Exact solution of a stochastic susceptible-infectious-recovered model

The susceptible-infectious-recovered (SIR) model describes the evolution of three species of individuals which are subject to an infection and recovery mechanism. A susceptible $S$ can become infectious with an infection rate $\beta$ by an infectious $I$ type provided that both are in contact. The $I$ type may recover with a rate $\gamma$ and from then on stay immune. Due to the coupling between the different individuals, the model is nonlinear and out of equilibrium. We adopt a stochastic individual-based description where individuals are represented by nodes of a graph and contact is defined by the links of the graph. Mapping the underlying master equation onto a quantum formulation in terms of spin operators, the hierarchy of evolution equations can be solved exactly for arbitrary initial conditions on a linear chain. In the case of uncorrelated random initial conditions, the exact time evolution for all three individuals of the SIR model is given analytically. Depending on the initial conditions and reaction rates $\beta$ and $\gamma$, the $I$ population may increase initially before decaying to zero. Due to fluctuations, isolated regions of susceptible individuals evolve, and unlike in the standard mean-field SIR model, one observes a finite stationary distribution of the $S$ type even for large population size. The exact results for the ensemble-averaged population size are compared with simulations for single realizations of the process and also with standard mean-field theory, which is expected to be valid on large fully connected graphs.

Figure 1

Time evolution of susceptibles $n_S$ and infectious $n_I$ individuals. Whereas the bold solid line corresponds to the simulation of $n_S\left(t\right)$, the bold dashed line represents the exact solution. The corresponding thin lines stand for $n_I\left(t\right)$. Different rates $\beta$ and $\gamma$ and different initial conditions are shown in (a) and (b). The population size is in both cases $N={10}^3$.

Figure 2

Time evolution of susceptibles and infectious individuals for different population size: The rates are in both cases $\beta =0.9$ and $\gamma =0.1$; the initial values are $n_S\left(0\right)=0.8$ and $n_I\left(0\right)=0.2$. The bold line (solid and dashed) corresponds to $n_S\left(t\right)$, and the thin ones represent $n_I\left(t\right)$.