Contain Epidemics with Stochastic Mixed-Integer Optimal Control

• Applicants: Professor Dr. Armin Fügenschuh; Professor Dr. Gerd Wachsmuth; Professor Dr. Ralf Wunderlich
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 458468407

The temporal and spatial course of a large-scale epidemic or pandemic can be described using mathematical models. These divide the total population into different compartments: the number of people not yet infected (susceptible to the disease), acutely infected (and contagious) people, and those who have recovered (with or without permanent immunity). The transition of a person from one of these compartments to another is described mathematically, e.g. by means of equations or probabilities that depend on parameters measuring the infection rates, incubation times or mortality of the disease and people's behavior.If one does not take the epidemic curve for granted, there are numerous possibilities for intervention in order to influence the parameters favorably. For example, exit restrictions can lower contact and mobility rates. Transporting medical supplies to hospitals in outbreak hotspots can lower the mortality rate. Such measures have financial or social costs. Furthermore, the available capacities for transport and the delivery quantities are limited. What is needed is a system optimum that describes an optimal containment of the epidemic through a mix of measures consisting of the distribution of goods and necessary behavioral restrictions.This problem is very challenging from a mathematical point of view, since several research areas interact: The dynamics of the epidemic are described using deterministic and stochastic differential equations. Classic SIR models are significantly expanded to cover different regions, genders, age groups and mobility behavior. Additional extensions of the classic models relate to asymptomatic disease courses with a high number of unreported cases of undetected infections and test strategies tailored to them. The data are therefore fraught with uncertainty. The influencing of the epidemic and the distribution of goods are treated using methods of optimal control and operations research (mixed-integer optimization), whereby uncertainties must also be taken into account and robust solutions are sought.In an interdisciplinary manner, three researchers from the respective mathematical fields will work together, use and further develop modern mathematical methods in order to be able to model and solve this problem together. The solution methods are to be implemented in the form of a prototype demonstrator. In this planning tool, after entering data to describe a spatial area and characteristic parameters of an infectious disease, a user should receive suggestions for containing it (e.g. distribution of medical goods, local lock-down measures), which were derived from the solution of the mathematical models.

• DFG Programme: Research Grants