A susceptible$S(t)$, infectious$I(t)$, and removed$R(t)$(SIR) family of deterministic, lumped-parameter models of directly transmitted infectious diseases is considered. A piecewise constant transmission rate is estimated via a Luenbergertype observer and used to adjust the recovery rate in an effort to maintain the basic reproduction number close to unity. The result is a mitigation strategy that avoids multiple infectious waves. A linear, discrete-time state-space model is used throughout with the transmission rate treated as an unknown but constant disturbance during a control design time window. The off-line simulations in discrete time may be used to produce heuristic policies implemented by public health and government organizations.

This paper presents a new method to design adaptive Proportional-Integral-Derivative (PID) controllers which optimize a performance criterion (integral absolute error or integral square error) subject to constraints such as controlled variable, manipulated variable and rate of change constraints. The design method is applicable to linear or nonlinear processes. Modeling of such processes is done using a linear ARX model. In addition, the design method is independent of the PID controller type and accounts for controllers with proportional or derivative action on the control error or the process variable. The proposed design method is not analytic in nature, but rather it is implemented using existing MATLAB optimization functions and demonstrated using simulation experiments for linear and nonlinear processes.