Research ArticleThe Effect of Impulsive Vaccination on Delayed SEIRSEpidemic Model Incorporating Saturation Recovery

Yongfeng Li1 Dongliang Xie1 and Jing-an Cui2

1 Department of Mathematics and Information Science Zhengzhou University of Light Industry Zhengzhou Henan 450002 China2 School of Science Beijing University of Civil Engineering and Architecture Beijing 100044 China

Correspondence should be addressed to Yongfeng Li yfli2003163com

Received 24 September 2013 Accepted 17 February 2014 Published 25 March 2014

Academic Editor Ryusuke Kon

Copyright copy 2014 Yongfeng Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A delay SEIRS model with pulse vaccination incorporating a general media coverage function and saturation recovery isinvestigated Using the discrete dynamical system determined by the stroboscopic map we obtain the existence of the disease-free periodic solution and its exact expression Further using the comparison theorem we establish the sufficient conditions ofglobal attractivity of the disease-free periodic solution Moreover we show that the disease is uniformly persistent if vaccinationrate is less than 120579

lowast Finally we discuss the effect of media coverage on controlling disease

1 Introduction

In recent years controlling infectious disease is a veryimportant issue vaccination is a commonly used method forcontrolling disease the study of vaccines against infectiousdisease has been a boon to mankind There are now vaccinesthat are effective in preventing such viral infections asrabies yellow fever poliovirus hepatitis B parotitis andencephalitis B Eventually vaccines will probably preventmalaria some forms of heart disease and cancer Vaccineshave been very important to people Theoretical results showthat pulse vaccination strategy can be distinguished fromthe conventional strategies leading to disease eradication atrelatively low values of vaccination [1] Theories of impulsivedifferential equations are found in the books [2 3] In recentyears their applications can be found in the domain ofapplied sciences [4ndash7] In this paper we consider impulsivevaccination to susceptible individuals

In the classical endemic models the incidence rate isassumed to be mass action incidence with bilinear interac-tions given by 120573119878119868 where 120573 is the probability of transmissionper contact and 119878 and 119868 represent the susceptible and infectedpopulations respectively If the population is saturated withinfective individuals there are three kinds of incidence formsthat are used in epidemiological model the proportionatemixing incidence 120573119878119868119873 [8] nonlinear incidence 120573119868119901119878119902 [9]

and saturation incidence 120573119868119878(1 + 120572119878) [10] or 120573119868119897119878(1 + 120572119868ℎ

)

[11] However some factors such as media coverage mannerof life and density of population may affect the incidencerate directly or indirectly nonlinear incidence rate can beapproximated by a variety of forms such as 120573(1 minus 119888119868)119868119878 (119888 gt0) (120573

1minus 1205732(119868(119898 + 119868)))119878119868 (120573

1gt 1205732gt 0119898 gt 0) which were

discussed by [12ndash14]In this paper we suggest a general nonlinear incidence

rate (1205731minus1205732(119868ℎ

(119888+119868ℎ

)))119878119868 (1205731gt 1205732gt 0 119888 gt 0 ℎ ge 1)which

reflects some characters of media coverage where 1205731= 1199011198881

1205732= 1199011198882 119901 is the transmission probability under contacts

in unit time 1198881is the usual contact rate 119888

2is the maximum

reduced contact rate through actual media coverage and 119888

is the rate of the reflection on the disease Again mediacoverage can not totally interrupt disease transmission sowe have 120573

1gt 1205732 We use 120573

2(119868ℎ

(119888 + 119868ℎ

)) to reflect theamount of contact rate reduced through media coverageWhen infective individuals appear in a region people reducetheir contact with others to avoid being infected and themore infective individuals being reported the less contactwith others hence we take the above form Few studies haveappeared on this aspect

In the classical disease transmission models the recoveryfrom infected class per unit of time is assumed to beproportional to the number of infective individuals (denoted

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 426456 7 pageshttpdxdoiorg1011552014426456

2 Discrete Dynamics in Nature and Society

by 119868) say 120574119868 where 120574 gt 0 is the removal rate This isa reasonable approximation to the truth when the numberof the infectious individuals is not too large and below thecapacity of health care settings If the number of illnessexceeds a fixed large size then the number of recovered isindependent of further changes in infectious size We adoptthe Verhulst-type function 119892(119868) = 120574119868(119889 + 119868) to modelthe recovered part which increases for small infectives andapproaches a maximum for large infectives Here 120574 gives themaximum recovery per unit of time and 119889 the infected sizeat which is 50 saturation (119892(119887) = 1198882) measures how soonsaturation occurs Cui et al studied this removal rate [15]

Cooke and Van den Diressche [16] investigated an SEIRSmodel with the latent period and the immune period themodel is as follows

119889119878 (119905)

119889119905= 119887119873 (119905) minus 119887119878 (119905) minus

120573119878 (119905) 119868 (119905)

119873 (119905)+ 120574119868 (119905 minus 120591) 119890

minus119887120591

119864 (119905) = int

119905

119905minus119908

120573119878 (119906) 119868 (119906)

119873 (119906)119890minus119887(119905minus119906)

119889119906

119889119868 (119905)

119889119905=120573119878 (119905 minus 119908) 119868 (119905 minus 119908)

119873 (119905 minus 119908)119890minus119887119908

minus (119887 + 120574) 119868 (119905)

119877 (119905) = int

119905

119905minus120591

120574119868 (119906) 119890minus119887(119905minus119906)

119889119906

119873 (119905) = 119878 (119905) + 119864 (119905) + 119868 (119905) + 119877 (119905)

(1)

where 119887 is the natural birth and death rate of the population120573 is average number of adequate contacts of an infectiousindividuals per unit time 120574 is the recovery rate of infectiousindividuals 119908 is the latent period of the disease and 120591 isimmune period of the population All coefficients are positiveconstants It is easy to obtain from system (1) that the totalpopulation is constant For convenience we assume that119873(119905) = 119878(119905) + 119864(119905) + 119868(119905) + 119877(119905) = 1 Based on the aboveassumptions we have the following SEIRS epidemic modelwith vaccination

119889119878 (119905)

119889119905= 119887 minus 119887119878 (119905) minus (120573

1minus

1205732119868ℎ

(119905)

119888 + 119868ℎ (119905)) 119878 (119905) 119868 (119905)

+120574119868 (119905 minus 120591)

119889 + 119868 (119905 minus 120591)119890minus119887120591

119864 (119905) = int

119905

119905minus119908

(1205731minus

1205732119868ℎ

(119906)

119888 + 119868ℎ (119906)) 119878 (119906) 119868 (119906) 119890

minus119887(119905minus119906)

119889119906

119889119868 (119905)

119889119905= (120573

1minus

1205732119868ℎ

(119905 minus 119908)

119888 + 119868ℎ (119905 minus 119908)) 119878 (119905 minus 119908) 119868 (119905 minus 119908) 119890

minus119887120591

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

119877 (119905) = int

119905

119905minus120591

120574119868 (119906)

119889 + 119868 (119906)119890minus119887(119905minus119906)

119889119906

119905 = 119896119879

119878 (119905+

) = (1 minus 120579) 119878 (119905minus

)

119864 (119905+

) = 119864 (119905minus

)

119868 (119905+

) = 119868 (119905minus

)

119877 (119905+

) = 119877 (119905minus

) + 120579119878 (119905minus

)

119905 = 119896119879

(2)

where 119896 isin 119885+ 119885+ = 0 1 2 119873(119905) = 119878(119905) + 119864(119905) + 119868(119905) +

119877(119905) = 1 Note that the variables 119864 and 119877 do not appear in thefirst and third equations of system (2) this allows us to attack(2) by studying the subsystem

119889119878 (119905)

119889119905= 119887 minus 119887119878 (119905) minus (120573

1minus

1205732119868ℎ

(119905)

119888 + 119868ℎ (119905)) 119878 (119905) 119868 (119905)

+120574119868 (119905 minus 120591)

119889 + 119868 (119905 minus 120591)119890minus119887120591

119889119868 (119905)

119889119905= (1205731minus

1205732119868ℎ

(119905 minus 119908)

119888 + 119868ℎ (119905 minus 119908)) 119878 (119905 minus 119908) 119868 (119905 minus 119908) 119890

minus119887120591

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

119905 = 119896119879

119878 (119905+

) = (1 minus 120579) 119878 (119905minus

)

119868 (119905+

) = 119868 (119905minus

)

119905 = 119896119879

(3)

The main purpose of this paper is to establish sufficientconditions that the disease dies out and show that the diseaseis uniformly persistent under some conditions

2 Notations and Definitions

We introduce some notations and definitions and state someresults which will be useful in subsequent sections

Let 119877+= [0infin) 119877

2

+= 119885 isin 119877

2

119885 ge 0 Denote 119891 =

(1198911 1198912)119879 and the map defined by the right hand of the first

and second equations of systems (3) Set 119897 = max120591 119908 Let 119862be the space of continuous functions on [minus119897 0] with uniformnormThe initial conditions for (3) are

(1206011(120577) 120601

2(120577)) isin 119862

+= 119862 ([minus119897 0] 119877

2

+) 120601

119894(0) gt 0 119894 = 1 2

(4)

The solution of system (3) is a piecewise continuousfunction 119885 119877

+rarr 119877

2

+ 119885(119905) is continuous on (119896119879 (119896 +

1)119879] 119896 isin 119885+ and 119885(119896119879

+

) = lim119905rarr119896119879

+119885(119905) existsObviously the smooth properties of 119891 guarantee the globalexistence and uniqueness of solution of system (3) (see [3]for details on fundamental properties of impulsive systems)Since 119878(119905)|

119878=0gt 0 and 119868(119905) = 0 whenever 119868(119905) = 0 for

Discrete Dynamics in Nature and Society 3

119905 = 119896119879 119896 isin 119885+ Moreover 119878(119896119879+) = (1 minus 120579)119878(119896119879

minus

) and119868(119896119879+

) = 119868(119896119879minus

) for 119896 isin 119885+Therefore we have the following

lemma

Lemma 1 Suppose 119885(119905) is a solution of system (3) with initialconditions (4) then 119885(119905) ge 0 for all 119905 ge 0

Denote that Ω = (119878 119868) isin 1198772

| 119878 ge 0 119868 ge 0 119878 + 119868 le 1Using the fact that 119878(119905) + 119864(119905) + 119868(119905) + 119877(119905) = 1 it is easy toshow that Ω is positively invariant with respect to (3)

Lemma 2 (see [10]) Consider the following impulsive system

(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879

119906 (119905+

) = (1 minus 120579) 119906 (119905minus

) 119905 = 119896119879

(5)

where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then there exists a uniquepositive periodic solution of system (5)

119890(119905) =

119886

119887+ (119906lowast

minus119886

119887) 119890minus119887(119905minus119896119879)

=119886

119887(1 minus

120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879) 119896119879 lt 119905 le (119896 + 1) 119879

(6)

which is globally asymptotically stable where 119906lowast = (119886119887)(1 minus

120579)(1 minus 119890minus119887119879

)(1 minus (1 minus 120579)119890minus119887119879

)

Lemma 3 (see [17]) Consider the following delayed differen-tial equation

1199091015840

(119905) = 1198861119909 (119905 minus 119908) minus 119886

2119909 (119905) (7)

where 1198861 1198862 119908 gt 0 119909(119905) gt 0 for minus119908 le 119905 le 0 We have

(i) if 1198861lt 1198862 then lim

119905rarrinfin119909(119905) = 0

(ii) if 1198861gt 1198862 then lim

119905rarrinfin119909(119905) = +infin

Definition 4 System (3) is said to be uniformly persistent ifthere is an 120578 gt 0 (independent of the initial conditions) suchthat every solution (119878(119905) 119868(119905)) with initial conditions (4) ofsystem (3) satisfies

lim119905rarrinfin

inf 119878 (119905) ge 120578 lim119905rarrinfin

inf 119868 (119905) ge 120578 (8)

Definition 5 System (3) is said to be permanent if thereexists a compact region Ω

0sub int1198772

+such that every

solution (119878(119905) 119868(119905)) of system (3) with initial conditions (4)will eventually enter and remain in regionΩ

0

3 Global Attractivity of Infection-FreePeriodic Solution

In this section we analyse the attractivity of infection-freeperiodic solution of system (3) If we let 119868(119905) = 0 then thegrowth of susceptible individuals must satisfy

119878 (119905) = 119886 minus 119887119878 (119905) 119905 = 119896119879

119878 (119905+

) = (1 minus 120579) 119878 (119905minus

) 119905 = 119896119879

(9)

By Lemma 2 we know that periodic solution of system(9)

119878119890(119905) = 1 minus

120579

1 minus (1 minus 120579) 119890minus119887119879119890minus119887(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879

(10)

is globally asymptotically stableAbout the global attractivity of infection-free periodic

solution (119878119890(119905) 0) of system (3) we have the following

theorem

Theorem 6 The infection-free periodic solution (119878119890(119905) 0) of

system (3) is globally attractivity provided that 119877lowast lt 1 where119877lowast

= 1205731119890minus119887119908

(119887+120574119890minus119887120591

(119889+1))(1minus119890minus119887119879

)(119887(119887minus1205732(119888+1))(1minus

(1 minus 120579)119890minus119887119879

))

Proof Since 119877lowast lt 1 we can choose 1205760gt 0 sufficiently small

such that

1205731120575119890minus119887119908

lt 119887 (11)

where 120575 = (119887+120574119890minus119887120591

(119889+1))(1minus119890minus119887119879

)(119887minus1205732(119888+1))(1minus(1minus

120579)119890minus119887119879

) + 1205760 From the first equation of system (3) we have

119878 (119905) lt 119887 minus 119887119878 (119905) + 1205732119878 (119905) 119868 (119905)

119868ℎ

(119905)

119888 + 119868ℎ (119905)+

120574119868 (119905 minus 120591)

119889 + 119868 (119905 minus 120591)119890minus119887120591

lt (119887 +120574119890minus119887120591

119889 + 1) minus (119887 minus

1205732

119888 + 1) 119878 (119905)

(12)

Then we consider the following comparison system withpulses

(119905) = (119887 +120574119890minus119887120591

119889 + 1) minus (119887 minus

1205732

119888 + 1) 119909 (119905) 119905 = 119896119879

119909 (119905+

) = (1 minus 120579) 119909 (119905minus

) 119905 = 119896119879

(13)

By Lemma 2 we know that there is a unique periodic solutionof system (13)

119909119890(119905) =

119887 + 120574119890minus119887120591

(119889 + 1)

119887 minus 1205732 (119888 + 1)

(1 minus120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879)

119896119879 lt 119905 le (119896 + 1) 119879

(14)

which is globally asymptotically stableLet (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and let 119878(0+) = 1198780gt 0 119909(119905) be the solution of

system (13) with initial values 119909(0+) = 1198780 By the comparison

theorem in impulsive differential equation [3] there exists aninteger 119896

1gt 0 such that 119878(119905) lt 119909

119890(119905) + 120576

0for 119905 gt 119896

1119879 thus

119878 (119905) lt(119887 + 120574119890

minus119887120591

(119889 + 1)) (1 minus 119890minus119887119879

)

(119887 minus 1205732 (119888 + 1)) (1 minus (1 minus 120579) 119890minus119887119879)

+ 1205760= 120575

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961

(15)

4 Discrete Dynamics in Nature and Society

Again from the second equation of system (3) we know that(15) implies that

119868 (119905) le (1205731minus

1205732119868ℎ

(119905 minus 119908)

119888 + 119868ℎ (119905 minus 119908)) 120575119868 (119905 minus 119908) 119890

minus119887119908

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

le 1205731120575119890minus119887119908

119868 (119905 minus 119908) minus 119887119868 (119905)

(16)

where 119905 gt 119896119879 + 119908 119896 gt 1198961

Consider the following comparison system

119910 (119905) = 1205731120575119890minus119887119908

119910 (119905 minus 119908) minus 119887119910 (119905) 119905 gt 119896119879 + 119908 119896 gt 1198961

(17)

According to (11) and Lemma 3 we have lim119905rarrinfin

119910(119905) = 0Let (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and let 119868(120577) = 120593(120577) gt 0 (120577 isin [minus119908 0]) 119910(119905)be the solution of system (17) with initial values 119910(120577) =

120593(120577) gt 0 (120577 isin [minus119908 0]) By the comparison theorem wehave lim

119905rarrinfinsup 119868(119905) le lim

119905rarrinfinsup119910(119905) = 0 Incorporating

into the positivity of 119868(119905) we know that lim119905rarrinfin

119868(119905) = 0Therefore there exists an integer 119896

2gt 1198961(where 119896

2119879 gt

1198961119879 + 119908) such that 119868(119905) lt 120576

0for all 119905 gt 119896

2119879

From the first equation of system (3) we have

119878 (119905) gt 119887 minus 119887119878 (119905) minus 1205731119878 (119905) 119868 (119905) gt 119887 minus (119887 + 120573

11205760) 119878 (119905)

119905 gt 1198962119879 + 120591

119878 (119905) lt 119887 minus 119887119878 (119905) + 1205732119878 (119905)

119868ℎ+1

(119905)

119888 + 119868ℎ (119905)+

120574119868 (119905 minus 120591)

119889 + 119868 (119905 minus 120591)119890minus119887120591

lt (119887 +1205741205760

119889 + 1205760

119890minus119887120591

) minus (119887 minus1205732120576ℎ+1

0

119888 + 120576ℎ0

)119878 (119905)

119905 gt 1198962119879 + 120591

(18)

Consider the following comparison impulsive differentialequations for 119905 gt 119896

2119879 + 120591 and 119896 gt 119896

2

1(119905) = 119887 minus (119887 + 120573

11205760) 1199111(119905) 119905 = 119896119879

1199111(119905+

) = (1 minus 120579) 1199111(119905minus

) 119905 = 119896119879

(19)

2(119905) = (119887 +

1205741205760

119889 + 1205760

119890minus119887120591

) minus (119887 minus1205732120576ℎ+1

0

119888 + 120576ℎ0

)1199112(119905) 119905 = 119896119879

1199112(119905+

) = (1 minus 120579) 1199112(119905minus

) 119905 = 119896119879

(20)

By Lemma 2 we have that the unique periodic solution ofsystem (19)

1119890(119905) =

119887

119887 + 12057311205760

(1 minus120579119890minus(119887+120573

11205760)(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879)

119896119879 lt 119905 le (119896 + 1) 119879

(21)

and the unique periodic solution of system (20)

2119890(119905) =

(119887 + (1205741205760 (119889 + 120576

0)) 119890minus119887120591

)

(119887 minus (1205732120576ℎ+1

0 (119888 + 120576ℎ

0)))

(1 minus120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879)

119896119879 lt 119905 le (119896 + 1) 119879

(22)

are globally asymptotically stableLet (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and 119878(0+

) = 1198780

gt 0 1199111(119905) and let 119911

2(119905) be the

solutions of system (19) and (20) with initial values 1199111(0+

) =

1199112(0+

) = 1198780 respectively By the comparison theorem in

impulsive differential equation there exists an integer 1198963gt 1198962

such that 1198963119879 gt 119896

2119879 + 120591 and

1119890(119905) minus 120576

0lt 119878 (119905) lt

2119890(119905) minus 120576

0

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198963

(23)

Because 1205760is arbitrarily small it follows from (23) that

119878119890(119905) = 1 minus

120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879 119896119879 lt 119905 le (119896 + 1) 119879 (24)

is globally attractive The proof is complete

Denote that 120579lowast = 1minus119890119887119879

+(1205731119890minus119887119908

(1+(120574119887)119890minus119887120591

)(119890119887119879

minus1))

119887(119887minus1205732(119888+1)) 120573lowast

2= (119888+1)(119887minus120573

1119890minus119887119908

(119887+120574119890minus119887120591

(119889+1))(1minus

119890minus119887119879

)119887(1minus(1minus120579)119890minus119887119879

)) and 119888lowast = 1205732119887(1minus(1minus120579)119890

minus119887119879

)(1198872

(1minus

(1 minus 120579)119890minus119887119879

) minus 1205731119890minus119887119908

(119887 + 120574119890minus119887120591

(119889 + 1))(1 minus 119890minus119887119879

)) minus 1According to Theorem 6 we can obtain the following

result

Corollary 7 The infection-free periodic solution (119878119890(119905) 0) of

system (3) is globally attractivity provided that 120579 gt 120579lowast or 120573

2lt

120573lowast

2or 119888 gt 119888

lowastFrom Corollary 7 we know that the disease will disappear

if the vaccination rate is larger than 120579lowast

4 Permanence

In this section we say the disease is endemic if the infectiouspopulation persists above a certain positive level for suffi-ciently large time

Denote that119877lowast= (1205731minus1205732(119888+1))119890

minus119887119908

(1minus120579)(1minus119890minus119887119879

)(119887+

120574119889)(1minus (1minus120579)119890minus119887119879

) and 119868lowast = (119887120573)[(1205731minus1205732(119888+1))119890

minus119887119908

(1minus

120579)(1 minus 119890minus119887119879

)(119887 + 120574119889)(1 minus (1 minus 120579)119890minus119887119879

) minus 1]

Theorem 8 If 119877lowast

gt 1 then there is a positive constant 119902such that each positive solution (119878(119905) 119868(119905)) of system (3) satisfies119868(119905) gt 119902 for 119905 large enough

Proof From 119877lowastgt 1 we easily know that 119868lowast gt 0 and there

exists sufficiently small 120576 gt 0 such that

(1205731minus

1205732

119888 + 1) 1205751119890minus119887119908

minus 119887 minus120574

119889gt 0 (25)

Discrete Dynamics in Nature and Society 5

where 1205751= (119887(119887 + 120573

1119868lowast

))(1 minus 120579(1 minus (1 minus 120579)119890minus(119887+120573

1119868lowast)119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt 119868

lowast forall 119905 ge 119905

0 Suppose that the claim is not valid Then there is

a 1199050gt 0 such that 119868(119905) lt 119868

lowast for all 119905 ge 1199050 It follows from the

first equation of (3) that for 119905 ge 1199050

119878 (119905) gt 119887 minus 119887119878 (119905) minus 1205731119878 (119905) 119868 (119905) gt 119887 minus (119887 + 120573

1119868lowast

) 119878 (119905) (26)

Consider the following comparison impulsive system for119905 ge 1199050

(119905) = 119887 minus (119887 + 1205731119868lowast

) 119906 (119905) 119905 = 119896119879

119906 (119905+

) = (1 minus 120579) 119906 (119905minus

) 119905 = 119896119879

(27)

By Lemma 2 we obtain that

119890(119905) =

119887

119887 + 1205731119868lowast+ (119906

lowast

minus119887

119887 + 1205731119868lowast) 119890minus(119887+120573

1119868lowast)(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879

(28)

is the unique positive periodic solution of (27) which isglobally asymptotically stable where 119906lowast = (119887(119887+120573

1119868lowast

))((1minus

120579)(1 minus 119890minus(119887+120573

1119868lowast)119879

)(1 minus (1 minus 120579)(1 minus 119890minus(119887+120573

1119868lowast)119879

)))Let (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and let 119878(0+) = 1198780gt 0 119906(119905) be the solution of

system (27) with initial values 119906(0+) = 1198780 By the comparison

theorem for impulsive differential equation there exists aninteger 119896

1(gt 1199050+ 119908) such that 119878(119905) gt

119890(119905) minus 120576 for 119905 ge 119896

1119879

thus

119878 (119905) gt 119906lowast

minus 120576 = 1205751 119905 ge 119896

1 (29)

The second equation of system (3) can be rewritten as

119868 (119905) = 1205731119890minus119887119908

119878 (119905) 119868 (119905) minus 1205731119890minus119887119908

119889

119889119905int

119905

119905minus119908

119878 (119906) 119868 (119906) 119889119906

minus 1205732119890minus119887119908

119878 (119905) 119868ℎ+1

(119905)

119888 + 119868ℎ (119905)

+ 1205732119890minus119887119908

119889

119889119905int

119905

119905minus119908

119878 (119906) 119868ℎ+1

(119906)

119888 + 119868ℎ (119906)119889119906

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

(30)

Let us consider any positive solution (119878(119905) 119868(119905)) of system (3)According to this solution we define

119881 (119905) = 119868 (119905) + 1205731119890minus119887119908

int

119905

119905minus119908

119878 (119906) 119868 (119906) 119889119906

minus 1205732119890minus119887119908

int

119905

119905minus119908

119878 (119906) 119868ℎ+1

(119906)

119888 + 119868ℎ (119906)119889119906

(31)

According to (30) we calculate the derivative of 119881 along thesolutions of system (3)

(119905) = 119868 (119905) (1205731119890minus119887119908

119878 (119905) minus1205732119878 (119905) 119868

ℎ

(119905)

119888 + 119868ℎ (119905)119890minus119887119908

minus 119887 minus120574

119889 + 119868 (119905))

(32)

By (25) and (29) for 119905 ge 1199051 we have

(119905) ge 119868 (119905) (1205731119890minus119887119908

119878 (119905) minus1205732119878 (119905)

119888 + 1119890minus119887119908

minus 119887 minus120574

119889)

gt 119868 (119905) [(1205731minus

1205732

119888 + 1) 119890minus119887119908

1205751minus 119887 minus

120574

119889] gt 0

(33)

Let 119868119897= min

119905isin[11990511199051+119908]

119868(119905) in the following we will showthat 119868(119905) ge 119868

119897for 119905 ge 119905

1 Suppose the contrary then there is a

1198790ge 0 such that 119868(119905) ge 119868

119897for 1199051le 119905 le 119905

1+ 119908 + 119879

0 119868(1199051+ 119908 +

1198790) = 119868119897and 119868(119905

1+119908+119879

0) le 0 However the second equation

of system (3) and (4) imply that

119868 (1199051+ 119908 + 119879

0) ge (120573

1minus

1205732

119888 + 1) 119878 (1199051+ 1198790) 119868 (1199051+ 1198790) 119890minus119887119908

minus 119887119868 (1199051+ 119908 + 119879

0) minus

120574

119889119868 (1199051+ 119908 + 119879

0)

ge (1205731minus

1205732

119888 + 1) 119878 (1199051+ 1198790) 119868119897119890minus119887119908

minus 119887119868119897minus120574

119889119868119897

gt [(1205731minus

1205732

119888 + 1) 1205751119890minus119887119908

minus 119887 minus120574

119889] 119868119897gt 0

(34)

This is a contradictionThus 119868(119905) ge 119868119897for 119905 ge 119905

1 So (33) leads

to

(119905) gt 119868119897[(1205731minus

1205732

119888 + 1) 119890minus119887119908

1205751minus 119887 minus

120574

119889] 119905 ge 119905

1 (35)

which implies that 119881(119905) rarr infin as 119905 rarr infin This contradictswith 119881(119905) le 1 + 120573

1119908119890minus119887119908 Hence the claim is proved From

the claim we will discuss the following two possibilities

(i) 119868(119905) ge 119868lowast for 119905 large enough

(ii) 119868(119905) oscillates about 119868lowast for 119905 large enough

Evidently we only need to consider the case (ii) Let 119905lowast gt 0

and 120585 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast

+ 120585) = 119868lowast and let 119868(119905) lt 119868

lowast for119905lowast

lt 119905 lt 119905lowast

+120585 where 119905lowast is sufficiently large such that 119878(119905) gt 1205751

for 119905lowast lt 119905 lt 119905lowast

+ 120585 Since 119868(119905) is continuous and ultimatelybounded and is not effected by impulses we conclude that119868(119905) is uniformly continuous Hence there exists a constant 120582(with 0 lt 120582 lt 119908 and 120582 is independent of the choice of 119905lowast)such that 119868(119905) gt 119868

lowast

2 for 119905lowast lt 119905 lt 119905lowast

+ 120582 If 120585 le 120582 our aim isobtained If 120582 lt 120585 le 119908 from the second equation of (3) wehave that 119868(119905) gt minus(119887 + 120574119889)119868(119905) and 119868(119905lowast) = 119868

lowast then we have119868(119905) ge 119902 for 119905 isin [119905

lowast

119905lowast

+ 119908] where 119902 = min119868lowast2 119902lowast 119902lowast =119868lowast

119890minus(119887+120574)119908) The same arguments can be continued and we

can obtain 119868(119905) ge 119902 for 119905 isin [119905lowast

+ 119908 119905lowast

+ 120585] Since the intervalis chosen in an arbitrary way we get that 119868(119905) ge 119902 for 119905 largeenough In view of our arguments above the choice of 119902 isindependent of the positive solution of (3) which satisfies that119868(119905) ge 119902 for sufficiently large 119905 This completes the proof

Denote that 120579lowast= [(1205731minus 1205732(119888 + 1))119890

minus119887119908

minus (119887 + 120574119889)](1 minus

119890minus119887119879

)((1205731minus 1205732(119888 + 1))119890

minus119887119908

(1 minus 119890minus119887119879

) + (119887 + 120574119889)119890minus119887119879

) 1205732lowast

=

(119888 + 1)(1205731+ (119887 + 120574119889)(1 minus (1 minus 120579)119890

minus119887119879

)119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

))

6 Discrete Dynamics in Nature and Society

and 119888lowast= 1205732119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

)(1205731119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

) +

(119887 + 120574119889)(1 minus (1 minus 120579)119890minus119887119879

)) minus 1It follows from Theorem 8 that the disease is uniformly

persistent provided that 120579 lt 120579lowastor 1205732lt 1205732lowast

or 119888 gt 119888lowast

Theorem 9 If 119877lowastgt 1 then system (3) is permanent

Proof Suppose that (119878(119905) 119868(119905)) is any positive solution ofsystem (3) with initial conditions (4) From the first equationof system (3) we have 119878(119905) ge 119887 minus (119887 + 120573

1)119878(119905) By similar

arguments as those in the proof of Theorem 6 we have that

lim119905rarrinfin

119878 (119905) ge 119901 (36)

where 119901 = (119887(119887 + 1205731))((1 minus 120579)(1 minus 119890

minus(119887+120573minus1)119879

)(1 minus (1 minus

120579)119890minus(119887+120573minus1)119879

)) minus 1205761 (1205761is sufficiently small)

Set Ω0= (119878 119868) isin 119877

2

| 119901 le 119878 119902 le 119868 119878 + 119868 le 1FromTheorem 8 and (36) we know that the setΩ

0is a global

attractor in Ω and of course every solution of system (3)with initial conditions (4) will eventually enter and remainin region Ω

0 Therefore system (3) is permanent The proof

is complete

FromTheorem 9 we can obtain the following result

Corollary 10 Assume that 120579 lt 120579lowastor 1205732lt 1205732lowast

or 119888 gt 119888lowast then

system (3) is permanent

5 Conclusion

In this paper we introduce media coverage and saturationrecovery in the delayed SEIRS epidemic model with pulsevaccination and analyze detailedly in theory that mediacoverage and pulse vaccination bring effects on infection-free and the permanence of epidemic disease We suggestthe probability of transmission per contact 120573(119868) = 120573

1minus

1205732(119868ℎ

(119888 + 119868ℎ

)) which reflects some characters of mediacoverage We can get that 120597120573(119868)120597120573

2= minus(119868

ℎ

(119888 + 119868ℎ

)) lt 0so 120573(119868) is a monotone decreasing function on 120573

2 that is

if 1205732(the reduced valid contact rate through actual media

coverage) is larger then infection rate of disease is smallerAgain 120597120573(119868)120597119888 = 120573

2119868ℎ

(119888 + 119868ℎ

)2

gt 0 so 120573(119868) is a monotoneincreasing function on 119888 that is if 119888 is smaller (refectionon the disease is quickly) then infection rate of disease issmallerWhen infective individuals appear in a region peoplereduce their contact with others to avoid being infectedand the more infective individuals being reported the lesscontact with others Fromabove analysis we know thatmediacoverage is very important on controlling disease and mediacoverage should be considered in incidence rate

By Theorem 6 the infection-free periodic solution(119878119890(119905) 0) of system (3) is globally attractivity provided that

119877lowast

lt 1 By Theorem 9 system (3) is permanent if 119877lowastgt 1

From Corollaries 7 and 10 we can choose the proportionof those vaccinated successfully to all of newborns such that120579 gt 120579

lowast in order to prevent the epidemic disease fromgenerating endemic and the epidemic is permanent if 120579 lt 120579

lowast

But for 120579 isin [120579lowast 120579lowast

] the dynamical behavior of model (3)

has not been studied and the threshold parameter for thevaccination rate between the extinction of the disease andthe uniform persistence of the disease has not been obtainedThese issues will be considered in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the NNSF of China (1137104811201433) NSF of Henan Province (112300410156122300410117) NSF of the Education DepartmentHenan Province (2011A110022) GGJS of Henan Province(2013GGJS-110) and XGGJS of Zhengzhou University ofLight Industry (2012XGGJS003)

References

[1] Z Agur L Cojocaru G Mazor R M Anderson and Y LDanon ldquoPulse mass measles vaccination across age cohortsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 90 no 24 pp 11698ndash11702 1993

[2] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificamp Technical New York NY USA 1993

[3] V Lakshmikantham D D Bainov and P S Simeonov Theoryof Impulsive Differential Equations World Scientific Singapore1989

[4] Y Li J Cui and X Song ldquoDynamics of a predator-prey systemwith pulsesrdquo Applied Mathematics and Computation vol 204no 1 pp 269ndash280 2008

[5] X Song and Y Li ldquoDynamic complexities of a Holling II two-prey one-predator systemwith impulsive effectrdquoChaos Solitonsamp Fractals vol 33 no 2 pp 463ndash478 2007

[6] X Liu ldquoImpulsive stabilization and applications to populationgrowth modelsrdquo Rocky Mountain Journal of Mathematics vol25 no 1 pp 381ndash395 1995

[7] A Lakmeche and O Arino ldquoBifurcation of non trivial periodicsolutions of impulsive differential equations arising chemother-apeutic treatmentrdquo Dynamics of Continuous Discrete andImpulsive Systems Series B vol 7 no 2 pp 265ndash287 2000

[8] W Wang ldquoGlobal behavior of an SEIRS epidemic model withtime delaysrdquoAppliedMathematics Letters vol 15 no 4 pp 423ndash428 2002

[9] J Hui and L-S Chen ldquoImpulsive vaccination of sir epidemicmodels with nonlinear incidence ratesrdquo Discrete and Continu-ous Dynamical Systems Series B vol 4 no 3 pp 595ndash605 2004

[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006

[11] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003

[12] W-M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

Discrete Dynamics in Nature and Society 7

[13] Y Li and J Cui ldquoThe effect of constant and pulse vaccination onSIS epidemic models incorporating media coveragerdquo Commu-nications inNonlinear Science andNumerical Simulation vol 14no 5 pp 2353ndash2365 2009

[14] Y Li C Ma and J Cui ldquoThe effect of constant and mixedimpulsive vaccination on SIS epidemic models incorporatingmedia coveragerdquo Rocky Mountain Journal of Mathematics vol38 no 5 pp 1437ndash1455 2008

[15] J Cui X Mu and H Wan ldquoSaturation recovery leads tomultiple endemic equilibria and backward bifurcationrdquo Journalof Theoretical Biology vol 254 no 2 pp 275ndash283 2008

[16] K L Cooke and P Van Den Driessche ldquoAnalysis of an SEIRSepidemic model with two delaysrdquo Journal of MathematicalBiology vol 35 no 2 pp 240ndash260 1996

[17] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press San Digo Calif USA1993

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

2 Discrete Dynamics in Nature and Society

by 119868) say 120574119868 where 120574 gt 0 is the removal rate This isa reasonable approximation to the truth when the numberof the infectious individuals is not too large and below thecapacity of health care settings If the number of illnessexceeds a fixed large size then the number of recovered isindependent of further changes in infectious size We adoptthe Verhulst-type function 119892(119868) = 120574119868(119889 + 119868) to modelthe recovered part which increases for small infectives andapproaches a maximum for large infectives Here 120574 gives themaximum recovery per unit of time and 119889 the infected sizeat which is 50 saturation (119892(119887) = 1198882) measures how soonsaturation occurs Cui et al studied this removal rate [15]

Cooke and Van den Diressche [16] investigated an SEIRSmodel with the latent period and the immune period themodel is as follows

119889119878 (119905)

119889119905= 119887119873 (119905) minus 119887119878 (119905) minus

120573119878 (119905) 119868 (119905)

119873 (119905)+ 120574119868 (119905 minus 120591) 119890

minus119887120591

119864 (119905) = int

119905

119905minus119908

120573119878 (119906) 119868 (119906)

119873 (119906)119890minus119887(119905minus119906)

119889119906

119889119868 (119905)

119889119905=120573119878 (119905 minus 119908) 119868 (119905 minus 119908)

119873 (119905 minus 119908)119890minus119887119908

minus (119887 + 120574) 119868 (119905)

119877 (119905) = int

119905

119905minus120591

120574119868 (119906) 119890minus119887(119905minus119906)

119889119906

119873 (119905) = 119878 (119905) + 119864 (119905) + 119868 (119905) + 119877 (119905)

(1)

where 119887 is the natural birth and death rate of the population120573 is average number of adequate contacts of an infectiousindividuals per unit time 120574 is the recovery rate of infectiousindividuals 119908 is the latent period of the disease and 120591 isimmune period of the population All coefficients are positiveconstants It is easy to obtain from system (1) that the totalpopulation is constant For convenience we assume that119873(119905) = 119878(119905) + 119864(119905) + 119868(119905) + 119877(119905) = 1 Based on the aboveassumptions we have the following SEIRS epidemic modelwith vaccination

119889119878 (119905)

119889119905= 119887 minus 119887119878 (119905) minus (120573

1minus

1205732119868ℎ

(119905)

119888 + 119868ℎ (119905)) 119878 (119905) 119868 (119905)

+120574119868 (119905 minus 120591)

119889 + 119868 (119905 minus 120591)119890minus119887120591

119864 (119905) = int

119905

119905minus119908

(1205731minus

1205732119868ℎ

(119906)

119888 + 119868ℎ (119906)) 119878 (119906) 119868 (119906) 119890

minus119887(119905minus119906)

119889119906

119889119868 (119905)

119889119905= (120573

1minus

1205732119868ℎ

(119905 minus 119908)

119888 + 119868ℎ (119905 minus 119908)) 119878 (119905 minus 119908) 119868 (119905 minus 119908) 119890

minus119887120591

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

119877 (119905) = int

119905

119905minus120591

120574119868 (119906)

119889 + 119868 (119906)119890minus119887(119905minus119906)

119889119906

119905 = 119896119879

119878 (119905+

) = (1 minus 120579) 119878 (119905minus

)

119864 (119905+

) = 119864 (119905minus

)

119868 (119905+

) = 119868 (119905minus

)

119877 (119905+

) = 119877 (119905minus

) + 120579119878 (119905minus

)

119905 = 119896119879

(2)

where 119896 isin 119885+ 119885+ = 0 1 2 119873(119905) = 119878(119905) + 119864(119905) + 119868(119905) +

119877(119905) = 1 Note that the variables 119864 and 119877 do not appear in thefirst and third equations of system (2) this allows us to attack(2) by studying the subsystem

119889119878 (119905)

119889119905= 119887 minus 119887119878 (119905) minus (120573

1minus

1205732119868ℎ

(119905)

119888 + 119868ℎ (119905)) 119878 (119905) 119868 (119905)

+120574119868 (119905 minus 120591)

119889 + 119868 (119905 minus 120591)119890minus119887120591

119889119868 (119905)

119889119905= (1205731minus

1205732119868ℎ

(119905 minus 119908)

119888 + 119868ℎ (119905 minus 119908)) 119878 (119905 minus 119908) 119868 (119905 minus 119908) 119890

minus119887120591

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

119905 = 119896119879

119878 (119905+

) = (1 minus 120579) 119878 (119905minus

)

119868 (119905+

) = 119868 (119905minus

)

119905 = 119896119879

(3)

The main purpose of this paper is to establish sufficientconditions that the disease dies out and show that the diseaseis uniformly persistent under some conditions

2 Notations and Definitions

We introduce some notations and definitions and state someresults which will be useful in subsequent sections

Let 119877+= [0infin) 119877

2

+= 119885 isin 119877

2

119885 ge 0 Denote 119891 =

(1198911 1198912)119879 and the map defined by the right hand of the first

and second equations of systems (3) Set 119897 = max120591 119908 Let 119862be the space of continuous functions on [minus119897 0] with uniformnormThe initial conditions for (3) are

(1206011(120577) 120601

2(120577)) isin 119862

+= 119862 ([minus119897 0] 119877

2

+) 120601

119894(0) gt 0 119894 = 1 2

(4)

The solution of system (3) is a piecewise continuousfunction 119885 119877

+rarr 119877

2

+ 119885(119905) is continuous on (119896119879 (119896 +

1)119879] 119896 isin 119885+ and 119885(119896119879

+

) = lim119905rarr119896119879

+119885(119905) existsObviously the smooth properties of 119891 guarantee the globalexistence and uniqueness of solution of system (3) (see [3]for details on fundamental properties of impulsive systems)Since 119878(119905)|

119878=0gt 0 and 119868(119905) = 0 whenever 119868(119905) = 0 for

Discrete Dynamics in Nature and Society 3

119905 = 119896119879 119896 isin 119885+ Moreover 119878(119896119879+) = (1 minus 120579)119878(119896119879

minus

) and119868(119896119879+

) = 119868(119896119879minus

) for 119896 isin 119885+Therefore we have the following

lemma

Lemma 1 Suppose 119885(119905) is a solution of system (3) with initialconditions (4) then 119885(119905) ge 0 for all 119905 ge 0

Denote that Ω = (119878 119868) isin 1198772

| 119878 ge 0 119868 ge 0 119878 + 119868 le 1Using the fact that 119878(119905) + 119864(119905) + 119868(119905) + 119877(119905) = 1 it is easy toshow that Ω is positively invariant with respect to (3)

Lemma 2 (see [10]) Consider the following impulsive system

(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879

119906 (119905+

) = (1 minus 120579) 119906 (119905minus

) 119905 = 119896119879

(5)

where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then there exists a uniquepositive periodic solution of system (5)

119890(119905) =

119886

119887+ (119906lowast

minus119886

119887) 119890minus119887(119905minus119896119879)

=119886

119887(1 minus

120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879) 119896119879 lt 119905 le (119896 + 1) 119879

(6)

which is globally asymptotically stable where 119906lowast = (119886119887)(1 minus

120579)(1 minus 119890minus119887119879

)(1 minus (1 minus 120579)119890minus119887119879

)

Lemma 3 (see [17]) Consider the following delayed differen-tial equation

1199091015840

(119905) = 1198861119909 (119905 minus 119908) minus 119886

2119909 (119905) (7)

where 1198861 1198862 119908 gt 0 119909(119905) gt 0 for minus119908 le 119905 le 0 We have

(i) if 1198861lt 1198862 then lim

119905rarrinfin119909(119905) = 0

(ii) if 1198861gt 1198862 then lim

119905rarrinfin119909(119905) = +infin

Definition 4 System (3) is said to be uniformly persistent ifthere is an 120578 gt 0 (independent of the initial conditions) suchthat every solution (119878(119905) 119868(119905)) with initial conditions (4) ofsystem (3) satisfies

lim119905rarrinfin

inf 119878 (119905) ge 120578 lim119905rarrinfin

inf 119868 (119905) ge 120578 (8)

Definition 5 System (3) is said to be permanent if thereexists a compact region Ω

0sub int1198772

+such that every

solution (119878(119905) 119868(119905)) of system (3) with initial conditions (4)will eventually enter and remain in regionΩ

0

3 Global Attractivity of Infection-FreePeriodic Solution

In this section we analyse the attractivity of infection-freeperiodic solution of system (3) If we let 119868(119905) = 0 then thegrowth of susceptible individuals must satisfy

119878 (119905) = 119886 minus 119887119878 (119905) 119905 = 119896119879

119878 (119905+

) = (1 minus 120579) 119878 (119905minus

) 119905 = 119896119879

(9)

By Lemma 2 we know that periodic solution of system(9)

119878119890(119905) = 1 minus

120579

1 minus (1 minus 120579) 119890minus119887119879119890minus119887(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879

(10)

is globally asymptotically stableAbout the global attractivity of infection-free periodic

solution (119878119890(119905) 0) of system (3) we have the following

theorem

Theorem 6 The infection-free periodic solution (119878119890(119905) 0) of

system (3) is globally attractivity provided that 119877lowast lt 1 where119877lowast

= 1205731119890minus119887119908

(119887+120574119890minus119887120591

(119889+1))(1minus119890minus119887119879

)(119887(119887minus1205732(119888+1))(1minus

(1 minus 120579)119890minus119887119879

))

Proof Since 119877lowast lt 1 we can choose 1205760gt 0 sufficiently small

such that

1205731120575119890minus119887119908

lt 119887 (11)

where 120575 = (119887+120574119890minus119887120591

(119889+1))(1minus119890minus119887119879

)(119887minus1205732(119888+1))(1minus(1minus

120579)119890minus119887119879

) + 1205760 From the first equation of system (3) we have

119878 (119905) lt 119887 minus 119887119878 (119905) + 1205732119878 (119905) 119868 (119905)

119868ℎ

(119905)

119888 + 119868ℎ (119905)+

120574119868 (119905 minus 120591)

119889 + 119868 (119905 minus 120591)119890minus119887120591

lt (119887 +120574119890minus119887120591

119889 + 1) minus (119887 minus

1205732

119888 + 1) 119878 (119905)

(12)

Then we consider the following comparison system withpulses

(119905) = (119887 +120574119890minus119887120591

119889 + 1) minus (119887 minus

1205732

119888 + 1) 119909 (119905) 119905 = 119896119879

119909 (119905+

) = (1 minus 120579) 119909 (119905minus

) 119905 = 119896119879

(13)

By Lemma 2 we know that there is a unique periodic solutionof system (13)

119909119890(119905) =

119887 + 120574119890minus119887120591

(119889 + 1)

119887 minus 1205732 (119888 + 1)

(1 minus120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879)

119896119879 lt 119905 le (119896 + 1) 119879

(14)

which is globally asymptotically stableLet (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and let 119878(0+) = 1198780gt 0 119909(119905) be the solution of

system (13) with initial values 119909(0+) = 1198780 By the comparison

theorem in impulsive differential equation [3] there exists aninteger 119896

1gt 0 such that 119878(119905) lt 119909

119890(119905) + 120576

0for 119905 gt 119896

1119879 thus

119878 (119905) lt(119887 + 120574119890

minus119887120591

(119889 + 1)) (1 minus 119890minus119887119879

)

(119887 minus 1205732 (119888 + 1)) (1 minus (1 minus 120579) 119890minus119887119879)

+ 1205760= 120575

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961

(15)

4 Discrete Dynamics in Nature and Society

Again from the second equation of system (3) we know that(15) implies that

119868 (119905) le (1205731minus

1205732119868ℎ

(119905 minus 119908)

119888 + 119868ℎ (119905 minus 119908)) 120575119868 (119905 minus 119908) 119890

minus119887119908

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

le 1205731120575119890minus119887119908

119868 (119905 minus 119908) minus 119887119868 (119905)

(16)

where 119905 gt 119896119879 + 119908 119896 gt 1198961

Consider the following comparison system

119910 (119905) = 1205731120575119890minus119887119908

119910 (119905 minus 119908) minus 119887119910 (119905) 119905 gt 119896119879 + 119908 119896 gt 1198961

(17)

According to (11) and Lemma 3 we have lim119905rarrinfin

119910(119905) = 0Let (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and let 119868(120577) = 120593(120577) gt 0 (120577 isin [minus119908 0]) 119910(119905)be the solution of system (17) with initial values 119910(120577) =

120593(120577) gt 0 (120577 isin [minus119908 0]) By the comparison theorem wehave lim

119905rarrinfinsup 119868(119905) le lim

119905rarrinfinsup119910(119905) = 0 Incorporating

into the positivity of 119868(119905) we know that lim119905rarrinfin

119868(119905) = 0Therefore there exists an integer 119896

2gt 1198961(where 119896

2119879 gt

1198961119879 + 119908) such that 119868(119905) lt 120576

0for all 119905 gt 119896

2119879

From the first equation of system (3) we have

119878 (119905) gt 119887 minus 119887119878 (119905) minus 1205731119878 (119905) 119868 (119905) gt 119887 minus (119887 + 120573

11205760) 119878 (119905)

119905 gt 1198962119879 + 120591

119878 (119905) lt 119887 minus 119887119878 (119905) + 1205732119878 (119905)

119868ℎ+1

(119905)

119888 + 119868ℎ (119905)+

120574119868 (119905 minus 120591)

119889 + 119868 (119905 minus 120591)119890minus119887120591

lt (119887 +1205741205760

119889 + 1205760

119890minus119887120591

) minus (119887 minus1205732120576ℎ+1

0

119888 + 120576ℎ0

)119878 (119905)

119905 gt 1198962119879 + 120591

(18)

Consider the following comparison impulsive differentialequations for 119905 gt 119896

2119879 + 120591 and 119896 gt 119896

2

1(119905) = 119887 minus (119887 + 120573

11205760) 1199111(119905) 119905 = 119896119879

1199111(119905+

) = (1 minus 120579) 1199111(119905minus

) 119905 = 119896119879

(19)

2(119905) = (119887 +

1205741205760

119889 + 1205760

119890minus119887120591

) minus (119887 minus1205732120576ℎ+1

0

119888 + 120576ℎ0

)1199112(119905) 119905 = 119896119879

1199112(119905+

) = (1 minus 120579) 1199112(119905minus

) 119905 = 119896119879

(20)

By Lemma 2 we have that the unique periodic solution ofsystem (19)

1119890(119905) =

119887

119887 + 12057311205760

(1 minus120579119890minus(119887+120573

11205760)(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879)

119896119879 lt 119905 le (119896 + 1) 119879

(21)

and the unique periodic solution of system (20)

2119890(119905) =

(119887 + (1205741205760 (119889 + 120576

0)) 119890minus119887120591

)

(119887 minus (1205732120576ℎ+1

0 (119888 + 120576ℎ

0)))

(1 minus120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879)

119896119879 lt 119905 le (119896 + 1) 119879

(22)

are globally asymptotically stableLet (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and 119878(0+

) = 1198780

gt 0 1199111(119905) and let 119911

2(119905) be the

solutions of system (19) and (20) with initial values 1199111(0+

) =

1199112(0+

) = 1198780 respectively By the comparison theorem in

impulsive differential equation there exists an integer 1198963gt 1198962

such that 1198963119879 gt 119896

2119879 + 120591 and

1119890(119905) minus 120576

0lt 119878 (119905) lt

2119890(119905) minus 120576

0

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198963

(23)

Because 1205760is arbitrarily small it follows from (23) that

119878119890(119905) = 1 minus

120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879 119896119879 lt 119905 le (119896 + 1) 119879 (24)

is globally attractive The proof is complete

Denote that 120579lowast = 1minus119890119887119879

+(1205731119890minus119887119908

(1+(120574119887)119890minus119887120591

)(119890119887119879

minus1))

119887(119887minus1205732(119888+1)) 120573lowast

2= (119888+1)(119887minus120573

1119890minus119887119908

(119887+120574119890minus119887120591

(119889+1))(1minus

119890minus119887119879

)119887(1minus(1minus120579)119890minus119887119879

)) and 119888lowast = 1205732119887(1minus(1minus120579)119890

minus119887119879

)(1198872

(1minus

(1 minus 120579)119890minus119887119879

) minus 1205731119890minus119887119908

(119887 + 120574119890minus119887120591

(119889 + 1))(1 minus 119890minus119887119879

)) minus 1According to Theorem 6 we can obtain the following

result

Corollary 7 The infection-free periodic solution (119878119890(119905) 0) of

system (3) is globally attractivity provided that 120579 gt 120579lowast or 120573

2lt

120573lowast

2or 119888 gt 119888

lowastFrom Corollary 7 we know that the disease will disappear

if the vaccination rate is larger than 120579lowast

4 Permanence

In this section we say the disease is endemic if the infectiouspopulation persists above a certain positive level for suffi-ciently large time

Denote that119877lowast= (1205731minus1205732(119888+1))119890

minus119887119908

(1minus120579)(1minus119890minus119887119879

)(119887+

120574119889)(1minus (1minus120579)119890minus119887119879

) and 119868lowast = (119887120573)[(1205731minus1205732(119888+1))119890

minus119887119908

(1minus

120579)(1 minus 119890minus119887119879

)(119887 + 120574119889)(1 minus (1 minus 120579)119890minus119887119879

) minus 1]

Theorem 8 If 119877lowast

gt 1 then there is a positive constant 119902such that each positive solution (119878(119905) 119868(119905)) of system (3) satisfies119868(119905) gt 119902 for 119905 large enough

Proof From 119877lowastgt 1 we easily know that 119868lowast gt 0 and there

exists sufficiently small 120576 gt 0 such that

(1205731minus

1205732

119888 + 1) 1205751119890minus119887119908

minus 119887 minus120574

119889gt 0 (25)

Discrete Dynamics in Nature and Society 5

where 1205751= (119887(119887 + 120573

1119868lowast

))(1 minus 120579(1 minus (1 minus 120579)119890minus(119887+120573

1119868lowast)119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt 119868

lowast forall 119905 ge 119905

0 Suppose that the claim is not valid Then there is

a 1199050gt 0 such that 119868(119905) lt 119868

lowast for all 119905 ge 1199050 It follows from the

first equation of (3) that for 119905 ge 1199050

119878 (119905) gt 119887 minus 119887119878 (119905) minus 1205731119878 (119905) 119868 (119905) gt 119887 minus (119887 + 120573

1119868lowast

) 119878 (119905) (26)

Consider the following comparison impulsive system for119905 ge 1199050

(119905) = 119887 minus (119887 + 1205731119868lowast

) 119906 (119905) 119905 = 119896119879

119906 (119905+

) = (1 minus 120579) 119906 (119905minus

) 119905 = 119896119879

(27)

By Lemma 2 we obtain that

119890(119905) =

119887

119887 + 1205731119868lowast+ (119906

lowast

minus119887

119887 + 1205731119868lowast) 119890minus(119887+120573

1119868lowast)(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879

(28)

is the unique positive periodic solution of (27) which isglobally asymptotically stable where 119906lowast = (119887(119887+120573

1119868lowast

))((1minus

120579)(1 minus 119890minus(119887+120573

1119868lowast)119879

)(1 minus (1 minus 120579)(1 minus 119890minus(119887+120573

1119868lowast)119879

)))Let (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and let 119878(0+) = 1198780gt 0 119906(119905) be the solution of

system (27) with initial values 119906(0+) = 1198780 By the comparison

theorem for impulsive differential equation there exists aninteger 119896

1(gt 1199050+ 119908) such that 119878(119905) gt

119890(119905) minus 120576 for 119905 ge 119896

1119879

thus

119878 (119905) gt 119906lowast

minus 120576 = 1205751 119905 ge 119896

1 (29)

The second equation of system (3) can be rewritten as

119868 (119905) = 1205731119890minus119887119908

119878 (119905) 119868 (119905) minus 1205731119890minus119887119908

119889

119889119905int

119905

119905minus119908

119878 (119906) 119868 (119906) 119889119906

minus 1205732119890minus119887119908

119878 (119905) 119868ℎ+1

(119905)

119888 + 119868ℎ (119905)

+ 1205732119890minus119887119908

119889

119889119905int

119905

119905minus119908

119878 (119906) 119868ℎ+1

(119906)

119888 + 119868ℎ (119906)119889119906

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

(30)

Let us consider any positive solution (119878(119905) 119868(119905)) of system (3)According to this solution we define

119881 (119905) = 119868 (119905) + 1205731119890minus119887119908

int

119905

119905minus119908

119878 (119906) 119868 (119906) 119889119906

minus 1205732119890minus119887119908

int

119905

119905minus119908

119878 (119906) 119868ℎ+1

(119906)

119888 + 119868ℎ (119906)119889119906

(31)

According to (30) we calculate the derivative of 119881 along thesolutions of system (3)

(119905) = 119868 (119905) (1205731119890minus119887119908

119878 (119905) minus1205732119878 (119905) 119868

ℎ

(119905)

119888 + 119868ℎ (119905)119890minus119887119908

minus 119887 minus120574

119889 + 119868 (119905))

(32)

By (25) and (29) for 119905 ge 1199051 we have

(119905) ge 119868 (119905) (1205731119890minus119887119908

119878 (119905) minus1205732119878 (119905)

119888 + 1119890minus119887119908

minus 119887 minus120574

119889)

gt 119868 (119905) [(1205731minus

1205732

119888 + 1) 119890minus119887119908

1205751minus 119887 minus

120574

119889] gt 0

(33)

Let 119868119897= min

119905isin[11990511199051+119908]

119868(119905) in the following we will showthat 119868(119905) ge 119868

119897for 119905 ge 119905

1 Suppose the contrary then there is a

1198790ge 0 such that 119868(119905) ge 119868

119897for 1199051le 119905 le 119905

1+ 119908 + 119879

0 119868(1199051+ 119908 +

1198790) = 119868119897and 119868(119905

1+119908+119879

0) le 0 However the second equation

of system (3) and (4) imply that

119868 (1199051+ 119908 + 119879

0) ge (120573

1minus

1205732

119888 + 1) 119878 (1199051+ 1198790) 119868 (1199051+ 1198790) 119890minus119887119908

minus 119887119868 (1199051+ 119908 + 119879

0) minus

120574

119889119868 (1199051+ 119908 + 119879

0)

ge (1205731minus

1205732

119888 + 1) 119878 (1199051+ 1198790) 119868119897119890minus119887119908

minus 119887119868119897minus120574

119889119868119897

gt [(1205731minus

1205732

119888 + 1) 1205751119890minus119887119908

minus 119887 minus120574

119889] 119868119897gt 0

(34)

This is a contradictionThus 119868(119905) ge 119868119897for 119905 ge 119905

1 So (33) leads

to

(119905) gt 119868119897[(1205731minus

1205732

119888 + 1) 119890minus119887119908

1205751minus 119887 minus

120574

119889] 119905 ge 119905

1 (35)

which implies that 119881(119905) rarr infin as 119905 rarr infin This contradictswith 119881(119905) le 1 + 120573

1119908119890minus119887119908 Hence the claim is proved From

the claim we will discuss the following two possibilities

(i) 119868(119905) ge 119868lowast for 119905 large enough

(ii) 119868(119905) oscillates about 119868lowast for 119905 large enough

Evidently we only need to consider the case (ii) Let 119905lowast gt 0

and 120585 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast

+ 120585) = 119868lowast and let 119868(119905) lt 119868

lowast for119905lowast

lt 119905 lt 119905lowast

+120585 where 119905lowast is sufficiently large such that 119878(119905) gt 1205751

for 119905lowast lt 119905 lt 119905lowast

+ 120585 Since 119868(119905) is continuous and ultimatelybounded and is not effected by impulses we conclude that119868(119905) is uniformly continuous Hence there exists a constant 120582(with 0 lt 120582 lt 119908 and 120582 is independent of the choice of 119905lowast)such that 119868(119905) gt 119868

lowast

2 for 119905lowast lt 119905 lt 119905lowast

+ 120582 If 120585 le 120582 our aim isobtained If 120582 lt 120585 le 119908 from the second equation of (3) wehave that 119868(119905) gt minus(119887 + 120574119889)119868(119905) and 119868(119905lowast) = 119868

lowast then we have119868(119905) ge 119902 for 119905 isin [119905

lowast

119905lowast

+ 119908] where 119902 = min119868lowast2 119902lowast 119902lowast =119868lowast

119890minus(119887+120574)119908) The same arguments can be continued and we

can obtain 119868(119905) ge 119902 for 119905 isin [119905lowast

+ 119908 119905lowast

+ 120585] Since the intervalis chosen in an arbitrary way we get that 119868(119905) ge 119902 for 119905 largeenough In view of our arguments above the choice of 119902 isindependent of the positive solution of (3) which satisfies that119868(119905) ge 119902 for sufficiently large 119905 This completes the proof

Denote that 120579lowast= [(1205731minus 1205732(119888 + 1))119890

minus119887119908

minus (119887 + 120574119889)](1 minus

119890minus119887119879

)((1205731minus 1205732(119888 + 1))119890

minus119887119908

(1 minus 119890minus119887119879

) + (119887 + 120574119889)119890minus119887119879

) 1205732lowast

=

(119888 + 1)(1205731+ (119887 + 120574119889)(1 minus (1 minus 120579)119890

minus119887119879

)119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

))

6 Discrete Dynamics in Nature and Society

and 119888lowast= 1205732119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

)(1205731119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

) +

(119887 + 120574119889)(1 minus (1 minus 120579)119890minus119887119879

)) minus 1It follows from Theorem 8 that the disease is uniformly

persistent provided that 120579 lt 120579lowastor 1205732lt 1205732lowast

or 119888 gt 119888lowast

Theorem 9 If 119877lowastgt 1 then system (3) is permanent

Proof Suppose that (119878(119905) 119868(119905)) is any positive solution ofsystem (3) with initial conditions (4) From the first equationof system (3) we have 119878(119905) ge 119887 minus (119887 + 120573

1)119878(119905) By similar

arguments as those in the proof of Theorem 6 we have that

lim119905rarrinfin

119878 (119905) ge 119901 (36)

where 119901 = (119887(119887 + 1205731))((1 minus 120579)(1 minus 119890

minus(119887+120573minus1)119879

)(1 minus (1 minus

120579)119890minus(119887+120573minus1)119879

)) minus 1205761 (1205761is sufficiently small)

Set Ω0= (119878 119868) isin 119877

2

| 119901 le 119878 119902 le 119868 119878 + 119868 le 1FromTheorem 8 and (36) we know that the setΩ

0is a global

attractor in Ω and of course every solution of system (3)with initial conditions (4) will eventually enter and remainin region Ω

0 Therefore system (3) is permanent The proof

is complete

FromTheorem 9 we can obtain the following result

Corollary 10 Assume that 120579 lt 120579lowastor 1205732lt 1205732lowast

or 119888 gt 119888lowast then

system (3) is permanent

5 Conclusion

In this paper we introduce media coverage and saturationrecovery in the delayed SEIRS epidemic model with pulsevaccination and analyze detailedly in theory that mediacoverage and pulse vaccination bring effects on infection-free and the permanence of epidemic disease We suggestthe probability of transmission per contact 120573(119868) = 120573

1minus

1205732(119868ℎ

(119888 + 119868ℎ

)) which reflects some characters of mediacoverage We can get that 120597120573(119868)120597120573

2= minus(119868

ℎ

(119888 + 119868ℎ

)) lt 0so 120573(119868) is a monotone decreasing function on 120573

2 that is

if 1205732(the reduced valid contact rate through actual media

coverage) is larger then infection rate of disease is smallerAgain 120597120573(119868)120597119888 = 120573

2119868ℎ

(119888 + 119868ℎ

)2

gt 0 so 120573(119868) is a monotoneincreasing function on 119888 that is if 119888 is smaller (refectionon the disease is quickly) then infection rate of disease issmallerWhen infective individuals appear in a region peoplereduce their contact with others to avoid being infectedand the more infective individuals being reported the lesscontact with others Fromabove analysis we know thatmediacoverage is very important on controlling disease and mediacoverage should be considered in incidence rate

By Theorem 6 the infection-free periodic solution(119878119890(119905) 0) of system (3) is globally attractivity provided that

119877lowast

lt 1 By Theorem 9 system (3) is permanent if 119877lowastgt 1

From Corollaries 7 and 10 we can choose the proportionof those vaccinated successfully to all of newborns such that120579 gt 120579

lowast in order to prevent the epidemic disease fromgenerating endemic and the epidemic is permanent if 120579 lt 120579

lowast

But for 120579 isin [120579lowast 120579lowast

] the dynamical behavior of model (3)

has not been studied and the threshold parameter for thevaccination rate between the extinction of the disease andthe uniform persistence of the disease has not been obtainedThese issues will be considered in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the NNSF of China (1137104811201433) NSF of Henan Province (112300410156122300410117) NSF of the Education DepartmentHenan Province (2011A110022) GGJS of Henan Province(2013GGJS-110) and XGGJS of Zhengzhou University ofLight Industry (2012XGGJS003)

References

[1] Z Agur L Cojocaru G Mazor R M Anderson and Y LDanon ldquoPulse mass measles vaccination across age cohortsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 90 no 24 pp 11698ndash11702 1993

[2] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificamp Technical New York NY USA 1993

[3] V Lakshmikantham D D Bainov and P S Simeonov Theoryof Impulsive Differential Equations World Scientific Singapore1989

[4] Y Li J Cui and X Song ldquoDynamics of a predator-prey systemwith pulsesrdquo Applied Mathematics and Computation vol 204no 1 pp 269ndash280 2008

[5] X Song and Y Li ldquoDynamic complexities of a Holling II two-prey one-predator systemwith impulsive effectrdquoChaos Solitonsamp Fractals vol 33 no 2 pp 463ndash478 2007

[6] X Liu ldquoImpulsive stabilization and applications to populationgrowth modelsrdquo Rocky Mountain Journal of Mathematics vol25 no 1 pp 381ndash395 1995

[7] A Lakmeche and O Arino ldquoBifurcation of non trivial periodicsolutions of impulsive differential equations arising chemother-apeutic treatmentrdquo Dynamics of Continuous Discrete andImpulsive Systems Series B vol 7 no 2 pp 265ndash287 2000

[8] W Wang ldquoGlobal behavior of an SEIRS epidemic model withtime delaysrdquoAppliedMathematics Letters vol 15 no 4 pp 423ndash428 2002

[9] J Hui and L-S Chen ldquoImpulsive vaccination of sir epidemicmodels with nonlinear incidence ratesrdquo Discrete and Continu-ous Dynamical Systems Series B vol 4 no 3 pp 595ndash605 2004

[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006

[11] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003

[12] W-M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

Discrete Dynamics in Nature and Society 7

[13] Y Li and J Cui ldquoThe effect of constant and pulse vaccination onSIS epidemic models incorporating media coveragerdquo Commu-nications inNonlinear Science andNumerical Simulation vol 14no 5 pp 2353ndash2365 2009

[14] Y Li C Ma and J Cui ldquoThe effect of constant and mixedimpulsive vaccination on SIS epidemic models incorporatingmedia coveragerdquo Rocky Mountain Journal of Mathematics vol38 no 5 pp 1437ndash1455 2008

[15] J Cui X Mu and H Wan ldquoSaturation recovery leads tomultiple endemic equilibria and backward bifurcationrdquo Journalof Theoretical Biology vol 254 no 2 pp 275ndash283 2008

[16] K L Cooke and P Van Den Driessche ldquoAnalysis of an SEIRSepidemic model with two delaysrdquo Journal of MathematicalBiology vol 35 no 2 pp 240ndash260 1996

[17] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press San Digo Calif USA1993

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

Discrete Dynamics in Nature and Society 3

119905 = 119896119879 119896 isin 119885+ Moreover 119878(119896119879+) = (1 minus 120579)119878(119896119879

minus

) and119868(119896119879+

) = 119868(119896119879minus

) for 119896 isin 119885+Therefore we have the following

lemma

Lemma 1 Suppose 119885(119905) is a solution of system (3) with initialconditions (4) then 119885(119905) ge 0 for all 119905 ge 0

Denote that Ω = (119878 119868) isin 1198772

| 119878 ge 0 119868 ge 0 119878 + 119868 le 1Using the fact that 119878(119905) + 119864(119905) + 119868(119905) + 119877(119905) = 1 it is easy toshow that Ω is positively invariant with respect to (3)

Lemma 2 (see [10]) Consider the following impulsive system

(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879

119906 (119905+

) = (1 minus 120579) 119906 (119905minus

) 119905 = 119896119879

(5)

where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then there exists a uniquepositive periodic solution of system (5)

119890(119905) =

119886

119887+ (119906lowast

minus119886

119887) 119890minus119887(119905minus119896119879)

=119886

119887(1 minus

120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879) 119896119879 lt 119905 le (119896 + 1) 119879

(6)

which is globally asymptotically stable where 119906lowast = (119886119887)(1 minus

120579)(1 minus 119890minus119887119879

)(1 minus (1 minus 120579)119890minus119887119879

)

Lemma 3 (see [17]) Consider the following delayed differen-tial equation

1199091015840

(119905) = 1198861119909 (119905 minus 119908) minus 119886

2119909 (119905) (7)

where 1198861 1198862 119908 gt 0 119909(119905) gt 0 for minus119908 le 119905 le 0 We have

(i) if 1198861lt 1198862 then lim

119905rarrinfin119909(119905) = 0

(ii) if 1198861gt 1198862 then lim

119905rarrinfin119909(119905) = +infin

Definition 4 System (3) is said to be uniformly persistent ifthere is an 120578 gt 0 (independent of the initial conditions) suchthat every solution (119878(119905) 119868(119905)) with initial conditions (4) ofsystem (3) satisfies

lim119905rarrinfin

inf 119878 (119905) ge 120578 lim119905rarrinfin

inf 119868 (119905) ge 120578 (8)

Definition 5 System (3) is said to be permanent if thereexists a compact region Ω

0sub int1198772

+such that every

solution (119878(119905) 119868(119905)) of system (3) with initial conditions (4)will eventually enter and remain in regionΩ

0

3 Global Attractivity of Infection-FreePeriodic Solution

In this section we analyse the attractivity of infection-freeperiodic solution of system (3) If we let 119868(119905) = 0 then thegrowth of susceptible individuals must satisfy

119878 (119905) = 119886 minus 119887119878 (119905) 119905 = 119896119879

119878 (119905+

) = (1 minus 120579) 119878 (119905minus

) 119905 = 119896119879

(9)

By Lemma 2 we know that periodic solution of system(9)

119878119890(119905) = 1 minus

120579

1 minus (1 minus 120579) 119890minus119887119879119890minus119887(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879

(10)

is globally asymptotically stableAbout the global attractivity of infection-free periodic

solution (119878119890(119905) 0) of system (3) we have the following

theorem

Theorem 6 The infection-free periodic solution (119878119890(119905) 0) of

system (3) is globally attractivity provided that 119877lowast lt 1 where119877lowast

= 1205731119890minus119887119908

(119887+120574119890minus119887120591

(119889+1))(1minus119890minus119887119879

)(119887(119887minus1205732(119888+1))(1minus

(1 minus 120579)119890minus119887119879

))

Proof Since 119877lowast lt 1 we can choose 1205760gt 0 sufficiently small

such that

1205731120575119890minus119887119908

lt 119887 (11)

where 120575 = (119887+120574119890minus119887120591

(119889+1))(1minus119890minus119887119879

)(119887minus1205732(119888+1))(1minus(1minus

120579)119890minus119887119879

) + 1205760 From the first equation of system (3) we have

119878 (119905) lt 119887 minus 119887119878 (119905) + 1205732119878 (119905) 119868 (119905)

119868ℎ

(119905)

119888 + 119868ℎ (119905)+

120574119868 (119905 minus 120591)

119889 + 119868 (119905 minus 120591)119890minus119887120591

lt (119887 +120574119890minus119887120591

119889 + 1) minus (119887 minus

1205732

119888 + 1) 119878 (119905)

(12)

Then we consider the following comparison system withpulses

(119905) = (119887 +120574119890minus119887120591

119889 + 1) minus (119887 minus

1205732

119888 + 1) 119909 (119905) 119905 = 119896119879

119909 (119905+

) = (1 minus 120579) 119909 (119905minus

) 119905 = 119896119879

(13)

By Lemma 2 we know that there is a unique periodic solutionof system (13)

119909119890(119905) =

119887 + 120574119890minus119887120591

(119889 + 1)

119887 minus 1205732 (119888 + 1)

(1 minus120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879)

119896119879 lt 119905 le (119896 + 1) 119879

(14)

which is globally asymptotically stableLet (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and let 119878(0+) = 1198780gt 0 119909(119905) be the solution of

system (13) with initial values 119909(0+) = 1198780 By the comparison

theorem in impulsive differential equation [3] there exists aninteger 119896

1gt 0 such that 119878(119905) lt 119909

119890(119905) + 120576

0for 119905 gt 119896

1119879 thus

119878 (119905) lt(119887 + 120574119890

minus119887120591

(119889 + 1)) (1 minus 119890minus119887119879

)

(119887 minus 1205732 (119888 + 1)) (1 minus (1 minus 120579) 119890minus119887119879)

+ 1205760= 120575

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961

(15)

4 Discrete Dynamics in Nature and Society

Again from the second equation of system (3) we know that(15) implies that

119868 (119905) le (1205731minus

1205732119868ℎ

(119905 minus 119908)

119888 + 119868ℎ (119905 minus 119908)) 120575119868 (119905 minus 119908) 119890

minus119887119908

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

le 1205731120575119890minus119887119908

119868 (119905 minus 119908) minus 119887119868 (119905)

(16)

where 119905 gt 119896119879 + 119908 119896 gt 1198961

Consider the following comparison system

119910 (119905) = 1205731120575119890minus119887119908

119910 (119905 minus 119908) minus 119887119910 (119905) 119905 gt 119896119879 + 119908 119896 gt 1198961

(17)

According to (11) and Lemma 3 we have lim119905rarrinfin

119910(119905) = 0Let (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and let 119868(120577) = 120593(120577) gt 0 (120577 isin [minus119908 0]) 119910(119905)be the solution of system (17) with initial values 119910(120577) =

120593(120577) gt 0 (120577 isin [minus119908 0]) By the comparison theorem wehave lim

119905rarrinfinsup 119868(119905) le lim

119905rarrinfinsup119910(119905) = 0 Incorporating

into the positivity of 119868(119905) we know that lim119905rarrinfin

119868(119905) = 0Therefore there exists an integer 119896

2gt 1198961(where 119896

2119879 gt

1198961119879 + 119908) such that 119868(119905) lt 120576

0for all 119905 gt 119896

2119879

From the first equation of system (3) we have

119878 (119905) gt 119887 minus 119887119878 (119905) minus 1205731119878 (119905) 119868 (119905) gt 119887 minus (119887 + 120573

11205760) 119878 (119905)

119905 gt 1198962119879 + 120591

119878 (119905) lt 119887 minus 119887119878 (119905) + 1205732119878 (119905)

119868ℎ+1

(119905)

119888 + 119868ℎ (119905)+

120574119868 (119905 minus 120591)

119889 + 119868 (119905 minus 120591)119890minus119887120591

lt (119887 +1205741205760

119889 + 1205760

119890minus119887120591

) minus (119887 minus1205732120576ℎ+1

0

119888 + 120576ℎ0

)119878 (119905)

119905 gt 1198962119879 + 120591

(18)

Consider the following comparison impulsive differentialequations for 119905 gt 119896

2119879 + 120591 and 119896 gt 119896

2

1(119905) = 119887 minus (119887 + 120573

11205760) 1199111(119905) 119905 = 119896119879

1199111(119905+

) = (1 minus 120579) 1199111(119905minus

) 119905 = 119896119879

(19)

2(119905) = (119887 +

1205741205760

119889 + 1205760

119890minus119887120591

) minus (119887 minus1205732120576ℎ+1

0

119888 + 120576ℎ0

)1199112(119905) 119905 = 119896119879

1199112(119905+

) = (1 minus 120579) 1199112(119905minus

) 119905 = 119896119879

(20)

By Lemma 2 we have that the unique periodic solution ofsystem (19)

1119890(119905) =

119887

119887 + 12057311205760

(1 minus120579119890minus(119887+120573

11205760)(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879)

119896119879 lt 119905 le (119896 + 1) 119879

(21)

and the unique periodic solution of system (20)

2119890(119905) =

(119887 + (1205741205760 (119889 + 120576

0)) 119890minus119887120591

)

(119887 minus (1205732120576ℎ+1

0 (119888 + 120576ℎ

0)))

(1 minus120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879)

119896119879 lt 119905 le (119896 + 1) 119879

(22)

are globally asymptotically stableLet (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and 119878(0+

) = 1198780

gt 0 1199111(119905) and let 119911

2(119905) be the

solutions of system (19) and (20) with initial values 1199111(0+

) =

1199112(0+

) = 1198780 respectively By the comparison theorem in

impulsive differential equation there exists an integer 1198963gt 1198962

such that 1198963119879 gt 119896

2119879 + 120591 and

1119890(119905) minus 120576

0lt 119878 (119905) lt

2119890(119905) minus 120576

0

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198963

(23)

Because 1205760is arbitrarily small it follows from (23) that

119878119890(119905) = 1 minus

120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879 119896119879 lt 119905 le (119896 + 1) 119879 (24)

is globally attractive The proof is complete

Denote that 120579lowast = 1minus119890119887119879

+(1205731119890minus119887119908

(1+(120574119887)119890minus119887120591

)(119890119887119879

minus1))

119887(119887minus1205732(119888+1)) 120573lowast

2= (119888+1)(119887minus120573

1119890minus119887119908

(119887+120574119890minus119887120591

(119889+1))(1minus

119890minus119887119879

)119887(1minus(1minus120579)119890minus119887119879

)) and 119888lowast = 1205732119887(1minus(1minus120579)119890

minus119887119879

)(1198872

(1minus

(1 minus 120579)119890minus119887119879

) minus 1205731119890minus119887119908

(119887 + 120574119890minus119887120591

(119889 + 1))(1 minus 119890minus119887119879

)) minus 1According to Theorem 6 we can obtain the following

result

Corollary 7 The infection-free periodic solution (119878119890(119905) 0) of

system (3) is globally attractivity provided that 120579 gt 120579lowast or 120573

2lt

120573lowast

2or 119888 gt 119888

lowastFrom Corollary 7 we know that the disease will disappear

if the vaccination rate is larger than 120579lowast

4 Permanence

In this section we say the disease is endemic if the infectiouspopulation persists above a certain positive level for suffi-ciently large time

Denote that119877lowast= (1205731minus1205732(119888+1))119890

minus119887119908

(1minus120579)(1minus119890minus119887119879

)(119887+

120574119889)(1minus (1minus120579)119890minus119887119879

) and 119868lowast = (119887120573)[(1205731minus1205732(119888+1))119890

minus119887119908

(1minus

120579)(1 minus 119890minus119887119879

)(119887 + 120574119889)(1 minus (1 minus 120579)119890minus119887119879

) minus 1]

Theorem 8 If 119877lowast

gt 1 then there is a positive constant 119902such that each positive solution (119878(119905) 119868(119905)) of system (3) satisfies119868(119905) gt 119902 for 119905 large enough

Proof From 119877lowastgt 1 we easily know that 119868lowast gt 0 and there

exists sufficiently small 120576 gt 0 such that

(1205731minus

1205732

119888 + 1) 1205751119890minus119887119908

minus 119887 minus120574

119889gt 0 (25)

Discrete Dynamics in Nature and Society 5

where 1205751= (119887(119887 + 120573

1119868lowast

))(1 minus 120579(1 minus (1 minus 120579)119890minus(119887+120573

1119868lowast)119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt 119868

lowast forall 119905 ge 119905

0 Suppose that the claim is not valid Then there is

a 1199050gt 0 such that 119868(119905) lt 119868

lowast for all 119905 ge 1199050 It follows from the

first equation of (3) that for 119905 ge 1199050

119878 (119905) gt 119887 minus 119887119878 (119905) minus 1205731119878 (119905) 119868 (119905) gt 119887 minus (119887 + 120573

1119868lowast

) 119878 (119905) (26)

Consider the following comparison impulsive system for119905 ge 1199050

(119905) = 119887 minus (119887 + 1205731119868lowast

) 119906 (119905) 119905 = 119896119879

119906 (119905+

) = (1 minus 120579) 119906 (119905minus

) 119905 = 119896119879

(27)

By Lemma 2 we obtain that

119890(119905) =

119887

119887 + 1205731119868lowast+ (119906

lowast

minus119887

119887 + 1205731119868lowast) 119890minus(119887+120573

1119868lowast)(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879

(28)

is the unique positive periodic solution of (27) which isglobally asymptotically stable where 119906lowast = (119887(119887+120573

1119868lowast

))((1minus

120579)(1 minus 119890minus(119887+120573

1119868lowast)119879

)(1 minus (1 minus 120579)(1 minus 119890minus(119887+120573

1119868lowast)119879

)))Let (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and let 119878(0+) = 1198780gt 0 119906(119905) be the solution of

system (27) with initial values 119906(0+) = 1198780 By the comparison

theorem for impulsive differential equation there exists aninteger 119896

1(gt 1199050+ 119908) such that 119878(119905) gt

119890(119905) minus 120576 for 119905 ge 119896

1119879

thus

119878 (119905) gt 119906lowast

minus 120576 = 1205751 119905 ge 119896

1 (29)

The second equation of system (3) can be rewritten as

119868 (119905) = 1205731119890minus119887119908

119878 (119905) 119868 (119905) minus 1205731119890minus119887119908

119889

119889119905int

119905

119905minus119908

119878 (119906) 119868 (119906) 119889119906

minus 1205732119890minus119887119908

119878 (119905) 119868ℎ+1

(119905)

119888 + 119868ℎ (119905)

+ 1205732119890minus119887119908

119889

119889119905int

119905

119905minus119908

119878 (119906) 119868ℎ+1

(119906)

119888 + 119868ℎ (119906)119889119906

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

(30)

Let us consider any positive solution (119878(119905) 119868(119905)) of system (3)According to this solution we define

119881 (119905) = 119868 (119905) + 1205731119890minus119887119908

int

119905

119905minus119908

119878 (119906) 119868 (119906) 119889119906

minus 1205732119890minus119887119908

int

119905

119905minus119908

119878 (119906) 119868ℎ+1

(119906)

119888 + 119868ℎ (119906)119889119906

(31)

According to (30) we calculate the derivative of 119881 along thesolutions of system (3)

(119905) = 119868 (119905) (1205731119890minus119887119908

119878 (119905) minus1205732119878 (119905) 119868

ℎ

(119905)

119888 + 119868ℎ (119905)119890minus119887119908

minus 119887 minus120574

119889 + 119868 (119905))

(32)

By (25) and (29) for 119905 ge 1199051 we have

(119905) ge 119868 (119905) (1205731119890minus119887119908

119878 (119905) minus1205732119878 (119905)

119888 + 1119890minus119887119908

minus 119887 minus120574

119889)

gt 119868 (119905) [(1205731minus

1205732

119888 + 1) 119890minus119887119908

1205751minus 119887 minus

120574

119889] gt 0

(33)

Let 119868119897= min

119905isin[11990511199051+119908]

119868(119905) in the following we will showthat 119868(119905) ge 119868

119897for 119905 ge 119905

1 Suppose the contrary then there is a

1198790ge 0 such that 119868(119905) ge 119868

119897for 1199051le 119905 le 119905

1+ 119908 + 119879

0 119868(1199051+ 119908 +

1198790) = 119868119897and 119868(119905

1+119908+119879

0) le 0 However the second equation

of system (3) and (4) imply that

119868 (1199051+ 119908 + 119879

0) ge (120573

1minus

1205732

119888 + 1) 119878 (1199051+ 1198790) 119868 (1199051+ 1198790) 119890minus119887119908

minus 119887119868 (1199051+ 119908 + 119879

0) minus

120574

119889119868 (1199051+ 119908 + 119879

0)

ge (1205731minus

1205732

119888 + 1) 119878 (1199051+ 1198790) 119868119897119890minus119887119908

minus 119887119868119897minus120574

119889119868119897

gt [(1205731minus

1205732

119888 + 1) 1205751119890minus119887119908

minus 119887 minus120574

119889] 119868119897gt 0

(34)

This is a contradictionThus 119868(119905) ge 119868119897for 119905 ge 119905

1 So (33) leads

to

(119905) gt 119868119897[(1205731minus

1205732

119888 + 1) 119890minus119887119908

1205751minus 119887 minus

120574

119889] 119905 ge 119905

1 (35)

which implies that 119881(119905) rarr infin as 119905 rarr infin This contradictswith 119881(119905) le 1 + 120573

1119908119890minus119887119908 Hence the claim is proved From

the claim we will discuss the following two possibilities

(i) 119868(119905) ge 119868lowast for 119905 large enough

(ii) 119868(119905) oscillates about 119868lowast for 119905 large enough

Evidently we only need to consider the case (ii) Let 119905lowast gt 0

and 120585 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast

+ 120585) = 119868lowast and let 119868(119905) lt 119868

lowast for119905lowast

lt 119905 lt 119905lowast

+120585 where 119905lowast is sufficiently large such that 119878(119905) gt 1205751

for 119905lowast lt 119905 lt 119905lowast

+ 120585 Since 119868(119905) is continuous and ultimatelybounded and is not effected by impulses we conclude that119868(119905) is uniformly continuous Hence there exists a constant 120582(with 0 lt 120582 lt 119908 and 120582 is independent of the choice of 119905lowast)such that 119868(119905) gt 119868

lowast

2 for 119905lowast lt 119905 lt 119905lowast

+ 120582 If 120585 le 120582 our aim isobtained If 120582 lt 120585 le 119908 from the second equation of (3) wehave that 119868(119905) gt minus(119887 + 120574119889)119868(119905) and 119868(119905lowast) = 119868

lowast then we have119868(119905) ge 119902 for 119905 isin [119905

lowast

119905lowast

+ 119908] where 119902 = min119868lowast2 119902lowast 119902lowast =119868lowast

119890minus(119887+120574)119908) The same arguments can be continued and we

can obtain 119868(119905) ge 119902 for 119905 isin [119905lowast

+ 119908 119905lowast

+ 120585] Since the intervalis chosen in an arbitrary way we get that 119868(119905) ge 119902 for 119905 largeenough In view of our arguments above the choice of 119902 isindependent of the positive solution of (3) which satisfies that119868(119905) ge 119902 for sufficiently large 119905 This completes the proof

Denote that 120579lowast= [(1205731minus 1205732(119888 + 1))119890

minus119887119908

minus (119887 + 120574119889)](1 minus

119890minus119887119879

)((1205731minus 1205732(119888 + 1))119890

minus119887119908

(1 minus 119890minus119887119879

) + (119887 + 120574119889)119890minus119887119879

) 1205732lowast

=

(119888 + 1)(1205731+ (119887 + 120574119889)(1 minus (1 minus 120579)119890

minus119887119879

)119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

))

6 Discrete Dynamics in Nature and Society

and 119888lowast= 1205732119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

)(1205731119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

) +

(119887 + 120574119889)(1 minus (1 minus 120579)119890minus119887119879

)) minus 1It follows from Theorem 8 that the disease is uniformly

persistent provided that 120579 lt 120579lowastor 1205732lt 1205732lowast

or 119888 gt 119888lowast

Theorem 9 If 119877lowastgt 1 then system (3) is permanent

Proof Suppose that (119878(119905) 119868(119905)) is any positive solution ofsystem (3) with initial conditions (4) From the first equationof system (3) we have 119878(119905) ge 119887 minus (119887 + 120573

1)119878(119905) By similar

arguments as those in the proof of Theorem 6 we have that

lim119905rarrinfin

119878 (119905) ge 119901 (36)

where 119901 = (119887(119887 + 1205731))((1 minus 120579)(1 minus 119890

minus(119887+120573minus1)119879

)(1 minus (1 minus

120579)119890minus(119887+120573minus1)119879

)) minus 1205761 (1205761is sufficiently small)

Set Ω0= (119878 119868) isin 119877

2

| 119901 le 119878 119902 le 119868 119878 + 119868 le 1FromTheorem 8 and (36) we know that the setΩ

0is a global

attractor in Ω and of course every solution of system (3)with initial conditions (4) will eventually enter and remainin region Ω

0 Therefore system (3) is permanent The proof

is complete

FromTheorem 9 we can obtain the following result

Corollary 10 Assume that 120579 lt 120579lowastor 1205732lt 1205732lowast

or 119888 gt 119888lowast then

system (3) is permanent

5 Conclusion

In this paper we introduce media coverage and saturationrecovery in the delayed SEIRS epidemic model with pulsevaccination and analyze detailedly in theory that mediacoverage and pulse vaccination bring effects on infection-free and the permanence of epidemic disease We suggestthe probability of transmission per contact 120573(119868) = 120573

1minus

1205732(119868ℎ

(119888 + 119868ℎ

)) which reflects some characters of mediacoverage We can get that 120597120573(119868)120597120573

2= minus(119868

ℎ

(119888 + 119868ℎ

)) lt 0so 120573(119868) is a monotone decreasing function on 120573

2 that is

if 1205732(the reduced valid contact rate through actual media

coverage) is larger then infection rate of disease is smallerAgain 120597120573(119868)120597119888 = 120573

2119868ℎ

(119888 + 119868ℎ

)2

gt 0 so 120573(119868) is a monotoneincreasing function on 119888 that is if 119888 is smaller (refectionon the disease is quickly) then infection rate of disease issmallerWhen infective individuals appear in a region peoplereduce their contact with others to avoid being infectedand the more infective individuals being reported the lesscontact with others Fromabove analysis we know thatmediacoverage is very important on controlling disease and mediacoverage should be considered in incidence rate

By Theorem 6 the infection-free periodic solution(119878119890(119905) 0) of system (3) is globally attractivity provided that

119877lowast

lt 1 By Theorem 9 system (3) is permanent if 119877lowastgt 1

From Corollaries 7 and 10 we can choose the proportionof those vaccinated successfully to all of newborns such that120579 gt 120579

lowast in order to prevent the epidemic disease fromgenerating endemic and the epidemic is permanent if 120579 lt 120579

lowast

But for 120579 isin [120579lowast 120579lowast

] the dynamical behavior of model (3)

has not been studied and the threshold parameter for thevaccination rate between the extinction of the disease andthe uniform persistence of the disease has not been obtainedThese issues will be considered in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the NNSF of China (1137104811201433) NSF of Henan Province (112300410156122300410117) NSF of the Education DepartmentHenan Province (2011A110022) GGJS of Henan Province(2013GGJS-110) and XGGJS of Zhengzhou University ofLight Industry (2012XGGJS003)

References

[1] Z Agur L Cojocaru G Mazor R M Anderson and Y LDanon ldquoPulse mass measles vaccination across age cohortsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 90 no 24 pp 11698ndash11702 1993

[2] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificamp Technical New York NY USA 1993

[3] V Lakshmikantham D D Bainov and P S Simeonov Theoryof Impulsive Differential Equations World Scientific Singapore1989

[4] Y Li J Cui and X Song ldquoDynamics of a predator-prey systemwith pulsesrdquo Applied Mathematics and Computation vol 204no 1 pp 269ndash280 2008

[5] X Song and Y Li ldquoDynamic complexities of a Holling II two-prey one-predator systemwith impulsive effectrdquoChaos Solitonsamp Fractals vol 33 no 2 pp 463ndash478 2007

[6] X Liu ldquoImpulsive stabilization and applications to populationgrowth modelsrdquo Rocky Mountain Journal of Mathematics vol25 no 1 pp 381ndash395 1995

[7] A Lakmeche and O Arino ldquoBifurcation of non trivial periodicsolutions of impulsive differential equations arising chemother-apeutic treatmentrdquo Dynamics of Continuous Discrete andImpulsive Systems Series B vol 7 no 2 pp 265ndash287 2000

[8] W Wang ldquoGlobal behavior of an SEIRS epidemic model withtime delaysrdquoAppliedMathematics Letters vol 15 no 4 pp 423ndash428 2002

[9] J Hui and L-S Chen ldquoImpulsive vaccination of sir epidemicmodels with nonlinear incidence ratesrdquo Discrete and Continu-ous Dynamical Systems Series B vol 4 no 3 pp 595ndash605 2004

[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006

[11] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003

[12] W-M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

Discrete Dynamics in Nature and Society 7

[13] Y Li and J Cui ldquoThe effect of constant and pulse vaccination onSIS epidemic models incorporating media coveragerdquo Commu-nications inNonlinear Science andNumerical Simulation vol 14no 5 pp 2353ndash2365 2009

[14] Y Li C Ma and J Cui ldquoThe effect of constant and mixedimpulsive vaccination on SIS epidemic models incorporatingmedia coveragerdquo Rocky Mountain Journal of Mathematics vol38 no 5 pp 1437ndash1455 2008

[15] J Cui X Mu and H Wan ldquoSaturation recovery leads tomultiple endemic equilibria and backward bifurcationrdquo Journalof Theoretical Biology vol 254 no 2 pp 275ndash283 2008

[16] K L Cooke and P Van Den Driessche ldquoAnalysis of an SEIRSepidemic model with two delaysrdquo Journal of MathematicalBiology vol 35 no 2 pp 240ndash260 1996

[17] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press San Digo Calif USA1993

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

4 Discrete Dynamics in Nature and Society

Again from the second equation of system (3) we know that(15) implies that

119868 (119905) le (1205731minus

1205732119868ℎ

(119905 minus 119908)

119888 + 119868ℎ (119905 minus 119908)) 120575119868 (119905 minus 119908) 119890

minus119887119908

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

le 1205731120575119890minus119887119908

119868 (119905 minus 119908) minus 119887119868 (119905)

(16)

where 119905 gt 119896119879 + 119908 119896 gt 1198961

Consider the following comparison system

119910 (119905) = 1205731120575119890minus119887119908

119910 (119905 minus 119908) minus 119887119910 (119905) 119905 gt 119896119879 + 119908 119896 gt 1198961

(17)

According to (11) and Lemma 3 we have lim119905rarrinfin

119910(119905) = 0Let (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and let 119868(120577) = 120593(120577) gt 0 (120577 isin [minus119908 0]) 119910(119905)be the solution of system (17) with initial values 119910(120577) =

120593(120577) gt 0 (120577 isin [minus119908 0]) By the comparison theorem wehave lim

119905rarrinfinsup 119868(119905) le lim

119905rarrinfinsup119910(119905) = 0 Incorporating

into the positivity of 119868(119905) we know that lim119905rarrinfin

119868(119905) = 0Therefore there exists an integer 119896

2gt 1198961(where 119896

2119879 gt

1198961119879 + 119908) such that 119868(119905) lt 120576

0for all 119905 gt 119896

2119879

From the first equation of system (3) we have

119878 (119905) gt 119887 minus 119887119878 (119905) minus 1205731119878 (119905) 119868 (119905) gt 119887 minus (119887 + 120573

11205760) 119878 (119905)

119905 gt 1198962119879 + 120591

119878 (119905) lt 119887 minus 119887119878 (119905) + 1205732119878 (119905)

119868ℎ+1

(119905)

119888 + 119868ℎ (119905)+

120574119868 (119905 minus 120591)

119889 + 119868 (119905 minus 120591)119890minus119887120591

lt (119887 +1205741205760

119889 + 1205760

119890minus119887120591

) minus (119887 minus1205732120576ℎ+1

0

119888 + 120576ℎ0

)119878 (119905)

119905 gt 1198962119879 + 120591

(18)

Consider the following comparison impulsive differentialequations for 119905 gt 119896

2119879 + 120591 and 119896 gt 119896

2

1(119905) = 119887 minus (119887 + 120573

11205760) 1199111(119905) 119905 = 119896119879

1199111(119905+

) = (1 minus 120579) 1199111(119905minus

) 119905 = 119896119879

(19)

2(119905) = (119887 +

1205741205760

119889 + 1205760

119890minus119887120591

) minus (119887 minus1205732120576ℎ+1

0

119888 + 120576ℎ0

)1199112(119905) 119905 = 119896119879

1199112(119905+

) = (1 minus 120579) 1199112(119905minus

) 119905 = 119896119879

(20)

By Lemma 2 we have that the unique periodic solution ofsystem (19)

1119890(119905) =

119887

119887 + 12057311205760

(1 minus120579119890minus(119887+120573

11205760)(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879)

119896119879 lt 119905 le (119896 + 1) 119879

(21)

and the unique periodic solution of system (20)

2119890(119905) =

(119887 + (1205741205760 (119889 + 120576

0)) 119890minus119887120591

)

(119887 minus (1205732120576ℎ+1

0 (119888 + 120576ℎ

0)))

(1 minus120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879)

119896119879 lt 119905 le (119896 + 1) 119879

(22)

are globally asymptotically stableLet (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and 119878(0+

) = 1198780

gt 0 1199111(119905) and let 119911

2(119905) be the

solutions of system (19) and (20) with initial values 1199111(0+

) =

1199112(0+

) = 1198780 respectively By the comparison theorem in

impulsive differential equation there exists an integer 1198963gt 1198962

such that 1198963119879 gt 119896

2119879 + 120591 and

1119890(119905) minus 120576

0lt 119878 (119905) lt

2119890(119905) minus 120576

0

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198963

(23)

Because 1205760is arbitrarily small it follows from (23) that

119878119890(119905) = 1 minus

120579119890minus119887(119905minus119896119879)

1 minus (1 minus 120579) 119890minus119887119879 119896119879 lt 119905 le (119896 + 1) 119879 (24)

is globally attractive The proof is complete

Denote that 120579lowast = 1minus119890119887119879

+(1205731119890minus119887119908

(1+(120574119887)119890minus119887120591

)(119890119887119879

minus1))

119887(119887minus1205732(119888+1)) 120573lowast

2= (119888+1)(119887minus120573

1119890minus119887119908

(119887+120574119890minus119887120591

(119889+1))(1minus

119890minus119887119879

)119887(1minus(1minus120579)119890minus119887119879

)) and 119888lowast = 1205732119887(1minus(1minus120579)119890

minus119887119879

)(1198872

(1minus

(1 minus 120579)119890minus119887119879

) minus 1205731119890minus119887119908

(119887 + 120574119890minus119887120591

(119889 + 1))(1 minus 119890minus119887119879

)) minus 1According to Theorem 6 we can obtain the following

result

Corollary 7 The infection-free periodic solution (119878119890(119905) 0) of

system (3) is globally attractivity provided that 120579 gt 120579lowast or 120573

2lt

120573lowast

2or 119888 gt 119888

lowastFrom Corollary 7 we know that the disease will disappear

if the vaccination rate is larger than 120579lowast

4 Permanence

In this section we say the disease is endemic if the infectiouspopulation persists above a certain positive level for suffi-ciently large time

Denote that119877lowast= (1205731minus1205732(119888+1))119890

minus119887119908

(1minus120579)(1minus119890minus119887119879

)(119887+

120574119889)(1minus (1minus120579)119890minus119887119879

) and 119868lowast = (119887120573)[(1205731minus1205732(119888+1))119890

minus119887119908

(1minus

120579)(1 minus 119890minus119887119879

)(119887 + 120574119889)(1 minus (1 minus 120579)119890minus119887119879

) minus 1]

Theorem 8 If 119877lowast

gt 1 then there is a positive constant 119902such that each positive solution (119878(119905) 119868(119905)) of system (3) satisfies119868(119905) gt 119902 for 119905 large enough

Proof From 119877lowastgt 1 we easily know that 119868lowast gt 0 and there

exists sufficiently small 120576 gt 0 such that

(1205731minus

1205732

119888 + 1) 1205751119890minus119887119908

minus 119887 minus120574

119889gt 0 (25)

Discrete Dynamics in Nature and Society 5

where 1205751= (119887(119887 + 120573

1119868lowast

))(1 minus 120579(1 minus (1 minus 120579)119890minus(119887+120573

1119868lowast)119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt 119868

lowast forall 119905 ge 119905

0 Suppose that the claim is not valid Then there is

a 1199050gt 0 such that 119868(119905) lt 119868

lowast for all 119905 ge 1199050 It follows from the

first equation of (3) that for 119905 ge 1199050

119878 (119905) gt 119887 minus 119887119878 (119905) minus 1205731119878 (119905) 119868 (119905) gt 119887 minus (119887 + 120573

1119868lowast

) 119878 (119905) (26)

Consider the following comparison impulsive system for119905 ge 1199050

(119905) = 119887 minus (119887 + 1205731119868lowast

) 119906 (119905) 119905 = 119896119879

119906 (119905+

) = (1 minus 120579) 119906 (119905minus

) 119905 = 119896119879

(27)

By Lemma 2 we obtain that

119890(119905) =

119887

119887 + 1205731119868lowast+ (119906

lowast

minus119887

119887 + 1205731119868lowast) 119890minus(119887+120573

1119868lowast)(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879

(28)

is the unique positive periodic solution of (27) which isglobally asymptotically stable where 119906lowast = (119887(119887+120573

1119868lowast

))((1minus

120579)(1 minus 119890minus(119887+120573

1119868lowast)119879

)(1 minus (1 minus 120579)(1 minus 119890minus(119887+120573

1119868lowast)119879

)))Let (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and let 119878(0+) = 1198780gt 0 119906(119905) be the solution of

system (27) with initial values 119906(0+) = 1198780 By the comparison

theorem for impulsive differential equation there exists aninteger 119896

1(gt 1199050+ 119908) such that 119878(119905) gt

119890(119905) minus 120576 for 119905 ge 119896

1119879

thus

119878 (119905) gt 119906lowast

minus 120576 = 1205751 119905 ge 119896

1 (29)

The second equation of system (3) can be rewritten as

119868 (119905) = 1205731119890minus119887119908

119878 (119905) 119868 (119905) minus 1205731119890minus119887119908

119889

119889119905int

119905

119905minus119908

119878 (119906) 119868 (119906) 119889119906

minus 1205732119890minus119887119908

119878 (119905) 119868ℎ+1

(119905)

119888 + 119868ℎ (119905)

+ 1205732119890minus119887119908

119889

119889119905int

119905

119905minus119908

119878 (119906) 119868ℎ+1

(119906)

119888 + 119868ℎ (119906)119889119906

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

(30)

Let us consider any positive solution (119878(119905) 119868(119905)) of system (3)According to this solution we define

119881 (119905) = 119868 (119905) + 1205731119890minus119887119908

int

119905

119905minus119908

119878 (119906) 119868 (119906) 119889119906

minus 1205732119890minus119887119908

int

119905

119905minus119908

119878 (119906) 119868ℎ+1

(119906)

119888 + 119868ℎ (119906)119889119906

(31)

According to (30) we calculate the derivative of 119881 along thesolutions of system (3)

(119905) = 119868 (119905) (1205731119890minus119887119908

119878 (119905) minus1205732119878 (119905) 119868

ℎ

(119905)

119888 + 119868ℎ (119905)119890minus119887119908

minus 119887 minus120574

119889 + 119868 (119905))

(32)

By (25) and (29) for 119905 ge 1199051 we have

(119905) ge 119868 (119905) (1205731119890minus119887119908

119878 (119905) minus1205732119878 (119905)

119888 + 1119890minus119887119908

minus 119887 minus120574

119889)

gt 119868 (119905) [(1205731minus

1205732

119888 + 1) 119890minus119887119908

1205751minus 119887 minus

120574

119889] gt 0

(33)

Let 119868119897= min

119905isin[11990511199051+119908]

119868(119905) in the following we will showthat 119868(119905) ge 119868

119897for 119905 ge 119905

1 Suppose the contrary then there is a

1198790ge 0 such that 119868(119905) ge 119868

119897for 1199051le 119905 le 119905

1+ 119908 + 119879

0 119868(1199051+ 119908 +

1198790) = 119868119897and 119868(119905

1+119908+119879

0) le 0 However the second equation

of system (3) and (4) imply that

119868 (1199051+ 119908 + 119879

0) ge (120573

1minus

1205732

119888 + 1) 119878 (1199051+ 1198790) 119868 (1199051+ 1198790) 119890minus119887119908

minus 119887119868 (1199051+ 119908 + 119879

0) minus

120574

119889119868 (1199051+ 119908 + 119879

0)

ge (1205731minus

1205732

119888 + 1) 119878 (1199051+ 1198790) 119868119897119890minus119887119908

minus 119887119868119897minus120574

119889119868119897

gt [(1205731minus

1205732

119888 + 1) 1205751119890minus119887119908

minus 119887 minus120574

119889] 119868119897gt 0

(34)

This is a contradictionThus 119868(119905) ge 119868119897for 119905 ge 119905

1 So (33) leads

to

(119905) gt 119868119897[(1205731minus

1205732

119888 + 1) 119890minus119887119908

1205751minus 119887 minus

120574

119889] 119905 ge 119905

1 (35)

which implies that 119881(119905) rarr infin as 119905 rarr infin This contradictswith 119881(119905) le 1 + 120573

1119908119890minus119887119908 Hence the claim is proved From

the claim we will discuss the following two possibilities

(i) 119868(119905) ge 119868lowast for 119905 large enough

(ii) 119868(119905) oscillates about 119868lowast for 119905 large enough

Evidently we only need to consider the case (ii) Let 119905lowast gt 0

and 120585 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast

+ 120585) = 119868lowast and let 119868(119905) lt 119868

lowast for119905lowast

lt 119905 lt 119905lowast

+120585 where 119905lowast is sufficiently large such that 119878(119905) gt 1205751

for 119905lowast lt 119905 lt 119905lowast

+ 120585 Since 119868(119905) is continuous and ultimatelybounded and is not effected by impulses we conclude that119868(119905) is uniformly continuous Hence there exists a constant 120582(with 0 lt 120582 lt 119908 and 120582 is independent of the choice of 119905lowast)such that 119868(119905) gt 119868

lowast

2 for 119905lowast lt 119905 lt 119905lowast

+ 120582 If 120585 le 120582 our aim isobtained If 120582 lt 120585 le 119908 from the second equation of (3) wehave that 119868(119905) gt minus(119887 + 120574119889)119868(119905) and 119868(119905lowast) = 119868

lowast then we have119868(119905) ge 119902 for 119905 isin [119905

lowast

119905lowast

+ 119908] where 119902 = min119868lowast2 119902lowast 119902lowast =119868lowast

119890minus(119887+120574)119908) The same arguments can be continued and we

can obtain 119868(119905) ge 119902 for 119905 isin [119905lowast

+ 119908 119905lowast

+ 120585] Since the intervalis chosen in an arbitrary way we get that 119868(119905) ge 119902 for 119905 largeenough In view of our arguments above the choice of 119902 isindependent of the positive solution of (3) which satisfies that119868(119905) ge 119902 for sufficiently large 119905 This completes the proof

Denote that 120579lowast= [(1205731minus 1205732(119888 + 1))119890

minus119887119908

minus (119887 + 120574119889)](1 minus

119890minus119887119879

)((1205731minus 1205732(119888 + 1))119890

minus119887119908

(1 minus 119890minus119887119879

) + (119887 + 120574119889)119890minus119887119879

) 1205732lowast

=

(119888 + 1)(1205731+ (119887 + 120574119889)(1 minus (1 minus 120579)119890

minus119887119879

)119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

))

6 Discrete Dynamics in Nature and Society

and 119888lowast= 1205732119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

)(1205731119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

) +

(119887 + 120574119889)(1 minus (1 minus 120579)119890minus119887119879

)) minus 1It follows from Theorem 8 that the disease is uniformly

persistent provided that 120579 lt 120579lowastor 1205732lt 1205732lowast

or 119888 gt 119888lowast

Theorem 9 If 119877lowastgt 1 then system (3) is permanent

Proof Suppose that (119878(119905) 119868(119905)) is any positive solution ofsystem (3) with initial conditions (4) From the first equationof system (3) we have 119878(119905) ge 119887 minus (119887 + 120573

1)119878(119905) By similar

arguments as those in the proof of Theorem 6 we have that

lim119905rarrinfin

119878 (119905) ge 119901 (36)

where 119901 = (119887(119887 + 1205731))((1 minus 120579)(1 minus 119890

minus(119887+120573minus1)119879

)(1 minus (1 minus

120579)119890minus(119887+120573minus1)119879

)) minus 1205761 (1205761is sufficiently small)

Set Ω0= (119878 119868) isin 119877

2

| 119901 le 119878 119902 le 119868 119878 + 119868 le 1FromTheorem 8 and (36) we know that the setΩ

0is a global

attractor in Ω and of course every solution of system (3)with initial conditions (4) will eventually enter and remainin region Ω

0 Therefore system (3) is permanent The proof

is complete

FromTheorem 9 we can obtain the following result

Corollary 10 Assume that 120579 lt 120579lowastor 1205732lt 1205732lowast

or 119888 gt 119888lowast then

system (3) is permanent

5 Conclusion

In this paper we introduce media coverage and saturationrecovery in the delayed SEIRS epidemic model with pulsevaccination and analyze detailedly in theory that mediacoverage and pulse vaccination bring effects on infection-free and the permanence of epidemic disease We suggestthe probability of transmission per contact 120573(119868) = 120573

1minus

1205732(119868ℎ

(119888 + 119868ℎ

)) which reflects some characters of mediacoverage We can get that 120597120573(119868)120597120573

2= minus(119868

ℎ

(119888 + 119868ℎ

)) lt 0so 120573(119868) is a monotone decreasing function on 120573

2 that is

if 1205732(the reduced valid contact rate through actual media

coverage) is larger then infection rate of disease is smallerAgain 120597120573(119868)120597119888 = 120573

2119868ℎ

(119888 + 119868ℎ

)2

gt 0 so 120573(119868) is a monotoneincreasing function on 119888 that is if 119888 is smaller (refectionon the disease is quickly) then infection rate of disease issmallerWhen infective individuals appear in a region peoplereduce their contact with others to avoid being infectedand the more infective individuals being reported the lesscontact with others Fromabove analysis we know thatmediacoverage is very important on controlling disease and mediacoverage should be considered in incidence rate

By Theorem 6 the infection-free periodic solution(119878119890(119905) 0) of system (3) is globally attractivity provided that

119877lowast

lt 1 By Theorem 9 system (3) is permanent if 119877lowastgt 1

From Corollaries 7 and 10 we can choose the proportionof those vaccinated successfully to all of newborns such that120579 gt 120579

lowast in order to prevent the epidemic disease fromgenerating endemic and the epidemic is permanent if 120579 lt 120579

lowast

But for 120579 isin [120579lowast 120579lowast

] the dynamical behavior of model (3)

has not been studied and the threshold parameter for thevaccination rate between the extinction of the disease andthe uniform persistence of the disease has not been obtainedThese issues will be considered in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the NNSF of China (1137104811201433) NSF of Henan Province (112300410156122300410117) NSF of the Education DepartmentHenan Province (2011A110022) GGJS of Henan Province(2013GGJS-110) and XGGJS of Zhengzhou University ofLight Industry (2012XGGJS003)

References

[1] Z Agur L Cojocaru G Mazor R M Anderson and Y LDanon ldquoPulse mass measles vaccination across age cohortsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 90 no 24 pp 11698ndash11702 1993

[2] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificamp Technical New York NY USA 1993

[3] V Lakshmikantham D D Bainov and P S Simeonov Theoryof Impulsive Differential Equations World Scientific Singapore1989

[4] Y Li J Cui and X Song ldquoDynamics of a predator-prey systemwith pulsesrdquo Applied Mathematics and Computation vol 204no 1 pp 269ndash280 2008

[5] X Song and Y Li ldquoDynamic complexities of a Holling II two-prey one-predator systemwith impulsive effectrdquoChaos Solitonsamp Fractals vol 33 no 2 pp 463ndash478 2007

[6] X Liu ldquoImpulsive stabilization and applications to populationgrowth modelsrdquo Rocky Mountain Journal of Mathematics vol25 no 1 pp 381ndash395 1995

[7] A Lakmeche and O Arino ldquoBifurcation of non trivial periodicsolutions of impulsive differential equations arising chemother-apeutic treatmentrdquo Dynamics of Continuous Discrete andImpulsive Systems Series B vol 7 no 2 pp 265ndash287 2000

[8] W Wang ldquoGlobal behavior of an SEIRS epidemic model withtime delaysrdquoAppliedMathematics Letters vol 15 no 4 pp 423ndash428 2002

[9] J Hui and L-S Chen ldquoImpulsive vaccination of sir epidemicmodels with nonlinear incidence ratesrdquo Discrete and Continu-ous Dynamical Systems Series B vol 4 no 3 pp 595ndash605 2004

[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006

[11] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003

[12] W-M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

Discrete Dynamics in Nature and Society 7

[13] Y Li and J Cui ldquoThe effect of constant and pulse vaccination onSIS epidemic models incorporating media coveragerdquo Commu-nications inNonlinear Science andNumerical Simulation vol 14no 5 pp 2353ndash2365 2009

[14] Y Li C Ma and J Cui ldquoThe effect of constant and mixedimpulsive vaccination on SIS epidemic models incorporatingmedia coveragerdquo Rocky Mountain Journal of Mathematics vol38 no 5 pp 1437ndash1455 2008

[15] J Cui X Mu and H Wan ldquoSaturation recovery leads tomultiple endemic equilibria and backward bifurcationrdquo Journalof Theoretical Biology vol 254 no 2 pp 275ndash283 2008

[16] K L Cooke and P Van Den Driessche ldquoAnalysis of an SEIRSepidemic model with two delaysrdquo Journal of MathematicalBiology vol 35 no 2 pp 240ndash260 1996

[17] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press San Digo Calif USA1993

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

Discrete Dynamics in Nature and Society 5

where 1205751= (119887(119887 + 120573

1119868lowast

))(1 minus 120579(1 minus (1 minus 120579)119890minus(119887+120573

1119868lowast)119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt 119868

lowast forall 119905 ge 119905

0 Suppose that the claim is not valid Then there is

a 1199050gt 0 such that 119868(119905) lt 119868

lowast for all 119905 ge 1199050 It follows from the

first equation of (3) that for 119905 ge 1199050

119878 (119905) gt 119887 minus 119887119878 (119905) minus 1205731119878 (119905) 119868 (119905) gt 119887 minus (119887 + 120573

1119868lowast

) 119878 (119905) (26)

Consider the following comparison impulsive system for119905 ge 1199050

(119905) = 119887 minus (119887 + 1205731119868lowast

) 119906 (119905) 119905 = 119896119879

119906 (119905+

) = (1 minus 120579) 119906 (119905minus

) 119905 = 119896119879

(27)

By Lemma 2 we obtain that

119890(119905) =

119887

119887 + 1205731119868lowast+ (119906

lowast

minus119887

119887 + 1205731119868lowast) 119890minus(119887+120573

1119868lowast)(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879

(28)

is the unique positive periodic solution of (27) which isglobally asymptotically stable where 119906lowast = (119887(119887+120573

1119868lowast

))((1minus

120579)(1 minus 119890minus(119887+120573

1119868lowast)119879

)(1 minus (1 minus 120579)(1 minus 119890minus(119887+120573

1119868lowast)119879

)))Let (119878(119905) 119868(119905)) be the solution of system (3) with initial

values (4) and let 119878(0+) = 1198780gt 0 119906(119905) be the solution of

system (27) with initial values 119906(0+) = 1198780 By the comparison

theorem for impulsive differential equation there exists aninteger 119896

1(gt 1199050+ 119908) such that 119878(119905) gt

119890(119905) minus 120576 for 119905 ge 119896

1119879

thus

119878 (119905) gt 119906lowast

minus 120576 = 1205751 119905 ge 119896

1 (29)

The second equation of system (3) can be rewritten as

119868 (119905) = 1205731119890minus119887119908

119878 (119905) 119868 (119905) minus 1205731119890minus119887119908

119889

119889119905int

119905

119905minus119908

119878 (119906) 119868 (119906) 119889119906

minus 1205732119890minus119887119908

119878 (119905) 119868ℎ+1

(119905)

119888 + 119868ℎ (119905)

+ 1205732119890minus119887119908

119889

119889119905int

119905

119905minus119908

119878 (119906) 119868ℎ+1

(119906)

119888 + 119868ℎ (119906)119889119906

minus 119887119868 (119905) minus120574119868 (119905)

119889 + 119868 (119905)

(30)

Let us consider any positive solution (119878(119905) 119868(119905)) of system (3)According to this solution we define

119881 (119905) = 119868 (119905) + 1205731119890minus119887119908

int

119905

119905minus119908

119878 (119906) 119868 (119906) 119889119906

minus 1205732119890minus119887119908

int

119905

119905minus119908

119878 (119906) 119868ℎ+1

(119906)

119888 + 119868ℎ (119906)119889119906

(31)

According to (30) we calculate the derivative of 119881 along thesolutions of system (3)

(119905) = 119868 (119905) (1205731119890minus119887119908

119878 (119905) minus1205732119878 (119905) 119868

ℎ

(119905)

119888 + 119868ℎ (119905)119890minus119887119908

minus 119887 minus120574

119889 + 119868 (119905))

(32)

By (25) and (29) for 119905 ge 1199051 we have

(119905) ge 119868 (119905) (1205731119890minus119887119908

119878 (119905) minus1205732119878 (119905)

119888 + 1119890minus119887119908

minus 119887 minus120574

119889)

gt 119868 (119905) [(1205731minus

1205732

119888 + 1) 119890minus119887119908

1205751minus 119887 minus

120574

119889] gt 0

(33)

Let 119868119897= min

119905isin[11990511199051+119908]

119868(119905) in the following we will showthat 119868(119905) ge 119868

119897for 119905 ge 119905

1 Suppose the contrary then there is a

1198790ge 0 such that 119868(119905) ge 119868

119897for 1199051le 119905 le 119905

1+ 119908 + 119879

0 119868(1199051+ 119908 +

1198790) = 119868119897and 119868(119905

1+119908+119879

0) le 0 However the second equation

of system (3) and (4) imply that

119868 (1199051+ 119908 + 119879

0) ge (120573

1minus

1205732

119888 + 1) 119878 (1199051+ 1198790) 119868 (1199051+ 1198790) 119890minus119887119908

minus 119887119868 (1199051+ 119908 + 119879

0) minus

120574

119889119868 (1199051+ 119908 + 119879

0)

ge (1205731minus

1205732

119888 + 1) 119878 (1199051+ 1198790) 119868119897119890minus119887119908

minus 119887119868119897minus120574

119889119868119897

gt [(1205731minus

1205732

119888 + 1) 1205751119890minus119887119908

minus 119887 minus120574

119889] 119868119897gt 0

(34)

This is a contradictionThus 119868(119905) ge 119868119897for 119905 ge 119905

1 So (33) leads

to

(119905) gt 119868119897[(1205731minus

1205732

119888 + 1) 119890minus119887119908

1205751minus 119887 minus

120574

119889] 119905 ge 119905

1 (35)

which implies that 119881(119905) rarr infin as 119905 rarr infin This contradictswith 119881(119905) le 1 + 120573

1119908119890minus119887119908 Hence the claim is proved From

the claim we will discuss the following two possibilities

(i) 119868(119905) ge 119868lowast for 119905 large enough

(ii) 119868(119905) oscillates about 119868lowast for 119905 large enough

Evidently we only need to consider the case (ii) Let 119905lowast gt 0

and 120585 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast

+ 120585) = 119868lowast and let 119868(119905) lt 119868

lowast for119905lowast

lt 119905 lt 119905lowast

+120585 where 119905lowast is sufficiently large such that 119878(119905) gt 1205751

for 119905lowast lt 119905 lt 119905lowast

+ 120585 Since 119868(119905) is continuous and ultimatelybounded and is not effected by impulses we conclude that119868(119905) is uniformly continuous Hence there exists a constant 120582(with 0 lt 120582 lt 119908 and 120582 is independent of the choice of 119905lowast)such that 119868(119905) gt 119868

lowast

2 for 119905lowast lt 119905 lt 119905lowast

+ 120582 If 120585 le 120582 our aim isobtained If 120582 lt 120585 le 119908 from the second equation of (3) wehave that 119868(119905) gt minus(119887 + 120574119889)119868(119905) and 119868(119905lowast) = 119868

lowast then we have119868(119905) ge 119902 for 119905 isin [119905

lowast

119905lowast

+ 119908] where 119902 = min119868lowast2 119902lowast 119902lowast =119868lowast

119890minus(119887+120574)119908) The same arguments can be continued and we

can obtain 119868(119905) ge 119902 for 119905 isin [119905lowast

+ 119908 119905lowast

+ 120585] Since the intervalis chosen in an arbitrary way we get that 119868(119905) ge 119902 for 119905 largeenough In view of our arguments above the choice of 119902 isindependent of the positive solution of (3) which satisfies that119868(119905) ge 119902 for sufficiently large 119905 This completes the proof

Denote that 120579lowast= [(1205731minus 1205732(119888 + 1))119890

minus119887119908

minus (119887 + 120574119889)](1 minus

119890minus119887119879

)((1205731minus 1205732(119888 + 1))119890

minus119887119908

(1 minus 119890minus119887119879

) + (119887 + 120574119889)119890minus119887119879

) 1205732lowast

=

(119888 + 1)(1205731+ (119887 + 120574119889)(1 minus (1 minus 120579)119890

minus119887119879

)119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

))

6 Discrete Dynamics in Nature and Society

and 119888lowast= 1205732119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

)(1205731119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

) +

(119887 + 120574119889)(1 minus (1 minus 120579)119890minus119887119879

)) minus 1It follows from Theorem 8 that the disease is uniformly

persistent provided that 120579 lt 120579lowastor 1205732lt 1205732lowast

or 119888 gt 119888lowast

Theorem 9 If 119877lowastgt 1 then system (3) is permanent

Proof Suppose that (119878(119905) 119868(119905)) is any positive solution ofsystem (3) with initial conditions (4) From the first equationof system (3) we have 119878(119905) ge 119887 minus (119887 + 120573

1)119878(119905) By similar

arguments as those in the proof of Theorem 6 we have that

lim119905rarrinfin

119878 (119905) ge 119901 (36)

where 119901 = (119887(119887 + 1205731))((1 minus 120579)(1 minus 119890

minus(119887+120573minus1)119879

)(1 minus (1 minus

120579)119890minus(119887+120573minus1)119879

)) minus 1205761 (1205761is sufficiently small)

Set Ω0= (119878 119868) isin 119877

2

| 119901 le 119878 119902 le 119868 119878 + 119868 le 1FromTheorem 8 and (36) we know that the setΩ

0is a global

attractor in Ω and of course every solution of system (3)with initial conditions (4) will eventually enter and remainin region Ω

0 Therefore system (3) is permanent The proof

is complete

FromTheorem 9 we can obtain the following result

Corollary 10 Assume that 120579 lt 120579lowastor 1205732lt 1205732lowast

or 119888 gt 119888lowast then

system (3) is permanent

5 Conclusion

In this paper we introduce media coverage and saturationrecovery in the delayed SEIRS epidemic model with pulsevaccination and analyze detailedly in theory that mediacoverage and pulse vaccination bring effects on infection-free and the permanence of epidemic disease We suggestthe probability of transmission per contact 120573(119868) = 120573

1minus

1205732(119868ℎ

(119888 + 119868ℎ

)) which reflects some characters of mediacoverage We can get that 120597120573(119868)120597120573

2= minus(119868

ℎ

(119888 + 119868ℎ

)) lt 0so 120573(119868) is a monotone decreasing function on 120573

2 that is

if 1205732(the reduced valid contact rate through actual media

coverage) is larger then infection rate of disease is smallerAgain 120597120573(119868)120597119888 = 120573

2119868ℎ

(119888 + 119868ℎ

)2

gt 0 so 120573(119868) is a monotoneincreasing function on 119888 that is if 119888 is smaller (refectionon the disease is quickly) then infection rate of disease issmallerWhen infective individuals appear in a region peoplereduce their contact with others to avoid being infectedand the more infective individuals being reported the lesscontact with others Fromabove analysis we know thatmediacoverage is very important on controlling disease and mediacoverage should be considered in incidence rate

By Theorem 6 the infection-free periodic solution(119878119890(119905) 0) of system (3) is globally attractivity provided that

119877lowast

lt 1 By Theorem 9 system (3) is permanent if 119877lowastgt 1

From Corollaries 7 and 10 we can choose the proportionof those vaccinated successfully to all of newborns such that120579 gt 120579

lowast in order to prevent the epidemic disease fromgenerating endemic and the epidemic is permanent if 120579 lt 120579

lowast

But for 120579 isin [120579lowast 120579lowast

] the dynamical behavior of model (3)

has not been studied and the threshold parameter for thevaccination rate between the extinction of the disease andthe uniform persistence of the disease has not been obtainedThese issues will be considered in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the NNSF of China (1137104811201433) NSF of Henan Province (112300410156122300410117) NSF of the Education DepartmentHenan Province (2011A110022) GGJS of Henan Province(2013GGJS-110) and XGGJS of Zhengzhou University ofLight Industry (2012XGGJS003)

References

[1] Z Agur L Cojocaru G Mazor R M Anderson and Y LDanon ldquoPulse mass measles vaccination across age cohortsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 90 no 24 pp 11698ndash11702 1993

[2] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificamp Technical New York NY USA 1993

[3] V Lakshmikantham D D Bainov and P S Simeonov Theoryof Impulsive Differential Equations World Scientific Singapore1989

[4] Y Li J Cui and X Song ldquoDynamics of a predator-prey systemwith pulsesrdquo Applied Mathematics and Computation vol 204no 1 pp 269ndash280 2008

[5] X Song and Y Li ldquoDynamic complexities of a Holling II two-prey one-predator systemwith impulsive effectrdquoChaos Solitonsamp Fractals vol 33 no 2 pp 463ndash478 2007

[6] X Liu ldquoImpulsive stabilization and applications to populationgrowth modelsrdquo Rocky Mountain Journal of Mathematics vol25 no 1 pp 381ndash395 1995

[7] A Lakmeche and O Arino ldquoBifurcation of non trivial periodicsolutions of impulsive differential equations arising chemother-apeutic treatmentrdquo Dynamics of Continuous Discrete andImpulsive Systems Series B vol 7 no 2 pp 265ndash287 2000

[8] W Wang ldquoGlobal behavior of an SEIRS epidemic model withtime delaysrdquoAppliedMathematics Letters vol 15 no 4 pp 423ndash428 2002

[9] J Hui and L-S Chen ldquoImpulsive vaccination of sir epidemicmodels with nonlinear incidence ratesrdquo Discrete and Continu-ous Dynamical Systems Series B vol 4 no 3 pp 595ndash605 2004

[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006

[11] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003

[12] W-M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

Discrete Dynamics in Nature and Society 7

[13] Y Li and J Cui ldquoThe effect of constant and pulse vaccination onSIS epidemic models incorporating media coveragerdquo Commu-nications inNonlinear Science andNumerical Simulation vol 14no 5 pp 2353ndash2365 2009

[14] Y Li C Ma and J Cui ldquoThe effect of constant and mixedimpulsive vaccination on SIS epidemic models incorporatingmedia coveragerdquo Rocky Mountain Journal of Mathematics vol38 no 5 pp 1437ndash1455 2008

[15] J Cui X Mu and H Wan ldquoSaturation recovery leads tomultiple endemic equilibria and backward bifurcationrdquo Journalof Theoretical Biology vol 254 no 2 pp 275ndash283 2008

[16] K L Cooke and P Van Den Driessche ldquoAnalysis of an SEIRSepidemic model with two delaysrdquo Journal of MathematicalBiology vol 35 no 2 pp 240ndash260 1996

[17] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press San Digo Calif USA1993

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 Discrete Dynamics in Nature and Society

and 119888lowast= 1205732119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

)(1205731119890minus119887119908

(1 minus 120579)(1 minus 119890minus119887119879

) +

(119887 + 120574119889)(1 minus (1 minus 120579)119890minus119887119879

)) minus 1It follows from Theorem 8 that the disease is uniformly

persistent provided that 120579 lt 120579lowastor 1205732lt 1205732lowast

or 119888 gt 119888lowast

Theorem 9 If 119877lowastgt 1 then system (3) is permanent

Proof Suppose that (119878(119905) 119868(119905)) is any positive solution ofsystem (3) with initial conditions (4) From the first equationof system (3) we have 119878(119905) ge 119887 minus (119887 + 120573

1)119878(119905) By similar

arguments as those in the proof of Theorem 6 we have that

lim119905rarrinfin

119878 (119905) ge 119901 (36)

where 119901 = (119887(119887 + 1205731))((1 minus 120579)(1 minus 119890

minus(119887+120573minus1)119879

)(1 minus (1 minus

120579)119890minus(119887+120573minus1)119879

)) minus 1205761 (1205761is sufficiently small)

Set Ω0= (119878 119868) isin 119877

2

| 119901 le 119878 119902 le 119868 119878 + 119868 le 1FromTheorem 8 and (36) we know that the setΩ

0is a global

attractor in Ω and of course every solution of system (3)with initial conditions (4) will eventually enter and remainin region Ω

0 Therefore system (3) is permanent The proof

is complete

FromTheorem 9 we can obtain the following result

Corollary 10 Assume that 120579 lt 120579lowastor 1205732lt 1205732lowast

or 119888 gt 119888lowast then

system (3) is permanent

5 Conclusion

In this paper we introduce media coverage and saturationrecovery in the delayed SEIRS epidemic model with pulsevaccination and analyze detailedly in theory that mediacoverage and pulse vaccination bring effects on infection-free and the permanence of epidemic disease We suggestthe probability of transmission per contact 120573(119868) = 120573

1minus

1205732(119868ℎ

(119888 + 119868ℎ

)) which reflects some characters of mediacoverage We can get that 120597120573(119868)120597120573

2= minus(119868

ℎ

(119888 + 119868ℎ

)) lt 0so 120573(119868) is a monotone decreasing function on 120573

2 that is

if 1205732(the reduced valid contact rate through actual media

coverage) is larger then infection rate of disease is smallerAgain 120597120573(119868)120597119888 = 120573

2119868ℎ

(119888 + 119868ℎ

)2

gt 0 so 120573(119868) is a monotoneincreasing function on 119888 that is if 119888 is smaller (refectionon the disease is quickly) then infection rate of disease issmallerWhen infective individuals appear in a region peoplereduce their contact with others to avoid being infectedand the more infective individuals being reported the lesscontact with others Fromabove analysis we know thatmediacoverage is very important on controlling disease and mediacoverage should be considered in incidence rate

By Theorem 6 the infection-free periodic solution(119878119890(119905) 0) of system (3) is globally attractivity provided that

119877lowast

lt 1 By Theorem 9 system (3) is permanent if 119877lowastgt 1

From Corollaries 7 and 10 we can choose the proportionof those vaccinated successfully to all of newborns such that120579 gt 120579

lowast in order to prevent the epidemic disease fromgenerating endemic and the epidemic is permanent if 120579 lt 120579

lowast

But for 120579 isin [120579lowast 120579lowast

] the dynamical behavior of model (3)

has not been studied and the threshold parameter for thevaccination rate between the extinction of the disease andthe uniform persistence of the disease has not been obtainedThese issues will be considered in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the NNSF of China (1137104811201433) NSF of Henan Province (112300410156122300410117) NSF of the Education DepartmentHenan Province (2011A110022) GGJS of Henan Province(2013GGJS-110) and XGGJS of Zhengzhou University ofLight Industry (2012XGGJS003)

References

[1] Z Agur L Cojocaru G Mazor R M Anderson and Y LDanon ldquoPulse mass measles vaccination across age cohortsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 90 no 24 pp 11698ndash11702 1993

[2] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificamp Technical New York NY USA 1993

[3] V Lakshmikantham D D Bainov and P S Simeonov Theoryof Impulsive Differential Equations World Scientific Singapore1989

[4] Y Li J Cui and X Song ldquoDynamics of a predator-prey systemwith pulsesrdquo Applied Mathematics and Computation vol 204no 1 pp 269ndash280 2008

[5] X Song and Y Li ldquoDynamic complexities of a Holling II two-prey one-predator systemwith impulsive effectrdquoChaos Solitonsamp Fractals vol 33 no 2 pp 463ndash478 2007

[6] X Liu ldquoImpulsive stabilization and applications to populationgrowth modelsrdquo Rocky Mountain Journal of Mathematics vol25 no 1 pp 381ndash395 1995

[7] A Lakmeche and O Arino ldquoBifurcation of non trivial periodicsolutions of impulsive differential equations arising chemother-apeutic treatmentrdquo Dynamics of Continuous Discrete andImpulsive Systems Series B vol 7 no 2 pp 265ndash287 2000

[8] W Wang ldquoGlobal behavior of an SEIRS epidemic model withtime delaysrdquoAppliedMathematics Letters vol 15 no 4 pp 423ndash428 2002

[9] J Hui and L-S Chen ldquoImpulsive vaccination of sir epidemicmodels with nonlinear incidence ratesrdquo Discrete and Continu-ous Dynamical Systems Series B vol 4 no 3 pp 595ndash605 2004

[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006

[11] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003

[12] W-M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

Discrete Dynamics in Nature and Society 7

[13] Y Li and J Cui ldquoThe effect of constant and pulse vaccination onSIS epidemic models incorporating media coveragerdquo Commu-nications inNonlinear Science andNumerical Simulation vol 14no 5 pp 2353ndash2365 2009

[14] Y Li C Ma and J Cui ldquoThe effect of constant and mixedimpulsive vaccination on SIS epidemic models incorporatingmedia coveragerdquo Rocky Mountain Journal of Mathematics vol38 no 5 pp 1437ndash1455 2008

[15] J Cui X Mu and H Wan ldquoSaturation recovery leads tomultiple endemic equilibria and backward bifurcationrdquo Journalof Theoretical Biology vol 254 no 2 pp 275ndash283 2008

[16] K L Cooke and P Van Den Driessche ldquoAnalysis of an SEIRSepidemic model with two delaysrdquo Journal of MathematicalBiology vol 35 no 2 pp 240ndash260 1996

[17] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press San Digo Calif USA1993

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

Discrete Dynamics in Nature and Society 7

[13] Y Li and J Cui ldquoThe effect of constant and pulse vaccination onSIS epidemic models incorporating media coveragerdquo Commu-nications inNonlinear Science andNumerical Simulation vol 14no 5 pp 2353ndash2365 2009

[14] Y Li C Ma and J Cui ldquoThe effect of constant and mixedimpulsive vaccination on SIS epidemic models incorporatingmedia coveragerdquo Rocky Mountain Journal of Mathematics vol38 no 5 pp 1437ndash1455 2008

[15] J Cui X Mu and H Wan ldquoSaturation recovery leads tomultiple endemic equilibria and backward bifurcationrdquo Journalof Theoretical Biology vol 254 no 2 pp 275ndash283 2008

[16] K L Cooke and P Van Den Driessche ldquoAnalysis of an SEIRSepidemic model with two delaysrdquo Journal of MathematicalBiology vol 35 no 2 pp 240ndash260 1996

[17] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press San Digo Calif USA1993

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