An Exact SIR Series Solution and an Exploration of the Related Parameter Space

TL;DR

The paper tackles the challenge of solving the SIR epidemic model analytically by constructing a convergent power-series solution via a gauge-based resummation. Beginning from a divergent direct time-series, it introduces a dominant-balance–informed shifted exponential gauge and a shifted gauge variable to produce a robust series for $V = \ln(\xi)$, with $\xi$ tied to the transformed susceptive population; the authors derive recursion relations and analyze the radius of convergence as a function of initial conditions. A key contribution is the identification of a Hershey-Kiss region in the $(\tilde S_0, \tilde I_0)$ plane within which the series converges over the full physical domain, along with a detailed mapping of singularities between the time domain and the gauge domain. The work demonstrates practical semi-analytical performance against RK4 across disease scenarios (Bubonic Plague, Ebola, Covid) and outlines future resummation strategies to enlarge the convergent region while also deepening the understanding of convergence boundaries. Collectively, this provides a framework for efficient, accurate semi-analytical solutions to nonlinear epidemic models and advances the theory of series resummation in dynamical systems.

Abstract

A convergent power series solution is obtained for the SIR model, using an asymptotically motivated gauge function. For certain choices of model parameter values, the series converges over the full physical domain (i.e., for all positive time). Furthermore, the radius of convergence as a function of nondimensionalized initial susceptible and infected populations is obtained via a numerical root test.

Figures (21)

• Figure 1: This intent of this flowchart is to briefly illustrate how the various equations of dynamical variables and their parameters connect to one another.
• Figure 2: For the direct time series contour plot of $\log_{10}(\rho)$ of the power series for $V$ given by (8), $N\boldsymbol{=} 300$ was used. In other words, $\rho\approx A_{300}^{-1/300}$ was used.
• Figure 3: This is a direct nondimensionalized time series comparison for the 1966 bubonic plague which took place in Eyam, England with the parameters $S_{0}\boldsymbol{=} 254$, $I_{0}\boldsymbol{=} 7$, $r\boldsymbol{=} 0.0178$, and $\alpha\boldsymbol{=} 2.73$BarlowWeinsteinKhan. This can be rewritten in the nondimensionalized quantities $(\tilde{S_{0}}, \tilde{I_{0}}) \boldsymbol{=} (1.656117, 0.045641)$.
• Figure 4: The above plots are the series predictions of V for the 1966 Bubonic Plague with $(\tilde{S_{0}}, \tilde{I_{0}}) \boldsymbol{=} (1.656117, 0.045641)$, $N\boldsymbol{=} 10$, $N\boldsymbol{=} 15$, and $N\boldsymbol{=} 35$.
• Figure 5: In the $\log_{10}(\rho)$ contour plot, $N\boldsymbol{=} 1000$ is used, and the fully convergent region (with $\rho\geq 1$) called the Hershey-Kiss region is the area shown in yellow. The blue asterisk is the Bubonic Plague at $(1.656117, 0.045641)$ with $\rho_{bubonic}\approx 1.048927$. With $r\boldsymbol{=} 0.2$, $\alpha\boldsymbol{=} 0.1$, $S_{0}\boldsymbol{=} 0.95$, and $I_{0}\boldsymbol{=} 0.05$Rachah, the red asterisk is Ebola at $(1.9, 0.1)$ with $\rho_{ebola}\approx 1.04929$. With $r\boldsymbol{=} 2.9236\times 10^{-5}$, $\alpha\boldsymbol{=} 0.0164$, $S_{0}\boldsymbol{=} 4206$, and $I_{0}\boldsymbol{=} 2$BarlowWeinsteincovid, Covid-19 in Japan is represented by the green asterisk located at $(7.497964, 0.003565)$ with $\rho_{covid}\approx 0.735302$. Note that since the basic reproductive number $r/alpha$ factors out of the analysis, it is not playing any role in the formation of the Herskey-Kiss region.