Stochastic epidemic dynamics on extremely heterogeneous networks

Networks of contacts capable of spreading infectious diseases are often observed to be highly heterogeneous, with the majority of individuals having fewer contacts than the mean, and a significant minority having relatively very many contacts. We derive a two-dimensional diffusion model for the full temporal behavior of the stochastic susceptible-infectious-recovered (SIR) model on such a network, by making use of a time-scale separation in the deterministic limit of the dynamics. This low-dimensional process is an accurate approximation to the full model in the limit of large populations, even for cases when the time-scale separation is not too pronounced, provided the maximum degree is not of the order of the population size.

Figure 1

Relative magnitudes of ${\mathrm{\Lambda }}^{\{K+1\}}$ and ${\mathrm{\Lambda }}^{\{2,...,K\}}, \epsilon =|\rho \varphi \left({\theta }^{*}\right)/\langle k^2\rangle -1|$, as a function of ${\theta }^{*}$ for $\alpha =-2.5$ and $K=10$ (blue squares), $K={10}^2$ (empty yellow circles), $K={10}^4$ (green triangles), and $K={10}^6$ (filled red circles).

Figure 2

Projection of the full epidemic dynamics onto (a) $(\theta ,i_1,i_2)$, and (b) $(\theta ,i_1,i_5)$, for $K=10, \alpha =-2.5, \rho =1.2, N={10}^5$, where $s_k=d_k{\theta }^k$ and $i_k$ are the fractions of susceptible and infected individuals of degree $k$, respectively. Red, dashed line: deterministic limit, Eqs. (36) and (37); blue, yellow, green lines: stochastic differential Eqs. (31) and (8). The initial condition corresponds to $i_{1,0}=0.005, i_{k,0}=0$ for $k=2,...,K$, and $G\left({\theta }_0\right)=1-i_{1,0}$. In both cases, the system is pushed back from the initial condition (lower right in both panels) toward the slow subspace described by Eq. (43), on which it seems to stay during the rest of the course of the epidemic. Note the different scales in the $z$ axes of both panels, so the fluctuations away from the slow subspace are small in both cases; the fluctuations on the slow subspace can, however, be large, as seen from the departure of the stochastic trajectories from the deterministic dynamics.

Figure 3

Time series of $\theta$ for $N=20000, \gamma =1$, and ${\theta }^{*}=0.7$, with (a, b) $K=10$, (c, d) $K=1000$, and (e, f) $K=19999$. Colored lines correspond to realizations of the full microscopic model (left panels) and the reduced mesoscopic model Eqs. (48) and (49) (right panels) started with a total number of five infectives with degrees sampled from a truncated Zipf distribution with $\alpha =-2.5$. The black, dashed line corresponds to the deterministic solution.

Figure 4

Time series of $\lambda$ for $N=20000, \gamma =1$, and ${\theta }^{*}=0.7$, with (a, b) $K=10$, (c, d) $K=1000$, and (e, f) $K=19999$. Colored lines correspond to realizations of the full microscopic model (left panels) and the reduced mesoscopic model Eqs. (48) and (49) (right panels) started with a total number of five infectives with degrees sampled from a truncated Zipf distribution with $\alpha =-2.5$. The black, dashed line corresponds to the deterministic solution.

Figure 5

Phase diagrams for $N=20000, \gamma =1$, and ${\theta }^{*}=0.7$, for (a)–(c) the full microscopic model, and (d)–(f) the reduced mesoscopic model Eqs. (48) and (49). Colored lines correspond to stochastic realizations started with a total number of five infectives with degrees sampled from a truncated Zipf distribution with $\alpha =-2.5$ and $K=10$ (left), $K=1000$ (center), and $K=19999$ (right). The black, dashed line corresponds to the deterministic solution.

Figure 6

Relative intensities of the noise terms in Eqs. (51) and (52) for (a)–(c) ${\theta }^{*}=0.7$, and (d)–(f) ${\theta }^{*}=0.9$, with $K=10$ (left), $K=1000$ (center), $K=19999$ (right). Blue, solid line: $f=f_2$; yellow, dashed line: $f=f_3$; green, dot-dashed line: $f=f_4$.

Figure 7

Phase diagrams from the semideterministic mesoscopic model with $N=20000, \gamma =1$, and ${\theta }^{*}=0.7$. Colored lines correspond to realizations of Eqs. (53) and (54), started with a total number of five infectives with degrees sampled from a truncated Zipf distribution with $\alpha =-2.5$ and (a) $K=10$, (b) $K=1000$, and (c) $K=19999$. The black, dashed line corresponds to the deterministic solution.

Figure 8

Distribution of epidemic sizes for $N=20000, \alpha =-2.5$, and $\gamma =1$, for (a, b) ${\theta }^{*}=0.7$, and (c, d) ${\theta }^{*}=0.9$, with $K=1000$ (left panels) and $K=19999$ (right panels). Histogram, full model; blue, solid line, reduced mesoscopic model Eqs. (48) and (49); red, dashed line, semideterministic mesoscopic model Eqs. (53) and (54).