Nonequilibrium statistical mechanics of a susceptible-infected-recovered epidemic model

The Liouville and stochastic Liouville equations have been investigated with respect to the susceptible-infected-recovered (SIR) epidemic model, described in terms of extended Nambu mechanics. It has been shown that, along with the equilibrium distribution functions, the Kubo formula can be derived for the SIR model, although the theory undergoes a few restrictions imposed by the nonlinear nature of the model.

Figure 1

Schematics of the $\sqrt{S}-\sqrt{I}-\sqrt{R}$ phase space. The solid circles and solid curves represent the particles and their trajectories, respectively. (a) The basic SIR model [Eq. (1)], (b) the many-body system in which particles on different spheres interact (dotted arrow), (c) the system involving fluctuation in the radius of the sphere, and (d) the system induced by an external field.

Figure 2

(a) Distribution functions for $S$, scaled with the peak height. The parameter values are $\beta =0.125$, $\gamma =0.10$, $N=10000$, and $D=50.0$. The numerical calculations are represented by circles and triangles for the solution of Eq. (9) and Eq. (10), respectively. The broken and solid curves denote Eq. (14) and the same equation with $D$ replaced by $\tilde{D}$, respectively. The inset depicts the enlarged view of the triangle and broken-line plots. (b) Values of the correction factor resulting from the nonlinearity, $\tilde{D}/D$. The solid curves and broken lines are the guide to eyes, representing that $\gamma$ and $\gamma /\beta$ are constant, respectively. The values of $\gamma$ and $\gamma /\beta$ are denoted by the appended numbers.