Impact of socio-economic and healthcare structure on epidemic dynamics: Mathematical modeling and optimal control

Abstract

We propose and analyze a deterministic SEIR-type epidemic model coupled with two dynamic capacity variables representing socio-economic support and healthcare resources. The resource variables reduce transmission and enhance recovery, whereas infection depletes both capacities, and healthcare operations impose an additional socio-economic cost. We establish positivity, boundedness, and global existence of solutions. The disease-free equilibrium is derived from the coupled resource subsystem, and the associated basic reproduction number R0 is obtained via the next-generation matrix method. The diseasefree equilibrium is locally asymptotically stable when R0 < 1 and unstable when R0 > 1. For R0 > 1, the endemic-equilibrium problem is reduced to a scalar equation, yielding the existence of at least one positive endemic equilibrium; a verifiable monotonicity condition that ensures uniqueness is also provided. We formulate an optimal control problem incorporating prevention, treatment, and healthcare-reinforcement controls, prove the existence of an optimal control, and derive the Pontryagin necessary conditions. For the illustrative baseline parameter set, the full endemic Jacobian has eigenvalues with negative real parts, and the numerical experiments illustrate lower exposed and infectious trajectories under the computed three-control solution of the optimality system.