Comparing Agent-Based and Differential Equation Models(2)

4

Strogatz 1998; Barabasi 2002; Rogers 2003), providing a strong test for differences between the two approaches. Third, the DE paradigm is well developed in epidemiology (for reviews see Anderson and May 1991 and Hethcote 2000); AB models also have a long history (e.g. Abbey 1952) and have recently gained momentum (for reviews see Newman 2002, 2003 and Watts 2004).

Finally, diffusion is a fundamental process in perse physical, biological, social and economic settings. Many diffusion phenomena in human systems involve processes of social contagion analogous to infectious disease, including word of mouth, imitation and network externalities. From the diffusion of innovations to rumors, financial panics and riots, contagion-

like dynamics, and formal models of them, have a rich history in the social sciences (Bass 1969; Watts and Strogatz 1998; Mahajan et al. 2000; Barabasi 2002; Rogers 2003). Insights into the advantages and disadvantages of AB and DE models in epidemiology can inform understanding of diffusion in many domains of concern to social scientists and managers.

Our procedure is as follows. We develop a stochastic AB version of the classic SEIR model, a widely used nonlinear deterministic DE model. The DE pides the population into four compartments: Susceptible (S), Exposed (E), Infected (I), and Removed (R). In the AB model, each inpidual is separately represented and must be in one of these four states. Both the AB and DE models use the same parameters. Therefore any differences in outcomes arise only from relaxing the restrictive assumptions of the DE model. In practice, DE modelers add compartments to capture heterogeneity in inpiduals and their contact networks, for example, disaggregating by biological or behavioral attributes (e.g., differences in age or contact frequencies), or by location (as in patch models; see e.g. Riley 2007). Here we use the classic SEIR model to maximize potential differences between the two approaches. We run the AB model under five widely used network topologies (fully connected, random, small world, scale-free, and lattice) and test each

with homogeneous and heterogeneous inpiduals. We compare the resulting diffusion dynamics

on a variety of metrics relevant to public health, including cumulative cases, peak prevalence, and the speed the disease spreads (the time available for health officials to respond).

The most obvious difference between the models we compare is that, for given parameters, the stochastic AB model generates a distribution of outcomes, while the deterministic DE generates a single path representing the expected trajectory under the mean-field approximation for

5

contacts between infectious and susceptible inpiduals. Further, due to chance events, the epidemic never takes off in some realizations of the stochastic model. Deterministic models, whether DE or AB, cannot generate this mode of behavior. Capturing outcome variability in the DE paradigm requires moving to a stochastic compartment model, an intermediate method between deterministic models and the full stochastic AB representation. More interesting are differences due to network topology and inpidual heterogeneity. On average, diffusion slows as contact networks become more tightly clustered compared to the DE. On average, heterogeneity accelerates the initial take-off, as highly connected inpiduals quickly spread the disease, but reduces overall diffusion as these same inpiduals quickly exit the infectious pool.

In a second set of tests, we also examine the ability of the DE model to capture the dynamics of each network structure in the realistic situation where parameters are poorly constrained by biological and clinical data. Epidemiologists often estimate potential diffusion, for both novel and established pathogens, by fitting models to the aggregate data as an outbreak unfolds (Dye and Gay 2003; Lipsitch et al. 2003; Riley et al. 2003). Calibration of innovation diffusion and new product marketing models is similar (Mahajan et al. 2000). We mimic this practice by treating the AB simulations as the “real world” and fitting the DE model to them. On average, the fitted models closely match the inpidual AB realizations under all network topologies and heterogeneity conditions tested. However, the estimated parameters are biased in the highly clustered and heterogeneous cases. Further, the ability to fit such data does not imply that the AB and calibrated DE models will respond to policy interventions in the same way, demanding caution in their use. When different models yield different inferences about policies it is important to identify the assumptions responsible to guide data collection, to improve the models and to select the most appropriate model for the purpose at hand.

The implications of the differences across models depend on the purpose of the analysis. Here we focus on the policy context. Policymakers face a world of time pressure, inadequate data and limited knowledge of parameters such as pathogen virulence, transmissibility, incubation latency, treatment efficacy, etc. Further, the appropriate boundary for analysis is often unclear: resources for vaccination and treatment may be limited; an outbreak, whether natural or triggered by bioterror, may alter the behavior of the public and first-responders, endogenously disrupting the

6

contact networks that feed back to condition disease spread through processes of risk amplification and attenuation (Kasperson et al. 1988, Glass and Schoch-Spana 2002). Hence we consider whether the differences in mean behavior between DE and AB models are large relative to the uncertainties policymakers face. We also consider how these differences in mean behavior might affect the assessment of the costs and benefits, and hence the optimality, of policies.

The mean behavior of different models may be significantly different in the statistical sense, yet be small relative to the variation in output caused by uncertainty about parameters, model boundary, and stochastic events (McCloskey and Ziliak 1996). For example, consider the variability in outcomes generated by a stochastic AB model. Each realization of the model will differ: some exhibiting fast diffusion, some slow; some with many inpiduals afflicted, some with fewer, depending on the chance nature of contacts between infectious and susceptible inpiduals. An ensemble of many simulations generates the distribution of possible epidemics, but only one will be observed in a particular outbreak. Several questions may now be asked.

One important question is whether the expected values of key metrics differ in different models. For example, does the mean value of peak prevalence under a scale-free network differ from the value generated by the corresponding deterministic compartment model? By running a sufficiently large number of simulations sampling error can be made arbitrarily small, and any differences in the mean behavior of the models will be highly statistically significant.

Another question is whether the differences among means are significant from the point of view of policymakers seeking appropriate responses to a potential outbreak. Models with similar “base case” behavior can have similar or different responses to policies, and, conversely, models with different base behaviors may nevertheless yield the same inferences about policy impacts. Differences in policy response across models can be statistically significant yet small relative to uncertainty in parameters, network structure, inpidual attributes, and model boundary. Policy-makers must assess the practical significance of each model assumption given the likely range of outcomes generated by all sources of uncertainty, not only uncertainty caused by random events.

The results document a number of differences between the DE and mean AB dynamics. The results, for both the base-case and calibrated DE models, also show that the differences between the deterministic compartment model, with its assumptions of homogeneous inpiduals

百度搜索“70edu”或“70教育网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，70教育网，提供经典知识文库Comparing Agent-Based and Differential Equation Models(2)在线全文阅读。