Age-structured epidemic models partition a population by both disease status (susceptible, infected, removed) and biological age. The infection force — the rate at which susceptibles become infected — depends nonlinearly on the age distribution of infected individuals, because different age groups mix at different rates. The time-dependent infection force makes the model a nonlinear, non-autonomous partial differential equation in age and time.
Malaguti and Perrotta (arXiv:2603.12943) prove existence and uniqueness for this model using topological degree theory for condensing maps — a different approach from the fixed-point methods previously applied. The shift from fixed points to topological degree is not just a technical alternative; it provides stronger results about the solution's dependence on initial data.
The semigroup formulation is the enabling step. The age-structured dynamics generate a semigroup — a family of operators indexed by time — on an appropriate function space. The semigroup is not compact (because the age variable is unbounded), but it is condensing: it shrinks the measure of non-compactness of bounded sets. For condensing maps, the topological degree is well-defined and shares the key properties (homotopy invariance, additivity, normalization) that make degree theory useful for existence proofs.
The result: a unique, global, nonnegative mild solution that depends continuously on initial data. “Mild” means the solution satisfies an integral formulation rather than the differential equation pointwise — weaker than a classical solution but strong enough for epidemiological applications. The topological degree approach handles the nonlinear, time-dependent infection force without requiring the contraction estimates that fixed-point methods need, making it applicable to a broader class of transmission functions.