Table 1.
Parameter values and descriptions.
Time units for each parameter are either time in days (d), the inverse of time (d-1) or unitless (-).
Fig 1.
Pathogen reproductive number, Rt, plotted against the ratio of contacts needing tracing to contact tracing capacity for variable delays (1/τIs) of 1–5 and 10 days between case symptom onset and the start of contact tracing (including getting tested and receiving result).
With testing, but no contact tracing, Rt increases 35% from 1.7 to 2.3 as the delay 1/τIs increases from 1 to 5 days, which is evident in the y-axis difference between black and green curves in the upper right of the graph where new case burdens are so high contact tracing is ineffective. The delays (1/τIs) are indicated by the small numbers on each curve in the left of the plot. Curves are horizontal where capacity exceeds contacts needing tracing. The number of contacts per case, Ncpc, was 10.
Fig 2.
Pathogen reproductive number, Rt, plotted against the number of cases per contact tracer calls per day, for four different numbers of contacts per case (5, 10, 20, 30; these reflect the range of contacts before and during restrictions on social gatherings [39,45,46]).
The number of contacts per case is indicated by the small numbers on each curve in the middle of the plot. The average delay between symptom onset and contact tracing (including getting tested and receiving result), 1/τIs, is set to 5 days; as a result, the green curve is identical to the green curve in Fig 1.
Fig 3.
Reduced contact tracing efficiency with increasing cases leads to accelerating epidemics.
Top panels (A, B, C) show the number of susceptible, infected (latent, pre-symptomatic and symptomatic combined), and recovered individuals. Bottom panels (D, E, F) show the reproductive number, Rt, over time. Left most panels (A, D) show dynamics with no contact tracing but social distancing (κ = 0.58) set to give same initial R0 (1.35) as with contact tracing. Middle panels (B, E) show dynamics with effectively unlimited contact tracing (1500 contact tracers making 12 calls/day; 10 contacts per case) but no social distancing (κ = 1), with an identical value of R0 as in panels A, D. Right panels (C, F) show dynamics with the same parameter values as (B, E) except with limited contact tracing (15 contact tracers). R0 is the same value as in panels D and E (R0 = 1.35), but Rt increases as cases increase and contact tracing becomes inefficient, which overwhelms the decrease in the fraction of the population that is susceptible. In all panels, the delay from symptom onset to receiving test results, 1/τIs, is 5d. All populations start with 100,000 individuals. Note the identical epidemic sizes (final fraction susceptible 0.53) for panels A (social distancing) and B (effectively unlimited contact tracing), but much larger epidemic size for limited contact tracing in panel C (final fraction susceptible 0.14).
Fig 4.
Variability in the timing and outcome of epidemics due to stochastic variation in individual transmission.
Lines show number of latently infected individuals in the E class over time for 1 year with moderate social distancing that reduces contact rates by 40% (κ = 0.6). Grey lines show runs from a single stochastic simulation and the black line shows the deterministic outcome. The fraction of epidemics that establish is the fraction of simulations where the maximum number of people infected at any time exceeds the starting number infected. The four scenarios shown include different starting numbers of latently infected individuals on day 0, E0 (A, C: 5; B, D: 50), and with (A, B) or without (C, D) contact tracing (CT) which lowered R0 from 1.57 to 1.33. The delay from symptom onset to testing and tracing 1/τIs was 10d. The modeled population of 100,000 people had 15 tracers making 12 calls/day, and each case had an average of 10 contacts which is intermediate between pre-lockdown and lockdown conditions; this scenario is the same as the yellow line in Fig 1.