Analysis of a reaction-diffusion SVEIR model for measles geographic spread

We formulate and analyze a deterministic reaction-diffusion SVEIR model for the geographic spread of measles, explicitly incorporating vaccination, vaccine efficacy, latency, recovery, demographic turnover, and spatial diffusion. The disease-free equilibrium and the basic reproduction number R₀ are obtained using the next-generation matrix, and linear stability analysis identifies the invasion threshold and the critical spatial mode. Because the critical wavenumber is zero, a long-wavelength analysis is used to characterize slow spatial variation near the invasion front. A mathematical consistency check shows that the common-diffusion formulation does not support a finite-wavenumber Turing instability; the roll, square, and hexagonal modes are therefore presented only as generic pattern-theory modes rather than as patterns proved to arise from the present model. Numerical simulations illustrate spatiotemporal propagation and quantify the effects of vaccine efficacy, vaccination rate, and transmission intensity.