On the control of a reaction–diffusion system: a class of SIR distributed parameter systems

Abstract

In this paper, two types of distributed control functions, vaccine and treatment have been applied to a spatiotemporal SIR model with no-flux boundary conditions. The spatiotemporal SIR epidemic model is formulated from existing SIR epidemic model by including a diffusion term in his different compartments to study the impact of spatial heterogeneity of disease transmission in dense regions. Our main objective to find the optimal control pair that minimizes the number of infected individuals, the corresponding vaccination and treatment costs. The existence of the positive solution for the state system and the existence of a distributed optimal control pair are proved. Techniques of optimal control are used to characterize optimal control pair in terms of state and adjoint functions. The optimality system is solved numerically; the numerical results show that the control effect is effective if the 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, [30]; see also [27, 31]). Consider the initial value problem

$$\begin{aligned} \left\{ \begin{array}{l} \dfrac{\partial y}{\partial t}=Ay\left( t\right) +g\left( t,y\left( t\right) \right) ,\qquad t\in \left[ 0,T\right] \\ y\left( 0\right) =y_{0} \end{array}\right. \end{aligned}$$

(36)

where A is a linear operator defined on a Banach space X, with the domain D(A) and $g:[0,T]\times X\rightarrow X$ 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 $(Ay,y)\le 0,\,\left( \forall y\in D(A)\right)$.

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 $g:[0,T]\times X\rightarrow X$ be a function measurable in t and Lipschitz continuous in $x\in X$, uniformly with respect to $t\in [0,T]$.

(i) If $y_{0}\in X$, then problem (36) admits a unique mild solution, i.e. a function $y\in C([0,T];X)$ which verifies the equality $y(t)=S(t)y_{0}+\int _{0}^{t}S(t-s)g(s,y(s))ds, (\forall t\in [0,T])$.

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