for birth-death processes exact results continuumasymptotics and the failure of the Fokker-Planck approxima-tionrdquo Multiscale Modeling amp Simulation vol 3 no 2 pp 283ndash299 2005

[6] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[7] O Ovaskainen ldquoThe quasistationary distribution of thestochastic logistic modelrdquo Journal of Applied Probability vol 38no 4 pp 898ndash907 2001

[8] D Clancy and P K Pollett ldquoA note on quasi-stationary distri-butions of birth-death processes and the SIS logistic epidemicrdquoJournal of Applied Probability vol 40 no 3 pp 821ndash825 2003

[9] TGKurtz ldquoExtensions of Trotterrsquos operator semigroup approx-imation theoremsrdquo Journal of Functional Analysis vol 3 pp354ndash375 1969

[10] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970

[11] R McVinish and P K Pollett ldquoThe deterministic limit of astochastic logistic model with individual variationrdquoMathemat-ical Biosciences vol 241 no 1 pp 109ndash114 2013

[12] R W R Darling and J R Norris ldquoDifferential equationapproximations for Markov chainsrdquo Probability Surveys vol 5pp 37ndash79 2008

[13] Y Shang ldquoOptimal control strategies for virus spreading ininhomogeneousepidemicdynamicsrdquo Canadian MathematicalBulletin In press

[14] T Gross C J D DrsquoLima and B Blasius ldquoEpidemic dynamicson an adaptivenetworkrdquo Physical Review Letters vol 96 no 20Article ID 208701 2006

[15] Y Shang ldquoDiscrete-time epidemic dynamics with awareness inrandomnetworksrdquo International Journal of Biomathematics vol6 no 2 Article ID 1350007 2013

[16] K Klemm M A Serrano V M Eguıluz and M S MiguelldquoA measure of individualrole in collective dynamicsrdquo ScientificReports vol 2 article 292 2012

[17] O Kallenberg Foundations of Modern Probability SpringerNew York NY USA 2nd edition 2002

[18] P Billingsley Convergence of Probability Measures John Wileyamp Sons New York NY USA 2nd edition 1999

[19] S N Ethier and T G KurtzMarkov Processes Characterizationand Convergence JohnWiley amp Sons Hoboken NJ USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of