A central parameter in epidemic models is the basic reproduction number, R0, the expected number of new cases each contagious inpidual generates before removal, assuming all others are susceptible. The base case parameters yield R0 = 4.125 (Table 1), similar to diseases like smallpox, R0≈ 3–6 (Gani and Leach 2001), and SARS, R0≈ 2-7 (Lipsitch et al. 2003; Riley et al. 2003). The base value provides a good opportunity to observe potential differences between DE and A
B models: diseases with R0 < 1 pose little risk to public health, while those with R0 >> 1, e.g., measles, cause a severe epidemic in (nearly) any network. The AB models use the same infectivities and expected residence times, and we choose inpidual contact frequencies so that mean total contact rates in each network and heterogeneity condition are the same as the DE model. We set the population N = 200, all susceptible except for two randomly chosen exposed inpiduals. Though small compared to settings of interest in policy design, e.g., cities, the effects of random events and network type are likely to be more pronounced in small populations (Gani and Yakowitz 1995), providing a stronger test for differences between the DE and AB models. A small population also reduces computation time, allowing more extensive sensitivity analysis. The DE has 4 state variables; computation time is negligible for all N. The AB model has 4N state variables and must also track interactions among the N inpiduals, implying that computation time can grow at rates up to O(N2), depending on the contact network. We report sensitivity analysis of R0 and N below. The supplement includes the models and full documentation. Experimental design: We vary both the network structure of contacts among inpiduals and the degree of inpidual heterogeneity in the AB model and compare the resulting dynamics to the DE. We implement a full factorial design with five network structures and two heterogeneity
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conditions. In each of the ten conditions we generate an ensemble of 1000 simulations of the AB model, each with different realizations of the random variables determining contacts, emergence, and recovery. Since the parameters in each simulation are identical to the DE model, differences in outcomes can only be due to differences in network topology, heterogeneity among inpiduals, or the discrete, stochastic treatment of inpiduals in the AB model.
Network topology: The DE model implemented here assumes perfect within-compartment mixing, implying any infectious inpidual can meet any susceptible inpidual with equal probability. Realistic networks are far more sparse and clustered. We explore five different network structures: fully connected, random (Erdos and Renyi 1960), small-world (Watts and Strogatz 1998), scale-free (Barabasi and Albert 1999), and lattice (where contact only occurs between neighbors on a ring). We parameterize the model so that all networks (other than the fully connected case) have the same mean number of links per node, k = 10 (Watts 1999).
The fully connected network corresponds to the perfect mixing assumption of the DE model. The random network is similar except people are linked with equal probability to a subset of the population. To test the network most different from the perfect mixing case, we also model a one-dimensional ring lattice with no long-range contacts. With k = 10 each person contacts only the five neighbors on each side. The small world and scale-free networks are intermediate cases with many local and some long-distance links. These widely-used networks characterize a number of real systems (Watts 1999; Barabasi 2002). We set the probability of long-range links in the small world networks to 0.05, in the range used by Watts (1999). We build the scale-free networks using the preferential attachment algorithm (Barabasi and Albert 1999) in which the probability a new node links to existing nodes is proportional to the number of links each node already has. Preferential attachment yields a power law for the probability that a node has k links, Prob(k)
=αk?γ. Empirical studies typically show 2 ≤γ≤ 3; the mean value of γ in our experiments is 2.6.
The fully connected and lattice networks are deterministic, so every simulation of these cases has the same network governing contacts among inpiduals. The Erdos-Renyi, small world, and scale-free cases are random networks. Each simulation of these cases uses a different realization of the network structure. In realistic networks the links among inpiduals change through time even as overall topology can remain stable (e.g., Kossinets and Watts 2006),
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introducing mixing that brings the AB model closer to the assumptions of the compartment model. To maximize the differences between the AB and DE conditions, however, we assume the network realization in each simulation is fixed. The supplement details the construction of each network. Inpidual Heterogeneity: Each inpidual has four relevant characteristics: expected contact rate, infectivity, emergence time, and disease duration. In the homogeneous condition (H=) each inpidual is identical with parameters set to the values of the DE model. In the heterogeneous condition (H≠) we vary inpidual contact frequencies.
Heterogeneity in contacts is modeled as follows. Given that two people are linked (that they can come into contact), the frequency of contact between them depends on two factors. First, how often does each use their links, on average: some people are gregarious; others shy. Second, time constraints may limit contacts. At one extreme, the frequency of link use may be constant, so that people with more links have more total contacts per day, a reasonable approximation for some airborne infections and easily communicated ideas: a professor may transmit an airborne virus or a simple concept to many people with a single sneeze or comment, (roughly) independent of class size. At the other extreme, if the time available to contact people is fixed, the chance of using a link is inversely proportional to the number of links, a reasonable assumption when transmission requires extended personal contact: the professor can only tutor a limited number of people each day. We capture these effects by assigning inpiduals different propensities to use their links,
λ[j], yielding the expected contact frequency for the link between inpiduals i and j, c[i,j]: c[i,j]=κ*λ[i]*λ[j]/ (k[i]*k[j])τ(5) where k[j] is the total number of links inpidual j has, τ captures the time constraint on contacts, and κ is a constant chosen to ensure that the expected contact frequency for the population equals the value used in the DE model. In the H= condition λ[j] = 1 for all j and τ = 1 so that expected contact frequencies are equal for all inpiduals, independent of how many links each has. In the H≠ condition λ[j] is a random variable and τ = 0: inpiduals have different contact rates and those with more links have more contacts per day. We use a uniform distribution, λ[j] ~ U[0.25, 1.75]. Calibrating the DE Model: In practice the parameters determining R0 are often poorly constrained by biological and clinical data. For emerging diseases such as vCJD, BSE and avian flu data are
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not available until the epidemic has already spread. Parameters are usually estimated by fitting models to aggregate data as an outbreak unfolds; SARS provides a typical example (Dye and Gay 2003; Lipsitch et al. 2003; Riley et al. 2003). Because R0 also depends on contact networks that are often poorly known, models of established diseases are commonly estimated the same way (e.g., Gani and Leach 2001). To mimic this protocol we treat each realization of the AB model as the “real world” and estimate the parameters of the DE to yield the best fit to the cumulative number of cases. We estimate infectivity (i ES and i IS), and incubation time (1/ε) by nonlinear least squares in a large set of inpidual AB realizations (see the supplement). Results assess whether calibrated compartment models can capture the behavior of heterogeneous inpiduals in realistic settings with different contact networks.
