Pre-Publicación 2025-10
Veronica Anaya, Gerardo Chowell, Felipe Jara, Mauricio Sepúlveda:
Optimal Control of Inter-Population Disease Spread via Reaction–Diffusion Models
Abstract:
In this work, we study the control of a reaction–diffusion system modeling the spread of an infectious disease between two interacting populations, H1 and H2, within a shared spatial domain Ω = Ω1 ∩ Ω2. The model incorporates constant-coefficient spatial diffusion and excludes non-local, nonlinear, or cross diffusion terms. The disease originates in population H1 and is transmitted to H2 through contact between infected individuals in H1 and susceptible individuals in H2. The transmission coefficient in H1 is timedependent and governed by control parameters a = (α, γ, tc) ∈ Q ⊂ R3, following an exponential decay. The objective is to minimize a cost functional associated with the attack rate and cumulative incidence in H2. We establish the existence and uniqueness of solutions to the reaction–diffusion system and the associated optimal control problem. Using a Lagrangian framework, we derive the continuous gradient of the cost functional and prove the well-posedness of the adjoint system, along with the necessary optimality conditions. Numerical experiments illustrate how changes in the intensity (α), rate (γ), and timing (tc) of interventions in H1 affect epidemic outcomes in H2. Specifically, lower values of α corresponding to stronger intervention efficacy—lead to greater reductions in transmission in H1 over time. The parameter γ regulates how quickly interventions take effect, modeling delays in behavior change or intervention rollout. Our findings show that early and sustained control strategies in H1 can substantially mitigate epidemic burden in H2, even without direct interventions in H2. This highlights the importance of targeting upstream sources of infection to achieve downstream public health benefits.
