Análisis matemático de un problema inverso para un sistema de reacción-difusión originado en epidemiología: Mathematical analysis of an inverse problem for a reaction-diffusion system originated in epidemiology

Aníbal Coronel; Fernando Huancas; Esperanza Lozada; Jorge Torres

doi:10.56754/0719-0646.2702.363

DOI:

https://doi.org/10.56754/0719-0646.2702.363

Abstract

In this article we focus our interest on the study of an inverse problem arising in the mathematical modeling of disease transmission. The mathematical model is given by an initial boundary value problem for a reaction diffusion system. Meanwhile, the inverse problem consists in the determination of the disease and recovery transmission rates from observed measurement of the direct problem solution at some fixed time. The unknowns of the inverse problem are 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. We establish the uniqueness of 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. Moreover, we develop a numerical approximation and simulations of the inverse problem.

Resumen

En este artículo centramos nuestro interés en el estudio de un problema inverso que surge en el modelamiento matemático de la transmisión de enfermedades infectocontagiosas. El modelo matemático viene dado por un problema con condiciones iniciales y en la frontera para un sistema de difusión-reacción. Mientras tanto, el problema inverso consiste en la determinación de las tasas de transmisión y de recuperación de la enfermedad, a partir de la medición observada de la solución del problema directo en un tiempo fijo. Las incógnitas del problema inverso aparecen en el modelo como coeficientes del término de reacción. Formulamos el problema inverso como un problema de optimización para un funcional de costo adecuado. Luego, se deduce la existencia de soluciones del problema inverso probando la existencia de un minimizador para el funcional de costo. Establecemos la unicidad del problema de identificación. La unicidad es una consecuencia de la condición necesaria de optimalidad de primer orden y una estabilidad de las incógnitas del problema inverso con respecto a las observaciones. Ademas se se realiza una aproximación numérica y simulaciones para el problema inverso.

Keywords

Identification problem , control problem , SIS , inverse problem

Mathematics Subject Classification:

35B45 , 35Q35 , 76B03 , 76D03

Aníbal Coronel GMA, Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Campus Fernando May, Chillán, Chile. https://orcid.org/0000-0001-6602-7378
Fernando Huancas Departamento de Matemática, Facultad de Ciencias Naturales, Matemáticas y del Medio Ambiente, Universidad Tecnológica Metropolitana, Las Palmeras 3360, Ñuñoa-Santiago 7750000, Chile. https://orcid.org/0000-0003-2794-029X
Esperanza Lozada GMA, Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Campus Fernando May, Chillán, Chile. https://orcid.org/0009-0000-7809-1342
Jorge Torres GMA, Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Campus Fernando May, Chillán, Chile. https://orcid.org/0009-0003-7786-603X

Pages: 363–390
Date Published: 2025-10-15
In Press

V. Akimenko, “An age-structured SIR epidemic model with fixed incubation period of infection,” Comput. Math. Appl., vol. 73, no. 7, pp. 1485–1504, 2017.

R. M. Anderson y R. M. May, Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, 1991, doi: 10.1093/oso/9780198545996.001.0001.

N. C. Apreutesei, “An optimal control problem for a pest, predator, and plant system,” Nonlinear Anal. Real World Appl., vol. 13, no. 3, pp. 1391–1400, 2012, doi: 10.1016/j.nonrwa.2011.11.004.

B. Armbruster y E. Beck, “An elementary proof of convergence to the mean-field equations for an epidemic model,” IMA J. Appl. Math., vol. 82, no. 1, pp. 152–157, 2017, doi: 10.1093/imamat/hxw010.

N. Bacaër, A short history of mathematical population dynamics. Springer-Verlag London, Ltd., London, 2011, doi: 10.1007/978-0-85729-115-8.

R. S. Cantrell y C. Cosner, Spatial ecology via reaction-diffusion equations, ser. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003, doi: 10.1002/0470871296.

L. Chang, S. Gao, y Z. Wang, “Optimal control of pattern formations for an SIR reaction-diffusion epidemic model,” J. Theoret. Biol., vol. 536, 2022, Art. ID 111003, doi: 10.1016/j.jtbi.2022.111003.

Q. Chen y J. Liu, “Solving an inverse parabolic problem by optimization from final measurement data,” J. Comput. Appl. Math., vol. 193, no. 1, pp. 183–203, 2006, doi: 10.1016/j.cam.2005.06.003.

I.-C. Chou y E. O. Voit, “Recent developments in parameter estimation and structure identification of biochemical and genomic systems,” Math. Biosci., vol. 219, no. 2, pp. 57–83, 2009, doi: 10.1016/j.mbs.2009.03.002.

A. Coronel, F. Huancas, C. Isoton, y A. Tello, “Optimal control problem and reaction identification term for carrier-borne epidemic spread with a general infection force and diffusion,” Electron. Res. Arch., vol. 33, no. 7, pp. 4435–4467, 2025, doi: 10.3934/era.2025202.

A. Coronel, F. Huancas, y M. Sepúlveda, “Identification of space distributed coefficients in an indirectly transmitted diseases model,” Inverse Problems, vol. 35, no. 11, 2019, Art. ID 115001.

A. Coronel, F. Huancas, y M. Sepúlveda, “A note on the existence and stability of an inverse problem for a SIS model,” Comput. Math. Appl., vol. 77, no. 12, pp. 3186–3194, 2019, doi: 10.1016/j.camwa.2019.01.031.

A. Coronel, F. Huancas, E. Lozada, y M. Rojas-Medar, “Results for a control problem for a sis epidemic reaction–diffusion model,” Symmetry, vol. 15, no. 6, 2023, Art. ID 1224, doi: 10.3390/sym15061224.

B. Dembele, A. Friedman, y A.-A. Yakubu, “Mathematical model for optimal use of sulfadoxine–pyrimethamine as a temporary malaria vaccine,” Bulletin of Mathematical Biology, vol. 72, no. 4, pp. 914–930, 2010, doi: 10.1007/s11538-009-9476-9.

Z.-C. Deng, L. Yang, J.-N. Yu, y G.-W. Luo, “An inverse problem of identifying the coefficient in a nonlinear parabolic equation,” Nonlinear Anal., vol. 71, no. 12, pp. 6212–6221, 2009, doi: 10.1016/j.na.2009.06.014.

O. Diekmann y J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases: Model building, analysis and interpretation, ser. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2000.

L. Djebara, R. Douaifia, S. Abdelmalek, y S. Bendoukha, “Global and local asymptotic stability of an epidemic reaction-diffusion model with a nonlinear incidence,” Math. Methods Appl. Sci., vol. 45, no. 11, pp. 6766–6790, 2022.

J. Ge, L. Lin, y L. Zhang, “A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment,” Discrete Contin. Dyn. Syst. Ser. B, vol. 22, no. 7, pp. 2763–2776, 2017, doi: 10.3934/dcdsb.2017134.

Q. Ge, Z. Li, y Z. Teng, “Probability analysis of a stochastic SIS epidemic model,” Stoch. Dyn., vol. 17, no. 6, 2017, Art. ID 1750041, doi: 10.1142/S0219493717500411.

M. D. Gunzburger, Perspectives in flow control and optimization, ser. Advances in Design and Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003, Control. vol. 5.

A. Kirsch, An introduction to the mathematical theory of inverse problems, 2nd ed., ser. Applied Mathematical Sciences. Springer, New York, 2011, vol. 120, doi: 10.1007/978-1-4419-8474-6.

M. Koivu-Jolma y A. Annila, “Epidemic as a natural process,” Math. Biosci., vol. 299, pp. 97–102, 2018, doi: 10.1016/j.mbs.2018.03.012.

N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, ser. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008, vol. 96, doi: 10.1090/gsm/096.

O. A. Ladyženskaja, V. A. Solonnikov, y N. N. Ural’ceva, Linear and quasilinear equations of parabolic type, ser. Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1968, vol. 23.

G. M. Lieberman, Second order parabolic differential equations. Co., Inc., River Edge, NJ, 1996, doi: 10.1142/3302. World Scientific Publishing

X. Lu, S. Wang, S. Liu, y J. Li, “An SEI infection model incorporating media impact,” Math. Biosci. Eng., vol. 14, no. 5-6, pp. 1317–1335, 2017, doi: 10.3934/mbe.2017068.

T. T. Marinov, R. S. Marinova, J. Omojola, y M. Jackson, “Inverse problem for coefficient identification in SIR epidemic models,” Comput. Math. Appl., vol. 67, no. 12, pp. 2218–2227, 2014, doi: 10.1016/j.camwa.2014.02.002.

A. Mummert y O. M. Otunuga, “Parameter identification for a stochastic SEIRS epidemic model: case study influenza,” J. Math. Biol., vol. 79, no.2, pp. 705–729, 2019, doi: 10.1007/s00285-019-01374-z.

A. Nwankwo y D. Okuonghae, “Mathematical analysis of the transmission dynamics of HIV syphilis co-infection in the presence of treatment for syphilis,” Bull. Math. Biol., vol. 80, no. 3, pp. 437–492, 2018, doi: 10.1007/s11538-017-0384-0.

A. Rahmoun, B. Ainseba, y D. Benmerzouk,“ Optimal control applied on an HIV-1within-host model,” Math. Methods Appl. Sci., vol. 39, no. 8, pp. 2118–2135, 2016, doi: 10.1002/mma.3628.

C. M. Saad-Roy, P. van den Driessche, y A.-A. Yakubu, “A mathematical model of anthrax transmission in animal populations,” Bull. Math. Biol., vol. 79, no. 2, pp. 303–324, 2017, doi: 10.1007/s11538-016-0238-1.

K. Sakthivel, S. Gnanavel, N. Barani Balan, y K. Balachandran, “Inverse problem for the reaction diffusion system by optimization method,” Appl. Math. Model., vol. 35, no. 1, pp. 571–579, 2011, doi: 10.1016/j.apm.2010.07.024.

D. Uciński, Optimal measurement methods for distributed parameter system identification, ser. Systems and Control Series. CRC Press, Boca Raton, FL, 2005.

V. M. Veliov, “Numerical approximations in optimal control of a class of hetero-geneous systems,” Comput. Math. Appl., vol. 70, no. 11, pp. 2652–2660, 2015, doi: 10.1016/j.camwa.2015.04.029.

A. Widder y C. Kuehn, “Heterogeneous population dynamics and scaling laws near epidemic outbreaks,” Math. Biosci. Eng., vol. 13, no. 5, pp. 1093–1118, 2016, doi: 10.3934/mbe.2016032.

H. Xiang y B. Liu, “Solving the inverse problem of an SIS epidemic reaction-diffusion model by optimal control methods,” Comput. Math. Appl., vol. 70, no. 5, pp. 805–819, 2015, doi: 10.1016/j.camwa.2015.05.025.

S. Zhi, H.-T. Niu, y Y. Su, “Global dynamics of a diffusive SIRS epidemic model in a spatially heterogeneous environment,” Appl. Anal., vol. 104, no. 3, pp. 390–418, 2025, doi: 10.1080/00036811.2024.2367667.