A skin microbiome model with AMP interactions and analysis of quasi-stability vs stability in population dynamics

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 analyze 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.