Optimal control of inter-population disease spread via reaction-diffusion models

Spatial spread and interactions between heterogeneous populations play an important role in the dynamics and control of infectious diseases. 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, representing scenarios such as zoonotic spillover or transmission from a reservoir population to a secondary host population. The transmission coefficient in H₁ is time-dependent and governed by control parameters a=(α,γ,t_c)∈Q⊂R³, following an exponential decay. The control strategy is represented through a small set of parameters governing the temporal evolution of the transmission coefficient, resulting in a low-dimensional optimization problem embedded in the reaction-diffusion epidemic model. To represent realistic operational constraints, the intervention timing parameter is fixed at the earliest feasible response time. Consequently, the optimization focuses on two parameters governing the intensity and speed of the intervention. 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 (t_c) of interventions in H₁ affect epidemic outcomes in H₂. In particular, lower values of α, corresponding to higher intervention efficacy, lead to greater reductions in transmission in H₁ over time. The parameter γ regulates how quickly interventions take effect, modeling delays in behavioral change or intervention rollout. Our results show that early and sustained control strategies in H₁ can substantially mitigate the epidemic burden in H₂, even without direct interventions in that population. These findings highlight the importance of targeting upstream sources of infection to achieve downstream public health benefits.