Modeling vaccination in a heterogeneous metapopulation system

• Dynamics of heterogeneous metapopulations that contain vaccinated individuals is proposed.
• Epidemic threshold only dependent on the diffusion coefficient of the infected population.
• With low diffusion coefficient, reproduction number can be calculated.
• Reproduction number dependent on the number of neighbors a city can have.

We present here a multicity SIS epidemic model with vaccination. The model describes the dynamics of heterogeneous metapopulations that contain imperfectly vaccinated individuals. The effect of vaccination on heterogeneous multicity models has not been previously studied. We show that under very generic conditions, the epidemic threshold does not depend on the diffusion coefficient of the vaccinated individuals, but it does depend on the diffusion coefficient of the infected population. We then show, using a novel methodology, that the reproduction number is determined by the homogeneous model parameters and by the maximal number of neighbors a city can have, when the diffusion coefficient of the infected population is low. Finally, we present numerical simulations to support the analytical results.