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A Mathematical Model of Dengue Transmission Incorporating Hospital Capacity and Threshold-Based Fogging Interventions

This paper presents a non-smooth ordinary differential equation model of dengue transmission that integrates finite hospital capacity and threshold-triggered fogging interventions, revealing how these state-dependent constraints and control policies generate complex dynamics like oscillatory outbreaks and bifurcations to guide the design of effective, resource-aware intervention strategies.

Original authors: Dipo Aldila, Joseph Páez Chávez, Aytül Gökçe, Thomas Götz, Burcu Gürbüz

Published 2026-07-21

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CC BY 4.0

Original authors: Dipo Aldila, Joseph Páez Chávez, Aytül Gökçe, Thomas Götz, Burcu Gürbüz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine a city as a giant, bustling party where a sneaky, invisible virus is the uninvited guest. This virus, called Dengue, doesn't knock on doors; it rides in on tiny, buzzing mosquitoes. When these mosquitoes bite people, they pass the virus along. Sometimes, the person feels fine and keeps dancing (asymptomatic), but other times, they get a terrible fever and need to sit down (symptomatic). To stop the virus from taking over the whole party, health officials usually do two things: they send sick people to the hospital to rest and recover, and they spray a special mist (fogging) to kill the mosquitoes.

But here's the tricky part: hospitals aren't infinite. They have a limited number of beds, just like a party has a limited number of couches. If too many people get sick at once, the hospital fills up, and the rest have to stay home, still feeling sick and still able to pass the virus to mosquitoes. Also, the fogging isn't sprayed all day, every day; it's usually turned on only when things get scary, like when a certain number of people are already in the hospital. Scientists use math to predict how this virus spreads, but many old math models pretend hospitals are magic (they never fill up) and that fogging is a constant, never-ending rain. This paper asks a better question: What happens when we use math that actually respects the limits of real life?

The authors of this paper built a new kind of math model to simulate Dengue outbreaks, but they added some very realistic "rules of the game." First, they gave the hospital a hard limit on how many patients it can hold. Second, they made the fogging spray turn on and off like a light switch, triggered only when the number of hospitalized patients hits a specific warning line. They found that this simple change creates a wild, bumpy ride for the disease.

When the number of sick people is low, everything is calm: no fogging is needed, and everyone gets a hospital bed. But once the number of patients crosses the "fogging trigger" line, the sprayers turn on to kill mosquitoes. This usually helps calm the outbreak. However, if the outbreak gets too big and the hospital fills up completely, things get chaotic. The extra sick people who can't get beds stay at home, still spreading the virus, while the fogging is still running. The authors discovered that this specific combination—hospitals filling up while fogging is active—can cause the disease to stop settling down and instead start bouncing up and down in regular, repeating waves. It's like the virus gets stuck in a loop, causing the same big outbreak to happen over and over again.

The researchers also played with the "knobs" of their model to see how to stop these loops. They found that there isn't just one "perfect" way to fog. If you fog too early (when only a few people are sick) or too hard, you might waste money and resources without stopping the virus. In fact, sometimes being too quick to spray can actually make the problem worse later because the mosquitoes bounce back quickly once the spraying stops. The model suggests there is a "sweet spot" for when to start spraying and how hard to spray, but it's a delicate balance. If you miss that spot, the disease might not die out; instead, it could settle into a permanent, recurring cycle of outbreaks that are hard to control.

In short, this paper shows that treating hospitals as if they have infinite space and fogging as if it's a constant background noise gives us a false sense of security. Real-world limits matter. The study suggests that to truly control Dengue, health officials need to carefully tune their response: knowing exactly when to start spraying and understanding that if the hospital gets too full, the virus might start a never-ending cycle of outbreaks that are much harder to stop. It's a reminder that in the battle against disease, timing and resources are just as important as the weapons themselves.

Technical Summary: A Mathematical Model of Dengue Transmission Incorporating Hospital Capacity and Threshold-Based Fogging Interventions

Problem Statement
Dengue fever remains a persistent public health challenge in tropical regions, with recurring outbreaks suggesting current intervention strategies are insufficient. Existing mathematical models often rely on idealized assumptions that fail to capture critical operational constraints: specifically, they typically assume unlimited hospital capacity and the continuous application of fogging (fumigation). In reality, hospital resources are finite, and fogging is rarely applied continuously due to financial, logistical, and environmental constraints. Instead, fogging is often implemented reactively, triggered only when surveillance indicators (such as reported infections) cross a specific threshold. This discrepancy between theoretical models and practical constraints motivates the development of a model that explicitly incorporates finite hospital capacity and threshold-triggered, non-smooth fogging interventions.

Methodology
The authors develop a non-smooth ordinary differential equation (ODE) model describing the transmission dynamics between human and mosquito populations. The human population is divided into susceptible ( $S$ ), asymptomatic infected ( $A$ ), symptomatic infected ( $I$ ), and recovered ( $R$ ) compartments, while the mosquito population consists of susceptible ( $U$ ) and infected ( $V$ ) compartments.

The model introduces three distinct operating regimes governed by two critical thresholds:

Low Infection Regime ($I < kC$): No fogging is applied ( $h(I)=0$ ). All symptomatic individuals are hospitalized, and recovery occurs at the hospital rate $\gamma_1$ .
Moderate Infection Regime ( $kC \le I < C$ ): Fogging is activated ( $h(I)=\eta$ ) once the number of hospitalized cases reaches a fraction $k$ of the total capacity $C$ . Hospitalization remains full ( $\gamma_1$ ), but mosquito mortality increases due to fogging.
Overload Regime ( $I \ge C$ ): Fogging remains active, but hospital capacity is exceeded. Symptomatic individuals beyond capacity $C$ receive home treatment with a lower recovery rate $\gamma_2$ ( $\gamma_2 < \gamma_1$ ). Crucially, these untreated individuals remain infectious and contribute to transmission, modeled by an increased infection rate $\beta_{m2}$ for the un-hospitalized fraction.

The system is treated as a piecewise-smooth dynamical system. The authors perform a rigorous analytical investigation of the existence and local stability of disease-free and endemic equilibria within each regime. Furthermore, they employ numerical continuation techniques using the COCO software package to construct bifurcation diagrams, analyzing the system's behavior as key parameters (transmission rate $\beta_h$ , fogging rate $\eta$ , and activation threshold $k$ ) vary.

Key Contributions and Results

Non-Smooth Dynamics and Bifurcations: The study establishes that the interaction between hospital capacity and threshold-based fogging creates a complex dynamical landscape. The analysis identifies boundary-equilibrium bifurcations occurring precisely at the switching thresholds ($I=kC$ and $I=C$ ). These bifurcations mark the transitions between the three operating regimes.
Hopf Bifurcation and Periodic Outbreaks: When the transmission rate is sufficiently high to push infections beyond hospital capacity ( $I > C$ ), the system undergoes a Hopf bifurcation. This leads to sustained periodic solutions, implying that outbreaks may not settle into a steady endemic state but rather recur in waves with high infection peaks.
Fold Bifurcation and Equilibrium Complexity: Near the epidemic threshold, a fold bifurcation is identified, producing additional unstable endemic equilibria even when the basic reproduction number ( $R_0$ ) is less than 1. This indicates a more intricate equilibrium structure than standard smooth models suggest, where the disease could potentially persist in unstable states before dying out.
Non-Monotonic Control Efficacy: Numerical continuation reveals that the relationship between control parameters and public health outcomes is non-monotonic.
- Fogging Rate ( $\eta$ ) and Threshold ( $k$ ): There exist locally optimal values for both the fogging intensity and the activation threshold that minimize peak infections ( $I_{Peak}$ ) and the fraction of asymptomatic cases ($AFrac$).
- Counter-intuitive Findings: Overly early fogging (low $k$ ) or excessively intense fogging does not necessarily yield proportional epidemiological benefits. Early activation keeps fogging active for longer periods, but due to the structural delays in human and mosquito dynamics, this can lead to a mismatch with the epidemic cycle, causing susceptible accumulation and a larger rebound once the threshold is no longer met. Similarly, high-intensity fogging suppresses mosquitoes strongly but briefly, leading to rapid population recovery once the intervention stops, potentially causing a sharper resurgence.

Significance
The paper argues that hospital capacity, fogging intensity, and the activation threshold are not merely operational details but fundamental drivers of dengue dynamics. The findings suggest that effective control policies require the joint calibration of these factors rather than treating early or intensive intervention as inherently optimal. The model provides quantitative guidance for designing reactive dengue control policies, highlighting that poorly timed or delayed fogging can prolong transmission or trigger larger resurgences. By capturing the discontinuous nature of real-world interventions, the model offers a more realistic framework for understanding how resource constraints and reactive policies shape the persistence and recurrence of dengue outbreaks.

Future Directions
The authors note that future work could involve a formal classification of the Hopf bifurcation criticality, sliding-mode analysis of the switching manifolds, and the incorporation of seasonal forcing, stochastic effects, and spatial heterogeneity. Additionally, formulating the intervention strategy as an optimal control problem calibrated against real data is proposed to enhance the model's utility as a decision-support tool.

Technical Summary: A Mathematical Model of Dengue Transmission Incorporating Hospital Capacity and Threshold-Based Fogging Interventions

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