PhD Math Defence: Controlling games, replicator dynamics and predator-prey models with transmission dynamics

CANDIDATE:   AHMED SHAWKI JABER

ABSTRACT:

The concept of optimal control is one of the significant techniques to observe the evolution of various dynamical systems that can be modeled in a mathematical framework. In optimal control problem, the aim is to minimize a performance measure function by defining a control and state trajectories for a dynamical system over a specified period. The context of optimal control is widely used in various disciplines such as engineering, economics, biomathematics, and ecology.

In this thesis, we use control theory methods to undertake the study of specific models of dynamical systems. For instance, we apply the structure of optimal control on the replicator dynamic systems associated with certain classes of games, to further study the game equilibria (or Nash equilibrium points). In essence, we aim to control the game model of population groups who use some pure and mixed strategies and to move their Nash choices to a newer Nash strategy choice with a different outcome. In our first game, we control the Nash strategies to minimize defectors from a social norm, in the second game, we minimize the nonvaccinators in a population contemplating vaccination against infectious disease. In a related way, we utilize classical control theory to analyze an epidemiological model with two different biological populations (species). The aim is to examine endemic equilibrium points of a susceptible-infections-susceptible (SIS) or a susceptible-infections-recovered (SIR) models and control them with a vaccine uptake rate in order to decrease the overall level of infection in both species. All the optimal control problems we introduce are treated with a numerical approach to get the required solutions and to comments on the effect of model parameters on the optimal system states.