Modeling Dengue-COVID-19 Co-Epidemics via Crossover Discrete Time Systems with Variable-Order Fractional Memory

This paper develops a comprehensive discrete-time mathematical framework to investigate the co-circulation dynamics of two dengue virus strains and COVID-19 by integrating integer-order modeling with advanced fractional and variable-order operators. We formulate four epidemiological models: a classical integer-order system, a fractional Caputo model with constant memory, and two novel crossover models in which the system transitions between fixed- and variable-order fractional operators to represent regime shifts in immunity, behavioral changes, and intervention strategies. This formulation captures nonlocal memory effects and provides a flexible mechanism to describe evolving epidemic phases. A rigorous analytical study is conducted, establishing positivity and boundedness of solutions and proving existence and uniqueness using Perov's fixed-point theorem in a generalized Banach space. The basic reproduction number is derived via the next-generation matrix approach, and local stability of the disease-free equilibrium is characterized. Furthermore, we show that the proposed systems undergo a forward (supercritical) transcritical bifurcation as the basic reproduction number crosses unity. The developed fractional and variable-order models are also shown to satisfy Ulam-Hyers and Lyapunov stability properties. Extensive numerical simulations validate the theoretical findings and demonstrate the role of fractional memory, variable-order dynamics, and crossover transitions in shaping co-epidemic trajectories. The results highlight the importance of incorporating time-varying memory effects in modeling real-world dengue-COVID-19 interactions and provide a robust mathematical framework to support public health planning during co-epidemic scenarios.