Global stability analysis, stabilization and synchronization of an SEIR epidemic model

In this paper, we investigate an SEIR epidemic model in which recruitment is assumed to be regulated by a constant function. The SEIR model is a widely-used epidemiological framework that categorises the population into four distinct groups: Susceptible, Exposed, Infectious, and Removed. It employs a system of differential equations to describe the transitions of individuals among these groups, making it a valuable tool for understanding and modelling the dynamics of infectious diseases. Mathematical analysis is used to study the dynamic behaviour of this model. We first determine the basic reproduction number, denoted as $ R_0 $ R0, which governs the dynamics of the SEIR model. Our findings reveal that when $ R_0 $ R0 is less than 1, the unique disease-free equilibrium is asymptotically stable, and there is no endemic equilibrium. However, when $ R_0 $ R0 exceeds 1, the disease-free equilibrium becomes unstable, and a unique, asymptotically stable endemic equilibrium emerges. Furthermore, by constructing appropriate Lyapunov functions, we have studied the global asymptotic stability of the two equilibrium points. On the other hand, we have used the backstepping method to design a control law that stabilises the SEIR model towards the disease-free equilibrium, which means the eradication of the disease from the target population. Next, we have used the same method to synchronise the SEIR drive system and the SEIR response system. This method is used to stabilise the synchronisation error dynamics arising from the mismatch between the drive and response systems. Finally, we presented numerical simulations to illustrate the theoretical results.