Stochastic fluctuations of the transmission rate in the susceptible-infected-susceptible epidemic model

We analyze the dynamics of the susceptible-infected-susceptible epidemic model when the transmission rate displays Gaussian white noise fluctuations around its mean value. We obtain analytic expressions for the final size distribution of infectives and the mean infected number of individuals. The model displays a variety of noise-induced transitions as a function of the basic reproductive rate and the noise intensity. We derive a threshold criterion for epidemic invasion in the presence of external noise and determine the mean epidemic duration and the mean epidemic extinction time.

Figure 1

Intrinsic growth rate as a function of $R_0$ for different values of the noise intensity $D$. Symbols represent the results of numerical simulations and solid curves correspond to (3.7).

Figure 2

Temporal behavior of the fraction of infected individuals $I\left(t\right)/N$ obtained by integrating (2.5) numerically. We plot ten different stochastic realizations in each panel. ${\beta }_0=1$ and $\gamma =0.5$ (i.e., $R_0=2$).

Figure 3

Stationary probability distribution (i.e., final size distribution, of the number of infected individuals for different values of $R_0$ and $\gamma =1)$. The symbols represent the results of the numerical simulations, and the solid curves correspond to the exact analytic results from (5.3) and (5.4).

Figure 4

Mean final size of the epidemic as function of the basic reproductive rate (left panel) and as function of the noise intensity (right panel). Symbols correspond to the results of numerical simulations and solid curves to the exact results (5.6). ($\gamma =1$).

Figure 5

Variance of the number of infected individuals. Solid curves represent the exact analytic result given by (5.7), the broken curves correspond to the approximation (5.8), and the symbols represent the results of numerical simulations. The circles correspond to $\gamma =0.5$ and ${\beta }_0=1$, and the triangles to $\gamma =1$ and ${\beta }_0=2$.

Figure 6

Diagram for the different shapes of the final size distribution.

Figure 7

Mean first passage time. In panels (a), (b), and (c) we plot the mean epidemic duration versus $\gamma$, $D$, and ${\beta }_0$, respectively. In panel (d) we plot the mean extinction time versus ${\beta }_0$. The symbols correspond to the results of numerical simulations, and the solid curves are obtained from the expressions (6.2) and (6.3) for the mean epidemic duration and mean epidemic extinction time, respectively.