【導(dǎo)讀】證明系統(tǒng)的全局動(dòng)力學(xué)特性完全由基本再生數(shù)0R所確定：當(dāng)01R?時(shí)，無(wú)病平衡點(diǎn)是。時(shí)，地方病平衡點(diǎn)是全局漸進(jìn)穩(wěn)定的.播也可通過(guò)媒介的胎盤轉(zhuǎn)移完成，如乙肝，風(fēng)疹，皰疹的病原體.對(duì)昆蟲或植物而言，入潛伏者類的新生兒為bqI，01q??.對(duì)于染病者類，我們假設(shè)δ比例的染病者具有。這里β是常規(guī)接觸率，參數(shù)μ是從E類到I類的轉(zhuǎn)換率.參數(shù)b，d，β，μ為是正數(shù)，θ，σ，r為非負(fù)數(shù).設(shè)x=S/N;y=E/N;z=I/N和ω=R/N分別表示S，E，I，R在總?cè)丝谥械谋壤?易證x，y，z，不出現(xiàn)在方程組的前三式中，這使我們減。少方程得到一個(gè)子式.本文的目的是要證明的動(dòng)力學(xué)行為由0R決定.在文獻(xiàn)[3]中提到的.若x是局部穩(wěn)定且在D中所有的軌跡收斂到x，則唯一的平衡點(diǎn)x是全局穩(wěn)定的.1H是等價(jià)于的一致持久性.矩陣值函數(shù)為1C的.假設(shè)當(dāng)x∈K，K為緊集時(shí)，性，如周期解，同宿軌和異宿軌，因而它蘊(yùn)含了x的局部穩(wěn)定性.中的最大緊不變集是單點(diǎn)集。時(shí)，0E的全局穩(wěn)定性由Lasalles不變集原理得到.

　　

【正文】 ing (13) into ( 11) and ( 14) into ( 12) ， we obtain， for t T 0 ， 1239。39。,yyg b g b? ? ? ? ? ? Therefore ? ? 39。/ ,B y y b? ?? for 0tT? .Along each solution ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?, , to w ith 0 , 0 , 0 , w h e r e x t y t z t x y z K K?（ 3 ） is the pact absorbing set， we have， for 0tT? ? ? ? ? ? ?? ?0 000 01 1 1 l n ,T yt tTB d s B d s bt t t t T t?? ?? ? ??? which in turn implies that 2 / 2 0,qb?? ? from ( 7) ， pleting the proof of Theorem 3. 3 The Dynamics of the Population Size We now turn to the dynamics of ? ? ? ? ? ? ? ?? ?, , ,S t E t I t R t and ? ? ? ? ? ? ? ? ? ?N t S t E t I t R t? ? ? ?， which are governed by systems ( 1) . The fact that R( t ) does not appear in the first three equations in ( 1) allows us to study the equivalent system： 39。,39。,39。,39。 39。 .SS bN I bqI dS s rINsE I bqI dE ENI E dI I rIN bN dN??????? ? ? ? ? ? ???? ? ? ? ???? ? ? ??? ??? (15) It is obvious that the total population size N (t ) may be increasing ， decreasing or constant ， depending on the growth rate r = b d . The proportions (x ， y ， z) might tend to ,0,0bb ???????? or the endemic equilibrium? ?*, *, *x y z ， but the behavior of the proportions does not g iv e us much insight on the behavior of the total number of infected individuals ( prise E class and I class) . In particular， even if the total number of infected individuals increases exponentially but at a lower rate than the total population size N ( t ) ， then the proportion 20 of the two tends to zero， however， the total number of infected individuals approach infinity. We can also imagine the reversed situation both the number of infected individuals and the total population decline to zero but the proportion remains almost constant ( and non zero) at all times. In this case the disease remains in the population as long as the population exist s. To describe the behavior of S( t) ， E( t ) ， I ( t ) ， one needs two additional threshold parameters which are introduced in reference [12] . The following are the pertinent threshold parameters： ? ?? ?? ?? ?? ?? ?? ?? ?02 *0, 1 , 1.b bq b Rb d d rRx bqRd b r? ? ?? ? ?????? ???? ? ? ? ???? ?? ?? ? ? ?? We derived the following Theorem. Theorem 4 (a) The number of susceptible individuals S(t) increase (decrease) with exponential asymptotic rate b d. (b) Assume that 0 1R? then ? ? ? ?? ? ? ? ? ?, 0 , 0 ,E t I t o r? ? ? if 2 1R? or 2 1R? . Proof (a) From Theorem 2， 0 1R? implies that ? ? ? ? ? ?? ?l im , , , 0 , 0tbx t y t z t b ?????? ????? ， from the first equation in (1) ， we have ? ?39。/S S b d?? . In the case 0 1R? ， from Theorem 3， ? ? ? ? ? ?? ? ? ?* * *l im , , , ,t x t y t z t x y z?? ?， we divide by S in the first equation in (1) and take the limit ： ? ? ****39。l imt S b zz d b q rS x x???? ? ? ? ? ? ? from the first equation in (3) it follows that the equilibrium *E satisfies Thus 39。limt S bdS?? ?? (b) To see the behavior of E(t) and I (t) ， consider the equations for E and I ? ? ? ?? ? ? ?? ?/ 0 /39。39。 00d b b bq x b bEEdrII? ? ? ? ?????? ? ? ? ? ???? ? ? ??? ????? ? ? ?? ? ?? ? ? ??? ?? (16) 21 which is a perturbation of a linear system. The solutions to the principal part of (16) behave as claimed in the theorem， as do those for the perturbed system (16) since the perturbation decays exponentially as t→∞ ( see reference [10] ， Chapter 3， Theorem ) . Ttheorem 5 Assume that 0 1R? ， (a) the number of individuals E(t) decreases if R 2 1 and increases if R2 1. Moreover， the exponential asymptotic rate of increase ( decrease) is ? ?? ?239。l im 1t E dRE ??? ? ? ? (17) (b) the number of individuals I (t) decreases if R2 1 and increases if R 2 1. Moreover， the exponential asymptotic rate of increase ( decrease) is ? ?? ?239。l im 1t I dRI ??? ? ? ? (18) Proof If 0 1R? ， from Theorem 3， we have ? ? ? ? ? ?? ? ? ?* * *l im , , , ,t x t y t z t x y z?? ?and ? ? ? ?? ?* * ***x bq z b yy b r z????? ? ? ???? ? ??? (19) Hence， **l im l imttI z zE y y b r? ?? ? ? ?? ? ? ?? (20) From the second equation in (1) and (19) ， (20) we obtain (17) . From the third equation in (1) and(20) we also obtain (18) . Acknowledgements We would like to thank the referees very much for their valuable ments suggestions. References： [1] 茍清明 .一類具有階段結(jié)構(gòu)和標(biāo)準(zhǔn)發(fā)生率的 SIS 模型 [J].西南大學(xué)學(xué)報(bào) ： 自然科學(xué)版 ， 2020， 29(9)： 613. [2] 茍清明 ， 王穩(wěn)地 .一類有遷移的傳染病模型的穩(wěn)定性 [J].西南師范大學(xué)學(xué)報(bào) ： 自然科學(xué)版 ， 2020， 31(1)： 1823. [3] Li M Y， Muldowney J S. A Geometric Approach to GlobalStability Problems [J]. SIAM J Math Anal， 1996， 27(4)： 10701083. [4] Thieme H R. Persistence Under Relaxed PointDissipativity (with an Application to an Endemic Model) [J]. SIAM J Math Anal， 1993， 24(2)： 407435. 22 [5] Driessche P Van den， Watmough J. Reproduction Numbers and SubThreshold Endemic Equilibria for Compartmental Models of Disease Transmission [J]. J Math Biosci， 2020(180)： 2948. [6] Busenberg S， Cooke K. Vertical Transmission Diseaes， Models and Dynamics [M]. Berlin： SpringerVerlag， 1993. [7] Li M Y， Smith H L， Wang L. Global Stability of an SEIR Model with Vertical Transmission [J]. SIAM J APPLMATH， 2020， 62(1)： 5869. [8] Li M Y， Wang L. A Criterion for Stability of Matrices [J].J Math Anal Appl， 1998(225)： 249264. [9] Muldowney J Matrice and Ordinary Differential Equations [J].Roky Moun J Math， 1990， 20(4)： 857872. [10] Hale J Differential Equations [M].New York： WileyInterscience， 1969： 296297. [11] Martin