A Pest Management Epidemic Model with Time Delay and Stage-Structure

Transcription

1 Applied Mahemaics,,, 5- doi:436/am36 Published Online Sepember (hp://wwwscirporg/journal/am) A Pes Managemen Epidemic Model wih ime Delay and Sage-Srucure Absrac Yumin Ding, Shujing Gao, Yujiang Liu, Yun Lan Key Laboraory of Jiangxi Province for Numerical Simulaion and Emulaion echniques, Gannan Normal Universiy, Ganzhou, China Jiangxi Environmenal Engineering Vocaional College, Ganzhou, China gaosjmah@6com Received June, ; revised July, ; acceped July 5, In his paper, an SI epidemic model wih sage srucure is invesigaed In his model, impulsive biological conrol which release infeced pes o he field a a fixed ime periodically is considered, and obained he sufficien condiions for he global araciviy of pes-exincion periodic soluion and permanence of he sysem We also prove ha all soluions of he model are uniformly ulimaely bounded he sensiive analysis on he wo hresholds and o he changes of he releasing amouns of infeced pes is shown by numerical simulaions Our resuls provide a reliable acic basis for he pracice of pes managemen Keywords: Pes Managemen, Sage Srucure, Impulsive Sysem, Permanence Inroducion Pess oubrea ofen cause serious ecological and economic problems, and he warfare beween human and pess has susained for housands of years Wih he developmen of sociey and he progress of science and echnology, a grea deal of pesicides were used o conrol pess, because hey can quicly ill a significan porion of pes populaion and someimes provide he only feasible mehod for prevening economic loss However, pesicide polluion is also recognized as a major healh hazard o human beings and beneficial insecs A presen, more and more people are concerned abou he effecs of pesicide residues on human healh and on he environmen [] In naural world, here are many insecs whose individual members have a life hisory ha aes hem hrough wo sages, larva and maure Pahogens may no be effecive agains laver, ha is, he disease only aacs he suscepible maure pes populaion For example, salcedar leaf beele is such a pes Pes conrol sraegies have been araced many expers over he pas years Recenly, sage-srucured models have received much aracion [,3] However, he epidemic models wih sage-srucure have been seldom sudied Zhang e al [4] inroduced he pes based on he sage-srucure model which incorporaes a discree delay and pulses in order o invesigae how epidemics influence he pes conrol process An SI model wih impulsive perurbaions on diseased pes and spraying pesicides a fixed momen is proposed and invesigaed in [5], which obained he sufficien condiions of he global araciviy of pes-exincion periodic soluion and permanence of he sysem Incidence plays a very imporan role in research of epidemic models, bilinear and sandard incidence raes have been frequenly used in classical epidemic models [6] Several differen incidence raes have been proposed by researchers Anderson e al poined ou ha sandard incidence is more suiable han bilinear incidence [7,8] Levin e al have adoped he incidence form lie q p q p S I or S I N [9] Lindsrom poined ou he crow- ed incidence S () ( as () bs()) [] However here are seldom auhors have concerned he sagesrucured models under he simulaneous effec of disease and crowed incidence A sage-srucure model wih he crowed inciden rae is considered in [] According o he facs of pes managemen, we ae he crowded effec as he incidence rae herefore, in his paper, a pes managemen epidemic model which wih ime delay and sage-srucure is considered he presen paper is organized as follows In he nex secion, we formulae he pes managemen model In Secion 3, some essenial lemmas which will be used o prove our main resuls are inroduced In Secion 4, global araciviy of he suscepible pes-eradicaion pe- Copyrigh SciRes

2 6 Y M DING E AL riodic soluion and he permanence of he model is analyzed In he final secion, we presen some numerical simulaions o illusrae he resuls and poin ou some fuure research direcions Model Formulaion In his paper, we sudy he pes managemen epidemic model: wih iniial condiions L() BS ( ()) S () L () e BS ( ( )) S ( ), SI () () S() e BS ( ( )) S ( ) S (), n, as( ) bs ( ) SI () () I() di(), as( ) bs ( ) I () I ( ) I (), n, n Z, ( ( ), ( ), 3( )) C3 for [,], i (), i,,3, r () e B( ( )) ( ) d, () () where all he coefficiens of model () are nonnegaive and L (), S (), I () represen he larva, maure suscepible and infeced pes populaion a ime, respecively he model is derived from he following assumpions (H ) he deah rae of larva populaion is proporional o he exising larva populaion wih proporionaliy consan, he deah rae of maure suscepible and infeced pes populaion is proporional o he exising maure suscepible and infeced pes populaion wih proporionaliy consans and d, respecively (H ) Only he suscepible pes populaion can reproduce BS ( ) is a birh rae funcion of he suscepible pes populaion for S (, ) wih BS ( ) is monoonically decreasing, lim BS ( ) B( ) exiss and B( ) S B( ), where min{,, d } (H 3 ) represens a consan ime o mauriy, he pro- duc erm e B( S( )) S( ) describes ha immaure pes who were laid a ime and survive a ime (H 4 ) he inciden rae is he crowded effec S () as( ) bs ( ) (H 5 ) is he releasing amouns of infeced pes a n, n,,, and is he period of he impulsive effec Before going ino any deail, we simplify model () and resric our aenion o he following model: SI () () S() e BS ( ( )) S ( ) S (), as( ) bs ( ) n, SI () () I() di(), as( ) bs ( ) I (), n (3) he iniial condiions for (3) are ( ( ), 3( )) C C([,], R ), i ()>, i,3 (4) 3 Some Useful Lemmas he soluion of sysem (), denoed by x() ( L(), S (), I ()) is a piecewise coninuous funcion 3 x : R R, x() is coninuous on ( n,( n ) ), n Z and x( n ) lim x( ) exiss Before demonsra- n ing he main resuls, we need o give some lemmas which will be used as follows Lemma (see []) Le ( ( ), ( ), 3( )) for hen any soluion of sysem () is sricly posiive Lemma Le he funcion m PC[ R, R] saisfies he inequaliies m() pm () () q (),,,,,, m ( ) dm( ) b,, where p, q PC[ R, R] and d, b are consans hen Copyrigh SciRes

3 Y M DING E AL 7 m () m ( ) d exp( psds () ) j s d j exp( p( s) ds) b d exp p( ) d q( s) ds, s he proof of his lemma is given in [] We now show ha all soluions of () are uniformly ulimaely bounded Lemma 3 Any soluion ( L ( ), S ( ), I ( )) of sysem () is uniformly ulimaely bounded ha is, here exiss e a consan M such ha L () M, e S( ) M, I( ) M for sufficienly large Proof Define V () L () S () I () By simple compuaion when n, we calculae he derivaive of V along he soluion of sysem () DV ( ) BSS ( ) LSdI BSS ( ) ( LS I), for ( n,( n ) ) hen we derive DV () V BSS ( ) S, ( n,( n ) ) Obviously, from condiions (H ) and (H ), we are easy o now ha here exiss a consan such ha DV () V, ( n,( n ) ), for n large enough When n, we ge V( n ) V( n) According o Lemma, we derive ( s) ( n) n V () V() e e ds e e M as e herefore by he definiion of V (), we obain ha each posiive soluion of sysem () is uniformly ulimaely bounded his complees he proof Lemma 4 Consider he following delay differenial equaion: x() ax( ) ax() (5) where a, a and are all posiive consans and x () for [,] We have: ) If a, a hen lim x ( ) ; x ) If a, a hen lim x ( ) x he proof of his lemma is given in [3] Lemma 5 (see []) Consider he following impulsive sysem: v( ) dv( ), n, (6) vn ( ) vn ( ), n, n,,, where d, hen here exiss a unique posiive periodic of sysem (6) d( n) v () ve, ( n,( n) ], nz, which is globally asympoically sable, where v d e 4 Main Resuls In his secion ha follows we deermine he global araciviy condiion of he suscepible pes-exincion periodic soluion and he permanence of he sysem (3) 4 Global Araciviy of he Suscepible Pes-Exincion Periodic Soluion Denoe R d B() e ( am bm )( e ) (7) e where M e heorem Le ( S ( ), I ( )) be any soluion of sysem (3), he suscepible pes-exincion periodic soluion (, I ( )) of (3) is globally aracive provided ha R Proof Since R, we can choose sufficienly small such ha d e B() e d (8) am bm e Noe ha I() di(), from Lemma and Lemma 5, we have ha for he given here exiss an ineger such ha for n ( n ), n d e I () I () d (9) e From condiion (H ), (3) and (9), we yield ds() B() e S( ) d S (), am bm for n, n Copyrigh SciRes

4 8 Y M DING E AL Consider he following comparison differenial sysem dy() B() e y( ) d () y (), am bm for n, n From (8), we have B() e am bm According o Lemma 4, we have lim y ( ) By he comparison heorem, we have lim sup S ( ) lim y ( ) Incorporaing ino he posiiviy of S (), we now ha lim S ( ) herefore, for any (sufficienly small), here exiss an ineger ( ) such ha S () for all Form sysem (3) and Lemma 5, we have di() di() ( d ) I() d hen we have z () I() z () and z () I (), z () I () as, while z () and z () are he soluions of z () dz(), n, z( ) z( ), n, z ( ) I( ), and z () ( d ) z(), n, z( ) z( ), n, z ( ) I( ), exp(( d )( n)) respecively, z () for n exp(( d ) ) ( n ) herefore, for any, here exiss an ineger 3, n such ha 3 I () I () z () for n Le, we ge z () I () Hence I() I () as his complees he proof 4 Permanence Persisence (or permanence) is an imporan propery of dynamical sysems, in his secion, we focus on he permanence of sysem (3) Denoe ( S d) ( B() e )( e ) R () where S d ln B() e heorem Suppose R hen here is a posiive consan q such ha each posiive soluion ( S ( ), I ( )) of (3) saisfies S () q, for sufficienly large Proof Le ( S ( ), I ( )) be he soluion of sysem (3) wih iniial condiion (4) Noe ha he firs equaion of (3) can be rewrien as ds() S() I() e BS ( ()) S () S () d as bs( ) d e B( S( )) S( ) d d In he following we define: W () S () e BS ( ( )) S( ) d hen he derivaive of W () wih respec o he soluion of sysem (3) is governed by dw I() e B( S( )) S( ) d as( ) bs ( ) Since R, we can choose sufficienly small d S ( ) and such ha ( ) e B S () ( S d) e We claim ha for any, i is impossible ha S () S for all Suppose ha he claim is no valid hen here is a such ha S () S for all I follows from he second equaion of sysem (3) ha for di() S() I() di() ( d S ) I () d as( ) bs ( ) Consider he comparison impulsive sysem for, z() ( d S ) z(), n, (3) z ( ) z ( ), n, According o Lemma, we ge he unique posiive periodic soluion of sysem (3) ( S d)( n) z () ze, n( n), is globally asympoically sable, where z S e ( d) Copyrigh SciRes

5 Y M DING E AL 9 By he comparison heorem in impulsive differenial equaions, we now ha for any sufficienly small, here exiss a ( ) such ha I () z for all I follows from () ha e Furher, Se, (4) B( S ) () ( ( ) ) () for W e B S S S min S( ), m [, ] (5) We will show ha S () S m for all Oherwise, here is a h such ha S () Sm for h, S ( h) Sm and S( h) Accordingly, from he firs equaion of (3) and he inequaliy (4) we yield S( h) B( S( h)) S( h) e S ( hi ) ( h) as( h) bs ( h) S ( h) ( BS ( ) e ) S which leads o a conradicion herefore S () Sm for all As a consequence, (5) leads o W B S e S m () ( ( ) ) for, which implies ha W () as his conradics W () M( B() e ) he claim is proved By he claim, we need o consider wo cases Case S () S for all large Case S () oscillaes abou S for ha is large enough Define m S q min, q ( ) where q S e We wan o show ha S () q for all large he conclusion is eviden in he firs case For he second case, le and saisfy S ( ) S ( ) S and S S (), for all (, ), where is sufficienly large such ha I ( ) for, S () is uniformly coninuous he posiive soluions of (3) are ulimaely bounded and S () is no affeced by he impulses Hence, here is a g ( g and g S is dependen of he choice of ) such ha S () for g If g, here is nohing o prove Le us consider he case g Since S() ( ) S() and S ( ) S, hence S () q for g If, i is obvious ha S () q for [, ] hen proceeding exacly as he proof for he above claim, we see ha S () q for [, ], because he ind of inerval [, ] is chosen in an arbirary way (we only need o be large) We concluded ha S () q for all large In he second case, in view of our above discussion, he choice of q is independen of he posiive soluion, and we proved ha any posiive soluion of (3) saisfies S () q for all sufficienly large his complees he proof heorem 3 Suppose R hen sysem () is permanen Proof Denoe ( L ( ), S ( ), I ( )) be any soluion of sysem () From he second equaion of sysem (3) and heorem, we have di() q I() d d am bm q Le A, i is easy o ge I () if am bm A d, so we can always obain he posiive lower boundary by he heorem of differenial equaions Oherwise, by he same argumen as hose in he proof of heorem, we have liminf I( ) p, where e p e ( Ad) ( Ad) In view of heorem, he firs equaion of sysem () becomes dl BMqMB e L d I is easy o obain liminf L ( ) BMq ( ) MB() e where By heorem and he above discussion, sysem () is permanen he proof of heorem 3 is complee 5 Numerical Analysis and Discussion We have sudied a delayed epidemic model wih sagesrucure and impulses, heoreically analyze he influ- Copyrigh SciRes

6 Y M DING ence of impulsive releasing for he infeced pes populaion, and also obained ha he pes-exincion periodic soluion of sysem () is globally aracive if he conrol variable R given by (7), and he sysem is permanen wih R which is given by () We now ha, besides he release amoun of infecious pess each ime, he period of impulsive vaccinaion and he effecive conac rae play an imporan role in he dynamical behavior of he sysem In he following, we will specially analyze he influence of he release amoun of infecious pess o he dynamical sysem We assume B ( S ) e S, and consider he hypoheical se of parameer values as 45,, a, b, 75, 5, d 6, 4, By heorem and heorem 3, we now ha when R =999, he pes-exincion periodic soluion of sysem (3) is globally aracive and he suscepible pes populaion becomes exinc (see Figure ) When he release amoun of infeced pes reaches a cerain value = such ha R 668, he sysem (3) is permanen (see Figure ) So far we have only discussed wo cases: R and R Bu for R R, he suscepible pes populaion eiher becomes exinc (see Figure 3) or coexiss o he infec pes populaion (see Figure 4) According o he above numerical simulaion, we hin here exiss a hreshold parameer o decide he exincion of he suscepible pes populaion and he permanence of he sysem hese issues will be considered in our fuure research From Figure 5(a), we can observe ha: R is sensiive o value as is small enough, whereas no sensiive o value as 5 We can also ge he similar phenomenon from Figure 5(b) We hope ha our resuls will provide an insigh o pes managemen pracicing Figure Dynamical behavior of sysem (3) wih 38, R 999 Copyrigh SciRes E AL Figure Dynamical behavzior of sysem (3) wih, R 668 Figure 3 Dynamical behavior of sysem (3) wih, R 669 and R 6 Figure 4 Dynamical behavior of sysem (3) wih, R 8343 and R 95

7 Y M DING E AL Figure 5 he sensiive analysis of o R and R ; (b) R 6 Acnowledgemens (a) (b) R (a) he research have been suppored by he Naural Science Foundaion of China (9737), he Naional Key echnologies R & D Program of China (8BAI68B), he Posgraduae Innovaion Fund of Jiangxi Province (YC9A4) Managemen, Communicaions in Nonlinear Science and Numerical Simulaion, Vol 5, No,, pp 4-49 [] J A Cui and X Y Song, Permanence of a Predaor-Prey Sysem wih Sage Srucure, Discree Coninuous Dynamical Sysems-Series B, Vol 4, No 3, 4, pp [3] Y N Xiao and L S Chen, Global Sabiliy of a Predaor-Prey Sysem wih Sage Srucure for he Predaor, Aca Mahemaica Sinica, Vol, No, 4, pp 63-7 [4] H Zhang, L S Chen and J J Nieo, A Delayed Epidemic Model wih Sage-Srucure and Pulses for Pes Managemen Sraegy, Nonlinear Analysis: Real World Applicaions, Vol 9, No 4, 8, pp [5] X Wang, Y D ao and X Y Song, Mahemaical Model for he Conrol of a Pes Populaion wih Impulsive Perurbaions on Diseased Pes, Applied Mahemaical Modelling, Vol 33, No 7, 9, pp [6] H W Hehcoe, he Mahemaics of Infecious Disease, Siam Review, Vol 4, No 4,, pp [7] R M Anderson, R M May and B Anderson, Infecious Diseases of Human: Dynamics and Conrol, Oxford Science Publicaions, Oxford, 99 [8] R M Anderson and R M May, Populaion Biological of Infecious Diseases, Springer Berlin-Heidelberg, New Yor, 98 [9] W M Liu, S A Levin and Y, Lwasa, Infuence of Nonlinear Incidence Raes upon he Behavior of SIRS Epidemiological Models, Journal of Mahemaical Biology, Vol 3, No, 986, pp 87-4 [] Lindsrom, A Generalized Uniqueness heorem for Limi Cycles in a Predaor-Prey Sysem, Aca Academic Aoensis, Series B, Vol 49, No, 989, pp -9 [] J J Jiao, X Z Meng ang L S Chen, Global Araciviy and Permanence of a Sage-Srucured Pes Managemen SI Model wih ime Delay and Diseased Pes Impulsive ransmission, Chaos, Solions and Fracals, Vol 38, No 3, 8, pp [] V Lashmianham, D D Bainov and P Simeonov, heory of Impulsive Differenial Equaions, World Scienific, Singapor, 989 [3] Y Kuang, Delay Differenial Equaion wih Applicaion in Populaion Dynamics, Academic Press, New Yor, References [] R Q Shi and L S Chen, An Impulsive Predaor-Prey Model wih Disease in he Prey for Inegraed Pes Copyrigh SciRes