Unifying incidence and prevalence under a time-varying general branching process

Area of Science:

• Epidemiology
• Mathematical Biology
• Infectious Disease Dynamics

Background:

• Renewal equations are standard for modeling infectious disease incidence.
• Existing models often assume constant parameters, limiting their applicability to real-world outbreaks with changing dynamics.

Purpose of the Study:

• To develop a flexible stochastic model for infectious disease outbreaks accommodating time-varying parameters.
• To derive and analyze renewal-like integral equations for incidence and prevalence within this new framework.
• To validate the model's ability to estimate transmission rates using historical and current epidemiological data.

Main Methods:

• Developed a time-varying Crump-Mode-Jagers branching process model.
• Derived renewal-like integral equations for incidence, cumulative incidence, and prevalence.
• Analyzed specific cases including Bellman-Harris and inhomogeneous Poisson processes.
• Implemented a numerical discretization scheme for solving the derived equations.

Main Results:

• The derived equations for incidence and prevalence align with the back-calculation relationship.
• The model's incidence equations are consistent with widely used renewal equations in infectious disease modeling.
• Successfully estimated SARS-CoV-2 transmission rates in the UK and historical data for Influenza, Measles, SARS, and Smallpox.

Conclusions:

• The developed stochastic branching process model provides a robust framework for analyzing infectious disease outbreaks with time-varying transmission dynamics.
• The derived integral equations and numerical methods enable accurate estimation of transmission rates from epidemiological data.
• This approach enhances our understanding of disease spread and informs public health interventions.