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A threshold theorem for the general epidemic in discrete time.

F de Hoog, J Gani, D J Gates

Journal of Mathematical Biology

|September 1, 1979

Summary

A discrete time epidemic model theorem, similar to the continuous time Threshold Theorem, was proven. This model allows for all susceptible individuals to become infected in a finite time, differing from continuous models.

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Area of Science:

Epidemiology
Mathematical Biology
Discrete Dynamical Systems

Background:

The Threshold Theorem is a fundamental concept in continuous-time epidemic modeling, notably established by Kermack and McKendrick.
Understanding epidemic dynamics in discrete time is crucial for analyzing real-world disease spread scenarios.
Previous discrete models often exhibited different behaviors compared to their continuous counterparts.

Purpose of the Study:

To establish a discrete time analogue of the continuous time Threshold Theorem.
To investigate the behavior of a specific discrete time epidemic model.
To identify conditions under which total population infection can occur in finite time.

Main Methods:

Development of a discrete time epidemic model.
Mathematical proof of a theorem analogous to the continuous time Threshold Theorem.
Analysis of model solutions to determine population-level infection dynamics.

Main Results:

A theorem analogous to the continuous time Threshold Theorem was successfully proven for the discrete time model.
The discrete time model exhibits solutions where the entire susceptible population can become infected within a finite time.
This finite-time total infection is a key distinction from typical continuous-time epidemic models.

Conclusions:

The established discrete time Threshold Theorem provides new insights into epidemic dynamics.
The finite-time total infection phenomenon highlights unique characteristics of discrete epidemic models.
This research contributes to the broader understanding of infectious disease modeling and control strategies.

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