Nonlinear variational inequality in reflexive Banach space with application in epidemiology

Omar Elamraoui; El-Hassan  Essoufi; Abderrahim Zafrar

doi:10.5269/bspm.67837

Omar Elamraoui Hassan 1 Univeristy, FST Settat
El-Hassan Essoufi Hassan 1 Univeristy, FST Settat
Abderrahim Zafrar Departement of Mathematics, Faculty of Sciences Dhar El Mahrz, Sidi Mohamed Ben Abdellah University

Abstract

This paper studies some nonlinear variational inequalities with operators defined on reflexive and separable Banach spaces and verifying pseudomonotone properties. This study comes up with some new results, such as the existence and the strong convergence of solutions. All these results are based on the Galerkin method. As an application, the established result is used for an epidemic model by taking the spatio-temporal SIR model, which is defined as a reaction-diffusion system with Signorini boundary condition to generate a variational inequality. In such application, the Signorini boundary condition is considered for infected individuals while the Neumann boundary condition is considered for susceptible and recovered individuals. The paper ends with a numerical analysis of the problem by applying a splitting method and the Uzawa algorithm based on the augmented Lagrangian method. Some tests are included to show the solutions.

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References

J. Aitchison, A. Lacey, and M. Shillor. A model for an electropaint process. IMA journal of applied mathematics, 33(1):17–31, 1984.

L. J. Allen, B. M. Bolker, Y. Lou, and A. L. Nevai. Asymptotic profiles of the steady states for an sis epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems, 21(1):1, 2008.

R. M. Anderson and R. M. May. Infectious diseases of humans: dynamics and control. Oxford university press, 1992.

T. Apel, J. Pfefferer, and A. Rosch. Finite element error estimates on the boundary with application to optimal control. Mathematics of Computation, 84(291):33–70, 2015.

E. Asano, L. J. Gross, S. Lenhart, and L. A. Real. Optimal control of vaccine distribution in a rabies metapopulation model. Mathematical Biosciences & Engineering, 5(2):219, 2008.

A. Bensoussan. Controle impulsionnel et inequations quasi variationelles. In International Congress of Mathematicians, page 329, 1975.

A. Bensoussan. Impulse control and quasi-variational inequalities. 1984.

D. P. Bertsekas. Constrained optimization and Lagrange multiplier methods. Academic press, 2014.

E. Bertuzzo, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe, and A. Rinaldo. On spatially explicit models of cholera epidemics. Journal of the Royal Society Interface, 7(43):321–333, 2010.

A. Bnouhachem, M. A. Noor, M. Khalfaoui, and Z. Sheng. On a new numerical method for solving general variational inequalities. International Journal of Modern Physics B, 25(32):4443–4455, 2011.

F. Brauer and C. Castillo-Chavez. Basic models in epidemiology. In Ecological time series, pages 410–447. Springer, 1995.

F. Brauer, C. Castillo-Chavez, and C. Castillo-Chavez. Mathematical models in population biology and epidemiology, volume 2. Springer, 2012.

H. Brezis and H. Brezis. Functional analysis, Sobolev spaces and partial differential equations, volume 2. Springer, 2011.

C. Brown. Differential equations: A modeling approach. Number 150. Sage, 2007.

R. S. Cantrell and C. Cosner. Spatial ecology via reaction-diffusion equations. John Wiley & Sons, 2004.

V. Capasso. Mathematical structures of epidemic systems, volume 97. Springer Science & Business Media, 2008.

C. Castillo-Chavez, S. Fridman, X. Luo, et al. Stochastic and deterministic models in epidemiology. 1993.

P. Chabrand, F. Dubois, and M. Raous. Various numerical methods for solving unilateral contact problems with friction. Mathematical and computer modelling, 28(4-8):97–108, 1998.

A. Chekroun and T. Kuniya. Global threshold dynamics of an infection age-structured sir epidemic model with diffusion under the dirichlet boundary condition. Journal of Differential Equations, 269(8):117–148, 2020.

O. Diekmann and J. A. P. Heesterbeek. Mathematical epidemiology of infectious diseases: model building, analysis and interpretation, volume 5. John Wiley & Sons, 2000.

Y. Du and J. Shi. A diffusive predator–prey model with a protection zone. Journal of Differential Equations, 229(1):63–91, 2006.

G. Duvaut and J.-L. Lions. Les inequations en mecanique et en physique. dunod, paris, 1972. Travaux et Recherches Mathematiques, 21, 1976.

A. Easton, M. Singh, and G. Cui. Solutions of two species reaction–diffusion systems. Canadian Applied Mathematics Quarterly, 5(4), 1997.

I. Ekeland and R. Temam. Convex analysis and variational problems. SIAM, 1999.

G. Fichera. Sul problema elastostatico di signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat, 34:138–142, 1963.

M. Fortin and R. Glowinski. Augmented Lagrangian methods: applications to the numerical solution of boundary-value problems. Elsevier, 2000.

R. Glowinski. Numerical methods for nonlinear variational problems, springer-verlag, berlin, heidelberg, 1984.

R. Glowinski and P. Le Tallec. Augmented Lagrangian and operator-splitting methods in nonlinear mechanics. SIAM, 1989.

P. E. Greenwood and L. F. Gordillo. Stochastic epidemic modeling. In Mathematical and statistical estimation approaches in epidemiology, pages 31–52. Springer, 2009.

W. Han and S. Zeng. On convergence of numerical methods for variational–hemivariational inequalities under minimal solution regularity. Applied Mathematics Letters, 93:105–110, 2019.

P. Hartman and G. Stampacchia. On some non-linear elliptic differential-functional equations. Acta mathematica, 115:271–310, 1966.

J.-H. Heegaard and A. Curnier. An augmented lagrangian method for discrete large-slip contact problems. International Journal for Numerical Methods in Engineering, 36(4):569–593, 1993.

H. W. Hethcote. The mathematics of infectious diseases. SIAM review, 42(4):599–653, 2000.

M. J. Keeling and K. T. Eames. Networks and epidemic models. Journal of the royal society interface, 2(4):295–307, 2005.

N. Keyfitz and H. Caswell. Applied mathematical demography, volume 47. Springer, 2005.

B. T. Kien, X. Qin, C.-F. Wen, and J.-C. Yao. The galerkin method and regularization for variational inequalities in reflexive banach spaces. Journal of Optimization Theory and Applications, 189(2):578–596, 2021.

K. Kim, W. Choi, and I. Ahn. Diffusive epidemiological predator–prey models with ratio-dependent functional responses and nonlinear incidence rate. Journal of Dynamical and Control Systems, 28:43–57, 2022.

D. Kinderlehrer and G. Stampacchia. An introduction to variational inequalities and their applications. SIAM, 2000.

H. Le Dret. Nonlinear elliptic partial differential equations. Springer, 2018.

A. L. Lloyd and V. A. Jansen. Spatiotemporal dynamics of epidemics: synchrony in metapopulation models. Mathematical biosciences, 188(1-2):1–16, 2004.

J. Manimaran, L. Shangerganesh, A. Debbouche, and J.-C. Cortes. A time-fractional hiv infection model with nonlinear diffusion. Results in Physics, 25:104293, 2021.

J. Matis and T. E. Wehrly. 17 compartmental models of ecological and environmental systems. Handbook of Statistics, 12:583–613, 1994.

J. Metz. The epidemic in a closed population with all susceptibles equally vulnerable; some results for large susceptible populations and small initial infections. Acta biotheoretica, 27(1):75–123, 1978.

J. D. Murray. MathematicalBiology I. An Introduction. Springer, 2002.

M. Z. Ndii and A. K. Supriatna. Stochastic mathematical models in epidemiology. Information, 20:6185–6196, 2017.

L. Rass, M. Lifshits, and J. Radcliffe. Spatial deterministic epidemics. American Mathematical Soc., 2003.

S. Ruan. Spatial-temporal dynamics in nonlocal epidemiological models. In Mathematics for life science and medicine, pages 97–122. Springer, 2007.

L. Scrimali. The financial equilibrium problem with implicit budget constraints. Central European Journal of Operations Research, 16(2):191–203, 2008.

G. Scutari, D. P. Palomar, F. Facchinei, and J.-s. Pang. Convex optimization, game theory, and variational inequality theory. IEEE Signal Processing Magazine, 27(3):35–49, 2010.

A. Signorini. Sopra alcune questioni di statica dei sistemi continui. Annali della Scuola Normale Superiore di PisaClasse di Scienze, 2(2):231–251, 1933.

J. C. Simo and T. Laursen. An augmented lagrangian treatment of contact problems involving friction. Computers & Structures, 42(1):97–116, 1992.

H. R. Thieme. Mathematics in population biology. In Mathematics in Population Biology. Princeton University Press, 2018.

J. Wang, R. Zhang, and T. Kuniya. A reaction–diffusion susceptible–vaccinated–infected–recovered model in a spatially heterogeneous environment with dirichlet boundary condition. Mathematics and Computers in Simulation, 190:848–865, 2021.

M. Wang. The diffusive eco-epidemiological prey-predator model with infectious diseases in prey. arXiv preprint arXiv:2211.02805, 2022.

P. Wriggers and G. Zavarise. Application of augmented lagrangian techniques for non-linear constitutive laws in contact interfaces. Communications in Numerical Methods in Engineering, 9(10):815–824, 1993.

E. Zeidler. Nonlinear Functional Analysis and Its Applications: II/B: Nonlinear Monotone Operators. Springer Science & Business Media, 2013.

C. Zhang, J. Gao, H. Sun, and J. Wang. Dynamics of a reaction–diffusion svir model in a spatial heterogeneous environment. Physica A: Statistical Mechanics and its Applications, 533:122049, 2019.