Fig 1.
Average susceptibility decreases over exposure events in a heterogeneous population.
The figure depicts individuals infected and not infected over two exposure events in a heterogeneous population with more susceptible (red) and less susceptible (blue) individuals. The pie charts show the composition of the not infected population. Average susceptibility in the not infected population decreases after each exposure event as the highly susceptible individuals are infected more frequently than the lowly susceptible individuals. Note that if the population lacked heterogeneity in susceptibility, all individuals would be either red or blue, and thus, susceptibility would not change.
Fig 2.
The figure summarizes the steps of our method from simulating contact tracing data to detecting heterogeneity in susceptibility to estimating the level of heterogeneity and predicting disease dynamics. The diagram depicts a toy example going through the steps of our method. At step 7, we show the log-likelihoods in terms of the R code that would be used to calculate them. With real contact tracing data, the simulation section would be skipped.
Table 1.
Descriptions of all parameters used in our method.
The parameters’ values are either set by input data, assumed, calculated from other parameters, or estimated via MCMC, which is specified in the Source column. Parameters in the first section are used universally across the cases, those in the second section are used for the discrete case, and those in the third section are used for the continuous case. Note that in the table, all probabilities of infection, risks of infection, and expected fractions infected are conditional on an individual showing up in a contact network and therefore do not depend on the overall force of infection in the population.
Table 2.
The 95% CIs, medians, and true values for parameters estimated from MCMC in the discrete and continuous cases with F = 1000 and N = 5.
Fig 3.
Increased heterogeneity in susceptibility (larger Cd and fA → 0.5), intermediate fractions of individuals infected (intermediate Ed), and increased sample sizes (larger F) enhance our power to detect heterogeneity in susceptibility in the discrete case.
The plots show the power to detect heterogeneity in susceptibility in the discrete case, calculated as described in the text, across different numbers of focal individuals F and fraction of the population that is type A and more susceptible fA. The areas above the gray dashed lines represent parameter space that gives computationally indistinguishable probabilities of infection pA and pB, and therefore power, to the parameter combination with the same Ed and highest Cd below the line. This occurs because risks of infection can be changed to increase Cd without bound, whereas probabilities are bounded. N = 5.
Fig 4.
Increased heterogeneity in susceptibility (larger Cc), greater fractions of individuals infected (larger Ec), and increased sample sizes (larger F) enhance our power to detect heterogeneity in susceptibility in the continuous case.
The plots show the power to detect heterogeneity in susceptibility in the continuous case, calculated as described in the text, across different numbers of focal individuals F. N = 5.
Fig 5.
Parameter estimates for pA, pB, and fA in the discrete case capture the true values and are highly correlated.
The plots show the correlation in the parameter estimates for a) pA vs. pB, b) pA vs. fA, and c) pB vs. fA with different numbers of focal individuals F. These are the parameters that determine the distribution of individuals’ susceptibilities in the discrete case. The red dots represent the true parameters used to generate our simulated data, and the gray dots depict 1, 000 parameter sets from our posterior distribution for F = 50 (light gray), 200 (medium gray), 1000 (dark gray), and 5000 (black). pA = 0.748, pB = 0.125, fA = 0.2, and N = 5.
Fig 6.
Parameter estimates for k and θ in the continuous case capture the true values and are highly correlated.
This plot shows the correlation in the parameter estimates for k and θ that determine the gamma distribution of individuals’ susceptibilities in the continuous case with different numbers of focal individuals F. The red dot represents the true parameters used to generate our simulated data, and the gray dots depict 1, 000 parameter sets from our posterior distribution for F = 50 (light gray), 200 (medium gray), 1000 (dark gray), and 5000 (black). k = 0.592, θ = 0.626, and N = 5.
Fig 7.
Parameter estimates for the coefficient of variation of risk (Cd, Cc) and expected fraction of naive individuals infected (Ed, Ec) capture the true values and become more precise with increasing numbers of focal individuals F.
The plots show the parameter estimates for C and E with different numbers of focal individuals F in a) the discrete case and b) the continuous case. The red dots represent the true parameters used to generate our simulated data, and the gray dots depict 1, 000 parameter sets from our posterior distribution for F = 50 (light gray), 200 (medium gray), 1000 (dark gray), and 5000 (black). Cd = Cc = 1.3, Ed = Ec = 0.25, fA = 0.2, and N = 5.
Fig 8.
Predicted SIR dynamics capture the true dynamics and the 95% CIs narrow as the number of focal individuals F increases.
The plots show the predicted SIR dynamics in a) the discrete case and b) the continuous case with different numbers of focal individuals F. Specifically, the fraction of susceptible individuals is shown over the course of an epidemic. Shaded regions represent 95% CIs determined from 1, 000 posterior samples for F = 50 (light gray), 200 (medium gray), 1000 (dark gray), and 5000 (black). The blue lines show the true dynamics for the parameters used to generate the contact tracing data, and the red lines show the corresponding dynamics if there is homogeneity in susceptibility. Cd = Cc = 1.3, Ed = Ec = 0.25, fA = 0.2, and N = 5.