Optimal control problem and reaction identification term for carrier-borne epidemic spread with a general infection force and diffusion

Anibal Coronel; Fernando Huancas; Camila Isoton; Alex Tello; Anibal Coronel; Fernando Huancas; Camila Isoton; Alex Tello

doi:10.3934/era.2025202

Electronic Research Archive

2025, Volume 33, Issue 7: 4435-4467. doi: 10.3934/era.2025202

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Optimal control problem and reaction identification term for carrier-borne epidemic spread with a general infection force and diffusion

1.
GMA, Departamento de Ciencias Básicas-Centro de Ciencias Exactas CCE-UBB, Facultad de Ciencias, Universidad del Bío-Bío, Campus Fernando May, Chillán 3780000, Chile
2.
Departamento de Matemática, Facultad de Ciencias Naturales, Matemáticas y del Medio Ambiente, Universidad Tecnológica Metropolitana, Las Palmeras No. 3360, Ñuñoa-Santiago 7750000, Chile
3.
FACET, Universidade Federal da Grande Dourados, Brasil
4.
Departamento de Matemática, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1270300, Chile

Received: 30 March 2025 Revised: 11 July 2025 Accepted: 29 July 2025 Published: 05 August 2025

In this paper, we simultaneously study reaction identification and the optimal control problem for a reaction-diffusion system modeling carrier-borne epidemics with a general transmission function and vaccination. The state equations are given by a susceptible-infected-recovered reaction-diffusion system with zero-flux boundary conditions and initial conditions. The reaction is modeled by three terms: a general transmission function modeling the force of the infection or the effective contact between susceptible and infected individuals, a linear function for transition between susceptible and infected individuals, and a function for control of vaccination of susceptible individuals. The cost function consists of two parts: two terms related to parameter identification, comprising a regularized least squares cost function, and five terms related to the control of the population through vaccination. The optimal control problem is analyzed by applying the Dubovitskii and Milyutin formalism. In the main results, we deduce the well-posedness of the state equation, the existence of the optimal control problem, the existence of solutions of the adjoint state, and a first-order optimality condition. We develop a numerical approximation for the optimal control problem by employing an IMEX method to approximate the state equations. In this approach, the coefficients of the reaction terms and the control functions depend on a finite set of parameters. We provide two numerical examples to demonstrate the agreement of our numerical solution with the measurement observations.
- optimal control,
- parameter identification,
- SIR with diffusion,
- carrier-borne epidemic,
- first-order optimality condition
Citation: Anibal Coronel, Fernando Huancas, Camila Isoton, Alex Tello. Optimal control problem and reaction identification term for carrier-borne epidemic spread with a general infection force and diffusion[J]. Electronic Research Archive, 2025, 33(7): 4435-4467. doi: 10.3934/era.2025202

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Abstract

In this paper, we simultaneously study reaction identification and the optimal control problem for a reaction-diffusion system modeling carrier-borne epidemics with a general transmission function and vaccination. The state equations are given by a susceptible-infected-recovered reaction-diffusion system with zero-flux boundary conditions and initial conditions. The reaction is modeled by three terms: a general transmission function modeling the force of the infection or the effective contact between susceptible and infected individuals, a linear function for transition between susceptible and infected individuals, and a function for control of vaccination of susceptible individuals. The cost function consists of two parts: two terms related to parameter identification, comprising a regularized least squares cost function, and five terms related to the control of the population through vaccination. The optimal control problem is analyzed by applying the Dubovitskii and Milyutin formalism. In the main results, we deduce the well-posedness of the state equation, the existence of the optimal control problem, the existence of solutions of the adjoint state, and a first-order optimality condition. We develop a numerical approximation for the optimal control problem by employing an IMEX method to approximate the state equations. In this approach, the coefficients of the reaction terms and the control functions depend on a finite set of parameters. We provide two numerical examples to demonstrate the agreement of our numerical solution with the measurement observations.

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