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Updated: Nov 11, 2025

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Epidemic models with discrete state structures.

Suli Liu1, Michael Y Li2

  • 1School of Mathematics, Jilin University, Changchun, Jilin Province, 130012, China.

Physica D. Nonlinear Phenomena
|March 30, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces a new epidemic model where infected individuals can switch between different infectivity states. The basic reproduction number determines if the disease dies out or persists, impacting disease control strategies.

Keywords:
Basic reproduction numberCOVID-19 pandemicEpidemic modelsGlobal stabilityState of infectionsState structures

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Area of Science:

  • Mathematical Epidemiology
  • Infectious Disease Dynamics
  • Network Theory

Background:

  • Infectious disease states reflect infectivity, susceptibility, and immunity, crucial for understanding disease progression.
  • Long disease progressions, like HIV, involve dynamic shifts between different states of infectivity.
  • Existing models often simplify the complexity of state transitions in infectious diseases.

Purpose of the Study:

  • To develop a novel epidemic model incorporating discrete infectivity states and transitions for infected individuals.
  • To analyze the role of the transmission-transfer network in disease dynamics.
  • To investigate the impact of state structures on disease persistence and control.

Main Methods:

  • Derivation of a compartmental epidemic model with discrete infectivity states and state-switching.
  • Incorporation of a general incidence function for new infections across different disease states.
  • Mathematical analysis of the basic reproduction number (R₀) as a threshold parameter.
  • Global stability analysis of disease-free and endemic equilibria under specific incidence conditions.

Main Results:

  • The basic reproduction number (R₀) acts as a sharp threshold, determining disease extinction or persistence.
  • If R₀ < 1, the disease-free equilibrium is globally asymptotically stable, leading to disease eradication.
  • If R₀ > 1, the disease-free equilibrium is unstable, and the disease becomes uniformly persistent.
  • A unique endemic equilibrium exists and is globally asymptotically stable for a restricted class of incidence functions when R₀ > 1.
  • Analysis reveals how different state structures influence R₀, equilibrium distributions, and control effectiveness.

Conclusions:

  • The developed model accurately captures the complexity of state transitions in infectious diseases.
  • The basic reproduction number is a critical determinant of disease persistence and epidemiological outcomes.
  • Understanding transmission-transfer networks and state structures is vital for effective disease control and prevention strategies, with implications for pandemics like COVID-19.