Solving the inverse problem of an SIS epidemic reaction–diffusion model by optimal control methods

Abstract

a b s t r a c tIn this contribution, a novel inverse problem on an SIS epidemic reaction-diffusion model is investigated by employing the optimal control methods. By setting a proper regular cost functional and choosing the coefficients to be identified as control variables, we embed the original inverse problem which is very difficult to solve into an optimal control problem. The existence of the optimal controls and the first order necessary optimality condition satisfied by the optimal controls are established, respectively. In particular, the local uniqueness and stability of the identification of the coefficients are proved by using analytical techniques.

On the existence and uniqueness of an inverse problem in epidemiology

“…Here, H m (Ω) and C α (Ω) denote the standard Sobolev and Hölder spaces W m,2 (Ω) and C 0,α (Ω), respectively; and Ran(f ) denote the range of function f . The construction of U ad (Ω) was recently developed in [8] and also we note that U ad (Ω) = A(Ω) when d = 1 and coincides with the admissible set considered by Xiang and Liu in [26].…”

“…Now, in [23] the authors study the inverse problem for a reaction-diffusion system with a linear reaction term and obtain existence and local uniqueness of the inverse problem. More recently, in [26] the authors have studied the one-dimensional version of the inverse problem considered in this paper. They obtain a result for existence and local uniqueness of the solution by assuming that the infection process is modeled by a frequency-dependent transmission function instead of the power law function.…”

Mathematical analysis of an optimal control problem for the predator‐prey model with disease in prey

“…Optimal analysis and stability of an epidemic model with transport‐related infection studied in Reference 22. The inverse problem of epidemic reaction‐diffusion type models studied in References 23-25. Recently, optimal control monographs precisely determined to the solution of an epidemic and biological models developed in References 26,27.…”

Inverse problem for a coupling model of reaction-diffusion and ordinary differential equations systems. Application to an epidemiological model

“…Even if this assumption seems to be less considered, some works can be found in this direction. An identifiability study has been proposed in [34] (see also reference [6] for a more recent reference) for a SIS (Susceptible -Infectious -Susceptible) epidemic reaction-diffusion model using an optimal control method. In [24], using input-output relationships, identifiability of a system of nonlinear integro-partial differential equations of transport type was proposed.…”