Transmission dynamics of dengue–Zika co-infection with antibody-dependent enhancement, cross-immunity, and vaccination: backward bifurcation analysis and data calibration

The co-circulation of dengue virus (DENV) and Zika virus (ZIKV) in Aedes aegypti-endemic regions creates a significantly more dangerous epidemiological landscape than either pathogen alone. Yet no existing model simultaneously incorporates antibody-dependent enhancement (ADE), dual Zika transmission routes, synergistic co-infection mortality, dengue vaccination, and calibration against confirmed co-endemic surveillance data. We address this gap by developing a sixteen-compartment deterministic model that captures ADE through susceptibility multipliers epsilon_1 and epsilon_2, partial cross-immunity, imperfect dengue vaccination, and both vector-borne and sexual Zika transmission. We derived the disease-free equilibrium and obtained closed-form expressions for the component reproduction numbers mathcal{R}_{0d} and mathcal{R}_{0z}. The overall reproduction number mathcal{R}_0 = rho(FV^{-1}), computed as the spectral radius of the 8 X 8 next-generation matrix, satisfies mathcal{R}_0 geq max{mathcal{R}_{0d}, mathcal{R}_{0z}}, with equality only when the linear cross-infection parameters beta_{z3}, beta_{vd2}, beta_{vz2} are zero. All co-infection effects, including ADE, are confined to the nonlinear regime. This confirms that single-disease models can systematically underestimate epidemic risk during co-circulation. Local asymptotic stability of the disease-free equilibrium when mathcal{R}_0 < 1, uniform persistence when mathcal{R}_0 > 1, and the existence of at least one endemic equilibrium are established. Centre manifold analysis reveals backward bifurcation in both submodels, implying that reducing \mathcal{R}_0 below unity is necessary but not sufficient for disease elimination. The model was calibrated using 36 weeks of SINAN surveillance data from Esp\'{i}rito Santo state, Brazil (January--September 2021), yielding key fitted parameters: b^* = 1.278~wk^{-1}, beta_d^* = 0.749, beta_{z2}^* = 0.161, beta_{vd1}^* = 0.118, and beta_{vz1}^* = 0.252, with Pearson correlations of r = 0.594 (dengue) and r = 0.739 (Zika). Sensitivity analysis identifies the mosquito biting rate b and vector mortality \mu_v as the dominant intervention targets. Simulations reveal super-linear amplification of co-infections with rising biting rates and a 120-fold suppression of peak dengue incidence when vector mortality doubles. These findings support an integrated control strategy in which sustained vector management is the most cost-effective lever, complemented by improved healthcare access and high-coverage dengue vaccination to mitigate ADE-driven amplification.