外文翻译---带有垂直传染和接种疫苗seirs流行病模型的全局稳定性(编辑修改稿)

外文翻译---带有垂直传染和接种疫苗seirs流行病模型的全局稳定性(编辑修改稿)内容摘要：

， the disease free equilibrium is globally asymptotically stable。 whereas if 0 1R ， the unique endemic equilibrium is globally asymptotically stable. Key words： epidemic model。 vertical transmission。 vaccination。 global stability CLC number： O175 13 Document code： A In many epidemic models， one assumes that infectious diseases transmit in a population through direct contact with infectious host s， or through disease vectors such as mosquitos or other biting sects， Viz. horizontal transmission . But many infectious diseases spread through not only horizontal transmission but also vertical transmission. Vertical transmission can be acplished through transplacental transfer of disease aents， such as Hepat it is B， rubella， herpes simplex . Among insect s or plant s， Vertical transmission is often through eggs or seeds. Busenbergetal and Lietal discussed the problem of diseases horizontal transmission and vertical transmission. In the present paper， the disease that transmits both horizontally and vertically are considered. We assume that the population which has a exponential birth can be divided into four homogeneneous partment s： susceptible (S) ， exposed (E) ， infectious (I) and immune (R) . So the host total population          N t S t E t I t R t    . We assume that the disease is not fatal and that per capita nature birth rate and per capita death rate are denoted by parameters b and d respectively. We assume that new born infants from the exposed class enter the susceptible class， while a fraction q of new born infants from the infectious class is infected. Consequently， the birth flux into the exposed class is given by bqI with01q . For the infectious class， we assume that a pro portion of infectious host s acquire permanent immunity and enter R class， while a proportion r o f infectious host s have no immunity 13 and enter S class. Our model includes vaccination at a rate for susceptible individuals. Based on the above assumptions， the following differential equations are derived： 39。 ,39。 ,39。 ,39。 .sS bN I bqI dS S rINSE I bqI dE ENI E dI I rIR I I dR                 (1) Here， is the adequate contact rate， the parameterμis the transfer rate from the E class to I class. The parameters b， d， β， μare positive， θ， σ， r， rare nonnegative. Let x = S / N， y = E / N ， z = I / N and w = R / N denote the fraction of the classes S ， E ， I ， R in the population， respectively. It is easy to verify that x， y， z， w satisfy the following differential equations： 39。 39。 39。 39。 x b bx x z bq z x rzy x z bq z by yz y bz z rzw z x bw                 (2) subject to the restriction x + y + z + w = 1. Because the variable w does not appear in the first three equations of ( 2) . This allow s us to reduce ( 2) to a subsystem： 39。 39。 39。 x b bx x z bq z x rzy x z bq z by yz y bz z rz            (3) From biological considerations， we study ( 3) in the feasible closed region   , , : 0 , 0 , 0 , 1V x y z x y z x y z       (4) The dynamical behavior of ( 3) in V and the fate of the disease is determined by the basic reproduction number      0B q bR b b b r         (5) The objective of this paper is to show that the dynamical behavior of ( 3) is characterized by 0R . 14 1 Mathematical Framework We briefly outline a general mathematical framework to prove the global stability o f a system of ordinary differential equations， which is proposed in reference [ 3] . Let   nx f x R be a 1C function for x in an open set nDR . Let us consider the system of differential equations 39。 x f x ( 6) We denote by  00,xxthe solution to ( 6) such that  000,x x x . A set K is said to be absorbing in D for (6)， if x ( t ， K 1 ) K for each pact K 1 D and sufficiently large t. We make two basic assumptions： ( H1 ) There exists a pact absorbing set K⊂D. ( H2 ) ( 6) has a unique equilibrium x in D. The unique equilibrium x is said to be globally stable in D if it is locally stable and all trajectories in D converge to x. For epidemic models w here the feasible region is abounded cone， ( H 1 ) is equivalent to the uniform persistence of ( 6). Let  x P x be an 22nn         matrixvalued function that is 1C For x∈ D Assume that  1Px exists and is continuous for x|∈ K， the pact set. A quantity 2q is defined as    02 001l im s u p s u p ,t x Rq B x s x d st     (7)  211f fB P P P Px  (8) The matrix Pf is obtained by replacing each entry pij of P by its derivative in the direct ion of f， ( pij ) f ， and 2fx is the second additive pound matrix of the Jacobian matrix fx of f ， and μ(B)is the Lozinski measure of B with respect to a vector norm  in  89NR  . 15 The following global stability result is proved in Theorem of reference [ 3] . Theorem 1 Assume that D is simply connected and that assumptions (H 1 ) ， (H 2 ) hold. Then the unique equilibrium x of ( 6) is globally stable in D if 2q 0. It is show in reference [3] that under the assumptions of Theorem 1， the condition 2q 0 rules out the presence of any orbit that g iv e rise to a simple closed rectifiable curve that is invariant for ( 6) ， such as periodic orbit s， homoclinic or bits and heteroc。