Renewal equations for delayed population behaviour adaptation coupled with disease transmission dynamics: A mechanism for multiple waves of emerging infections

There are many plausible reasons for recurrent outbreaks of emerging infectious diseases. In this paper, we develop a mathematical model to illustrate how population behavioural adaption and adaptation implementation delay, in response to the perceived infection risk, can lead to recurrent outbreak patterns. We consider the early phase of an infection outbreak when herd immunity is not reached, pathogen mutation is not considered, and seasonality is ruled out as a major contributor. We derive a transmission dynamics model coupled with the renewal equation for the disease transmission effective contacts (contact rate per unit time multiplied by the transmission probability per contact). The model incorporates two critical parameters: the population behavioural adaptation flexibility index and the behavioural change implementation delay. We show that when the behavioural change implementation delay reaches a critical value, the number of infections starts to oscillate in an equilibrium that is determined by the population behavioural adaptation flexibility. We also show that the numbers of infections at the subsequent peaks can exceed that of the first peak. This was an oblique observation globally during the early phase of the COVID-19 pandemic before variants of concern emerged, and it was an observed phenomena with the Omicron variant induced wave in areas where early interventions were successful in preventing the large outbreaks. Our model and analyses can provide partially explanation for these observations.