Dengue Fever Model Predicts Stable Outbreaks Without Needing Prior Conditions

Researchers investigated the global stability of dengue fever transmission dynamics within a spatially homogeneous environment, addressing limitations present in previous modelling approaches. Xue Ren and Ran Zhang, both from Heilongjiang University, demonstrate global attractivity of the positive constant steady state using a Lyapunov functional, crucially removing the need for a sufficient condition previously required by Xu and Zhao (2015). This refinement significantly enhances existing understanding, proving global attractivity when the basic reproduction number is precisely greater than unity and expanding upon the findings detailed in Xu and Zhao (2015) section 3.3. The work provides a more robust and generally applicable framework for predicting dengue fever prevalence and informing public health interventions.

Global stability of a dengue fever model with reduced reproduction number constraints is demonstrated through Lyapunov analysis

Scientists have achieved a significant advancement in understanding the global stability of a dengue fever model, resolving a long-standing condition required for predicting disease stabilization. Recent work established the global attractivity of a positive constant steady state, but only if the basic reproduction number, denoted as R0, exceeded a specific threshold.
This study eliminates that restrictive condition, demonstrating that the positive constant steady state is globally attractive when R0 is precisely greater than unity, thereby refining previous findings. The research centers on a spatially homogeneous model describing dengue transmission, incorporating nonlocal delays to account for incubation periods within both mosquito and human populations.

Researchers employed a Lyapunov functional, a mathematical tool for assessing stability, to prove this enhanced result. This approach bypasses the need for a sufficient condition previously required to guarantee disease stabilization, offering a more precise understanding of the system’s dynamics. The model incorporates parameters representing diffusion rates of mosquitoes and humans, rates of new individuals entering the population, biting rates, infection rates, recovery rates, and the lengths of incubation delays.

Specifically, the study investigates a reaction-diffusion model represented by a system of three partial differential equations governing the densities of infectious mosquitoes, susceptible and infectious humans. The analysis utilizes a Lyapunov functional, V(t), defined with components L1, L2, and L3, along with weighting functions W1 and W2, to assess the system’s stability.

Derivatives of these components are calculated to demonstrate that V(t) is non-increasing, indicating convergence towards the endemic steady state. This breakthrough has implications for more accurate predictions of dengue outbreaks and the development of effective control strategies. By removing the need for a restrictive condition on R0, the research provides a more robust framework for understanding dengue dynamics and potentially improving public health interventions. The findings contribute to a deeper understanding of epidemic modeling and offer a valuable tool for forecasting disease prevalence in vulnerable populations.

Global attractivity of dengue steady states via Lyapunov functional analysis is demonstrated rigorously

A spatially homogeneous reaction-diffusion model describing dengue transmission forms the basis of this work. Researchers investigated the global attractivity of positive constant steady states, building upon prior studies by Xu and Zhao (2015). The study employed a Lyapunov functional approach to demonstrate global attractivity without requiring a sufficient condition on the basic reproduction number, R0.

This contrasts with previous work which established global attractivity subject to the condition that R0 exceeds a certain threshold. The model consists of three coupled partial differential equations describing the densities of infectious mosquitoes, susceptible humans, and infectious humans. These equations incorporate diffusion terms with coefficients dm and dh representing mosquito and human dispersal rates, respectively.

New individuals are added to both populations at rates A and H, while mosquito bites initiate transmission between humans and mosquitoes at a rate of b, with infection probabilities p and q. The model also includes incubation delays, τa and τb, representing the time between infection and infectiousness for mosquitoes and humans, respectively.

To analyze the system’s dynamics, the researchers considered a bounded domain Ω with Neumann boundary conditions, ensuring no flux across the boundaries. Solutions were sought within the space C, comprising bounded and uniformly continuous functions from the closure of Ω to R3, defined over a time interval including the delays.

A mild solution, u(t, ·, φ), was established for all t ≥0, originating from an initial condition φ ∈ CM, a subset of C with non-negative function values. The existence of a global compact attractor was leveraged, providing a foundation for proving global attractivity of the positive steady state. A key innovation lies in the elimination of the restrictive condition on R0, proving global attractivity when R0 is exactly greater than unity.

This was achieved through careful construction of a Lyapunov functional, a scalar function that decreases along solution trajectories, guaranteeing convergence to the steady state. The basic reproduction number, R0, is calculated as q βhβmAHe−μhτb μhμmρh, where parameters define transmission and removal rates.

Global convergence to a positive steady state via Lyapunov functional analysis is demonstrated

The research demonstrates that a unique constant steady state, u∗, is achieved within system (1) when the basic reproduction number, R0, exceeds 1. This finding establishes a globally attractive positive constant steady state, resolving a previously existing condition requiring R0 to be greater than unity.

The study employed a Lyapunov functional, V(t), defined as the integral over Ω of 3Li(t, x) + 2Wi(t, x) dx, where L1(t, x) = βhu∗ 1u∗ 2μm g(u1/u∗ 1), L2(t, x) = u∗ 2g(u2/u∗ 2), and W1 and W2 represent additional terms related to delayed transmission. Analysis of dV(t)/dt reveals a non-increasing map, indicating that the solution u(t, φ) converges to u∗ as time approaches infinity.

Specifically, the derivative dV(t)/dt is shown to be negative, incorporating terms representing spatial gradients of u1, u2, and u3, alongside integral terms involving the function g and the Neumann boundary condition. The calculation of dV(t)/dt incorporates integrals over both space (Ω) and time, accounting for the delayed transmission dynamics.

The Lyapunov functional’s components, including |∇u1|², |∇u2|², and |∇u3|², contribute to the overall negative rate of change, ensuring stability. The integral terms involving g(u3/u∗ 3) and g(u1u2/u∗ 1u∗ 2) further reinforce the convergence towards the steady state. This rigorous mathematical framework confirms the global stability of the model under the specified conditions, providing insights into the long-term behavior of the system.

Global stability established via Lyapunov functional analysis for a dengue diffusion model is demonstrated

Researchers have demonstrated the global attractivity of a positive constant steady state in a diffusive dengue disease model, refining previous findings by removing a sufficient condition previously thought necessary. Employing a Lyapunov functional approach, this work establishes that the steady state is globally attractive when a key parameter exceeds unity, improving upon earlier results detailed in Xu and Zhao (2015).

The analysis centres on a detailed examination of the rate of change of a Lyapunov functional, denoted as V(t), revealing its non-increasing nature over time. This achievement signifies a more robust understanding of the long-term behaviour of the dengue disease model, indicating that regardless of initial conditions, the system will converge to a unique, stable equilibrium when the specified parameter is sufficiently large.

The mathematical framework utilized provides a powerful tool for analysing similar reaction-diffusion epidemic models, potentially informing public health strategies and disease control measures. However, the authors acknowledge that the analysis relies on specific assumptions regarding the functional forms of certain terms within the model and the properties of the spatial domain considered.

Future research could focus on extending these results to more complex models incorporating additional factors such as seasonality, vector-host interactions, or spatial heterogeneity. Investigating the sensitivity of the steady state to variations in model parameters would also be valuable, as would exploring the implications of these findings for the design of effective intervention strategies. The current work provides a solid foundation for further exploration of the dynamics of infectious diseases in spatially distributed populations.