When Hospitals Overflow: The Mathematical Threshold That Turns Dengue Into a Recurring Crisis

Every year, dengue fever pushes millions of people into hospitals across the tropics—and when those hospitals fill up, the disease doesn't wait politely. It keeps spreading. In 2024 alone, the world recorded more than 14 million dengue cases, over 52,000 severe infections, and more than 11,000 deaths, the highest toll in recorded history. Yet most mathematical models used to plan dengue control assume hospitals have infinite beds and that mosquito-killing fogging campaigns run forever—simplifications that make the math tractable but the conclusions unreliable for actual public health decisions.

A team of mathematicians from Indonesia, Ecuador, Turkey, and Germany has built a more honest model. Their work, published in July 2025 on arXiv, incorporates two constraints that every dengue response team knows intimately but models routinely ignore: hospitals that run out of beds, and fogging campaigns that turn on and off based on how bad things look. The result is not just a more realistic picture of dengue dynamics—it's a set of specific, actionable numbers: an optimal fogging intensity, a trigger threshold, and a clear-eyed warning about what happens when hospitals overflow.

The study uses a branch of mathematics called non-smooth dynamical systems, which is exactly what you need when a policy flips a switch—fogging turns on, say, when hospitalized cases hit a certain number. The model tracks six interacting populations: susceptible humans, asymptomatically infected humans, symptomatically infected humans, recovered humans, susceptible mosquitoes, and infected mosquitoes. Each population changes according to differential equations that describe birth, infection, recovery, and death. What makes this model different is that three of those equations have different forms depending on whether symptomatic infections are below hospital capacity, above it, or at the trigger threshold that activates fogging.

When hospitals have room, symptomatic patients get admitted, treated, and removed from the infectious pool. When hospitals overflow—which they do in major outbreaks—some patients go home to recover on their own. Those patients stay infectious longer. They also infect more mosquitoes, because the model assumes that only hospitalized patients are effectively isolated from mosquito bites. This creates a feedback loop: more infections overwhelm hospitals, which leaves more people untreated, which feeds more infections back into the mosquito population.

The model also incorporates a fogging threshold, denoted k, which represents the fraction of hospital capacity that triggers a fogging response. If k = 0.6, for example, fogging begins when hospitalized cases reach 60% of maximum capacity—a policy choice reflecting how aggressively a health authority wants to respond. Below that threshold, no fogging occurs. Above it, a constant fogging rate η kicks in, killing mosquitoes at a steady clip until the epidemic subsides below the trigger point.

This threshold-based approach mirrors how governments actually operate. Fogging isn't free—it's expensive, environmentally contentious, and generates insecticide resistance when used excessively. No health authority fogs continuously. They fog when the data suggest they must. By modeling this on-off behavior explicitly, the researchers capture dynamics that smoother models miss entirely.

The mathematics of the model are sophisticated but the epidemiological intuition is clear. The human population is split into four compartments. Susceptible individuals (S) acquire infection when bitten by infected mosquitoes. A fraction p of newly infected people—approximately 27%, based on the literature—develop no symptoms and move into the asymptomatic compartment (A). They recover naturally at rate γ₀. The remaining 73% develop symptoms; some transition from asymptomatic to symptomatic at rate α as their illness progresses, while the rest move directly into the symptomatic compartment (I). Symptomatic individuals either recover with hospital treatment at rate γ₁, recover with home treatment at rate γ₂ (which is slower, since γ₂ < γ₁), or die from dengue at rate δ. Recovered individuals (R) have temporary immunity.

Mosquitoes are simpler: they're either susceptible (U) or infected (V). New mosquitoes are born at rate Λₘ, all into the susceptible class. They become infected when they bite an asymptomatic person or an untreated symptomatic person—the model assumes hospitalized patients don't contribute to transmission because they're isolated from mosquito contact. The infection rate from symptomatic individuals is higher than from asymptomatic ones, reflecting the higher viral loads in people with active symptoms.

The recovery function f(I) for symptomatic individuals is piecewise: when I < C (hospital capacity), all symptomatic patients receive hospital treatment at rate γ₁. When I ≥ C, only C patients can be hospitalized, and the rest recover at home at rate γ₂. The infection term for mosquitoes g(U, A, I) is similarly piecewise: only asymptomatics and non-hospitalized symptomatics contribute to mosquito infection.

The fogging term h(I) is the simplest piecewise function: zero below the threshold kC, a constant η above it. This on-off structure is what makes the system non-smooth—mathematically, the derivatives can jump at the threshold.

The model is parameterized for Jakarta, Indonesia, where dengue is endemic and hospital data are relatively well-documented. The human population is approximately 11.25 million, with a life expectancy of 64.7 years, giving a natural death rate μₕ ≈ 1/(64.7 × 365) per day and a birth rate Λₕ set to balance this. Mosquitoes live much shorter lives—between 7.7 and 40 days depending on conditions—yielding death rates in the range μₘ ∈ [1/40, 1/7.7] per day. The mosquito-to-human population ratio is assumed to be 2:1 in the absence of fogging.

The transmission parameters come from the dengue modeling literature. The mosquito-to-human transmission rate βₕ is the product of mosquito biting rate and the probability of transmission per bite; in the baseline scenario, the researchers use βₕ ≈ 0.472/Nₕ, where Nₕ is the human population size. The human-to-mosquito rates are βₘ₁ ≈ 0.063/Nₕ for asymptomatic transmission and βₘ₂ = 5βₘ₁ for symptomatic transmission, reflecting higher viral loads in symptomatic patients.

Hospital capacity C is set to 15,000 beds in the baseline scenario, derived from Jakarta's approximately 2.6 hospital beds per 1,000 people, with about 5% allocated to dengue patients. The trigger threshold k is set to 0.6 in the baseline, meaning fogging begins when hospitalized cases reach 9,000. The baseline fogging rate η is 0.01 per day—a relatively modest intervention—though the researchers explore a wide range in their numerical experiments.

The case fatality rate δ is perhaps the most morally significant parameter. The researchers cite a range from 0.4% with treatment (consistent with Indonesia's reported CFR) to 20% without it. In the model, this rate applies uniformly to all symptomatic individuals regardless of treatment setting, which is a simplifying assumption that errs on the side of caution.

With these parameters, the researchers first establish the basic reproduction number ℛ₀—the average number of secondary infections generated by a single infected individual in a fully susceptible population. For dengue, ℛ₀ typically ranges from 2 to 5 in endemic areas. The model's structure yields a more complex expression than the classic Ross-Macdonald formulas, because the transmission rates depend on which compartment infects mosquitoes and whether hospitals are overwhelmed.

The equilibrium analysis reveals three possible states, corresponding to the three operating regimes: a disease-free equilibrium (DFE) where dengue cannot sustain itself, an endemic equilibrium with full hospitalization (EE1), and an endemic equilibrium with hospital overflow (EE2). The existence and stability of each depends on ℛ₀ and on the policy parameters k and η.

In the disease-free equilibrium, both human symptomatic infections I and infected mosquitoes V equal zero. This equilibrium always exists. It is locally asymptotically stable when ℛ₀ < 1 and unstable when ℛ₀ > 1—the familiar threshold behavior. However, the researchers show that the threshold ℛ₀ = 1 does not coincide exactly with the bifurcation point where the endemic equilibria emerge; instead, there is a region where multiple equilibria can coexist, a phenomenon called backward bifurcation. This occurs because the model incorporates nonlinear recovery terms that saturate hospital capacity.

When ℛ₀ exceeds 1, the system can settle into EE1 (endemic equilibrium with full hospitalization) if the basic reproduction number stays below a second threshold that depends on the hospital capacity. In this regime, the disease persists at a constant level, hospitals keep up with demand, and fogging may or may not be active depending on whether the equilibrium I* falls below the trigger threshold kC.

As transmission increases further, or as the initial outbreak overwhelms hospital capacity, the system can transition to EE2, the overflow regime. Here, I* > C, meaning hospitals cannot accommodate all symptomatic patients. Untreated individuals recover more slowly and remain infectious longer, which paradoxically sustains higher transmission. The fogging rate η, which was calibrated to work in the full-hospitalization regime, may now be insufficient to bring the epidemic under control. The equilibrium can be stable or unstable depending on parameters, and the model predicts that oscillatory dynamics become possible.

The researchers use numerical continuation methods to trace how equilibria change as parameters vary—essentially drawing bifurcation diagrams that reveal the full landscape of possible dynamics. A one-parameter continuation with respect to the mosquito-to-human transmission rate βₕ reveals a rich structure with several bifurcation points. At very low transmission (βₕ ≈ 2.05 × 10⁻⁸), a branching point (BP) marks where the endemic equilibrium branch emerges from the disease-free equilibrium. Two boundary equilibrium bifurcations (BE1 at βₕ ≈ 2.62 × 10⁻⁸ and BE2 at βₕ ≈ 3.84 × 10⁻⁸) occur as the system crosses the fogging threshold kC and the hospital capacity threshold C, respectively. At each boundary, the vector field changes discontinuously, producing a non-smooth fold (NSF1 at βₕ ≈ 4.70 × 10⁻⁸) where the equilibrium structure suddenly shifts. A classical fold (F) occurs near the epidemic threshold, generating additional unstable equilibria even at transmission rates where the disease might be expected to die out. Finally, a Hopf bifurcation (H) at βₕ ≈ 7.67 × 10⁻⁷ marks where the stable endemic equilibrium loses stability and gives way to sustained periodic oscillations.

This last finding is perhaps the most dramatic: when hospitals overflow, the dengue system doesn't settle into a steady state. Instead, it enters a regime of recurrent outbreaks, with infections rising and falling in predictable but persistent cycles. The oscillations arise because of a feedback loop between hospital saturation and transmission. When infections spike, hospitals overflow, more people stay home and remain infectious, mosquito infection rates increase, more mosquitoes emerge infected, and infections spike again. Fogging provides a dampening force, but only if it's strong enough and triggered early enough.

The Hopf bifurcation at H separates two fundamentally different epidemiological futures. Below H, a sufficiently aggressive outbreak will burn itself out or settle into a manageable endemic steady state. Above H, the system exhibits limit cycles—infinite sequences of outbreak peaks and troughs that persist without any external forcing. For public health, this means that once transmission crosses a certain threshold and hospitals fill up, dengue transitions from a problem you can potentially manage with steady effort to a problem that demands continuous, adaptive intervention.

The fold bifurcation at F reveals another subtler danger: near the epidemic threshold, the model generates multiple unstable equilibria. This means that even when conditions seem barely favorable for transmission, the system can exhibit hysteresis—history-dependent behavior where the same parameter values can support either disease extinction or sustained transmission depending on how you got there. An outbreak that tips the system into the unstable region can self-amplify, while a system that just barely avoids that region may persist indefinitely in a precarious disease-free state.

Having characterized the equilibrium structure, the researchers turn to the periodic solutions that arise after the Hopf bifurcation. Using numerical continuation with respect to the fogging rate η and the activation threshold k, they trace how periodic outbreak peaks respond to these policy parameters. The results are counterintuitive in some respects and reassuring in others.

When they vary the fogging rate η while holding all other parameters fixed in the oscillatory regime, they find that peak infections first decrease sharply as η increases from zero—the obvious benefit of killing more mosquitoes. But then the curve bottoms out at η ≈ 0.0155 per day (point P_opt1 in the paper's notation) and begins to rise again at higher fogging rates. The researchers also identify a second local minimum at η ≈ 0.01505 per day (P_opt2), separated by a small hump. Beyond this hump, further increases in fogging intensity actually correlate with larger outbreak peaks.

This non-monotonic behavior is surprising at first glance. Why would more fogging sometimes mean worse outcomes? The answer lies in the interaction between fogging and the threshold trigger. In the model, fogging activates when I exceeds kC and deactivates when I falls below kC. With very aggressive fogging, infections drop rapidly below the threshold, and fogging turns off. But the system is now in a regime where the remaining infected population can regrow because the equilibrium is unstable. Mosquitoes rebloom, infections rise again, fogging reactivates briefly, then shuts off, leaving the system cycling around an unstable endemic equilibrium. Moderate fogging, by contrast, keeps the epidemic suppressed just enough that the system can settle into the stable manifold of the endemic equilibrium rather than oscillating around it.

The researchers also examine the cost implications of fogging intensity. Cost in their model increases linearly with η, reflecting the operational expense of fogging campaigns. At low η, the high infection burden drives total cost up despite low fogging expenditure. At very high η, fogging costs dominate. Between these extremes lies a cost-minimizing region that aligns roughly with the epidemiological optima. But the researchers are careful to note that their cost model is simplified; a fuller analysis would need to incorporate environmental costs, insecticide resistance risks, and community compliance effects.

The continuation with respect to the activation threshold k tells a similar story. When k is very small—meaning fogging begins very early, when hospitals are barely stressed—peak infections are low but costs are high because fogging activates frequently and for long durations. As k increases, triggering fogging later, peak infections initially decrease slightly (because the system saves its fogging resource for when it's more needed), then begin to rise sharply once k crosses approximately 0.82 (P_opt3). Two local optima appear: k ≈ 0.82 (P_opt3) and k ≈ 0.845 (P_opt4). Below these values, earlier triggering provides diminishing returns; above them, the delay is too long, and the outbreak has already peaked by the time fogging activates.

The researchers also track the fraction of asymptomatic infections A_Frac,% across the continuation, finding that it is relatively insensitive to k in the optimal range but increases sharply when fogging is triggered very late. This makes sense: late fogging allows more transmission, and a larger fraction of infections are asymptomatic by virtue of the longer time course of the epidemic.

The two-parameter continuation, which simultaneously varies fogging rate and transmission rate, produces a landscape with three distinct regions: a disease-free region where dengue cannot persist regardless of fogging; a region where the endemic equilibrium is stable, so fogging can drive the system to that equilibrium; and a region where periodic oscillations dominate, requiring sustained adaptive intervention. The boundary between the second and third regions is defined by the Hopf bifurcation surface, which the researchers approximate as a curve ℓ₁ in the (η, βₕ) plane. A second boundary curve ℓ₂ separates the stable endemic region from the disease-free region.

These bifurcation boundaries have direct policy interpretations. If transmission is low enough (βₕ below ℓ₂), dengue will die out on its own; intervention is unnecessary. If transmission is moderate (between ℓ₂ and ℓ₁), a sufficiently aggressive fogging campaign can drive the system to a stable endemic equilibrium, effectively managing dengue at a constant level. If transmission is high (above ℓ₁), the system is in the oscillatory regime, and no static fogging policy can achieve equilibrium; instead, authorities must continuously adjust their response to the epidemic's phase.

The model makes several assumptions that are worth acknowledging. It assumes homogeneous mixing—both humans and mosquitoes encounter each other at random, which overestimates transmission in sparse rural settings and underestimates it in densely packed urban environments. It assumes a single dengue serotype, ignoring the complex cross-immunity and antibody-dependent enhancement that arise from sequential infections with different serotypes. It assumes that hospitalized patients are perfectly isolated from mosquito contact, which is an idealization even in well-run hospitals. And it assumes a constant human population size in the absence of disease mortality, which may not hold in high-CFR scenarios.

The fogging model itself is simplified: a constant additional mortality rate for mosquitoes, regardless of fogging frequency, spatial coverage, or mosquito resistance. Real fogging campaigns involve complex logistics, diminishing returns from repeated applications, and potential community resistance. The threshold trigger is based on hospitalized cases, which requires accurate surveillance data that many dengue-endemic countries lack.

Despite these limitations, the model offers genuine insight. The most important is the existence and consequences of the Hopf bifurcation at the hospital overflow threshold. This is not a theoretical curiosity—it describes what dengue-endemic cities actually experience when an outbreak peaks. Hospitals fill up. Emergency departments divert patients. Treatment is delayed. And then, often, the epidemic subsides, only to return months later. The model explains this cycle as a deterministic consequence of the interaction between hospital capacity and transmission dynamics, not merely as a result of seasonal forcing or random fluctuations.

The optimal fogging parameters—η ≈ 0.0155 per day, k ≈ 0.82—are specific enough to inform policy, though they should not be taken as universal prescriptions. They depend on all the other parameters in the model, which vary by location and over time. What the model provides is a framework: given local estimates of transmission rates, hospital capacity, and case fatality, officials can compute their own optimal trigger threshold and fogging intensity.

The finding that excessively early fogging can be counterproductive is particularly relevant for cost-conscious health ministries. If fogging too aggressively depletes budgets, breeds insecticide resistance, and generates community fatigue without proportionally reducing transmission, then targeting fogging to the right moment—neither too early nor too late—becomes a form of efficiency as well as a epidemiological strategy.

The backward bifurcation near the epidemic threshold is a warning about the fragility of near-elimination states. When transmission is barely below the threshold for persistence, small perturbations—seasonal increases in mosquito abundance, imported cases, lapses in surveillance—can push the system into the endemic regime. And once there, reversing course requires more than simply returning to the pre-perturbation parameters; the system has multiple equilibria, and getting back to disease-free may require a sustained reduction in transmission well below the level that originally caused extinction. This has implications for dengue elimination campaigns, which should build in buffer margins rather than targeting the theoretical minimum intervention.

What does this mean for the world? Dengue is expanding. Climate change is extending the geographical range of Aedes aegypti mosquitoes, and urbanization is concentrating susceptible human populations in cities where transmission is efficient. The 2024 record of 14 million cases is not an anomaly; it is likely a preview of coming decades. Countries that currently see dengue only occasionally will need to develop response capacity, and those that already struggle with it will face growing pressure.

The model developed in this paper offers a more realistic planning tool than previous approaches. By explicitly incorporating hospital capacity constraints, it captures the feedback loop between epidemic severity and healthcare system strain that determines whether an outbreak becomes a crisis. By modeling threshold-based fogging, it aligns with how control programs actually operate. And by identifying optimal intervention parameters and bifurcation boundaries, it provides quantitative guidance for policy.

Several directions for future research follow naturally from this work. Extending the model to incorporate seasonal forcing—rainfall patterns that drive mosquito abundance, temperature effects on transmission efficiency—would make it more applicable to specific regions. Adding spatial heterogeneity, so that the model distinguishes urban and suburban areas or different neighborhoods within a city, would enable targeted rather than uniform intervention strategies. Incorporating multiple dengue serotypes would allow analysis of how hospital capacity constraints affect the probability of severe secondary infections, which tend to occur when someone infected with one serotype is later infected with another.

A particularly valuable extension would be to replace the fixed threshold trigger with a more realistic surveillance model. Real data streams are noisy, reporting is delayed, and the fraction of cases that are hospitalized varies with healthcare-seeking behavior. A model that incorporates these uncertainties would be more directly applicable to operational decision-making.

Finally, the optimal control theory framework used in previous dengue modeling studies could be applied to this non-smooth model, seeking not just static optimal parameters but dynamic intervention strategies that adjust to the evolving state of the epidemic. Such strategies might activate fogging at the threshold, as in the current model, but also vary fogging intensity in response to real-time surveillance data.

The broader lesson of this work is that effective dengue control is not a single intervention but a system of interacting constraints. Hospital capacity shapes how many people receive treatment and how long they remain infectious. Fogging intensity determines how quickly the mosquito population can be suppressed. The activation threshold reflects how aggressively a health authority is willing to act before an outbreak becomes unmanageable. And underlying transmission dynamics set the stage on which all of these interventions play out.

Understanding these interactions quantitatively—not just qualitatively—is the first step toward designing control strategies that are both effective and efficient. The model presented here is a significant step in that direction.