A dengue outbreak is like a wildfire in the savanna: the flames leap from person to person via mosquito bites. Authorities respond in two ways — sending the sick to hospitals and spraying chemicals that kill the insects. But in reality, a hospital has only a limited number of beds, and fogging doesn't start immediately; it begins only when the number of infected crosses a critical mark.
The authors built a mathematical model that, like a sensitive barometer, tracks these thresholds. The model revealed three possible scenarios. If the number of patients is low, the system copes. When the influx of patients overwhelms hospitals and fogging lags behind,
In the most severe cases, new unstable states appear — the slightest nudge, and the situation sharply deteriorates.
The scientists didn't just describe the problem; they also found optimal intervention regimes. It turns out you can choose the frequency of fogging and the threshold for activating it so as to shave off epidemic peaks and avoid spraying chemicals needlessly. This is like fine-tuning a medication: the dose must be enough to bring down the fever, but not so high as to cause harm.
as written by the authors arXiv:2607.18140
Dengue remains a major public health challenge in tropical regions, and recurring outbreaks suggest that current intervention strategies are not yet fully effective. Existing mathematical models typically assume unlimited hospital capacity and continuously applied fogging, neglecting practical constraints that strongly influence disease control. We develop a non-smooth ordinary differential equation model of dengue transmission that incorporates finite hospital capacity and a threshold-triggered fogging strategy activated when reported infections exceed a prescribed fraction of the available capacity. The model exhibits three epidemiologically relevant operating regimes, reflecting changes in hospitalization and vector-control policies as the epidemic progresses. We establish the existence and local stability of the disease-free and endemic equilibria. Numerical continuation confirms the analytical results and reveals boundary-equilibrium bifurcations at the switching thresholds, a Hopf bifurcation after hospital capacity is exceeded leading to sustained oscillatory outbreaks, and a fold bifurcation near the epidemic threshold that generates additional unstable equilibria. We further investigate periodic solutions with respect to the fogging rate and activation threshold, identifying locally optimal intervention regimes that reduce epidemic peaks while avoiding unnecessarily intensive control efforts. The results demonstrate that hospital capacity, reactive fogging, and intervention thresholds fundamentally shape dengue dynamics and provide quantitative insights for designing effective state-dependent control strategies under limited healthcare resources.
The dengue virus is carried by Aedes aegypti mosquitoes, which bite primarily during the day, unlike malaria mosquitoes, which are active at night.