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Semi-Parametric Estimation of Incubation and Generation Times by Means of Laguerre Polynomials

Alexander Kreiss, Ingrid Van Keilegom

Abstract

In epidemics many interesting quantities, like the reproduction number, depend on the incubation period (time from infection to symptom onset) and/or the generation time (time until a new person is infected from another infected person). Therefore, estimation of the distribution of these two quantities is of distinct interest. However, this is a challenging problem since it is normally not possible to obtain precise observations of these two variables. Instead, in the beginning of a pandemic, it is possible to observe for infection pairs the time of symptom onset for both people as well as a window for infection of the first person (e.g. because of travel to a risk area). In this paper we suggest a simple semi-parametric sieve-estimation method based on Laguerre-Polynomials for estimation of these distributions. We provide detailed theory for consistency and illustrate the finite sample performance for small datasets via a simulation study.

Semi-Parametric Estimation of Incubation and Generation Times by Means of Laguerre Polynomials

Abstract

In epidemics many interesting quantities, like the reproduction number, depend on the incubation period (time from infection to symptom onset) and/or the generation time (time until a new person is infected from another infected person). Therefore, estimation of the distribution of these two quantities is of distinct interest. However, this is a challenging problem since it is normally not possible to obtain precise observations of these two variables. Instead, in the beginning of a pandemic, it is possible to observe for infection pairs the time of symptom onset for both people as well as a window for infection of the first person (e.g. because of travel to a risk area). In this paper we suggest a simple semi-parametric sieve-estimation method based on Laguerre-Polynomials for estimation of these distributions. We provide detailed theory for consistency and illustrate the finite sample performance for small datasets via a simulation study.

Paper Structure

This paper contains 17 sections, 11 theorems, 76 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Methodology
  4. Theory
  5. Discussion of Theorem \ref{['thm:consistency']}
  6. Proof of Theorem \ref{['thm:consistency']}
  7. Empirical Studies
  8. Simulation Study
  9. Real Data Application
  10. Conclusion and Related Questions
  11. Appendix
  12. A Consistency Result and Approximation Through Laguerre Polynomials
  13. Further Empirical Results
  14. Simulation Study
  15. Real Data Application
  16. ...and 2 more sections

Key Result

Lemma 3.1

For any $m\in\mathbb{N}_0$ and any $\theta\in\mathbb{R}^{m+1}$ with $\|\theta\|_2=1$ we have that $\varphi_{\theta}$ as defined in eq:def_phi_theta is a density function on $[0,\infty)$. Suppose moreover that $\varphi$ is an arbitrary density on $[0,\infty)$ such that $p(x):=\sqrt{e^x\varphi(x)}$ is Then, for any sequence $m_n\to\infty$ there are $\theta_n\in\mathbb{R}^{m_n+1}$ with $\|\theta_n\|=

Figures (13)

  • Figure 1: Schematic depiction of the important quantities and their relation.
  • Figure 2: Best approximations of $\varphi_I$ (left) and $\varphi_G$ (right) through Laguerre densities of the form \ref{['eq:def_phi_theta']} for various choices of $m_1$ and $m_2$.
  • Figure 3: Estimation results for simulated data with $n=40$ observations. True densities are shown as solid blue lines and the dashed lines show the Laguerre densities which come closest to the true densities. The shaded areas show the point-wise simulated confidence areas from $99\%$ (lightest gray) over $95\%$ in $5\%$-steps to $60\%$ in black.
  • Figure 4: Histograms of squared Hellinger distances of estimates to true densities.
  • Figure 5: Histograms of estimated basic reproduction numbers of fictional pandemic. The dotted line indicates the true $R_0$ and the solid line is the density of a normal distribution with the sample mean and standard deviation.
  • ...and 8 more figures

Theorems & Definitions (26)

  • Remark 2.1
  • Remark 2.2: Serial Interval vs. Generation Time
  • Remark 2.3: Asymptomatic Patients
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • ...and 16 more