  Facing a global epidemic of new infectious diseases such as COVID-19,
non-pharmaceutical interventions (NPIs), which reduce transmission rates
without medical actions, are being implemented around the world to mitigate
spreads. One of the problems in assessing the effects of NPIs is that different
NPIs have been implemented at different times based on the situation of each
country; therefore, few assumptions can be shared about how the introduction of
policies affects the patient population. Mathematical models can contribute to
further understanding these phenomena by obtaining analytical solutions as well
as numerical simulations. In this study, an NPI was introduced into the SIR
model for a conceptual study of infectious diseases under the condition that
the transmission rate was reduced to a fixed value only once within a finite
time duration, and its effect was analyzed numerically and theoretically. It
was analytically shown that the maximum fraction of infected individuals and
the final size could be larger if the intervention starts too early. The
analytical results also suggested that more individuals may be infected at the
peak of the second wave with a stronger intervention. This study provides
quantitative relationship between the strength of a one-shot intervention and
the reduction in the number of patients with no approximation. This suggests
the importance of the strength and time of NPIs, although detailed studies are
necessary for the implementation of NPIs in complicated real-world environments
as the model used in this study is based on various simplifications.
