Epidemic spreading

TL;DR

This paper analyzes six deterministic epidemic models using differential equations, focusing on the basic reproduction number and critical thresholds for outbreak, highlighting their relation to phase transitions and stability analysis.

Contribution

It introduces a unified analysis of six epidemic models, emphasizing the eigenvalue approach to the basic reproduction number and the phase transition analogy.

Findings

The basic reproduction number relates to the largest eigenvalue of the stability matrix.

Epidemic outbreaks occur at a critical population density, akin to phase transitions.

Models show how immunity affects epidemic dynamics.

Abstract

We present an analysis of six deterministic models for epidemic spreading. The evolution of the number of individuals of each class is given by ordinary differential equations of the first order in time, which are set up by using the laws of mass action providing the rates of the several processes that define each model. The epidemic spreading is characterized by the frequency of new cases, which is the number of individuals that are becoming infected per unit time. It is also characterized by the basic reproduction number, which we show to be related to the largest eigenvalue of the stability matrix associated with the disease-free solution of the evolution equations. We also emphasize the analogy between the outbreak of an epidemic with a critical phase transition. When the density of the population reaches a critical value the spreading sets in, a result that was advanced by Kermack and McKendrick in their study of a model in which the recovered individuals acquire permanent immunization, which is one of the models analyzed here.