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and perfect mixing, and the mean behavior of the stochastic AB models are often small compared to the variability in AB outcomes caused by chance encounters among inpiduals, at least for the public health metrics examined here. However, cost/benefit assessments of policy interventions, and hence the optimal policy, can depend on network structure and model boundary, underscoring the importance of sensitivity analysis across these dimensions.
The next section reviews the literature comparing AB and DE approaches. We then describe the models, the design of the simulation experiments, and results, closing with implications and directions for future work.
A spectrum of aggregation assumptions: A
B and DE models should be viewed as regions in a space of modeling assumptions, not as incompatible modeling paradigms. Aggregation is one dimension of that space. Models can range from lumped deterministic differential equations (also called deterministic compartmental models), to stochastic compartmental models, in which the continuous variables of the DE are replaced by counts of discrete inpiduals, to event history models, where the states of inpiduals are tracked but their network of relationships is ignored, to models with explicit contact networks linking inpiduals (e.g., Koopman et al. 2001; Riley 2007).
A few studies compare A
B and DE models. Axtell et al. (1996) call for “model alignment” or “docking” and illustrate with the Sugarscape model. Edwards et al. (2003) contrast an AB model of innovation diffusion with an aggregate model, finding that the two can perge when multiple attractors exist in the deterministic model. In epidemiology, Jacquez and O'Neill (1991) and Jacquez and Simon (1993) analyze the effects of stochasticity in inpidual-level SIS and SI models, finding some differences in mean behavior for small populations. However, the differences practically vanish for homogeneous populations above 100. Similarly, Gani and Yakowitz (1995) examine deterministic approximations to stochastic disease diffusion processes, and find a high correspondence between the two for larger populations. Greenhalgh and Lewis (2001) compare a stochastic model with the deterministic DE version in the case of AIDS spread through needle-sharing, and find similar behavior for those cases in which the epidemic takes off.
Heterogeneity has also been explored in models with different mixing sites for population subgroups. Anderson and May (1991, Chapter 12) show that the immunization fraction required to quench an epidemic rises with heterogeneity if immunization is implemented uniformly but falls if
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those most likely to transmit the pathogen are the focus of immunization. Ball et al. (1997) and Koopman et al. (2002) find expressions for cumulative cases and epidemic thresholds in stochastic SIR and SIS models with global and local contacts, finding that the behavior of deterministic and stochastic DE models can perge for small populations, low basic reproduction rates (R0), or highly clustered contact networks where transmission occurs in mixing sites such as schools and offices. Keeling (1999) formulates a DE model that approximates the effects of spatial structure when networks are highly clustered. Chen et al. (2004) develop AB models of smallpox, finding the dynamics generally consistent with DE models. In sum, AB and DE models of the same phenomenon sometimes agree and sometimes perge, especially when compartments contain smaller populations. Multiple network topologies and heterogeneity conditions have not been compared, and the practical significance of differences in mean behavior relative to uncertainties in stochastic events, parameters and model boundary has not been explored.
Model Structure: The SEIR model is a deterministic nonlinear differential equation model in which all members of a population are in one of four states—Susceptible, Exposed, Infected, or Removed. Contagious inpiduals can infect susceptibles before they are “removed” (i.e., recover or die). The exposed compartment captures latency between infection and the emergence of symptoms. Depending on the disease, exposed inpiduals may become infectious before symptoms emerge, and can be called early-stage infectious. Typically, such inpiduals have more contacts than those in later stages because they are asymptomatic.
SEIR models have been successfully applied to many diseases. Additional compartments are often introduced to capture more complex disease lifecycles, diagnostic categories, therapeutic protocols, population heterogeneity and mixing patterns, birth or recruitment of new susceptibles, loss of immunity, etc. (see Anderson and May 1991 and Murray 2002). In this study we maintain the standard assumptions of the classic SEIR model (four stages, fixed population, permanent immunity). The DE implementation of the model imposes several additional assumptions, including perfect mixing and homogeneity of inpiduals within each compartment and mean field aggregation (the flows between compartments equal the expected value of the sum of the underlying probabilistic rates for each inpidual). To derive the differential equations, consider the rate at which each infectious inpidual generates new cases:
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10 c is *Prob (Contact with Susceptible)*Prob (Transmission|Contact with Susceptible) (1) where the contact frequency c is is the expected number of contacts between infectious inpidual i and susceptible inpidual s; homogeneity implies c is is a constant, denoted c IS , for all inpiduals i, s. If the population is well mixed, the probability of contacting a susceptible inpidual is simply the proportion of susceptibles in the total population, S/N. Denoting the probability of transmission given contact between inpiduals i and s, or infectivity, as i is (which, under homogeneity, equals i IS for all i, s), and summing over the infectious population yields the total flow of new cases generated by contacts between the I and S populations, c IS *i IS *I*(S/N). The number of new cases generated by contacts between the exposed and susceptibles is formulated analogously, yielding the total Infection Rate, f ,
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f = (c ES i ES E + c IS i IS I)(S/N). (2) To model emergence and recovery, consider these to be Markov processes with certain transition probabilities. In the classic SEIR model each compartment is assumed to be well mixed so that the probability of emergence (or recovery) is independent of how lon
g an inpidual has been in the E (or I) state. Denoting the inpidual hazard rates for emergence and recovery as ε and δ, the mean emergence time and disease duration are then 1/ε and 1/δ, respectively. Summing over the E and I populations and taking expected values yields the flows of emergence and recovery:
e =εE and r = δI .
(3) The full model is thus:
f dt dS !=, e f dt dE !=, r e dt dI !=, r dt dR =. (4) Equation (3) implies the probabilities of emergence and recovery are independent of how lon
g an inpidual has been in the E or I states, respectively, and results in exponential distributions for the residence times in these states. Exponential residence times are not realistic for most diseases, where the probability of emergence and recovery is initially low, then rises, peaks and falls. Note, however, that any lag distribution can be captured throug
h the use of partial differential equations, approximated in the ODE paradigm by adding additional compartments within the exposed and infectious categories (Jacquez and Simon 2002). For simplicity we maintain the assumption of a single compartment per disease stage of the classic SEIR model.
The AB model relaxes the perfect mixing and homogeneity assumptions of the DE. Each inpidual j ∈ (1, …, N) is in one of the four states S, E, I, or R. The inpidual state transitions
f[j], e[j], and r[j] equal 1 at the moment of infection, emergence, and recovery, respectively, and 0 otherwise, and depend on inpidual attributes such as contact frequencies and on the chances of interaction with others as specified by the contact network. The aggregate flows f,e, and r over any interval dt are the sum of the inpidual transitions during that interval. The online supplement details the formulation of the AB model and shows how the DE model can be derived from it by assuming homogeneous agents and applying the mean-field approximation.
