Epidemic interventions: insights from classic results

Abstract

Analytical expressions and approximations from simple models have performed a pivotal role in our understanding of infectious disease epidemiology. During the current COVID-19 pandemic, while there has been proliferation of increasingly complex models, still the most basic models have provided the core framework for our thinking and interpreting policy decisions. Here, classic results are presented that give insights into both the role of transmission-reducing interventions (such as social distancing) in controlling an emerging epidemic, and also what would happen if insufficient control is applied. Though these are simple results from the most basic of epidemic models, they give valuable benchmarks for comparison with the outputs of more complex modelling approaches. This article is part of the theme issue 'Modelling that shaped the early COVID-19 pandemic response in the UK'.

Keywords: SIR model; epidemic; non-pharmacutical interventions; pandemic.

Figure 1.

Baseline peak prevalence and final size, as functions of the reproduction ratio, R₀. Peak prevalence (a) and final size (b) against reproduction ratio R₀, both given as a proportion of the population. The dotted line on the final size plot gives the susceptible proportion when peak prevalance is achieved, which is also the herd immunity threshold. (Online version in colour.)

Figure 2.

Effect of intervention, by strength of intervention. The relative peak prevalence (a) and relative final size (b) as function of the transmission reduction of the intervention (ϕ) for R₀ = 3. In each, the different coloured lines represent different trigger values for application of the intervention θ.

Figure 3.

Effect of intervention, by strength and timing of intervention. The relative peak prevalence (a) and relative final size (b) as function of both the transmission reduction of the intervention (ϕ) and trigger for application of the intervention (θ) for R = 3. For the peak prevalence plot (a), the additional dotted red and blue lines separate the different cases for the timing of peak incidence relative to intervention timing: below the red line, cases are rising at the time intervention is applied and the intervention is not strong enough to immediately turn over the epidemic; between the red and blue lines, the intervention is enough to immediately turn over the epidemic (and hence the contours are horizontal in this region); above the blue dotted line, the epidemic has already peaked before intervention is applied, so the intervention does not affect the peak prevalence.

Figure 4.

Effects of intervention, matched to values seen in SAGE meeting 10. The percentage reduction in final size (a) and peak prevalence (b) as function of the transmission reduction of the intervention (now also as percentage) for R = 2.0, 2.2 and 2.4, chosen for comparison to table 1 in Imperial College paper [1].

Figure 5.

Effect of prior immunity, by strength of intervention. The relative peak prevalence (a) and relative final size (b) as function of the transmission reduction of the intervention (ϕ) for R = 3, assuming intervention applied as soon as the epidemic starts. In each, the different coloured lines represent different levels of prior immunity.

Figure 6.

Effect of intervention, by strength of intervention and level of prior immunity. The relative peak prevalence (a) and relative final size (b) as function of both the transmission reduction of the intervention (ϕ) and level of prior immunity for R = 3, assuming interventions are applied as soon as the epidemic starts. Here, interventions are applied quickly, so they entirely stop any subsequent epidemics if they are strong enough. Some prior immunity means that a lower strength of intervention will be sufficient to entirely stop a subsequent outbreak. Even an imperfect intervention may be enough to dramatically reduce the outbreak with a modest level of prior immunity.