Waning immunity and sub-clinical infection in an epidemic model: implications for pertussis in The Netherlands

In The Netherlands, an epidemic outbreak of pertussis took place in 1996–1997. Understanding of the causes of the epidemic is hampered by the fact that many cases of infection with Bordetella pertussis go by unnoticed, and by the fact that immunity against infection does not last lifelong. Motivated by these observations, we develop and analyze an age-structured epidemic model that takes these factors into account. A distinction is made between infection in immunologically naive individuals, and infection in individuals whose immune system has been primed before by infection or vaccination. While the former often lead to severe symptoms and thus are more often diagnosed and notified, the latter are largely sub-clinical. The main questions are: (1) to what extent do sub-clinical infections contribute to the circulation of B. pertussis; and (2) what might be the causes for the recent epidemic? To answer these questions, we first present a new method to estimate the force of infection from notification data. The method is applied to the 1988–1995 case notification data from The Netherlands. Estimates of the force of infection vary greatly, depending on the rate at which immunity is lost, and on the fraction of sub-clinical infections. For the 1988–1995 period, our analysis indicates that if immunity is lost at a small rate and if a majority of infections is sub-clinical, the contribution of infection in adults to the transmission process cannot be neglected. Our results furthermore indicate that a decrease in the duration of protection after vaccination due to a change in the pathogen is the most likely factor to account for the 1996–1997 epidemic.