On the existence and uniqueness of an inverse problem in epidemiology

ABSTRACT

In this paper, we introduce the functional framework and the necessary conditions for the well-posedness of an inverse problem arising from the mathematical modeling of disease transmission. The direct problem is given by an initial boundary value problem for a reaction-diffusion system. The inverse problem consists in the determination of the disease and recovery transmission rates from observed measurement of the direct problem solution at the final time. The unknowns of the inverse problem are the coefficients of the reaction term. We formulate the inverse problem as an optimization problem for an appropriate cost functional. Then, the existence of solutions of the inverse problem is deduced by proving the existence of a minimizer for the cost functional. Moreover, we establish the uniqueness up an additive constant of the identification problem. The uniqueness is a consequence of the first order necessary optimality condition and a stability of the inverse problem unknowns with respect to the observations.

COMMUNICATED BY:

• X. U. Dinghua

KEYWORDS:

• Inverse problem
• SIS
• identification problem
• control problem

AMS CLASSIFICATIONS:

• 49N45
• 49K20
• 92D30
• 92D25

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors acknowledge the support of CONICYT (Chile) through the program “Becas de Doctorado” and Universidad del Bío-Bío(Chile) through the research project DIUBB GI 172409/C, DIUBB 183309 4/R, DIUBB 181409 3/R, FAPEI, and Postdoctoral Program as a part of the project “Instalación del Plan Plurianual UBB 2016–2020”. AC thanks the support of Fundacion Carolina (Spain) through the program “Estancias Cortas Postdoctorales 2018”.