Approximating reproduction numbers: a general numerical method for age-structured models

TL;DR

A general numerical method to approximate the reproduction numbers of a large class of multi-group, age-structured, population models with a finite age span and an equivalent formulation for the age-integrated state within the extended space framework is proposed.

Abstract

In this paper, we introduce a general numerical method to approximate the reproduction numbers of a large class of multi-group, age-structured, population models with a finite age span. To provide complete flexibility in the definition of the birth and transition processes, we propose an equivalent formulation for the age-integrated state within the extended space framework. Then, we discretize the birth and transition operators via pseudospectral collocation. We discuss applications to epidemic models with continuous and piecewise continuous rates, with different interpretations of the age variable (e.g., demographic age, infection age and disease age) and the transmission terms (e.g., horizontal and vertical transmission). The tests illustrate that the method can compute different reproduction numbers, including the basic and type reproduction numbers as special cases.

Figures (6)

• Figure 4.1: Model \ref{['TSI']}. Left: $R_c$ varying the multiplicative parameter $R_0$, for different values of the fraction of symptomatic individuals ($\epsilon$) and assuming no delay from symptoms to diagnosis ($D=0$). Right: $R_c$ varying $\epsilon$ ($x$-axis) and $D$ ($y$-axis), for $R_0 = 1.2$.
• Figure 4.3: Model \ref{['multistrain']}. Basic reproduction numbers $R_{0,1}$ (left) and $R_{0,2}$ (right) varying the parameters $c_1, c_2, m_1$ and $m_2$. When not varied, the parameters are fixed at: $c_1=1$, $c_2 = 0.06$, $m_1 = 0.1$, and $m_2 = 0.06$.
• Figure 4.5: Model \ref{['TSI']}. Log-log plot of the absolute error of approximation for $R_{0,1}\approx 5.24$ (blue) and $R_{0,2}\approx 16.88$ (red) for increasing $N$ with $c_1=1,\ c_2=0.06,\ m_1=0.1,\ m_2=0.06$.
• Figure 4.6: Model \ref{['asymptomatic']}. Left: $R_0$ and $T_S$ as functions of $r:= b_{11}/b_{21}$. Right: $T_S$ and the spectral radius of the NGO relevant to the asymptomatic individuals as functions of $r$.
• Figure 4.7: Model \ref{['rubella']}. Estimated WAIFW matrices ($k_{ij}$) for case a (upper row) and case b (lower row). Numerical values reported in \ref{['tab:rubellaKi']}.