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

Hyun Mo Yang|arXiv (Cornell University)|Jun 11, 2020

Mathematical and Theoretical Epidemiology and Ecology Models9 references5 citations

TL;DR

This paper revisits a drug-sensitive and drug-resistant tuberculosis transmission model to demonstrate that next generation matrices can simultaneously predict both the initial outbreak intensity (via the gross reproduction number) and the final epidemic size (via the asymptotic fraction of susceptible individuals). By constructing next generation matrices in different ways, the study shows that the sum of coefficients in the characteristic equation yields the gross reproduction number, while the spectral radius relates to the final susceptible fraction—offering a unified framework for modeling both epidemic onset and end phases.

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.

Motivation & Objective

• To investigate whether next generation matrices can describe both the beginning and ending phases of epidemics in a single framework.
• To resolve the discrepancy between the basic reproduction number (R₀) and the final susceptible fraction (s*) in models with multiple transmission routes.
• To demonstrate that different constructions of the next generation matrix yield distinct thresholds: one for epidemic initiation (gross reproduction number) and one for epidemic termination (asymptotic susceptible fraction).
• To validate the theoretical framework using a drug-sensitive and drug-resistant tuberculosis transmission model with complex transmission dynamics.
• To clarify the role of characteristic equation coefficients and spectral radius in determining epidemic thresholds, especially when multiple transmission pathways exist.

Proposed method

• Revisits a previously numerically analyzed drug-resistant tuberculosis model with compartments for susceptible (s), exposed (e₁, e₂), and infectious (i₁, i₂) individuals.
• Constructs two distinct next generation matrices: one emphasizing transmission routes (FV⁻¹) and another for stability analysis via the spectral radius.
• Applies the method of summing coefficients of the characteristic equation of the next generation matrix to derive the gross reproduction number (Rg = R₀ + Ra).
• Uses the spectral radius of the next generation matrix to determine the global stability of the disease-free equilibrium and relates it to the inverse of the final susceptible fraction.
• Compares results from different matrix constructions to show that Rg ≠ 1/s* when multiple transmission routes exist, unlike in simple SIR models.
• Validates findings by comparing with known results from SEIR and dengue models with transovarial transmission, where s* = 1/R₀ holds only under single-route transmission.

Experimental results

Research questions

• RQ1 Can next generation matrices simultaneously predict the initial intensity and final size of an epidemic?
• RQ2 How do different constructions of the next generation matrix affect the interpretation of epidemic thresholds?
• RQ3 Why does the relationship s* = 1/R₀ break down in models with multiple transmission routes?
• RQ4 What is the role of the sum of coefficients in the characteristic equation of the next generation matrix compared to the spectral radius?
• RQ5 Under what conditions does the gross reproduction number Rg correspond to the final susceptible fraction s*?

Key findings

• The sum of the coefficients of the characteristic equation of the next generation matrix yields the gross reproduction number Rg, which captures the initial epidemic potential.
• The spectral radius of the next generation matrix corresponds to the inverse of the asymptotic fraction of susceptible individuals, s*⁻¹, thus predicting the final epidemic size.
• In models with multiple transmission routes (e.g., drug-sensitive and drug-resistant TB), Rg ≠ 1/s*, invalidating the simple s* = 1/R₀ relationship.
• When only one transmission route exists, the standard relationship s* = 1/R₀ is recovered, confirming consistency with classical SIR models.
• Different matrix constructions lead to different thresholds: one for epidemic onset (via coefficient sum) and one for epidemic ending (via spectral radius).
• The method of summing coefficients provides a unified approach to derive both epidemic thresholds, especially in complex models with multiple infection pathways.