[Paper Review] Analyzing the qualitative properties of white noise on a family of infectious disease models in a highly random environment

Divine Wanduku|arXiv (Cornell University)|Aug 29, 2018

Mathematical and Theoretical Epidemiology and Ecology Models44 references4 citations

TL;DR

This paper develops a stochastic SEIRS model with nonlinear incidence and three distributed delays to analyze the impact of white noise on vector-borne disease dynamics, particularly for dengue and malaria. It demonstrates that increasing noise intensity in transmission or death rates induces stronger oscillations and can lead to disease extinction, while the disease-free equilibrium remains stochastically stable under low noise levels.

ABSTRACT

A class of stochastic vector-borne infectious disease models is derived and studied. The class type is determined by a general nonlinear incidence rate of the disease. The disease spreads in a highly random environment with variability from the disease transmission and natural death rates. Other sources of variability include the random delays of disease incubation inside the vector and the human being, and also the random delay due to the period of effective acquired immunity against the disease. The basic reproduction number and other threshold conditions for disease eradication are computed. The qualitative behaviors of the disease dynamics are examined under the different sources of variability in the system. A technique to classify the different levels of the intensities of the noises in the system is presented, and used to investigate the qualitative behaviors of the disease dynamics in the infection-free steady state population under the different intensity levels of the white noises in the system. Moreover, the possibility of population extinction, whenever the intensities of the white noises in the system are high is examined. Numerical simulation results are presented to elucidate the theoretical results.

Motivation & Objective

• To model the complex dynamics of vector-borne diseases like dengue and malaria under environmental randomness using a stochastic SEIRS framework.
• To examine how white noise in disease transmission and natural death rates affects disease persistence and extinction.
• To investigate the role of distributed delays—incubation in vectors, incubation in humans, and immunity duration—in shaping disease dynamics under stochasticity.
• To establish threshold conditions for disease eradication based on the basic reproduction number and noise intensity.
• To classify noise intensity levels and analyze their impact on stochastic stability and oscillatory behavior near the disease-free equilibrium.

Proposed method

• Formulates a system of Itô-Doob type stochastic differential equations with a general nonlinear incidence function $ G $ to capture complex transmission dynamics.
• Incorporates three distributed delays representing incubation in vectors, incubation in humans, and duration of effective immunity post-recovery.
• Applies Lipschitz continuity, stopping times, and energy functions to prove existence of unique global positive solutions.
• Uses Lyapunov functional techniques to analyze asymptotic stochastic stability of the disease-free equilibrium.
• Classifies noise intensity levels and evaluates their effects on oscillatory behavior and population extinction.
• Employs numerical simulations to validate theoretical results under varying noise intensities in transmission and death rates.

Experimental results

Research questions

• RQ1 How does white noise in disease transmission rates affect the stability and oscillatory behavior of the disease-free equilibrium in vector-borne SEIRS models?
• RQ2 What is the impact of noise intensity in natural death rates on the long-term dynamics and potential extinction of the disease population?
• RQ3 How do distributed delays—incubation in vectors, incubation in humans, and immunity duration—affect disease persistence under stochastic perturbations?
• RQ4 Under what conditions does increased noise intensity lead to disease extinction rather than sustained oscillations?
• RQ5 What threshold conditions, including the basic reproduction number, determine disease eradication in the presence of environmental white noise?

Key findings

• Increasing the intensity of white noise in the disease transmission rate ($ \sigma_\beta $) from 0.5 to 9 leads to stronger oscillations in susceptible and infectious populations, reducing their average values over time.
• When noise intensity in natural death rates ($ \sigma_i $, for $ i = S,E,I,R $) increases from 0.5 to 9, the system exhibits strong oscillations across all compartments, with a rapid decline in average population sizes.
• High noise intensity in natural death rates causes the disease population to go extinct over time, as shown in numerical simulations with $ \sigma_i = 9 $.
• The disease-free equilibrium remains stochastically stable under low noise intensity, but higher noise levels disrupt this stability and induce persistent oscillations.
• The basic reproduction number serves as a threshold for disease eradication, but its effectiveness is modulated by the intensity of white noise in transmission and death processes.
• Numerical results confirm that noise intensity level is a critical determinant of whether the system exhibits sustained oscillations or leads to population extinction, with distinct behavioral shifts observed at high intensities.