Pertussis is well controlled in the United States by routine childhood immunization. In contrast, this disease is endemic and epidemic in Germany because routine immunization has not been implemented. To gain information relating to the epidemiology of Bordetella pertussis infections, we examined the prevalence and magnitude of B. pertussis agglutinins and of IgG and IgA antibodies (detected by enzyme-linked immunosorbent assay) to four B. pertussis antigens--lymphocytosis-promoting factor, filamentous hemagglutinin, pertactin, and fimbriae-2--in the sera of 119 American university students and 119 German military recruits of similar age. Geometric mean titers of agglutinins and geometric mean values for IgG antibodies to the four antigens were two- to threefold higher in sera from the American students than in sera from German recruits. In contrast, the geometric mean IgA values and the percentage of subjects with detectable IgA antibodies to the four antigens were similar in the two populations. Since IgA antibody results mainly from infection and not from immunization, our data suggest that B. pertussis infections are common among both American and German young adults despite the marked difference in rates of clinical pertussis in the two countries.

We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertex-edge random graphs. Edge random graphs are Erdős-Rényi random graphs [8,9], vertex random graphs are generalizations of geometric random graphs [20], and vertexedge random graphs generalize both. The names of these three types of random graphs describe where the randomness in the models lies: in the edges, in the vertices, or in both. We show that vertex-edge random graphs, ostensibly the most general of the three models, can be approximated arbitrarily closely by vertex random graphs, but that the two categories are distinct.

ABSTRACT. We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertex-edge random graphs. Edge random graphs are Erdős-Rényi random graphs [8,9], vertex random graphs are generalizations of geometric random graphs [20], and vertexedge random graphs generalize both. The names of these three types of random graphs describe where the randomness in the models lies: in the edges, in the vertices, or in both. We show that vertex-edge random graphs, ostensibly the most general of the three models, can be approximated arbitrarily closely by vertex random graphs, but that the two categories are distinct.

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