Mathematical Model for Hashimoto Autoimmune Thyroiditis Disease With steady state

Hashimoto’s disease (HT) is the first autoimmune disease that was discovered by Japanese physician Hakaru Hashimoto in 1912. This disease targets the thyroid gland, produces antibodies to it, and then causes deficiency and atrophy in it due to the attack of immune cells on it. The main goal of this work is to study the stability of pre-designed mathematical model described interactions between thyroid cells, lymphocytes 〖TH〗_17,〖TH〗_1 and Treg cells and the gut microbiota, hence the model is a nonlinear system consisting of four nonlinear ordinary differential equations. To achieve this goal, all active equilibrium points of the system are found, and hence studied one by one to get the conditions under which the studying point is stable, that is to find a stability point for the system that limits disease complications and immune system activity. Calculating the values of eigenvalues, however, indicates that at least one of them is zero for each point which means the equilibrium points are non-hyperbolic, in this case the linearization method is not effective, therefore Lyapunov method is used. A different Lyapunov function suggested for each point in order to enrich the study. Keywords: Hashimoto’s disease, Mathematical modelling, Autoimmune disease, Thyroid gland, Stability, Lyapunov function.