*2.1. Brief Review of the R0, the Basic Reproductive Number*

A central concept in the epidemiology of infectious disease transmission is the basic reproductive number: also called the basic reproduction number or basic reproduction ratio. It is the expected number of infected individuals by a single infected individual in a susceptible population. Denoted as *R*0, this number is a measure of the pathogen's fitness but is also a threshold that characterizes the epidemic potential of the disease because: if *R*<sup>0</sup> > 1, the pathogen continues its propagation through susceptible hosts, with the number of infected hosts increasing exponentially; if *R*<sup>0</sup> < 1, the number of infected hosts decreases exponentially, and the pathogen is extinct. The number of infected hosts is stable in the unlikely case where *R0* equals one [18,19].

*R0* is a dimensionless number defined as *R*<sup>0</sup> = *β*·*c*·*d*, where *β* is the probability of infection if an infected individual contacts a susceptible one, *c* is the number of contacts between infected and susceptible individuals per time unit, and *d* is the duration of the infectious period. Its mathematical formula shows that *R0* is proportional to the mean duration of infections *d*. This period decreases when the intrinsic mortality (i.e., mortality not caused by the pathogen) increases, the pathogen-induced mortality increases, and the rate at which the host recovers (e.g., through immunity) increases. When the colonizing agents are pathogens, typically, either the hosts die or manage to clear the pathogen. If the host's death or recovery is quick, the *R*<sup>0</sup> of the pathogen is lower than if the pathogen remains a long time in the host. Therefore, there is something peculiar about non-pathogenic microorganisms with public-health consequences.

Non-pathogenic agents do not cause host death, and the hosts do not need to eliminate them. Consequently, commensal or mutualistic bacteria, including those harboring drug resistance or even a few virulence genes, may colonize their hosts for longer, so their *R*<sup>0</sup> is prone to be larger than one. This conclusion may impact public health because an *R*<sup>0</sup> larger than one implies that the microorganism may spread through the host population, and as mentioned above, non-pathogenic bacteria may harbor non-housekeeping genes and mobile genetic elements such as virulence and resistance genes.

Even if newly arrived non-pathogenic bacterial cells cannot persist in a microbiome for more than a few days, its mobile genetic elements can have several opportunities to transfer to one of the other established cells in that microbiome [20–22], and may thus remain there for a long time. This is possible due to the presence of hundreds of bacterial species in many microbiomes, including the human gut, and because bacteria can receive foreign DNA through three primary mechanisms: (i) transformation, where bacteria directly uptakes DNA from the surroundings [23]; (ii) transduction, where bacteriophages bring DNA from their previous hosts [24]; and (iii) bacterial conjugation, where bacteria receive conjugative plasmids or integrative conjugative elements from neighboring cells [25]. A bacterial cell can uptake DNA from phylogenetically distant bacteria cells, and conjugative plasmids can transfer between cells belonging to different bacterial species [26]. Therefore, both pathogenic and non-pathogenic bacteria can share a gene pool that includes virulence and resistance genes.

Moreover, some bacterial populations containing neither mobile genetic elements nor virulence or resistance genes can become great "amplifiers" of these genes after receiving some plasmids. Some bacterial strains are excellent donors of conjugative plasmids and are able to "amplify" it among them while quickly transmitting the plasmid to other cells in a microbiome [7,27]. These amplifiers are present among strains of *Escherichia coli* and other enterobacterial species [7], among soil bacteria [27], and most probably in the majority of microbiomes.
