| Titre : | Uncertainty-Aware Control of Epidemic Processes |
| Auteurs : | Chabane Djamila, Auteur ; Mekkaoui Imen, Directeur de thèse ; Kebiri Omar, Directeur de thèse |
| Type de document : | texte manuscrit |
| Editeur : | University of Saida - Dr Moulay Tahar. Faculty: Mathematics, Computer Science and Telecommunications Department: Mathematics, 2025/2026 |
| Format : | 63ص |
| Accompagnement : | CD |
| Langues: | Anglais |
| Index. décimale : | BUC-M 008497 |
| Catégories : |
Master Mathématiques Spécialité: Analyse stochastique, statistique des processus et applications |
| Mots-clés: | Epidemic model, Lévy process, optimal control, stochastic maximum principle, forward-backward SDEs, LSMC, Milstein scheme, immigration control, preventive measures |
| Résumé : |
This thesis develops and analyzes a stochastic epidemic model incorporating Lévy
jump processes to capture the discontinuous, which classical deterministic and Brownian motion driven models fail to represent. The work begins by formulating a deterministic compartmental system of ordinary differential equations (ODEs), then progressively introduces continuous random fluctuations via a Wiener process, and finally adds discontinuous jumps through a Lévy process. The core contribution lies in the optimal control problem, where the objective is to minimize the number of infected individuals while simultaneously reducing intervention costs. Two controls are considered: immigration control and preventive measures. Using the stochastic maximum principle extended to Lévy processes, we derive necessary optimality conditions expressed through a Hamiltonian function and a coupled forward-backward system of stochastic differential equations (FBSDEs) for the state and adjoint variables. Because analytical solutions are infeasible, we employ a numerical scheme based on the Least Squares Monte Carlo (LSMC) method combined with the Milstein method. The simulation results are presented through graphs that demonstrate the effectiveness of the optimal control strategy. The findings confirm that incorporating Lévy jumps and solving the resulting optimal control problem yields robust, practical intervention policies for epidemic management |
| Note de contenu : |
Contents
List of Figures List of Tables List of Algorithms Introduction 2 1 Deterministic and stochastic epidemic models 5 1.1 Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Deterministic epidemic models . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Hamer’s simple deterministic model (SI model) . . . . . . . . 6 1.2.2 The SIS model . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 The SIR model Kermack–McKendrick epidemic model . . . . 7 1.2.3.1 The SIR model without demographic factors : . . . . 8 1.2.3.2 SIR model with varying total population size: . . . . 8 1.2.4 SIRS model (SIR with temporary immunity) . . . . . . . . . . 9 1.2.5 SEIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Basic reproduction number R0 . . . . . . . . . . . . . . . . . . . . . . 10 1.3.1 Basic reproduction number for SIRS model . . . . . . . . . . . 11 1.3.2 Basic reproduction number for COVID 19 . . . . . . . . . . . 11 1.4 Equilibrium in epidemic models . . . . . . . . . . . . . . . . . . . . . 13 1.4.1 Disease-free equilibrium (DFE) . . . . . . . . . . . . . . . . . 13 1.4.2 Endemic equilibrium . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Stochastic epidemic model . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5.1 Stochastic SIRS Model with Brownian noise . . . . . . . . . . 14 1.5.2 Incorporating jump processes in SIRS model . . . . . . . . . . 16 1.6 Existence and uniqueness of the positive solution . . . . . . . . . . . 21 CHAPTER 0. INTRODUCTION 2 Optimal control for system of stochastic differential equations with Lévy jumps 24 2.1 Optimal control background . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Optimal control for the SIRS model with Lévy jumps . . . . . . . . . 25 2.3 Maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3.1 Hamiltonian of the model . . . . . . . . . . . . . . . . . . . . 26 2.3.2 Derivation of the adjoint model . . . . . . . . . . . . . . . . . 27 2.3.3 Optimal controls u∗(t), v∗(t) of the model . . . . . . . . . . . . 28 3 Numerical simulations for the controlled epidemic model 30 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Preliminaries on numerical methods for stochastic differential equations 30 3.2.1 Euler–Maruyama method . . . . . . . . . . . . . . . . . . . . 31 3.2.2 Milstein method . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.3 Numerical example: Linear stochastic differential equation . . 32 3.3 Least-Squares Monte Carlo method . . . . . . . . . . . . . . . . . . . 38 3.3.1 General framework . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.2 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.3 Algorithm of the LSMC method . . . . . . . . . . . . . . . . . 41 3.3.4 Convergence and complexity of LSMC method . . . . . . . . . 42 3.3.5 Advantages of LSMC method . . . . . . . . . . . . . . . . . . 43 3.4 Numerical simulation of the controlled epidemic model with jumps . . 43 3.4.1 Simulation of stochastic SIRS model with jump . . . . . . . . 43 3.4.2 The Lévy jumps effect on the epidemic dynamics . . . . . . . 44 3.4.3 Numerical Approximation of optimal control for SIRS models 45 3.4.4 Impact of control measures on SIRS model dynamics . . . . . 46 3.4.5 Effect of different levels of randomness on the epidemic model 48 3.4.6 Stochastic threshold for the SIRS model with Lévy jumps . . . 49 3.5 Python code for the SIRS epidemic model . . . . . . . . . . . . . . . 50 Conclusion 60 Bibliography 61 1 |
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