Optimal Control of a Reaction-Diffusion Epidemic Model with Noncompliance

TL;DR

This paper develops a model wherein non-pharmaceutical intervention methods are implemented, but a portion of the population does not comply with them, and this non-compliance affects the spread of the disease.

Abstract

In this paper, we consider an optimal distributed control problem for a reaction-diffusion-based SIR epidemic model with human behavioral effects. We develop a model wherein non-pharmaceutical intervention methods are implemented, but a portion of the population does not comply with them, and this noncompliance affects the spread of the disease. Drawing from social contagion theory, our model allows for the spread of noncompliance parallel to the spread of the disease. The quantities of interest for control are the reduction in infection rate among the compliant population, the rate of spread of noncompliance, and the rate at which non-compliant individuals become compliant after, e.g., receiving more or better information about the underlying disease. We prove the existence of global-in-time solutions for fixed controls and study the regularity properties of the resulting control-to-state map. The existence of optimal control is then established in an abstract framework for a fairly general class of objective functions. Necessary first--order optimality conditions are obtained via a Lagrangian based stationarity system. We conclude with a discussion regarding minimization of the size of infected and non-compliant populations and present simulations with various parameters values to demonstrate the behavior of the model.

Figures (7)

• Figure 1: A list of state variables, control maps, and parameters for \ref{['PDE_SYS']}.
• Figure 2: The flow diagram for \ref{['PDE_SYS']}. Any arrow flowing out of a population indicates flow proportional to the population it leaves. Here $I_M = (1-\alpha)I + I^*$ denotes the actively mixing infectious population (i.e., those who contribute to disease spread).
• Figure 3: Dynamics of the model with baseline parameters (table \ref{['tab:baselineParams']}) in the absence (top) and presence (bottom) of controls. We notice that in the controlled case, the optimal $\alpha(\cdot,t)$ is primarily used near the beginning of the dynamics to decrease the first wave of infections. The optimal $\mu(\cdot,t)$ is hardly used at all, and the optimal $\nu(\cdot,t)$ is used throughout. This has the effect that the total noncompliant population settles at a lower portion of the population. The variation in the controls as the end of the dynamics should be seen as artificial: they are there because the policy-maker is aware of the time-horizon $T = 200$, and can slightly decrease costs by drastically altering controls for the final few time steps. Overall, with these values of $\lambda_1,\lambda_2,\zeta$, the optimal controls achieve a $9.64\%$ relative cost reduction against the uncontrolled scenario, reducing the cost from $\mathcal{J}(y,\underline \alpha,0,0) = 1.4071$ to $\mathcal{J}(y,\alpha^\circ,\mu^\circ,\nu^\circ) = 1.2715$. Snapshots of the control maps $\alpha(x,t),\mu(x,t),\nu(x,t)$ at different times are displayed in figure \ref{['fig:2']}.
• Figure 4: Snapshots of the optimal control maps $\alpha(x,t),\mu(x,t),\nu(x,t)$ for different times $t$ for simulation with baseline parameters The time $t = 1.75$ corresponds to the first peak in infections seen in figure \ref{['fig:1']}. The control efforts are concentrated near the origin because $b(x,y)$ and $S_0(x,y)$ are Gaussians centered at the origin, meaning this is where the bulk of the population is. Note that as the infection dies out over time, $\alpha(x,t)$ and $\mu(x,t)$ seem to decrease. However, $\nu(x,t)$ decreases at its peak, but grows elsewhere. This is the primary mechanism used to decrease the final asymptote for the noncompliant population, and thus achieve a decreased total cost, despite the increase in the cost of the control.
• Figure 5: When we decrease $\zeta$ to $0.1$ (top), the controls are cheap enough to implement that the optimal strategy is now to significantly suppress the initial outbreak, and to suppress noncompliance initially. Eventually noncompliance spreads and a small, more gradual outbreak occurs. In this case, the relative cost reduction against the uncontrolled scenario is $17.32\%$. When we increase $\zeta$ to $0.4$ (bottom), the results look qualitatively similar to the baseline case except the control maps are significantly scaled down because controls are more expensive to implement. In this case, the relative cost reduction against the uncontrolled scenario is only $4.66\%$

Theorems & Definitions (33)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.1
• proof
• Lemma 2.3
• proof
• Theorem 2.1
• proof