Modeling the dynamics of innate and adaptive immune response to Parkinson's disease with immunotherapy

A mathematical model was built using delay differential equations to investigate the effect of active and passive immunotherapies in delaying the progression of Parkinson's Disease. The model described the dynamics between healthy and infected neurons and alpha-synuclein with innate and adaptive immune responses. The model was examined qualitatively and numerically. The qualitative analysis produced two equilibrium points. The local stability of the free and endemic equilibrium points was established depending on the basic reproduction number, $R_0$. Numerical simulations were executed to show the agreement with the qualitative results. Moreover, a sensitivity analysis on $R_0$ was conducted to examine the critical parameters in controlling $R_0$. We found that if treatment is administered in the early stages of the disease with short time delays, alpha-synuclein is combated, inhibiting activated microglia and T cells and preserving healthy neurons. It can be concluded that administering time of immunotherapies plays a significant role in hindering the advancement of Parkinson's disease.