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J Theor Biol. 2014 Sep 7;356:123-32. doi: 10.1016/j.jtbi.2014.04.020. Epub 2014 Apr 23.

Modeling rates of infection with transient maternal antibodies and waning active immunity: application to Bordetella pertussis in Sweden.

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  • 1Department of Mathematics, Purdue University, West Lafayette, IN, USA.
  • 2Centers for Disease Control and Prevention, Atlanta, GA, USA. Electronic address:
  • 3Centers for Disease Control and Prevention, Atlanta, GA, USA.
  • 4Swedish Institute for Communicable Disease Control(5), Solna, Sweden.
  • 5Scientific Advice Unit, European Centre for Disease Prevention and Control, Solna, Sweden.


Serological surveys provide reliable information from which to calculate forces (instantaneous rates) of infection, but waning immunity and clinical consequences that depend on residual immunity complicate interpretation of results. We devised a means of calculating these rates that accounts for passively acquired maternal antibodies that decay or active immunity that wanes, permitting re-infection. We applied our method to pertussis (whooping cough) in Sweden, where vaccination was discontinued from 1979 to 1995. A national cross-sectional serosurvey of antibodies to pertussis toxin, which peak soon after infection and then decay, was conducted shortly after vaccination resumed. Together with age-specific contact rates in Finland, contemporary forces of infection enable us to evaluate the recent assertion that the probability of infection upon contact is age-independent. We find elevated probabilities among children, adolescents and young adults, whose contacts may be more intimate than others. Products of contact rates and probabilities of infection permit transmission modeling and estimation of the intrinsic reproduction number. In contrast to another recent estimate, ours approximates the ratio of life expectancy and age at first infection. Our framework is sufficiently general to accommodate more realistic sojourn distributions and additional lifetime infections.

Published by Elsevier Ltd.


Antibodies to pertussis toxin; Catalytic modeling; Force of infection; Reproduction number; Waning immunity

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