Discontinuous epidemic transition due to limited testing

Abstract

High impact epidemics constitute one of the largest threats humanity is facing in the 21^st century. In the absence of pharmaceutical interventions, physical distancing together with testing, contact tracing and quarantining are crucial in slowing down epidemic dynamics. Yet, here we show that if testing capacities are limited, containment may fail dramatically because such combined countermeasures drastically change the rules of the epidemic transition: Instead of continuous, the response to countermeasures becomes discontinuous. Rather than following the conventional exponential growth, the outbreak that is initially strongly suppressed eventually accelerates and scales faster than exponential during an explosive growth period. As a consequence, containment measures either suffice to stop the outbreak at low total case numbers or fail catastrophically if marginally too weak, thus implying large uncertainties in reliably estimating overall epidemic dynamics, both during initial phases and during second wave scenarios.

Fig. 1. Spatial epidemic model with testing and quarantining.

The base model is illustrated on a square grid. Every day each infectious individual (agents are represented here as a tile on a lattice) interacts with their neighbors and with a randomly selected individual, and transmits the disease (arrows in figure) with constant probability if the individuals they interact with are susceptible. The tiles with yellow and white stripes denote the potential contacts that can be exposed. Upon identification of a positive case (red tile) all the neighbors are put into quarantine and tested (blue dashed border). Weak-symptom cases (brown tile with blue dashed border) can only be identified if they are neighbors of a known positive case.

Fig. 2. Discontinuity in flattening of epidemic curves.

a Daily new cases for continuously decreasing values of R₀ (3 > R₀ > 0), mimicking mitigation measures of increasing strength. Testing and quarantining are carried out at the same time with a capacity limit of N_T = 1000 tested individuals per day. Initially the peak reduces continuously in response to mitigation (red curves from left to right), however, once R₀ is reduced below 2.5 the epidemic curve drops to very small numbers of new cases (curves not visible in the figure scale). b Daily new cases for decreasing values of R₀ (1.5 > R₀ > 0) without testing and quarantining. Decreasing progressively R₀ gradually flattens the epidemic curve (blue curves from left to right) until a very low number of cases is reached (curves not visible in the figure scale). c Final fraction of infected (N_F/P) as a function of R₀ corresponding to the curves shown in a (red dots) and b (blue dots). The black dots denote outbreaks that have been effectively suppressed. When testing and quarantining are active the epidemic transition becomes discontinuous and happens at a higher basic reproduction number (R₀ ≈ 2.5) in comparison to the usual continuous epidemic transition observed at R₀ = 1. d Evolution of the reproduction number with testing and quarantining active. Such measures efficiently reduce the reproduction number R_t below unity for R₀ < 2.3 (black dots). For larger values of R₀ testing and quarantining can initially reduce the reproduction number to a constant level, however, R_t remains above one (red dots). Owing to the continuing spread the number of suspects will eventually exceed the daily test limit and hence Δ_Test changes sign (red dashed line in d). At this point the spread accelerates and R₀ increases. The values of R_t have been averaged over 400 and 100 simulations for R₀ = 2.3 and R₀ = 2.7, respectively. In all these cases the population size is P = 3162 × 3162 ≈ 10⁷ people and epidemics start with 100 initial infectious.

Fig. 3. Period of faster than exponential growth during the second wave of COVID-19 in Italy.

a Number of new daily cases per million (logarithmic axis) reported in Italy. The red dotted lines indicate time intervals over which the new cases double, while the black dashed line represents an exponential fit to the data from day 180 to day 200. After a period of relatively weak exponential growth (days until 200) new cases surge suddenly leading to an intermediate phase of faster than exponential growth similar to what is observed in our simulations (cf. Supplementary Fig. 3). b The positivity rate, the fraction of tests that resulted positive in a given day, during the same time interval. As the number of new cases rapidly increases the contact tracing and testing strategy is put under strain and the positivity rate starts to increase. In the inset we compare the daily increment of positivity rate measured in Italy (blue curve) with the one predicted by a simulation (red curve). In both panels the data represent a 7-days moving average. The simulation parameters are the same ones used to generate the red curve of Fig. 2d.

Fig. 4. Discontinuous nature of lockdown scenarios.

Cumulative total number of cases for R₀ = 2 starting from 10⁴ infectious agents in a population P = 3162 × 3162 ≈ 10⁷ and subjected to gradually stronger lockdowns. Mitigation measures are simulated by a reduction in the basic reproduction number in the range of 0 < ΔR₀ < 2.0. The duration is 30 days in all scenarios. Testing here is limited to N_T = 1000 individuals per day. Mild interventions (0 < ΔR₀ < 0.5, red curves) only result in an initial drop (see inset) in daily new cases but ultimately cannot prevent a subsequent rise in numbers and eventually a high proportion of the population becomes infected. Stronger interventions (0.5 < ΔR₀ < 2, blue curves) on the other hand efficiently bring the epidemic under control. Inset, corresponding epidemic curves of daily new cases. Reducing continuously R₀ during lockdown produces a family of epidemic curves that ultimately result in a discontinuous outcome: Either the outbreak is suppressed (blue curves), or containment fails catastrophically leading to a high proportion of the population being infected (red curves).