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# A martingale formulation for stochastic compartmental susceptible-infected-recovered (SIR) models to analyze finite size effects in COVID-19 case studies

This work is supported by NSF grant DMS-2027438, ARO MURI Grant W911NF1810208, NSF grant DMS-2027277, and Simons Math + X Investigator Award number 510776

• Deterministic compartmental models for infectious diseases give the mean behaviour of stochastic agent-based models. These models work well for counterfactual studies in which a fully mixed large-scale population is relevant. However, with finite size populations, chance variations may lead to significant departures from the mean. In real-life applications, finite size effects arise from the variance of individual realizations of an epidemic course about its fluid limit. In this article, we consider the classical stochastic Susceptible-Infected-Recovered (SIR) model, and derive a martingale formulation consisting of a deterministic and a stochastic component. The deterministic part coincides with the classical deterministic SIR model and we provide an upper bound for the stochastic part. Through analysis of the stochastic component depending on varying population size, we provide a theoretical explanation of finite size effects. Our theory is supported by quantitative and direct numerical simulations of theoretical infinitesimal variance. Case studies of coronavirus disease 2019 (COVID-19) transmission in smaller populations illustrate that the theory provides an envelope of possible outcomes that includes the field data.

Mathematics Subject Classification: Primary: 60G51; Secondary: 35Q91.

 Citation: • • Figure 1.  Plots of infected compartment fractions $\textbf{i}_{N}(t)$ for the stochastic SIR and $s(t),i(t)$, and $r(t)$ for deterministic SIR. For both models, $p_1 = 0.5$, $p_2 = 0.5$, $\gamma = 1$, $T = 23$, ${s}_0 = \textbf{s}_{0N} = 0.96$, ${i}_0 = \textbf{i}_{0N} = 0.04$, ${r}_0 = \textbf{r}_{0N} = 0$, and In Figs. 1b and 1a, 1d and 1c, 1f and 1e, and 1h and 1g, $\beta = 0.95, 1.1, 1.2, 1.3$, respectively. For the SIR-IPC model displayed in Figs. 1b, 1d, 1f, and 1h, $N = 10^{2.5}$, $10^{3}$, $10^{3.5}, 10 ^4$ in every panel

Figure 2.  Examples of the infinitesimal standard deviation for $\sigma _{N }^{(l)} \left ( t \right )$, $l = 1, 2, 3$. In both cases, $p_1 = 0.5$, $p_2 = 0.5$, $\gamma = 1$, $T = 23$, ${s}_0 = \textbf{s}_{0N} = 0.96$, ${i}_0 = \textbf{i}_{0N} = 0.04$, ${r}_0 = \textbf{r}_{0N} = 0$. In the cases with $\mathcal R_0 = 0.95<1$ in Figs 2a, 2c, and 2e, $\beta = 0.95$. And the blue, orange, green, and red lines show results with the same realization from simulations with the corresponding colors in Fig 1b, respectively. In the cases with $\mathcal R_0 = 1.3>1$ in Figs 2b, 2d, and 2f, $\beta = 0.95$. And the blue, orange, green, and red lines show results with the same realization from simulations with the corresponding colors in Fig 1h, respectively

Figure 3.  Comparisons of the log–log plot of the theoretical and empirical scaling for $\overline{ \mathcal V _{N}^{(l)} }$, $l = 1, 2, 3$, as in (38) - (40). The vertical bars are the error bars of the empirical scaling of the infinitesimal variance from the same realization as in Figs. 1b, and 1h. We set $[T_1, T_2]$ as $[0, 10], [1, 11], [2, 12],\cdots$ and $[13, 23]$. The minimum and maximum values of $y = \log\left (\overline{\mathcal {V}_{ N }^{(l)}}\right)$, $l = 1, 2, 3$, taken over all such intervals are set as the limits of the error bars. The straight lines with slope $-1$ show the theoretical scaling with the $x$-intercepts as $\log ( p_1 \beta { {i}_0} { s_0})$, $\log ( p_1 \beta { {i}_0} { s_0} + p_2 \gamma { {i}_0})$, and $\log (p_2 \gamma { {i}_0} )$, for $l = 1, 2, 3$, respectively. In all the figures, $p_1 = 0.5$, $p_2 =$, $\gamma = 1$, $T = 23$, ${s}_0 = \textbf{s}_{0N} = 0.96$, ${i}_0 = \textbf{i}_{0N} = 0.04$, and ${r}_0 = \textbf{r}_{0N} = 0$. In Figs. Figs. 3a, 3c, and 3e, $\beta = 0.95$. In Figs. 3a, 3c, and 3e, $\beta = 1.3$

Figure 4.  Comparison of field data (solid black line) for daily confirmed case percentages with 30 realisations of the stochastic SIR model for Churchill County, NV and the Diamond Princess Cruise Ship

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