Learning contact policies for SEIR epidemics on networks: A mean-field game approach

Weinan Wang; Weinan Wang

doi:10.3934/mbe.2026076

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 7: 2083-2109. doi: 10.3934/mbe.2026076

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Research article

Learning contact policies for SEIR epidemics on networks: A mean-field game approach

Weinan Wang ^,

Department of Mathematics, University of Oklahoma, Norman, OK, USA

Received: 22 April 2026 Revised: 29 May 2026 Accepted: 23 June 2026 Published: 13 July 2026

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.
- mean-field games,
- SEIR epidemic model,
- heterogeneous networks,
- Nash equilibrium,
- strategic delay,
- behavioral epidemiology
Citation: Weinan Wang. Learning contact policies for SEIR epidemics on networks: A mean-field game approach[J]. Mathematical Biosciences and Engineering, 2026, 23(7): 2083-2109. doi: 10.3934/mbe.2026076

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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.

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