Are the beginning and ending phases of epidemics provided by next generation matrices? -- Revisiting drug sensitive and resistant tuberculosis model

TL;DR

This paper revisits tuberculosis models to analyze how next generation matrices can predict epidemic initiation and progression, focusing on the reproduction number and susceptible fraction.

Contribution

It demonstrates how next generation matrices can be used to understand both the start and the progression of tuberculosis epidemics, including drug resistance.

Findings

Reproduction number calculated from next generation matrix indicates epidemic onset.

Fraction of susceptible individuals predicts epidemic progression.

The approach applies to both drug sensitive and resistant tuberculosis models.

Abstract

In epidemiological modelings, the spectral radius of the next generation matrix evaluated at the trivial equilibrium was considered as the basic reproduction number. Also, the global stability of the trivial equilibrium point was determined by the left eigenvector associated to that next generation matrix. More recently, the fraction of susceptible individuals was also obtained from the next generation matrix. By revisiting drug sensitive and resistant tuberculosis model, the gross reproduction number and the fraction of susceptible individuals are calculated. Hence, the next generation matrices shed light to the evolution of the dynamics: the beginning of the epidemics via the reproduction number and the approaching to the epidemics level via the asymptotic fraction of susceptible individuals.