3 times t$_{r}$ ≪ t ≪ W − 1 e , where t$_{r}$ is the characteristic relax- ation time for the noise-free motion described by Eq. (3). We note that, even though the state x$_{S}$ is a saddle point in the mean-field approximation, it differs from the saddle-point states encountered in the problem of switching between metastable states of reaction systems. In the case of interstate switching, the rates of elemen- tary transitions W ( X , r ) in the unstable direction are nonzero, and ultimately fluctuations drive the system away from the saddle point. In the case of extinction, fluctuations around x$_{S}$ occur only in the extinction hy- perplane, whereas the probability of exiting this hyper- plane is zero. extinction rate, W$_{e}$ ∝ exp( −Q ), at N ≫ 1. This bar- rier is entropic in nature, as it results from an unlikely sequence of elementary transitions. It can be found by either solving the mean first passage time problem for reaching x$_{S}$ ( t ) [7–9] or by calculating the tail of the quasi- stationary probability distribution P ( X , t ) for x close to x$_{S}$ ( t ) [11, 13, 14]. Here we choose the latter strategy and determine, to the leading order in N , the logarithm of the distribution tail. We seek the solution of Eq. (1) in eikonal form, P ( X , t ) = exp[ − Ns ( x , t )] [28–30]. In the limit of large N , from Eq. (1) we obtain the following equation for s ( x , t ): Here, we have taken into account that, typically, | r | ≪ N , and W ( X , r , t ) depends on X polynomially, whereas P is exponential in X . Therefore we expanded P ( X + r , t ) ≈ P ( X , t ) exp( − r ∂$_{x}$S ) and replaced, to the leading order in 1 /N , w ( x − N − $^{1}$r , r , t ) by w ( x , r , t ). If the system has, in the mean-field approximation, more than one steady state away from the extinction hy- perplane, extinction can go in steps: from the endemic state to another steady state and, ultimately to the ex- tinction hyperplane. In particular, if the only additional steady state is a saddle point at the boundary between the basins of attraction of x$_{A}$ and x$_{S}$ , the problem of extinction can be reduced to the problem of escape over this saddle point [23]. Equation (4) has the form of the Hamilton-Jacobi equation for an auxiliary Hamiltonian system with Hamiltonian H ( x , p , t ); s ( x , t ) is the action of this sys- tem. The Hamilton equations of motion are The above picture can be extended to the case where the transition rates are periodic functions of time, W ( X , r , t + T ) = W ( X , r , t ). Periodicity of some of the rates in time is a natural way of modeling sea- sonal variations of epidemics [18]. The attracting so- lution of Eqs. (3), which describes the endemic state, is no longer stationary. We will assume that this solu- tion, x$_{A}$ ( t ), is periodic in time with the same period T , x$_{A}$ ( t + T ) = x$_{A}$ ( t ). The asymptotic disease extinction state x$_{S}$ ( t ) is also periodic in time; it lies in the hyper- plane x$_{E}$ = 0. These trajectories determine, in turn, the most probable, or optimal, trajectories that the system follows in a fluc- tuation to a given state x at time t . We will calculate action s ( x , t ) using these trajectories and thus find the exponent in the distribution P ( X , t ). The most probable fluctuation which causes extinction of the disease corresponds to a transition from x$_{A}$ ( t ) to x$_{S}$ ( t ) [20]. An important characteristic of such a tran- sition is the period-averaged disease extinction rate W$_{e}$ . It can be introduced if the modulation period T ≪ W − 1 e and, in addition, t$_{r}$ ≪ W − 1 e . In this case, for time t such that t$_{r}$, T ≪ t ≪ W − 1 e , a quasi-stationary time-periodic probability distribution is formed, centered at x$_{A}$ ( t ). The probability current from x$_{A}$ ( t ) to x$_{S}$ ( t ) is also periodic in time, and the period-averaged value of this current gives W$_{e}$ [20], in a direct analogy with the problem of switch- ing between metastable states in noise-driven dynamical systems [24–27]. To find the boundary conditions for Hamiltonian tra- jectories (5), we note that the quasi-stationary distribu- tion P ( X , t ) has a Gaussian maximum at X$_{A}$ ( t ). This means that, close to attractor x$_{A}$ ( t ), action s ( x , t ) is quadratic in | x − x$_{A}$ ) | for stationary systems, whereas for periodically modulated systems s ( x , t ) = s ( x , t + T ) is quadratic in the distance from trajectory x$_{A}$ ( t ) [31]. On the Hamiltonian trajectories that give such action, the momentum p ≡ ∂$_{x}$s → 0 for x → x$_{A}$ ( t ), and since x = x$_{A}$ ( t ) , p = 0 is a fixed point (a periodic trajectory) of the Hamiltonian dynamics, the trajectories of interest start at t → −∞ , In the Lagrangian L , Eq. (6), p should be expressed in terms of ˙ x , x using Eq. (5). Since w ≥ 0, we have L ≥ 0. We will be interested in evaluating the disease extinc- tion barrier Q which gives the exponent in the disease B. Eikonal approximation 1. Equations of motion 2. Boundary conditions for the optimal extinction trajectory 2. Periodically modulated systems ∂$_{t}$s = − H ( x , ∂$_{x}$s, t ) , (4) H ( x , p , t ) = ∑ r w ( x , r , t ) [exp( pr ) − 1] . ˙ x = ∑ $_{r}$r w ( x , r , t ) e $^{pr}$, ˙ p = − r ∂$_{x}$w ( x , r , t ) ( e pr − 1) . (5) s ( x , t ) = ∫ t −∞ dt L ( ˙ x , x , t ) , (6) L ( ˙ x , x , t ) = ∑ r w ( x , r , t ) [( pr − 1) e pr + 1] .