A Two-Patch SEIRV Model with Waning Vaccine Immunity for Measles Transmission Dynamics in Urban and Rural Bangladesh: Mathematical Analysis and Vaccination Scenario Projections for the 2026 Outbreak

Bangladesh is experiencing a severe measles resurgence, following the collapse of first-dose vaccination coverage from 97% (2023) to 57% (2025). Existing models treat Bangladesh as spatially homogeneous, ignoringthe pronounced urban–rural disparity in immunity and transmission intensity. We develop a two-patchdeterministic SEIRV compartmental model for the full urban and rural populations, with cross-patch exposure represented by a residence-time contact-mixing term rather than physical relocation, so each patch’spopulation remains exactly at its target value at equilibrium. Vaccination is calibrated from reported dosecoverage, and the disease-free-equilibrium protected fraction is derived as an explicit function of coverage andvaccine efficacy. The basic reproduction number R0, the Perron eigenvalue of the next-generation matrix,is 2.712 under 2025 coverage (79% urban, 50% rural). The disease-free equilibrium is locally asymptoticallystable when R_0 < 1; when cross-patch mixing is present and R-0 > 1, the disease is uniformly persistent and an endemic equilibrium exists, though global stability remains open. The disease-free-equilibriumprotected fraction, R_0, and herd immunity threshold (HIT) are exactly invariant to the assumed vaccinewaning rate. Sensitivity analysis shows rural vaccination coverage is by far the most actionable lever onR_0 (Υ_covR = −0.968), while the urban index is small (Υ_covU = −0.023). HIT is 85.2% (urban) and 82.0%(rural). An exploratory single-patch comparison illustrates the importance of urban–rural heterogeneityfor projected burden and peak timing, though it does not isolate spatial aggregation from accompanyingparameter changes. Restoration of 95% coverage in both patches drives R_0 = 0.341, below the eliminationboundary. [Under Review]