On the Reproduction Number of a Gut Microbiota Model

Abstract

A spatially structured linear model of the growth of intestinal bacteria is analysed from two generational viewpoints. Firstly, the basic reproduction number associated with the bacterial population, i.e. the expected number of daughter cells per bacterium, is given explicitly in terms of biological parameters. Secondly, an alternative quantity is introduced based on the number of bacteria produced within the intestine by one bacterium originally in the external media. The latter depends on the parameters in a simpler way and provides more biological insight than the standard reproduction number, allowing the design of experimental procedures. Both quantities coincide and are equal to one at the extinction threshold, below which the bacterial population becomes extinct. Optimal values of both reproduction numbers are derived assuming parameter trade-offs.

A practical approach to R0 in continuous‐time ecological models

“…As it has been already pointed out, the classification into birth terms (B) and nonbirth terms (M) is not uniquely determined, and so the resulting expression of the basic reproduction number  0 depends on what is considered as a birth event; see previous studies. 1,5,6 As the extinction threshold, one can use either the basic reproduction number or the exponential growth rate of the population (Malthusian parameter), but we prefer the former because it typically results in a more explicit expression. This statement is particularly true when the birth operator B has a finite-dimensional range since then the computation of  0 reduces to an eigenvalue problem for matrices (see Section 3).…”

“…Again, from the second term in (6) we define the next-generation operator as B ∫ ∞ 0 T( ) d = BM −1 , which turns out to be a positive-bounded linear operator on X. Next, we can compute the basic reproduction number for model 4 as the spectral radius of the next-generation operator 3 :…”

“…7 Notice that the derivation of  0 for structured models of type I is based on the fact that Be −M , with ∈ X such that || || X = 1, gives the expected density of newborns produced per unit of time by each individual born units of time ago. See Barril et al 6 for an example of a structured model of type I, ie, system 4 with specific operators B and M, where the standard and an alternative basic reproduction numbers are given. Let us briefly recall that the Malthusian parameter for the infinite-dimensional system 4 is the spectral bound of the complete generator r = s(B − M) and that the well-known relation between r and  0 holds: s(B − M) has the same sign as (BM −1 ) − 1; see previous studies.…”

“…Let us illustrate an infinite-dimensional example in the line of the cell population model introduced in Barril et al 6 Let us consider a population of bacteria proliferating and moving along a very long intestine of an animal host. The evolution of u(x, t), the spatial density of bacteria along the intestine at time t is given by t u(…”

“…Here, the classification into birth and nonbirth terms is not uniquely determined, and so different interpretations of what a birth event is give rise to different expressions and results for R 0 . 1,5,6 Typically, the second approach results in a more explicit expression allowing the study of evolutionary issues of the model, and thus, it provides more biological insight. Both approaches give a threshold determining the asymptotic behaviour of the solutions, either an ever-growing population or the extinction of population, for the generic irreducible case.…”