  The skin microbiome plays an important role in the maintenance of a healthy
skin. It is an ecosystem, composed of several species, competing for resources
and interacting with the skin cells. Imbalance in the cutaneous microbiome,
also called dysbiosis, has been correlated with several skin conditions,
including acne and atopic dermatitis. Generally, dysbiosis is linked to
colonization of the skin by a population of opportunistic pathogenic bacteria.
Treatments consisting in non-specific elimination of cutaneous microflora have
shown conflicting results. In this article, we introduce a mathematical model
based on ordinary differential equations, with 2 types of bacteria populations
(skin commensals and opportunistic pathogens) and including the production of
antimicrobial peptides to study the mechanisms driving the dominance of one
population over the other. By using published experimental data, assumed to
correspond to the observation of stable states in our model, we reduce the
number of parameters of the model from 13 to 5. We then use a formal
specification in quantitative temporal logic to calibrate our model by global
parameter optimization and perform sensitivity analyses. On the time scale of 2
days of the experiments, the model predicts that certain changes of the
environment, like the elevation of skin surface pH, create favorable conditions
for the emergence and colonization of the skin by the opportunistic pathogen
population, while the production of human AMPs has non-linear effect on the
balance between pathogens and commensals. Surprisingly, simulations on longer
time scales reveal that the equilibrium reached around 2 days can in fact be a
quasi-stable state followed by the reaching of a reversed stable state after 12
days or more. We analyse the conditions of quasi-stability observed in this
model using tropical algebraic methods, and show their non-generic character in
contrast to slow-fast systems. These conditions are then generalized to a large
class of population dynamics models over any number of species.
