Results: For each experimental condition we examine three measures relevant to public health. The maximum symptomatic infectious population (peak prevalence, I max) indicates the peak load on public health infrastructure including health workers, immunization resources, hospitals and quarantine facilities. The time from initial exposure to the maximum of the infected population (the peak time, T p) measures how quickly the epidemic spreads and therefore how long officials have to deploy those resources. The fraction of the population ultimately infected (the final size, F) measures the total burden of morbidity and mortality. To illustrate, figure 1 compares the base case DE model with a typical simulation of the AB model (in the heterogeneous scale-free case). The sample scale-free epidemic grows faster than the DE (T p = 37 vs. 48 days), has similar peak prevalence (I max = 27%), and ultimately afflicts fewer people (F = 85% vs. 98%).
In this study we focus on the practical significance of differences between the mean output of AB and DE models. Specifically, we explore whether the differences among models are large relative to the variability in outcomes for which policymakers should plan and whether the differences alter the choice of optimal policies. To begin, we conservatively consider outcome variability arising only from stochastic interactions among inpiduals. Specifically, suppose policymakers planning for a possible outbreak know with certainty mean infectivity, incubation period, disease duration, network type, and all other parameters conditioning contagion and diffusion, and that these characteristics are unaffected by the course of the epidemic. In short, assume policymakers possess a perfect agent-based model of the situation, and lack only
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knowledge of which inpiduals will, by chance, encounter each other at any moment and transmit the disease. As an example, suppose the contact network is characterized by a scale-free degree distribution with known parameters, and that inpiduals are heterogeneous in their behavior (but with known distribution). For the hypothetical disease we examine, prevalence peaks on average after 44 days at a mean of 23.9% of the population. In the deterministic compartment model with the same parameters, prevalence peaks after 48 days at 27.1% of the population. Given the large sample of AB realizations, these differences are statistically significant (p < .001), but they are not practically significant. Unobservable stochastic interactions among inpiduals means policymakers, to be, for example, 95% confident resources will be sufficient, must plan to handle an epidemic peaking between 4 and 75 days after introduction, with peak prevalence between 4% and 31.5% of the population. Of course, the deterministic model yields a single trajectory representing the expected path under the mean-field approximation. No responsible policymaker should plan for the mean epidemic without considering uncertainty. To assess the range of outcomes arising from the random nature of inpidual interactions, policymakers using compartment models would have to estimate the impact of uncertainty by, for example, moving to a stochastic DE representation. Such a model would be computationally efficient relative to the full AB model, but would still assume within-compartment mixing and homogeneous agents.
Policymakers should also consider how model assumptions affect the optimality of interventions. Consider, for example, a quarantine policy. Quarantine should be implemented if its benefit/cost ratio (e.g. the value of QALYs or DALYs saved and avoided health costs relative to the costs of quarantine implementation), is favorable and higher than that of other policy options (including no action). Two models may yield similar estimates of epidemic diffusion, yet respond differently to policies. In such cases the differences between the models may be of great practical significance even if their base case behavior is similar. We provide an example below.
Figure 2 shows the symptomatic infectious population, I, in 1000 AB simulations for each network and heterogeneity condition. Also shown are the mean of the ensemble and the trajectory of the base case DE model. Table 2 reports results for the fitted DE models; Tables 3-4 compare the means of T p, I max, and F for each condition with both the base and fitted DE models. Except for the lattice, the DE and mean AB dynamics are qualitatively similar. Initial diffusion is driven
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by positive feedback as contagious inpiduals spread the infection. The epidemic peaks when the susceptible population is sufficiently depleted that the mean number of new cases generated by contagious inpiduals is less than the rate at which they are removed from the contagious pool.
Departures from the DE model increase from the connected to the random, scale free, small world, and lattice structures (Figure 2; tables 3-4). The degree of clustering explains some of these variations. In the fully connected and random networks the chance of contacts in distal regions is the same as for neighbors. The positive contagion feedback is strongest in the connected network because an infectious inpidual can contact everyone else, minimizing local contact overlap. In contrast, the lattice has maximal clustering. When contacts are localized in a small region of the network, infectious inpiduals contact their common neighbors repeatedly. As these people become infected the chance of contacting a susceptible and generating a new case declines, slowing diffusion on average, even if the total susceptible population remains high.
In the deterministic DE model there is always an epidemic if R0 > 1. Due to the stochastic nature of interactions in the AB model, it is possible that no epidemic occurs or that it ends early if, by chance, the few initially contagious inpiduals recover before generating new cases. As a measure of early burnout, table 3 reports the fraction of cases where cumulative cases remain below 10%. (Except for the lattice, the results are not sensitive to the 10% cutoff. The online appendix shows the histogram of final size for each network and heterogeneity condition.) Early burnout ranges from 1.8% in the homogeneous connected case to 6.8% in the heterogeneous scale-free case. Heterogeneity raises the incidence of early burnout in each network since there is a higher chance that the first cases will have few contacts and recover before spreading the disease. Network structure also affects early burnout. Greater contact clustering increases the probability that the epidemic burns out in a local patch of the network before it can jump to other regions, slowing diffusion and increasing the probability of global quenching.
Heterogeneity results in smaller final size, F, in all conditions: the mean reduction over all
ten conditions is 0.10, compared to a mean standard deviation across all conditions,
!", of 0.19.
Similarly, heterogeneity reduces T p in all conditions (by a mean of 9.5 days, with
!" = 26 days).
Maximum prevalence also falls in all conditions (by 1.5%,
!" = 5.1%). In the H≠ condition high-
contact inpiduals tend to become infected sooner, causing, on average, faster take-off compared
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to the H= case (hence earlier peak times). These inpiduals are also removed sooner, reducing mean contact frequency, and hence the reproduction rate, among those who remain compared to the H= case. Subsequent diffusion is slower, peak prevalence is smaller, and the epidemic ends sooner, yielding fewer cumulative cases.
Consider now the differences between the DE and AB cases by network type.
Fully Connected: The fully connected network corresponds closely to the perfect mixing assumption of the DE. As expected, the base DE model closely tracks the mean of the AB simulations. In the H= condition, T p, I max, and F in the base DE model fall well within the 95% confidence interval defined by the ensemble of AB simulations. In the H≠ case, T p and I max also fall within the 95% range, but F lies just outside the range encompassing 95% of the ensemble. Random: The random network behaves much like the connected case. The DE values of T p and I max fall within the 95% outcome range for both heterogeneity conditions. The value of F in the DE falls outside the 95% range for both H= and H≠, because the sparse contact network means more people escape contact with infectious inpiduals compared to the perfect mixing case. Scale-Free: The scale free network departs substantially from perfect mixing. Most nodes have few links, so initial takeoff is slower, but once the infection reaches a hub it spreads quickly. The base DE values of T p and I max fall well within the 95% outcome interval for both heterogeneity conditions. However, as the hubs are removed from the infectious pool, the remaining nodes have lower average contact rates, causing the epidemic to burn out at lower levels of diffusion; the 95% range for final size is 2% to 98% for H= and 1% to 92% for H≠, while the base DE value is 98%. Small World: Small world networks are highly clustered and lack highly-connected hubs. Nevertheless, the presence of a few long-range links is sufficient to seed the epidemic throughout the population (Watts and Strogatz 1998). Diffusion is slower on average compared to the DE and the connected, random, and scale-free networks. The existence of a few randomly placed long-range links also increases the variability in outcomes. The 95% range for T p is 22 to 154 days for H= (7 to 176 days for H≠), easily encompassing the base DE value. Slower diffusion relative to the DE causes peak prevalence in the DE to fall outside the 95% interval of AB outcomes for both H= and H≠. The main impact of heterogeneity is greater dispersion and a reduction in final size.
