<p>Mechanistic infectious disease models are crucial for understanding the forces driving epidemics. Many epidemic models naturally produce equilibrium or stationary behaviors readily seen in epidemiological surveillance. However, respiratory viruses in tropical locations exhibit non-equilibrium behavior without consistent timing. This prevents standard models that commonly produce cycles and equilibria from describing their true dynamics. To evaluate deterministic model structures that can create asynchronous nonannual epidemic patterns seen with tropical influenza, we parameterized 30 variations of a respiratory disease model incorporating three influenza (sub)types and evaluated their abilities to emulate nonannual behaviors. Variations across models included immune-waning dynamics, subpopulations, and periodic case introduction. We defined seven criteria describing irregular epidemic behaviors and conducted parameter-space searches to find parameter sets meeting these criteria. Small parameter perturbations (1%) often led to violations of these criteria, indicating a lack of robustness in the parameter sets’ abilities to predict nonannual influenza dynamics. The models were also unstable to traditional perturbation in state space. We were unable to find evidence of a deterministic epidemic model with a stable parameterization or structure that can robustly produce asynchronous behaviors. It remains unknown whether a canonical deterministic epidemic model exists for the observed asynchronous nonstationary dynamics of tropical influenza.</p>