Seasonal forcing in stochastic epidemiology models

Abstract

The goal of this paper is to motivate the need and lay the foundation for the analysis of stochastic epidemiological models with seasonal forcing. We consider stochastic SIS and SIR epidemic models, where the internal noise is due to the random interactions of individuals in the population. We provide an overview of the general theoretic framework that allows one to understand noise-induced rare events, such as spontaneous disease extinction. Although there are many paths to extinction, there is one path termed the optimal path that is probabilistically most likely to occur. By extending the theory, we have identified the quasi-stationary solutions and the optimal path to extinction when seasonality in the contact rate is included in the models. Knowledge of the optimal extinction path enables one to compute the mean time to extinction, which in turn allows one to compare the effect of various control schemes, including vaccination and treatment, on the eradication of an infectious disease.

Appendix A: Iterative Action Minimizing Method (IAMM)

To analyze the dynamics of spontaneous escape from an endemic state, we numerically compute the optimal path, which is a zero-energy curve for the Hamiltonian that connects two steady state saddle points. We use the Iterative Action Minimizing Method (IAMM) [38], a numerical scheme based on Newton’s method. The IAMM is useful in the general situation where a path connecting steady states $C_a$ and $C_b$ starts at $C_a$ at $t=-\infty$ and ends at $C_b$ at $t=+\infty$. A time parameter t exists such that $-\infty<t<\infty$. For this method, we require a numerical approximation of the time needed to leave the region of $C_a$ and arrive in the region of $C_b$. Therefore, we define a time $T_{\epsilon }$ such that $-\infty<-T_{\epsilon }<t<T_{\epsilon }<\infty$. Additionally, $C(-T_{\epsilon }) \approx C_a$ and $C(T_{\epsilon }) \approx C_b$. In other words, the solution stays very near the equilibrium $C_a$ for $-\infty <t\le -T_{\epsilon }$, has a transition region from $-T_{\epsilon }<t<T_{\epsilon }$, and then stays near $C_b$ for $T_{\epsilon }<t<+\infty$. The interval $[-T_{\epsilon },T_{\epsilon }]$ is discretized into n segments using a uniform step size $h=(2T_{\epsilon })/n$ or a suitable non-uniform step size $h_k$. The corresponding time series is $t_{k+1}=t_k+h_k$.

The derivative of the function value $\mathbf {q}_k$ is approximated using central finite differences by the operator $\delta _h$ given as

$$\begin{aligned} \frac{d}{dt}\mathbf {q}_k \approx \delta _h\mathbf {q}_k \equiv \frac{h^2_{k-1}\mathbf {q}_{k+1} + (h^2_k - h^2_{k-1})\mathbf {q}_k - h^2_k\mathbf {q}_{k-1} }{h_{k-1}h^2_k + h_kh^2_{k-1}}, \quad k=0,\ldots ,n. \end{aligned}$$

(39)

Clearly, if a uniform step size is chosen then Eq. (39) simplifies to the familiar form given as

$$\begin{aligned} \frac{d}{dt}\mathbf {q}_k \approx \delta _h\mathbf {q}_k \equiv \frac{\mathbf {q}_{k+1} - \mathbf {q}_{k-1} }{2h}, \quad k=0,\ldots ,n. \end{aligned}$$

(40)

Thus, one can develop the system of nonlinear algebraic equations

$$\begin{aligned} \delta _h\mathbf {x}_k - \frac{\partial H(\mathbf {x}_k,\mathbf {p}_k)}{\partial \mathbf {p}}=0, \quad \delta _h\mathbf {p}_k + \frac{\partial H(\mathbf {x}_k,\mathbf {p}_k)}{\partial \mathbf {x}}=0, \quad k=0,\ldots ,n, \end{aligned}$$

(41)

which is solved using a general Newton’s method. We let

$$\begin{aligned} \mathbf {q}_j(\mathbf {x,p})=\{\mathbf {x}_{1,j}...\mathbf {x}_{n,j},\mathbf {p}_{1,j}...\mathbf {p}_{n,j}\}^T \end{aligned}$$

(42)

be an extended vector of 2nN components that contains the $j\mathrm{th}$ Newton iterate, where N is the number of populations. When $j=0$, $\mathbf {q}_0(\mathbf {x,p})$ provides the initial “guess” as to the location of the path that connects $C_a$ and $C_b$. Given the $j\mathrm{th}$ Newton iterate $\mathbf {q}_j$, the new $\mathbf {q}_{j+1}$ iterate is found by solving the linear system

$$\begin{aligned} \mathbf {q}_{j+1}=\mathbf {q}_{j}-\frac{\mathbf {F}\left( \mathbf {q}_{j}\right) }{\mathbf {J}\left( \mathbf {q}_{j}\right) }, \end{aligned}$$

(43)

where $\mathbf {F}$ is the function defined by Eq. (41) acting on $\mathbf {q}_{j}$, and $\mathbf {J}$ is the Jacobian. Equation (43) is solved using LU decomposition with partial pivoting.