英语原文Schistosomiasis models with density dependenceand age of infection in snail dynamicsZhilan Feng *, Cheng-Che Li, Fabio A. MilnerDepartment of Mathematics, Purdue University, West Lafayette, IN 47907-1395, USAReceived 31 December 2000; received in revised form 6 September 2001; accepted 28 September 20011.IntroductionIn modeling our environment one of the most difficult choices we haveto make is the amount of detail we are willing to put into the model or,rather, the amount of detail we are capable of modeling due to our limitedability to gather information. This decision is especially crucial whentrying to describe biotic interactions, and it requires a delicate balancebetween the need for enough detail to resolve important characteristicsof the dynamics of the interactions and our ability to obtain data insufficient detail to use the model.When modeling any ecosystem in some detail, one should begin with asimple model. If this model captures the essential features of theinteractions being modeled, simulations should reflect those aspects ofreallife behavior. Hopefully, a simple model can be thoroughly analyzedand then built upon to derive increasingly more complex models. At eachstage these require the same type of validation concerning the newfeatures being added and the extent to which the numbers predicted by themodel compare with real data.Several studies have addressed the dynamics of schistosomiasis andother helminth infection of humans 14,9,10,1315,18,19,25,26 usingsystems similar to the one we are interested in. We shall consider hereSchistosoma mansoni, a human blood fluke which causes schistosomiasis.The freshwater snail Biomphalaria glabrata serves as the main interme-diate host. Anderson and May1 introduced models for macroparasite-host interactions when the parasites have direct lifecycles involvingonly a single host population and one stage of parasites. In 19 afree-living stage of the parasite was considered in the model. Inter-actions between schistosome infection and molluscan intermediate hosts(snails) were studied in 2. Multiple stages of parasites and two hostpopulations were considered in 10 which, however, assumes a constantpopulation size of the human host and no density dependence and age ofinfection in the snail population. Campaigns against S.mansoni frequentlyfocus on treatment of infected humans with Praziquantal or other drugsthat kill the parasites in the treated humans. Mathematical models havebeen used to assess community chemotherapy programs for schistosomiasisthrough simulations 6,7. These models also do not include explicit snaildynamics. We shall introduce in this paper models with more features describing the dynamics of schistosomes, snails, and humans; we shall study their qualitative and quantitative mathematical properties and deduce biologi-cal and ecological consequences. In particular, we shall model treatmentof humans and establish an explicit treatment rate threshold above whichparasites will die out or the infection in the population will remain belowa certain level. We shall also study the sensitivity of the mean parasiteload per human host to the changes in the two transmission rates: the mansnail transmission rate and the snailman transmission rate. We shallintroduce an age structure in the class of infected snails,since theircercarial production seems to be periodic in time and there is a prepatentperiod after initial infection (see Fig. 1, 21). The manschistosomeinteraction is modeled as a macroparasite infection and we assume anegative binomial distribution for the parasite distribution among thehuman hosts as in 1. The population sizes of both human and snail hostsare variable by allowing disease-induced mortality.We shall also compare the prediction of the new models with that ofa simpler model proposed earlier by other authors 25,26, to which ourmodel reduces if we assume no parasiteinduced additional mortality inhumans as well as constant cercarial production by infected snails. Weshall show that this mean parasite load does not depend linearly on thetransmission rate from snail to human as the simpler model predicts, butrather on the square root of this transmission rate. We shall also show(numerically) that the new model may produce a bifurcation at which theunique endemic equilibrium changes its stability and stable periodicsolutions exist.Fig. 1. This graph shows how the number of cercariae released by a snailchanges with time since infection. The prepatent period is about 35 daysafter initial infection. After that the number fluctuates with a periodof about 30 days until the snail dies.汉语翻译1.引言在模拟环境中，我们不得不做出的一个最艰难的抉择之一是，由于我们收集数据的能力有限，我们将投入大量细节到模型中，或者相反，我们只能掌握我们力所能及的模型。当试图描述生物间相互作用的时候，这个抉择就显得尤为重要了，它需要有一个微妙的平衡，即对于尽力解决相互作用间动力主要特征的需要同我们为了使用模型而尽可能获取数据能力之间的平衡。当我们在某些细节上模拟任何生态系统时，起初都应该从一个简单的模型开始。如果这个模型捕捉到了模拟过的相互作用的基本功能，模拟就应该反映现实生活习惯中的那些方面。我们希望，一个简单的模型能够用来深入分析，然后建立在提取更加复杂的模型之上。在每一阶段，这些都需要被添加新的功能的相同类型