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Partial health status observability and time horizon uncertainty in mean-field game epidemiological models

Carlos Doebeli, Alexander Vladimirsky

Abstract

We introduce Mean-Field Game (MFG) epidemiological models, in which immunity either wanes with time in a fully observable way or disappears instantaneously with no direct observation (making a previously recovered individual fully susceptible again without realizing it). Both interpretations create computational challenges for rational noninfected individuals deciding on their contact rates based on their personal current immunity state and the changing epidemiological situation. Both require solving a forward-backward MFG system that includes PDEs (an advection-reaction equation for the immunity-structured population and a Hamilton-Jacobi-Bellman equation for the corresponding value function). We show how this can be done efficiently by solving a two-point boundary value problem for a system of approximating ODEs. We also show how the same approach can be extended to handle an initial uncertainty in the planning horizon.

Partial health status observability and time horizon uncertainty in mean-field game epidemiological models

Abstract

We introduce Mean-Field Game (MFG) epidemiological models, in which immunity either wanes with time in a fully observable way or disappears instantaneously with no direct observation (making a previously recovered individual fully susceptible again without realizing it). Both interpretations create computational challenges for rational noninfected individuals deciding on their contact rates based on their personal current immunity state and the changing epidemiological situation. Both require solving a forward-backward MFG system that includes PDEs (an advection-reaction equation for the immunity-structured population and a Hamilton-Jacobi-Bellman equation for the corresponding value function). We show how this can be done efficiently by solving a two-point boundary value problem for a system of approximating ODEs. We also show how the same approach can be extended to handle an initial uncertainty in the planning horizon.

Paper Structure

This paper contains 10 sections, 1 theorem, 28 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. Basic SIRSD model
  3. MFG-STYLE MODELS
  4. Fully observable absolute immunity model (MFG-SIRSD)
  5. Fully observable waning immunity model
  6. Unobserved disappearing immunity model
  7. Uncertain Planning Horizons
  8. NUMERICAL METHODS
  9. COMPUTATIONAL EXPERIMENTS
  10. CONCLUSIONS

Key Result

Proposition 1

Suppose a noninfected individual has last recovered at the time $t{}_{{\hbox{$\mathrm{R}$}}}$ and now follows some chosen continuous contact rate policy $c(t)$ for $t \geq t{}_{{\hbox{$\mathrm{R}$}}}$ until a possible future re-infection. If they are fully rational and aware of the changing epidemio $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure D1: Comparison of MFG-SIRSD and "SIRSD with myopic contact rates" models. TOP: The epidemic trajectories for these two models are shown by solid and dashed lines respectively. Susceptibles (S) are plotted in green, Infected (I) - in red, and Recovered/Immune (R) - in blue. Dead (D) are not plotted to simplify the figure. Performance statistics for both models: (Peak I$\,\approx0.3117,$ Mean I$\,\approx0.0823,$ Final D$\,\approx0.0247$) in MFG-SIRSD; (Peak I$\,\approx0.6000,$ Mean I$\,\approx0.1869,$ Final D$\,\approx0.0318$) in myopic/baseline SIRSD. BOTTOM: Nash-optimal contact rates in the MFG-SIRSD.
  • Figure D2: Dynamics and contact rates under (a) the fully observed waning immunity model and (b) the unobserved disappearing immunity model; both computed for nine $p$-bands (i.e., $m=8$). TOP: Fractions of noninfected individuals $N_j(t)$ shown by stacked green area plots, with the shades of green corresponding to different $p$-bands (interpreted as different bands of immunity in subfigure (a) on the left vs. different bands of immunity confidence in subfigure (b) on the right). The darkest green corresponds to the highest susceptibility ($p \in [0,\frac{1}{16})$); the lightest green is the highest immunity ($p \in [\frac{15}{16}, 1]$). The fraction of infected $I(t)$ is plotted on top in red. BOTTOM: Nash-optimal contact rates for each $p$-band in each model.
  • Figure D3: Uncertain horizon MFG-SIRSD example with $T \in \{150, 300\}$ and $\mathbb{P}(T=150) = \theta,$ shown for three different values of $\theta.$ TOP: Infection dynamics. BOTTOM: Nash-optimal contact rates for Susceptibles.
  • Figure D4: Horizon-uncertainty implemented in the unobserved disappearing immunity model. Terminal time $T \in \{50, 100, 200, 285, 300\}$ with the associated probabilities $(0.2, 0.1, 0.05, 0.5, 0.15).$ Computed and plotted with only five $p$-bands ($m=4$) for the sake of visual interpretability. TOP: dynamics of Infected (red) and Noninfected (green) fractions, with the latter shown using stacked area plots and shades of green corresponding to different immunity confidence bands. Performance statistics: (Peak I $\approx$ 0.2782, Mean I $\approx$ 0.1156, Final D $\approx$ 0.0222). BOTTOM: Nash-optimal contact rates for Noninfected, exhibiting jump discontinuities at all possible "early termination" times.

Theorems & Definitions (2)

  • Proposition
  • proof