Analytical estimation of maximum fraction of infected individuals with one-shot non-pharmaceutical intervention in a hybrid epidemic model

TL;DR

This paper analytically investigates how a single, one-time non-pharmaceutical intervention affects the maximum infection rate and epidemic size in a simplified SIR model, revealing counterintuitive effects of early intervention and strength of measures.

Contribution

It provides an analytical framework for understanding the impact of a one-shot NPI on epidemic dynamics, highlighting the importance of timing and strength of interventions.

Findings

Early intervention can lead to larger maximum infected fractions.

Stronger interventions may increase infections at the second wave peak.

Analytical relationships between intervention strength and epidemic size are established.

Abstract

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.