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Finite horizon optimal control of reaction-diffusion SIV epidemic system with stochastic environment

Zong Wang

TL;DR

An algorithm to approximate the Hamilton-Jacobi Bellman (HJB) equation is given and the necessary and sufficient conditions for the near-optimal control of a stochastic reaction diffusion SIV epidemic model are obtained.

Abstract

This contribution mainly focuses on the finite horizon optimal control problems of a susceptible-infected-vaccinated(SIV) epidemic system governed by reaction-diffusion equations and Markov switching. Stochastic dynamic programming is employed to find the optimal vaccination effort and economic return for a stochastic reaction diffusion SIV epidemic model. To achieve this, a key step is to show the existence and uniqueness of invariant measure for the model. Then, we obtained the necessary and sufficient conditions for the near-optimal control. Furthermore, we give an algorithm to approximate the Hamilton-Jacobi Bellman (HJB) equation. Finally, some numerical simulations are presented to confirm our analytic results.

Finite horizon optimal control of reaction-diffusion SIV epidemic system with stochastic environment

TL;DR

An algorithm to approximate the Hamilton-Jacobi Bellman (HJB) equation is given and the necessary and sufficient conditions for the near-optimal control of a stochastic reaction diffusion SIV epidemic model are obtained.

Abstract

This contribution mainly focuses on the finite horizon optimal control problems of a susceptible-infected-vaccinated(SIV) epidemic system governed by reaction-diffusion equations and Markov switching. Stochastic dynamic programming is employed to find the optimal vaccination effort and economic return for a stochastic reaction diffusion SIV epidemic model. To achieve this, a key step is to show the existence and uniqueness of invariant measure for the model. Then, we obtained the necessary and sufficient conditions for the near-optimal control. Furthermore, we give an algorithm to approximate the Hamilton-Jacobi Bellman (HJB) equation. Finally, some numerical simulations are presented to confirm our analytic results.
Paper Structure (16 sections, 11 theorems, 108 equations, 5 figures)

This paper contains 16 sections, 11 theorems, 108 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Model and Preliminaries
  3. Model
  4. Preliminaries
  5. Existence and uniqueness of invariant measures
  6. Necessary and sufficient conditions for near-optimal controls
  7. Some priori estimates of the susceptible, infected and vaccinated
  8. Necessary conditions for near-optimal controls
  9. Sufficient conditions for near-optimal controls
  10. Optimal control and Numerical Algorithm
  11. Optimal control
  12. Approximation of HJB equation
  13. Numerical examples
  14. The invariant measure
  15. Finite horizon optimal control
  16. ...and 1 more sections

Key Result

Lemma 2.1

$($3$)$ Let $N<\infty$ and assume further that $\sum_{i=1}^{N}\mu_{i}\rho_{i}<0,$ where $\mu_{i}$ is the stationary distribution of Markov chain $\{\Lambda_{t}\}_{t\geq 0}$. Then $(1)$$\eta_{p}>0$ if $\max\limits_{i\in \mathbb{S}}\rho_{i}\leq0$; $(2)$$\eta_{p}>0$ for $p<\max\limits_{i\in \mathbb{S}

Figures (5)

  • Figure 1: Density plots (a)-(c) based on 10 000 stochastic simulations for group susceptible, infected and vaccinated at time t = 25, 28 and 30. Here we choose $\beta=0.2,~\sigma=0.05$. The simulations confirm the existence of the unique ergodic invariant measure for system \ref{['adj']}.
  • Figure 2: Density plots (a)-(c) based on 10 000 stochastic simulations for group susceptible, infected and vaccinated with and without control. Here we choose $\beta=0.2,~\sigma=0.05$. The simulations confirm the existence of the unique ergodic invariant measure for system \ref{['adj']}.
  • Figure 3: The path of $S(x,t), I(x,t)$ for the stochastic SIV model \ref{["1'q1"]} with initial $S_0 = 0.6, I_0 = 0.1,V_0 = 1.0$.
  • Figure 4: The path of $p_{1}(x,t), p_{2}(x,t)$ and $p_{3}(x,t)$ of the adjoint equations \ref{['adj']}.
  • Figure 5: The path of $u_{1}(x,t), u_{2}(x,t)$.

Theorems & Definitions (21)

  • Remark 2.1
  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 11 more