Fig 1.
Schematic of an SEIR model with diagnosis described by testing and contact-tracing.
SEIR is a compartmentalised model describing susceptible (S), exposed (E—infected but not infectious), infectious (I) and removed (R) population cohorts. Individuals move between these compartments in sequence as they become exposed, infected and infectious during disease progression until recovery. The novelty here is that each compartment comprises diagnosed and undiagnosed individuals with diagnosis leading to isolation. We assume that diagnosis happens through testing or putatively through tracing. Tracing is mediated through contact, and the intersection with CI represents contact with an infectious individual. Non-infectious individuals having been isolated through contact tracing have, in effect, been misdiagnosed. Individuals transition between compartments X and Y at rates ΔX→Y which we derive in the text.
Fig 2.
The effect of testing and isolation alone in a hypothetical population.
The dynamics represented here are for a scenario with normal contact, c = 13, and an initial number of infected individuals, I(0) = 100, 000. Individuals who test positive are isolated for the duration of their illness. The top plot shows the total infections (exposed and infectious individuals) over time for various testing rates ranging from none, θ = 0, to testing all infectious individuals every two days, θ = 0.55. The bottom plot shows the reproduction number over time for these same scenarios. Observe that even fairly frequent testing, e.g every five days, θ = 0.2, this is only sufficient to reduce peak infections by one order of magnitude from about 20 million to about two million. In the infrequent testing regimes, θ ∈ [0.05, 0.25], we can also observe that the curve described by Re(t)R(t) is not a sigmoid but instead first falls to a value above R(t) = 1 before stabilising and then falling again. This is because though testing and isolating does have an effect at those rates, it is not sufficiently frequent to identify all of those who are infectious.
Fig 3.
Reproduction number after 30 days for various values of the contact rate, c and the testing rate, θ.
The red line is is given by the equation . As above,
and γ = 0.1429.
Fig 4.
The effect of testing, tracing in a hypothetical population.
The dynamics presented here are the same as those of Fig 2 with a testing rate θ = 14−1days−1, meaning testing of infectious individuals on average once per two weeksonce per week. The rate of contact tracing is set at χ = 2−1days−1, meaning that it takes on average two days to trace a contact. A variety of values of tracing success rate, η are explored. Under these conditions, even a modest success rate of 30-40% enhances testing and results in a maximum of infectious individuals that is about half the magnitude with testing alone. The lower panel is, as above, the corresponding time-series for the reproduction number.
Fig 5.
Reproduction number after 30 days for various values of the testing rate, θ and the contact tracing success η.
Fig 6.
Comparison of differential equation and agent-based model.
Here the population size is 10000 individuals with 100 initially infected. The testing rate is θ = 7−1days−1, and tracing rate and success probability are χ = 0.5, η = 0.5. The plots show the time-series for each compartment. The top row are the compartments representing unconfined individuals, those in SU, EU, IU and RU. The bottom row are those representing isolated—diagnosed or distanced—individuals, SD, ED, ID, RD. The heavy orange curves are the output of the ODE-based simulation. The teal curves are the average output of the agent-based simulation, with envelopes for one and two standard deviations.
Fig 7.
Pathological parameter values.
This plot shows the effect of very low levels of testing, θ = 50−1days−1. In this circumstance, the number of traceable susceptible individuals takes on unphysically high values, shown by the red line in the top left panel. This results in an overestimation of the maximum number of unconfined exposed and infectious individuals and a corresponding underestimation of the effect of contact tracing in preventing infection in this scenario.
Fig 8.
Comparison of differential equation and agent-based model with correlated transitions.
Here the population size is 10000 individuals with 100 initially infected. The testing rate is θ = 7−1days−1, the base testing rate is θ0 = 2θ, and tracing rate and success probability are χ = 0.5, η = 0.5. The plots show the time-series for each compartment. This choice of parameter values is such that, because the transitions are strongly correlated, individuals are only tested at the end of their infectious period and are effectively never isolated. The top row are the compartments representing unconfined individuals, those in SU, EU, IU and RU. The bottom row are those representing isolated—diagnosed or distanced—individuals, SD, ED, ID, RD. The heavy orange curves are the output of the ODE-based simulation. The violet curves are the average output of the agent-based simulation with correlated transitions, with envelopes for one and two standard deviations.
Fig 9.
Comparison of differential equation and agent-based model with correlated transitions.
As in Fig 8, the population size is 10000 individuals with 100 initially infected. The testing rate is θ = 2.5−1days−1, the base testing rate is θ0 = θ, and tracing rate and success probability are χ = 0.5, η = 0.5. This choice of parameters exhibits a good match between the models because the testing rate, though strongly correlated, is faster than the recovery rate. The plots show the time-series for each compartment. The top row are the compartments representing unconfined individuals, those in SU, EU, IU and RU. The bottom row are those representing isolated—diagnosed or distanced—individuals, SD, ED, ID, RD. The heavy orange curves are the output of the ODE-based simulation. The violet curves are the average output of the agent-based simulation with correlated transitions, with envelopes for one and two standard deviations.
Table 1.
Parameters used in the SEIR-TTI model.