On the Regional Control of a Reaction–Diffusion System SIR

Abstract

This paper presents a study of regional optimal control strategies of a spatiotemporal SIR epidemic model which is formulated from existing SIR epidemic models by including a diffusion term. Our main objective is to characterize the two optimal controls that minimize the number of infected individuals, the corresponding vaccination and treatment costs. For that matter, we prove the existence of a pair of control and provide a characterization of optimal controls in terms of state and adjoint functions. Finally, we present numerical simulations on data concerning the evolution of the zoonotic Ebola virus in Africa. Results show that control is effective if regional treatment and vaccine strategies are used simultaneously.

Appendix

First recall a general existence result which we use in the sequel (Proposition 1.2, p. 175, (Barbu 2012); see also (Pazy 2012; Vrabie 2003). Consider the initial value problem

$$\begin{aligned} \left\{ \begin{array}{l} \dfrac{\partial z}{\partial t}=Az\left( t\right) +f\left( t,z\left( t\right) \right) ,\qquad t\in \left[ 0,T\right] \\ z\left( 0\right) =z_{0} \end{array}\right. \end{aligned}$$

(38)

where A is a linear operator defined on a Banach space X, with the domain D(A) and $f:[0,T]\times X\rightarrow X$ is a given function. If X is a Hilbert space endowed with the scalar product $(\cdot ,\cdot )$, then the linear operator A is called dissipative if $(Az,z)\le 0,\,\left( \forall z\in D(A)\right)$.

Theorem 5

X be a real Banach space, $A:D(A)\subseteq X\rightarrow X$ be the infinitesimal generator of a $C_{0}-$semigroup of linear contractions ${S(t),\,t\ge 0}$ on X, and $f:[0,T]\times X\rightarrow X$ be a function measurable in t and Lipschitz continuous in $x\in X$, uniformly with respect to $t\in [0,T]$.

• (i) If $z_{0}\in X$ , then problem (38) admits a unique mild solution, i.e., a function $z\in C([0,T];X)$ which verifies the equality $z(t)=S(t)z_{0}+\int _{0}^{t}S(t-s)f(s,z(s))\mathrm{d}s,(\forall t\in [0,T]$ .

• (ii) If X is a Hilbert space, A is self-adjoint and dissipative on X and $z_{0}\in D(A)$, then the mild solution is in fact a strong solution and $z\in W^{1,2}(\left[ 0,T\right] ;X)\cap L^{2}(0,T;D(A))$