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

Verónica Anaya1, Gerardo Chowell2, Felipe Jara3

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Area of Science:

  • Mathematical modeling of infectious diseases
  • Epidemiology
  • Control theory

Background:

  • Infectious disease dynamics are influenced by spatial spread and population interactions.
  • Modeling disease transmission between heterogeneous populations (H1 and H2) is crucial for understanding and controlling outbreaks.
  • Zoonotic spillover or reservoir-to-host transmission dynamics can be modeled using reaction-diffusion systems.

Purpose of the Study:

  • To investigate the optimal control of a reaction-diffusion model for infectious disease spread between two interacting populations.
  • To analyze the impact of time-dependent transmission control parameters in population H1 on epidemic outcomes in population H2.
  • To minimize the attack rate and cumulative incidence in population H2 through interventions in population H1.

Main Methods:

  • Developed a reaction-diffusion system modeling disease spread between populations H1 and H2.
  • Formulated an optimal control problem with a low-dimensional parameter set governing transmission control in H1.
  • Utilized a Lagrangian framework to derive the cost functional gradient and establish necessary optimality conditions.
  • Performed numerical experiments to assess the effects of intervention intensity (α), rate (γ), and timing (tc).

Main Results:

  • Established the existence and uniqueness of solutions for the reaction-diffusion system and optimal control problem.
  • Demonstrated that lower intervention intensity (α) in H1 leads to greater transmission reduction over time.
  • Showed that the parameter γ influences the speed of intervention effectiveness, reflecting implementation delays.
  • Confirmed that early and sustained control in H1 substantially mitigates epidemic burden in H2.

Conclusions:

  • Targeting upstream infection sources (population H1) is a highly effective strategy for downstream public health benefits (population H2).
  • Optimal control strategies focusing on intervention intensity and speed in the source population can significantly reduce epidemic spread.
  • Mathematical modeling provides valuable insights into designing effective public health interventions for inter-population disease dynamics.