Master's Thesis : A systematic analysis of mechanistic models for Vaccines-Adjuvant Induced Immune Response

Abstract

The objectives of this Master thesis are to perform a detailed exploratory analysis of a mechanistic model describing a part of the immune response induced by Vaccine-Adjuvant and to carry out numerical sim- ulations and parametric studies to investigate different scenarios. The main aim of working with a mathematical model is that it can be used to perform ”in silico” ex- periments. This kind of experiments are carried out using computer simulations. There are some major advantages to carry out such experiments. First, simulations are much faster than real experiments. In- deed it takes from a few seconds to a few hours to run a simulation depending on the complexity of the model while actual experiments take from days to months to give results. Second, ”in silico” experiments do not lead to ethics concerns since they do not involve any living organism. Indeed this is a great way to avoid wasting both animals lives and time solving ethic issues. Third, this kind of experiments also allows to test some conditions that may be infeasible or really complicated to achieve in reality. Obviously, there are also some drawbacks. Indeed, as a full understanding of all the underlying mechanisms and their interactions is never achieved, the results of these simulations are not an exact representation of the reality. However, even if it can not fully replace them, the simulations allow to reduce the number of real experiments needed. The exploratory analysis includes an equilibrium point stability analysis and the assessment of asymptotic properties. An equilibrium point was found in a particular case in which one of the model parameters is equal to 0. This equilibrium point has a direct dependency on some of the initial conditions and it was also found to be non-hyperbolic which means that depending on the direction of approach it can be either stable or not. Non-hyperbolic points are usually challenging to analyse, in our case this challenge was increased by the number of variables and parameters in the model that is why, eventually, a compu- tational approach was used. Contrary to the stability analysis, the assessment of asymptotic properties was straightforward. This assessment provides that as time tends toward infinity, most of the variables drop to 0, some of them stabilise at a constant value and two of them tends toward infinity. These two variables going to infinity explain why there is no equilibrium point in the general case. The numerical simulations includes some simulations to validate or investigate further the results of the exploratory analysis, some simulations of particular biological cases. The first series of simulations vali- dated the asymptotic properties and proved the bi-stability of the equilibrium point. The second series of simulations includes a simulation in normal conditions, some simulations under knock-out conditions, a simulation under bacterial infection conditions and simulations with addition of noise to account for variability. The latter showed that the model is rather robust to noise. Indeed, there are some variations but the overall behaviour remains the same which is quite representative of what happens in reality.