OSCILLATIONS IN AN ENDEMIC MODEL
FOR PERTUSSIS

HERBERT W. HETHCOTE

Abstract. An epidemiologic model for the spread of pertussis (whooping cough) is built from a demographic model with an equilibrium age distribution. Computer simulations of the mathematical model are used to study pertussis in the United States. Both before and after vaccination began in 1940, the reported incidence of pertussis has been cyclical with a period of 3 to 4 years. Because a seasonal contact rate in models for measles leads to subharmonic oscillations with a period of two years, it is plausible that a seasonal contact rate could produce similar multiyear oscillations in the pertussis model. But this does not happen, since the resulting oscillations in incidence have a 1-year period. However, yearly stochasticity in the contact rate does lead to irregular oscillations in the pertussis simulation model with a period of about 4 years. Thus stochasticity is a new possible mechanism leading to multiyear oscillations.