Inference of a Susceptible-Infectious stochastic model

Abstract

We consider a time-inhomogeneous diffusion process able to describe the dynamics of infected people in a susceptible-infectious epidemic model in which the transmission intensity function is time-dependent. Such a model is well suited to describe some classes of micro-parasitic infections in which individuals never acquire lasting immunity and over the course of the epidemic everyone eventually becomes infected. The stochastic process related to the deterministic model is transformable into a non homogeneous Wiener process so the probability distribution can be obtained. Here we focus on the inference for such process, by providing an estimation procedure for the involved parameters. We point out that the time dependence in the infinitesimal moments of the diffusion process makes classical inference methods inapplicable. The proposed procedure is based on Generalized Method of Moments in order to find suitable estimate for the infinitesimal drift and variance of the transformed process. Several simulation studies are conduced to test the procedure, these include the time homogeneous case, for which a comparison with the results obtained by applying the MLE is made, and cases in which the intensity function are time dependent with particular attention to periodic cases. Finally, we apply the estimation procedure to a real dataset.

Figures (6)

• Figure 1: On the top: Box plot of MLE and GMM estimates of the parameters $\lambda$ (on the left) and of $\sigma^2$ (on the right) based on 500 replicates. On the bottom: Gaussian kernel density estimates of $\lambda$ (on the left) and of $\sigma^2$ (on the right). The smoothing bandwidth is also indicated. Black curve is the GMM density, while red curve is the MLE density.
• Figure 2: Estimates of $\lambda(t)$ (on the left) and $\sigma^2(t)$ (on the rigth) for $\lambda(t)=0.4+\sin t$ and $\sigma^2(t)=\sigma^2=0.1$. The red curve is the true function, the red curve is the mean of the 500 obtained estimates. The black lines define the observed confidence interval.
• Figure 3: As in Figure \ref{['Fig2']} with $\lambda(t)=0.4+\sin t,$ and $\sigma^2(t)=0.1+0.01(1-e^{2 t})^2$.
• Figure 4: As in Figure \ref{['Fig2']} with $\lambda(t)=0.4$ and $\sigma^2(t)=0.01(1.2+\sin t)$.
• Figure 5: Sample paths of the infected population (on the left) and related normalized sample paths (on the right).