State Estimation of Kermack–McKendrick PDE Model With Latent Period and Observation Delay

In this article, we study the problem of estimating the state of the linearized Kermack–McKendrick partial differential equation (PDE) model in real time. Especially, we assume that the model contains two kinds of delays. One of them is contained in the nonlocal boundary condition, which expresses the latent period of infection. The other is an observation delay, which corresponds to the time needed for counting the number of infected people at an infection elapsed time. The element of time lags can be expressed by a transport equation. As a result, the system with two delays is equivalently written by a
$3\times 3$
hyperbolic system. In this article, we construct observers with three gain functions, using a backstepping method of PDEs. Then, the triple of the designed gain belongs to the domain of the generator governing the state evolution of the error system. Furthermore, based on this fact and the semigroup theory, it is shown that the error system is
$L^2$
-stable in the Hilbert space.