17
Lattice: In the lattice inpiduals only contact their k nearest neighbors, so the epidemic advances roughly linearly in a well-defined wave of new cases trailed by symptomatic and then recovered inpiduals. Such waves are observed in the spread of plant pathogens, where transmission is mostly local, though in two dimensions more complex patterns are common (Bjornstad et al. 2002; Murray 2002). Because the epidemic wave front reaches a stochastic steady state in which removal balances new cases, the probability of burnout is roughly constant over time, and I max is lower, with the base DE value falling outside the 99% range. For the same reason, mean final size is much lower and peak time longer than the base DE. Interestingly, the variance is higher as well, so that in the H= condition the DE values of F and T p fall within the 95% range of AB outcomes. In sum, peak time in the uncalibrated base DE model falls within the envelope encompassing 95% of the AB simulations in all ten network and heterogeneity conditions. Peak prevalence falls within the 95% range in all but the small world and lattice. Final size, however, is sensitive to clustering and heterogeneity, falling within the 95% range in only three cases.
Calibrated DE Model: In practice parameters such as R0 and incubation times are poorly constrained and are estimated by fitting models to aggregate data. Table 2 summarizes the results of fitting the DE model to 200 randomly selected AB simulations in each experimental condition, a total of 2000 calibrations. The median R2 for the fit to cumulative cases exceeds 0.985 in all scenarios. The mean values of F, T p, and I max in the calibrated DE fall within the range encompassing 95% of the AB outcomes in all network and heterogeneity conditions. The DE model fits well even though it is clearly mis-specified in all but the homogeneous fully connected network. Why? As the network becomes increasingly clustered and diffusion slows, the estimated parameters adjust accordingly. Specifically, in deterministic SEIR compartment models, R0 and final size are related by R0 = –ln(1 – F)/F (Anderson and May 1991). Consequently, when contact clustering leads to smaller F, the estimated incubation time or transmission rates must shift to yield a smaller estimate of R0. The parameter estimates are biased because deviations from their underlying values are the only way the DE, with its within-compartment homogeneity and mixing assumptions, can capture the impact of heterogeneity and network structure. Further, the close fit of the compartment model does not imply that its response to policies will be the same as that of the underlying clustered and heterogeneous network. The supplement provides further details.
18
Sensitivity to Population Size: We repeated the analysis for N = 50 and 800 (see the supplement). The results change little over this factor of 16. For most conditions, the rate of early burnout falls in the larger population, so the final fraction of the population infected is slightly larger (and therefore closer to the value in the DE). Population size has little impact on the other metrics. Sensitivity to R0: We varied R0 from 0.5 to 2 times the base value; detailed results are reported in the supplement. Naturally, diffusion is strongly affected by R0. Somewhat surprisingly, however, over the range tested the differences between the DE and mean AB outcomes remain small relative to the 95% outcome range for most of the metrics. Changes in R0 have two offsetting effects. First, the smaller the value of R0, the larger are the differences between the DE and means of the AB trajectories. Second, however, the smaller R0, the greater the variation in outcomes within a given network and heterogeneity condition caused by chance encounters among inpiduals. Small values of R0 reduce the expected number of new cases each infectious inpidual generates before removal. In effect, the fraction of the contact network sampled by each infectious inpidual is smaller, so the probability that the epidemic will be seeded at multiple points in the network decreases. In highly clustered and heterogeneous networks, the lower representativeness of these small samples increases the difference between the DE and the mean of the AB trajectories (for example, more cases of early quenching will be observed). For the same reason, however, inpidual realizations of the same network and heterogeneity condition will differ more with small R0, increasing the variance in outcomes for which policy makers must prepare. Similarly, larger R0 reduces the differences between the DE model and the means of the AB models but also reduces variability in outcomes because each infectious inpidual samples the network many times before recovering. These offsetting effects imply that, over the range examined here, the differences between the DE and the mean behavior of the AB models are relatively insensitive to variations in R0.
Sensitivity to disease natural history: In many diseases the exposed gradually become more infectious prior to becoming symptomatic. This progression can be modeled by adding additional compartments to the exposed stage with different infectivities in each. In the classic SEIR model used here, with only one compartment per stage, pre-symptomatic infectivity is approximated by
19
assuming the exposed are contagious, though with i ES < i IS. To test the impact of this assumption we set i ES = 0, adjusting i IS to keep R0 at its base value. Results are reported in the supplement. As expected, diffusion slows and the probability of early quenching grows. However, the differences in the mean values of the metrics across models generally remain small relative to the 95% range of AB outcomes. Assuming that exposed inpiduals are not contagious has little impact on the differences between the DE and mean AB behavior relative to the variability in AB outcomes. Policy Analysis and Sensitivity to Model Boundary: Another important question is whether the behavior of the models differs in response to policy interventions and expansion of the model boundary. While comprehensive treatment of these questions is beyond the scope of this paper, we illustrate by examining the impact of actions that reduce contact rates. For example, the 2003 SARS epidemic appears to have been quenched through contact reduction (Wallinga and Teunis 2004, Riley et al. 2003, Lipsitch et al. 2003). Contact reduction can arise from policies, e.g., quarantine (including mandatory isolation and travel restrictions), and from behavioral feedbacks, e.g., social distancing, where inpiduals who fear infection reduce contacts with others. For simplicity we assume contact rates fall linearly to a minimum value as the total number of confirmed cases (cumulative prevalence P = I + R) rises.1 Specifically, we model the contact
frequency c js between infectious persons, j
!" {E, I}, and susceptibles, s
!
" {S}, as a weighted
average of the initial frequency, c*js, and the minimum achieved under quarantine, c q js:
!c
js
=(1"q)c
js
*+qc
js
q(6)
!
q=MIN[1,MAX(0,(P"P
)/(P
q
"P
))](7) The impact of contact reduction, q, rises linearly as cumulative prevalence, P, rises from a threshold, P0, to the level at which the effect saturates, P q. We set P0 = 2 and P q = 10 cases. Neither social distancing nor quarantine are perfect; we set the minimum contact frequency, c q js = 0.15c*js. This value gradually reduces R0 in the DE model from 4.125 to ≈ 0.6, roughly similar to the reduction Wallinga and Teunis (2004) estimate for the SARS epidemic.
1 Other policies, such as targeted immunization, can exploit the structure of the contact network, if it is known, and generally require an AB model, though some such policies can be approximated in DE models (e.g., Kaplan et al. 2003).
20
As expected, contact reduction quenches the epidemic earlier. In the DE model, prevalence peaks 17 days sooner, I max falls from 27% to 4.4%, and F falls from 98% to 19% of the population, greatly easing the burden on public health resources. Contact reduction has similar benefits in the AB cases. The differences between the means of the metrics in the AB models and their DE value are small relative to the variation in outcomes caused by stochastic interactions in the AB models. The DE results fall within the 95% outcome range for all three metrics in all network and heterogeneity conditions, with one exception: the value of F in the lattice (Table 5). However, as in the base case, clustering and heterogeneity cause some differences between the DE and mean AB outcomes. Under contact reduction heterogeneity increases mean F because high-contact inpiduals tend to be infected first, increasing the exposed population relative to H= before contact reduction is triggered. In the base case, however, heterogeneity lowers F because early high-contact cases are also removed early, lowering the reproduction rate. Therefore the mean reduction in F under contact reduction is smaller in the heterogeneous cases.
