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

Source: arXiv:2607.18140 · Published 2026-07-20 · By Dipo Aldila, Joseph Páez Chávez, Aytül Gökçe, Thomas Götz, Burcu Gürbüz

TL;DR

This paper addresses the practical limitations in dengue transmission modeling by incorporating finite hospital capacity and a threshold-based fogging intervention into a non-smooth ordinary differential equation framework. Traditional dengue models typically assume unlimited healthcare resources and continuous vector control, which are unrealistic in epidemic management. The authors develop a compartmental host-vector model that segments the human population into susceptible, asymptomatic, symptomatic, and recovered groups, while modeling mosquito populations as susceptible and infected. Hospital capacity constrains symptomatic patient treatment, reducing recovery rates once capacity is exceeded, and fogging is activated only when hospitalizations surpass a specified fraction of capacity. The model exhibits three distinct dynamical regimes corresponding to no fogging with full hospitalization, fogging with full hospitalization, and fogging with exceeded capacity and partial hospitalization. Analytical results prove existence and local stability of disease-free and endemic equilibria, while numerical bifurcation analysis reveals complex dynamics including boundary-equilibrium and fold bifurcations, and Hopf bifurcations leading to sustained oscillations once hospital capacity is breached. By varying fogging intensity and activation thresholds, the study identifies locally optimal control policies that minimize peak infection levels and asymptomatic cases without excessive intervention costs. Overall, the work quantitatively demonstrates how healthcare constraints and adaptive vector control jointly shape dengue outbreak dynamics and offers a framework for designing reactive control strategies sustainable under limited medical resources.

Key findings

• Inclusion of finite hospital capacity creates three distinct regimes: (I) no fogging with full hospitalization when infections are low, (II) fogging with full hospitalization once infections reach a threshold kC, and (III) fogging with exceeded capacity where partial hospitalization leads to reduced recovery and longer infectious periods.
• Basic reproduction number R0 is modified by fogging via R0η = R0 / (1 + η/µm), indicating fogging reduces transmission potential proportionally to mosquito mortality increase (Eq. 10).
• Disease-free equilibrium is locally stable if R0 < 1; endemic equilibria exist and are stable when R0 > 1, with explicit endemic states derived in closed form for each regime (Theorems 1-4).
• Numerical continuation detects boundary-equilibrium bifurcations exactly at switching thresholds kC and C, and a Hopf bifurcation emerges when infections exceed hospital capacity, causing sustained oscillatory outbreaks.
• A fold bifurcation arises near the epidemic threshold generating extra unstable equilibria, implying more complex disease persistence scenarios than classical smooth models.
• Optimal fogging intensity η and activation threshold k can be identified, with intermediate values minimizing epidemic peak sizes and asymptomatic cases while avoiding unnecessary control costs.
• Hospital capacity limits recovery and increases the infectious pool when exceeded, feeding back into vector infection dynamics and enabling oscillations absent in unconstrained models.
• Fogging interventions triggered below full hospital capacity (k < 1) allow earlier vector control activation, which can suppress outbreak magnitude without excessive fogging duration.

Threat model

Not a security paper. However, the implicit adversary is the dengue virus transmission pathway enabled by mosquitoes acting as vectors in a human population with constrained healthcare resources. The 'capability' modeled is disease transmission combined with limited hospitalization and delayed or triggered fogging interventions. The model assumes no external vector or human population migration and no vertical transmission, focusing on local transmission dynamics under capacity and intervention constraints.

Methodology — deep read

The authors develop a compartmental dengue transmission model incorporating human and mosquito populations under practical constraints. The threat model assumes an endemic dengue transmission environment with limited hospital resources and vector-control interventions activated upon threshold triggers. The human population is divided into susceptible (S), asymptomatic infected (A), symptomatic infected (I), and recovered (R) compartments, while mosquitoes are modeled as susceptible (U) and infected (V). Dengue virus transmission occurs via mosquito bites with distinct infection rates from asymptomatic and symptomatic humans to mosquitoes, reflecting viral load differences. Hospital capacity C restricts symptomatic hospitalization such that only min(I,C) receive hospital treatment with higher recovery rate γ1, while the remainder (if I>C) self-treat at home with slower recovery γ2. A fogging intervention reduces the mosquito population at rate η, triggered when hospitalized symptomatic cases exceed a fraction k of capacity (kC). This threshold-based fogging creates piecewise-smooth dynamical regimes modeled by switching terms in the ODE system.

Parametrization uses demographic and epidemiological data from Indonesian contexts such as Jakarta, including human birth and death rates, mosquito lifespan and birth rates, transmission parameters βh, βm1, βm2, recovery rates γ0, γ1, γ2, hospitalization threshold k, and death rate δ induced by dengue. Parameter intervals align with literature values and produce realistic population dynamics.

The model is mathematically formed as a system of six non-smooth ordinary differential equations (ODEs), with nonlinear functions defining recovery f(I), mosquito infection g(U,A,I), and fogging intervention h(I). Each function is piecewise-defined depending on hospital occupancy I relative to thresholds kC and C.

Analytical techniques derive disease-free and endemic equilibria for each regime (I < kC, kC ≤ I < C, I ≥ C), calculating the basic reproduction number (R0 and R0η) and using linear stability analysis via Jacobians and characteristic polynomials to establish local stability conditions. The Routh-Hurwitz criteria are employed to confirm stability of equilibria where analytical expressions permit.

Numerical continuation studies implement bifurcation analysis to explore complex dynamics induced by switching thresholds. These computations track equilibria stability changes, detect boundary-equilibrium bifurcations at switching points, identify Hopf bifurcations generating sustained periodic oscillations once hospital capacity is exceeded, and locate fold bifurcations near the epidemic threshold that induce multiple unstable equilibria.

The authors further investigate periodic solution branches by parametrically varying fogging intensity η and activation threshold k, revealing optimal intervention zones that balance epidemiological benefit (e.g., reduced infection peaks, lower asymptomatic prevalence) against operational costs.

Code availability and reproducibility details are not explicitly mentioned. The model is constructed following well-documented classical host-vector dengue models extended with piecewise-smooth vector fields driven by hospital capacity and fogging threshold parameters. Baseline parameters are drawn from multiple literature sources ensuring epidemiological realism. One concrete example includes demonstrating local stability of the disease-free equilibrium for R0 < 1 and how exceeding hospital capacity induces oscillatory outbreaks via Hopf bifurcation (see Figure references in the source).

Technical innovations

• Introduction of a piecewise-smooth ODE dengue transmission model incorporating finite hospital capacity as a dynamic constraint on symptomatic recovery rates.
• Modeling fogging intervention as a threshold-triggered, state-dependent vector control activated when hospitalized cases cross a fraction k of hospital capacity, leading to a non-smooth dynamical system.
• Analytical derivation of local stability and equilibria existence in multiple operation regimes corresponding to no fogging, fogging with full hospitalization, and fogging with exceeded hospital capacity.
• Identification of boundary-equilibrium, fold, and Hopf bifurcations induced by interaction of hospital capacity constraints and threshold-based fogging, capturing oscillatory outbreak phenomena absent in smooth models.

Baselines vs proposed

• R0 without fogging: calculated via next-generation matrix spectral radius; baseline used for stability thresholds.
• Effective R0 with fogging R0η = R0 / (1 + η/µm): demonstrates proportional reduction in transmission potential due to fogging-induced mosquito mortality.
• Comparison of disease-free equilibrium stability: stable if R0 <1; unstable and endemic equilibrium emerges above this threshold under baseline parameters.
• Numerical bifurcation results show Hopf bifurcation occurs for I > C triggering periodic outbreaks, a dynamic absent in unconstrained classical models.

Limitations

• Model assumes homogeneous mixing between humans and mosquitoes, neglecting spatial heterogeneity or localized transmission clusters.
• Recovery and transmission parameters are averaged and do not explicitly model serotype variation or antibody-dependent enhancement effects.
• The fogging intervention is simplified as an instantaneous rate once threshold kC is exceeded, ignoring realistic delays, spatial coverage, or fading effects of insecticide.
• Hospital capacity is represented as a fixed scalar without accounting for dynamic adjustments, patient triage, or differential allocation of beds over time.
• Model excludes human mobility, vector resistance development, weather seasonality, and other ecological factors influencing dengue dynamics.
• No explicit validation against real outbreak or intervention data is reported to test model predictive performance.

Open questions / follow-ons

• How do spatial heterogeneity and human mobility patterns affect the dynamics under limited hospital capacity and threshold-based fogging?
• What are the quantitative impacts of adding serotype interactions, antibody-dependent enhancement, or cross-immunity into a capacity- and threshold-aware model?
• How do delays, partial coverage, and diminishing returns in fogging campaigns alter the optimal activation threshold and intensity?
• Can data-driven approaches be integrated to calibrate hospital capacity thresholds and fogging parameters for real-time adaptive dengue control?

Why it matters for bot defense

Though this work is epidemiological modeling rather than bot-defense or CAPTCHA focused, it provides an instructive example of how state-dependent threshold-triggered interventions can induce non-smooth dynamics with critical bifurcations. Bot-defense engineers interested in mitigation strategies under resource constraints can learn from the approach of modeling capacity limits and threshold-based control activation to avoid system overloads and unintended oscillatory behaviors. The bifurcation analysis insights emphasize the need for carefully designed activation criteria to optimize intervention effectiveness without excessive effort. Concepts of threshold-triggered state changes and piecewise dynamics may inspire analogous frameworks in adaptive bot-detection or CAPTCHA challenge deployment triggered by suspicious traffic levels surpassing infrastructure capacity thresholds.

Cite

@article{arxiv2607_18140,
  title={ A Mathematical Model of Dengue Transmission Incorporating Hospital Capacity and Threshold-Based Fogging Interventions },
  author={ Dipo Aldila and Joseph Páez Chávez and Aytül Gökçe and Thomas Götz and Burcu Gürbüz },
  journal={arXiv preprint arXiv:2607.18140},
  year={ 2026 },
  url={https://arxiv.org/abs/2607.18140}
}

Read the full paper

• arXiv:2607.18140 (abstract)
• arXiv:2607.18140 (PDF)