【正文】 9。 2039。2 , 2 ,z yzB b B r b kyz? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?這里 min( , )k ??? 由引理 2知，當(dāng) 0tT? 時(shí)，有 ? ?12 zB x bq y???，21 yB z??. 因此，當(dāng) 0tT? 時(shí)， ? ? ? ?1 1 1 1 1 2 2 zg B B z b x b q y? ? ? ? ?? ? ? ? ? ? ? ? ? (11) ? ?2 2 1 1 2 2 39。yz brzz? ?? ? ? ? (14) 8 將 (13)代入到 (11)， (14)代入 (12)，我們可得 ， 當(dāng) 0tT? 時(shí)， 1239。,39。 .SS bN I bqI dS s rINsE I bqI dE ENI E dI I rIN bN dN??????? ? ? ? ? ? ???? ? ? ? ???? ? ? ??? ??? (15) 顯然 ， 人口總 量 N(t)可能增加、減少或?yàn)槌?shù)，其完全依賴于增長(zhǎng)率 r=b?d. 比例 (x ， y ， z)可能趨向于 ,0,0bb ????????或地方病平衡點(diǎn) ? ?*, *, *x y z ，但是感染者比例的變化并不能給提供我們關(guān)于染病者 (包括 E類和 I類 )行為變化的信息 .特別地，即使被感染的個(gè)體的總數(shù)呈指數(shù)增加 ， 但以比人口總量 ??Nt增長(zhǎng)率低，那么這兩者所占的比重將趨向于零，然而，被感染個(gè)體的總數(shù)趨于無(wú)窮 .我們也能夠想象出相反的情況 —— 感染者的數(shù)量和人口總量的下降到零 ， 但是二者比例一直保持 (非零 )常量不變 .在這種情況下 ， 只要人口總量非零，就一定存在感染者 .為了描述 ? ? ? ? ? ?,S t I t E t 9 的變動(dòng)情況，我們需要另外兩個(gè)閾值參數(shù) (文獻(xiàn) [12]中有介紹 ).以下是相關(guān)的閾值參數(shù)： ? ?? ?? ? ? ? ? ?? ?? ? ? ?02 *0, 1 , 1.b bq bRb d d rRx bqRd b r? ? ?? ? ?????? ???? ? ? ? ???? ?? ?? ? ? ?? 我們得到以下兩個(gè)定理 . 定理 4 (a) 易感者的數(shù)量 ??St以指數(shù)漸近率 b?d增加 (減小 ). (b) 假設(shè) 0 1R? ，如果 2 1R? 或 2 1R? ，則 ? ? ? ?? ? ? ? ? ?, 0 , 0 ,E t I t o r? ? ?. 證明 (a)由定理 2知 ， 0 1R? 意味著 ? ? ? ? ? ?? ?l im , , , 0 , 0tbx t y t z t b ?????? ?????由 (1)的第一個(gè)方程 ， 我們有 ? ?39。39。 whereas if 0 1R? ， the unique endemic equilibrium is globally asymptotically stable. Key words： epidemic model。,39。39。39。if R0 1， a unique endemic equilibrium ? ?* * * *,E x y z? ?V( the interior of V ) exists. Theorem 2 The diseasefree equilibrium 0 , 0, 0bE b ???? ?????of ( 3) is globally asymptotically stable in V if 0 1R? 。bx b ?? ? ； whereas， if 16 bx b ?? ? ， then y=z=0 in V. Therefore， the largest pact invariant set in is the singleton? ?0E .The global stability o f 0E w hen 0 1R? 時(shí)， follow s from the LaSalles invariance principle[ 10] . If 0 1R? ， then 39。0bbx b q rb????? ??? ? ? ?????? when 0 xx?? It follow s that ??x t x? for all larget. The conclusion of the lemma now follows. Theorem 3 Assume that 0 1R? . T hen the unique endemic equilibrium *E is globally asymptotically stable in V. Proof From the discussion in section 1 and Lemma 1， we see that system ( 3) satisfies the assumptions? ?1H and ? ?2H . The Jacobian matrix J associated with a general solution ? ? ? ? ? ?? ?,x t y t z tto ( 3) is 00b z b q rz b x b qbr? ? ?? ? ???? ? ? ? ? ?????? ? ?? ? ??? Its second additive pound matrix is： ? ?2 2 2002z b x b q x b q rJ z r bz r b? ? ? ? ?? ? ? ?? ? ?? ? ? ? ? ? ?????? ? ? ? ? ?? ? ? ??? (9) For detailed discussions of pound matrix and their properties we refer the reader to reference [9] . Set the function P (x ) in ( 8) as ? ?( , , ) 1 , / , /P x y z d ia g y z y z? ， then ? ?1 0 , 39。/fP P dia g y y z z y y z z? ? ? ? and the matrix ? ?211fB P P P J P???? in ( 8) can be written in block form 11 1221 22BBB BB??????? where 11 2B z b? ? ?? ? ? ? ? ? ? ? ?12 ,zzB x b q x b q ryy????? ? ? ?????， 210yB z??????????， 18 2239。 2yz z r byzByzz r byz? ? ?? ? ???? ? ? ? ? ????? ? ? ? ??? The vector norm? in 3 32RR??? ???? is chosen as ? ? ? ?, , m a x ,u v w u v w??. Let μ(.) denote the Lozinski measure with respect to this no rm. U sing the method of estimating μ(.) in reference [ 11] ， we have ? ? ? ?12sup ,B g g? ? (10) where ? ? ? ?1 1 1 1 1 2 2 2 1 1 2 2,g B B g B B??? ? ? ? ? ?1 22B? denote the Lozinski measure with respect to the 1l norm in 2R . Since 11B is scalar， its Lozinski measure with respect to any norm in R1 is equal to 11B . 12B ， 21B are matrix norms with respect to 1l vector norm. therefore， ? ? ? ?1 1 2 2 39。 2y z yg B B r b ky z z???? ? ? ? ? ? ? ? ? (12) Rewriting ( 3) ， we have ? ? 39。,yyg b g b? ? ? ? ? ? Therefore ? ? 39。,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。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。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： ? ?? ?? ?? ??