Mathematical control theory, the immune system, and cancer

I use simple ideas, endowed from mathematical control theory and from qualitative theory of differential equations, in order to analyze in general grounds the human immune system. The simplest model, in which the response is proportional to the pathogen load, leads to the condition that the annihilation rate of pathogens in any tissue should be greater than the pathogen’s average rate of growth. When nonlinearities are added, a reference value for the number of pathogens and a stability condition, relating strength of regular threats, barrier height and annihilation rate, emerge. The stability condition allows a qualitative comparison between tissues. On the other hand, in cancer immunity, the linear model leads to an expression for the lifetime risk, which accounts for both the effects of carcinogens (endogenous or external) and the immune response. Finally, an integral term is added to the model in order to simulate the process of recruitment of immune cells from blood circulation.