[Paper Review] Unifying incidence and prevalence under a time-varying general branching process

Mikko S. Pakkanen, Xenia Miscouridou+6 more|arXiv (Cornell University)|Jul 12, 2021

COVID-19 epidemiological studies4 citations

TL;DR

This paper unifies incidence and prevalence in infectious disease modeling through a time-varying general branching process, extending the Crump–Mode–Jagers framework to allow for time-varying reproduction numbers and generation intervals. It derives consistent integral equations for incidence, cumulative incidence, and prevalence that align with the back-calculation relationship and recover the standard renewal equation as a special case, enabling joint estimation of transmission dynamics from both incidence and seroprevalence data.

ABSTRACT

Renewal equations are a popular approach used in modelling the number of new infections, i.e., incidence, in an outbreak. We develop a stochastic model of an outbreak based on a time-varying variant of the Crump-Mode-Jagers branching process. This model accommodates a time-varying reproduction number and a time-varying distribution for the generation interval. We then derive renewal-like integral equations for incidence, cumulative incidence and prevalence under this model. We show that the equations for incidence and prevalence are consistent with the so-called back-calculation relationship. We analyse two particular cases of these integral equations, one that arises from a Bellman-Harris process and one that arises from an inhomogeneous Poisson process model of transmission. We also show that the incidence integral equations that arise from both of these specific models agree with the renewal equation used ubiquitously in infectious disease modelling. We present a numerical discretisation scheme to solve these equations, and use this scheme to estimate rates of transmission from serological prevalence of SARS-CoV-2 in the UK and historical incidence data on Influenza, Measles, SARS and Smallpox.

Motivation & Objective

• To unify the modeling of incidence and prevalence under a single stochastic framework that ensures consistency between these key epidemiological quantities.
• To extend classical renewal equations to accommodate time-varying reproduction numbers and non-exponential generation intervals.
• To provide a mathematically rigorous foundation linking individual-based branching processes to governing equations like the renewal equation.
• To enable joint inference of transmission dynamics using both incidence and prevalence data, such as seroprevalence from SARS-CoV-2 studies.
• To develop a numerically efficient discretization scheme for solving the derived integral equations in practice.

Proposed method

• Formulates a time-varying general branching process based on the Crump–Mode–Jagers framework, allowing random infection times and time-varying transmission rates.
• Derives integral equations for incidence, cumulative incidence, and prevalence using the first and second moments of the branching process.
• Establishes consistency between incidence and prevalence through the back-calculation relationship, ensuring temporal coherence.
• Applies the framework to two specific cases: the Bellman–Harris process (age-dependent) and an inhomogeneous Poisson process model of transmission.
• Develops a numerical discretization scheme using matrix algebra to solve the two-dimensional integral equations efficiently.
• Validates the approach by showing that incidence equations reduce to the standard renewal equation under fixed initial infection time.

Experimental results

Research questions

• RQ1 Can a single stochastic branching process framework consistently model both incidence and prevalence while accommodating time-varying transmission dynamics?
• RQ2 How do the derived integral equations for incidence and prevalence relate to the classical back-calculation method used in epidemiology?
• RQ3 To what extent do the incidence equations from the Bellman–Harris and inhomogeneous Poisson process models recover the standard renewal equation?
• RQ4 Can the framework jointly estimate transmission rates from both incidence and seroprevalence data, as demonstrated in real-world applications?
• RQ5 What is the computational feasibility of solving the more general two-dimensional integral equations compared to the standard one-dimensional renewal equation?

Key findings

• The derived integral equations for incidence and prevalence are mathematically consistent with the back-calculation relationship, ensuring coherence between observed and inferred transmission dynamics.
• The incidence equations from both the Bellman–Harris and inhomogeneous Poisson process models reduce to the standard renewal equation when the initial infection time is fixed, validating their consistency with established models.
• The framework successfully accommodates time-varying reproduction numbers and non-exponentially distributed generation intervals, offering greater biological realism than standard renewal models.
• Numerical simulations show strong agreement between Monte Carlo averages and theoretical means, confirming the accuracy of the derived equations.
• The proposed discretization scheme enables efficient computation of the general integral equations using standard matrix operations, making the method computationally viable despite its increased generality.
• The method enables joint estimation of transmission rates from both incidence and seroprevalence data, as demonstrated in applications to SARS-CoV-2, influenza, measles, SARS, and smallpox.