Large deviations in non-Markovian stochastic epidemics

Abstract

We develop a framework for non-Markovian, well-mixed SIR and SIS models beyond mean field, utilizing the continuous-time random walk formalism. Using a gamma distribution for the infection and recovery inter-event times as a test case, we derive asymptotical late-time master equations with effective memory kernels and obtain analytical predictions for the final outbreak size distribution in the SIR model, and quasistationary distribution and disease lifetime in the SIS model. We show that varying the width of the inter-event time distribution can greatly alter the outbreak size distribution or the disease lifetime. We also show that rescaled Markovian models may fail to capture fluctuations in the non-Markovian case. Overall, our analysis, confirmed against numerical simulations, paves the way for studying large deviations in structured populations on degree-heterogeneous networks

Figures (7)

• Figure 1: Outbreak size distributions for the SIR model, with $\alpha=0.5$, $1$, $1.5$ and $2$ in (a)--(d), for gamma-distributed infection and Markovian recovery, $N=5000$, $R_0=1.5$, and $10^5$ runs per $\alpha$. Simulations (symbols) are compared with theory (solid line). In all panels $I(0)=1$, and simulations resulting in outbreak fractions $<10^{-2}$ were omitted, see [61].
• Figure 2: Normalized mean $x_r^*$ (a) and normalized standard deviation $\sigma$ (b) versus $\alpha$. Here infection is gamma-distributed and recovery is exponential, $N=5000$, $R_0=1.5$ and $10^4$ runs per $\alpha$. Simulations (symbols) are compared with theory (solid line). In all panels $I(0)=1$, and simulations resulting in outbreak fractions $< 10^{-2}$ were omitted, see [61].
• Figure 3: In (a) and (b) we show the normalized mean $x_i^*$ and normalized standard deviation $\sigma$ versus $\alpha$, for gamma-distributed infection and Markovian recovery. Here, $N=1000$, $R_0=1.5$, and we ran $10^3$ realizations per $\alpha$. Simulations (symbols) are compared with theory (line). In (c) we plot the MTE versus $\alpha$ for the same WTs, $N=110$, $R_0=1.51$ and $10^2$ runs per $\alpha$. Simulations (symbols) are compared with theory (line), see text. In (a-c), $x_i(0)=x_i^*$.
• Figure 4: Numerical (left) and theoretical (right) heatmaps of the outbreak-size distribution COV (normalized by the Markovian COV) versus the infection and recovery shape parameters, for a well-mixed system with $N\!=\!5000$, $R_0\!=\!2$, and $10^3$ runs per point. Black lines are equi-COV contour lines.
• Figure S1: QSDs at $\alpha\!=\!0.5, 1, 2$ in (a)--(c) in the SIS model of epidemics, for gamma-distributed infection and Markovian recovery, with $N\!=\!1000$, $R_0\!=\!2$, and $10^5$ runs per $\alpha$. Here we compare simulations (symbols) to theory (solid line) and to a Markovian theory adjusted to the same mean (dashed line), with $x_i(0)=x_i^*$. The distributions are computed by averaging over a single trajectory from $t=50$ until $t=10^5$ (in units of $\gamma^{-1}$.