Control of rare events in reaction and population systems by deterministically imposed transitions

We consider control of reaction and population systems by imposing transitions between states with different numbers of particles or individuals. The transitions take place at predetermined instants of time. Even where they are significantly less frequent than spontaneous transitions, they can exponentially strongly modify the rates of rare events, including switching between metastable states or population extinction. We also study optimal control of rare events. Specifically, we are interested in the optimal control of disease extinction for a limited vaccine supply. A comparison is made with control of rare events by modulating the rates of elementary transitions rather than imposing transitions deterministically. It is found that, unexpectedly, for the same mean control parameters, controlling the transitions rates can be more efficient.

Figure 1

Two types of transitions with a change of the number of molecules or individuals $\mathbf{X}\to \mathbf{X}+\mathbf{r}$. Spontaneous transitions happen at random and are characterized by rates $W(\mathbf{X},\mathbf{r},t)$. In addition, transitions may be imposed externally in a deterministic way. We assume that the duration of a transition is small compared to the interval between successive transitions, which is necessary for the system to be described by vector $\mathbf{X}$ with integer-valued components.

Figure 2

(a) The Fourier transform of the logarithmic susceptibility with respect to deterministic vaccination for the SVIR model, Eq. (25). The parameters are $\beta /\mu =80$ and $\gamma /\mu =50$. The susceptibility spectrum displays a sharp peak at the characteristic frequency ${\omega }_0$ of decaying oscillations near the mean-field endemic state. (b) The change of the scaled extinction barrier $s^{\prime }=\mu s_{\mathrm{ext}}^{(1)}/\Xi$ with vaccination period $T$. The solid line shows $s^{\prime }$ where there is no limit on vaccine accumulation. The dashed lines refer to the case of a limited amount of accumulated vaccine $M$, where the actual mean speed of vaccination is ${\Xi }_a=min(\Xi ,M/T)$. The scaled accumulation limit is $M^{\prime }=\mu M/\Xi$. The locations of resonances of $s^{\prime }$ are independent of $M$.