On final and peak sizes of an epidemic with latency and effect of behaviour change

In this paper, we use the renewal equation approach to explore the impact of behaviour change and/or non-pharmaceutical interventions (NPIs) on the final size and peak size of an infectious disease without demography. To this end, we derive the renewal equations (REs) for the force of infection (both instantaneous and cumulative) that have reflected the NPIs and/or behaviour change by the notion of practically susceptible population. A novelty in these REs is that they contain time-varying kernels arising from the incorporation of effect of behaviour change. We then build the new REs into the Kermack-McKendrick model to obtain a general full model. Following Breda et al. (J Biol Dyn 6(sup2):103-117, 2012) and Diekmann et al. (Proc Natl Acad Sci USA 118(39):e2106332118, 2021), we analyze this new model to derive a general formula for the final size relation, by which we find that the final size relation depends not only on the basic reproduction number [Formula: see text] but also on other associated values that reflect the impact of behaviour change. Specifically, we demonstrate that behaviour change can reduce the infection peak and herd immunity threshold in some specific models.