Learning Contact Policies for SEIR Epidemics on Networks: A Mean-Field Game Approach

TL;DR

A mean-field game model for SEIR epidemics on heterogeneous contact networks, where individuals choose state-dependent contact effort to balance infection losses against the social and economic costs of isolation, and establishes existence of equilibrium via a fixed-point argument and proves the uniqueness under a suitable monotonicity condition.

Abstract

In this paper, we develop a mean-field game model for SEIR epidemics on heterogeneous contact networks, where individuals choose state-dependent contact effort to balance infection losses against the social and economic costs of isolation. The Nash equilibrium is characterized by a coupled Hamilton--Jacobi--Bellman/Kolmogorov system across degree classes. An important feature of the SEIR setting is the exposed compartment: the incubation period separates infection from infectiousness and changes incentives after infection occurs. In the baseline formulation, exposed agents optimally maintain full contact, while susceptible agents reduce contact according to an explicit best-response rule driven by infection pressure and the value gap. We also discuss extensions that yield nontrivial exposed precaution by introducing responsibility or compliance incentives. We establish existence of equilibrium via a fixed-point argument and prove the uniqueness under a suitable monotonicity condition. The analysis identifies a delay in the onset of precaution under longer incubation, which can lead to weaker behavioral responses and larger outbreaks. Numerical experiments illustrate how network degree and the cost exponent shape equilibrium policies and epidemic outcomes.

Figures (5)

• Figure 5.1: The Dual Role of Incubation ($\sigma$). The incubation rate $\sigma$ affects the epidemic through two distinct channels. Biologically (bottom arrow), it controls the flow rate from exposed to infectious ($\dot{I} \propto \sigma E$). Strategically (top arrow), it determines the immediacy of the infection cost, altering the value gap $\Delta U_k$ and thus the contact effort $n_k^{S*}$.
• Figure 6.2: Nash equilibrium dynamics for SEIR-MFG (solid) and SIR-MFG (dashed) on homogeneous networks with $k\in\{4,6,8,12,20\}$. Top row: infected fraction $I(t)$ (left) and effort $n^{S*}(t)$ (right) for SEIR-MFG. Bottom row: same quantities for SIR-MFG. SEIR epidemics peak later and exhibit weaker behavioral responses, consistent with Theorem 6.3 (Delayed Peak and Attenuated Response).
• Figure 6.3: Effect of the incubation rate $\sigma$ on the SEIR-MFG Nash equilibrium ($k=6$). Decreasing $\sigma$ (longer latency) delays the epidemic peak and attenuates the optimal effort, quantifying the "strategic delay" effect. The SIR-MFG limit ($\sigma\to\infty$, dashed black) exhibits the strongest and earliest behavioral response.
• Figure 6.4: Detailed SEIR-MFG analysis for $k=8$, $\sigma=2$. Left: All four compartments $S,E,I,R$. Center: Optimal effort $n^{S*}(t)$ and infected fraction $I(t)$ for SEIR-MFG (solid) vs. SIR-MFG (dashed). Right: Cumulative infection probability $\varphi(t)=1-S(t)$; the larger final value for SEIR-MFG confirms Theorem 6.3 (Larger Final Size).
• Figure 6.5: Quantitative comparison of peak timing $t_{\mathrm{peak}}$ (left) and final outbreak size $R(\infty)$ (right) for SEIR-MFG vs. SIR-MFG across all degree classes $k\in\{4,6,8,12,20\}$. SEIR-MFG peaks later and yields a larger final size for $k\leq 12$. For $k=20$, both models saturate at $n^{S*}=\mathfrak{n}_{\min}$, removing strategic differentiation; see Table in Section \ref{['sec:numerical']}.

Theorems & Definitions (33)

• Remark 2.1: Why $n^I \equiv 1$
• Remark 3.1: Infection lump-sum cost via the compensator identity
• Theorem 3.2: HJB Derivation for SEIR-MFG
• proof
• Definition 3.3: Degree-scaled infection pressure
• Theorem 3.4: Closed-Form Optimal Control
• proof
• Remark 3.5: Nash equilibrium value of $n_k^{E*}$ and extensions
• Theorem 3.6: Time-varying effective reproduction number as a spectral radius
• proof