@phdthesis{MbouandiNjiasse2025, author = {Mbouandi Njiasse, Ibrahim}, title = {Stochastic models and optimal control of epidemics under partial information}, doi = {10.26127/BTUOpen-7127}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-71272}, school = {BTU Cottbus - Senftenberg}, year = {2025}, abstract = {In this thesis, we develop a mathematical framework to estimate the compartment sizes of an epidemic model with some unobserved compartments (i.e., a model with partial information) and formulate a social planner's stochastic optimal control problem for such an epidemic. Including asymptomatic and unreported infectious individuals in the model allows a more realistic description of the course of an epidemic and avoids misleading estimation. In order to estimate the unobservable states, a suitable description of the dynamics of the epidemic progression by a diffusion approximation is achieved using counting processes. The estimation of the unobservable states is attributed to a stochastic filter problem. To solve this problem, we use the extended Kalman filter, which provides approximate solutions for such non-standard filter problems. This approach is applied to the state estimation of a model with incomplete information for a disease with lifelong immunity after recovery or vaccination. Simulations are carried out to demonstrate the effectiveness of this approach in practice. The main part of this work addresses a social planner's stochastic optimal control problem for an epidemic model with a partially observed state process. It decomposes the population into susceptible, detected and non-detected infected, detected and non-detected recovered, and hospitalized individuals. Control measures include social distancing, testing, and vaccination. We apply a filtering argument to transform the partially observable stochastic optimal control problem into a completely observable control problem by replacing the hidden state process with the associated Kalman filter processes. This transformed problem with an eight-dimensional state process is treated as a Markov decision process after a time discretization. The associated Bellman equation is solved numerically using a backward recursion algorithm combined with state discretization and quantization techniques to mitigate the curse of dimensionality. In particular, two approaches are implemented. The first involves state discretization and linear interpolation of the value function between grid points, while the second utilizes state discretization and parameterization of the value function with educated ansatz functions. Extensive numerical experiments are presented to demonstrate the effectiveness of both approximate optimal controls in containing the epidemic. Finally, we generalize the results of Picard (1991) on the efficiency of the continuous-time extended Kalman filter with small noise scaled by a parameter ε>0. First, we show that when the observation drift coefficient is strongly injective and the signal and observation drift become linear for ε → 0, the estimation error is of order √ε. Subsequently, we establish conditions under which the error in the initial filter estimate decays exponentially fast.}, subject = {Compartmental model; Partial information; Extended Kalman filter; Stochastic optimal control; Dynamic programming; Kompartimentmodell; Teilinformationen; Erweiterter Kalman-Filter; Stochastische optimale Steuerung; Dynamische Programmierung; Dynamische Optimierung; Kompartimentmodell; Kalman-Filter; Stochastische optimale Kontrolle}, language = {en} }