[Paper Review] The stochastic permanence of malaria, and the existence of a stationary distribution for a class of malaria models

Divine Wanduku|arXiv (Cornell University)|Sep 8, 2018

Mathematical and Theoretical Epidemiology and Ecology Models33 references4 citations

TL;DR

This paper proposes a class of stochastic differential equation models for malaria dynamics, incorporating random fluctuations in transmission and death rates via independent white noise processes. Using Lyapunov functional techniques and local martingale characterizations, it establishes conditions for stochastic permanence and the existence of a unique stationary distribution, showing that higher noise intensities—especially from natural death rates—can destabilize the system and increase extinction risk, while numerical simulations approximate the stationary distribution of disease states near endemic equilibrium.

ABSTRACT

This paper investigates the stochastic permanence of malaria and the existence of a stationary distribution for the stochastic process describing the disease dynamics over sufficiently longtime. The malaria system is highly random with fluctuations from the disease transmission and natural deathrates, which are expressed as independent white noise processes in a family of stochastic differential equation epidemic models. Other sources of variability in the malaria dynamics are the random incubation and naturally acquired immunity periods of malaria. Improved analytical techniques and local martingale characterizations are applied to describe the character of the sample paths of the solution process of the system in the neighborhood of an endemic equilibrium. Emphasis of this study is laid on examination of the impacts of (1) the sources of variability- disease transmission and natural death rates, and (2) the intensities of the white noise processes in the system on the stochastic permanence of malaria, and also on the existence of the stationary distribution for the solution process over sufficiently long time. Numerical simulation examples are presented to illuminate the persistence and stochastic permanence of malaria, and also to numerically approximate the stationary distribution of the states of the solution process.

Motivation & Objective

• To analyze the stochastic permanence of malaria under environmental noise in transmission and natural death rates.
• To investigate the existence of a stationary distribution for the stochastic malaria process over long time horizons.
• To examine how noise intensities from disease transmission and natural death rates affect disease persistence and distribution.
• To numerically approximate the stationary distribution of susceptible, exposed, infectious, and removed populations in the model.
• To assess the impact of nonlinear incidence rates and random delays on long-term disease dynamics.

Proposed method

• Formulates a family of Itô-Doob type stochastic differential equations with a general nonlinear incidence function G to model malaria transmission.
• Introduces independent white noise processes to represent random fluctuations in disease transmission and natural death rates.
• Applies Lyapunov functional techniques and local martingale characterizations to analyze sample path behavior near the potential endemic equilibrium.
• Uses numerical simulations to track trajectories of the stochastic process under varying noise intensities and parameter sets.
• Employs ensemble simulations to approximate the stationary distribution of the disease states across multiple realizations.
• Analyzes the influence of noise sources and intensities on long-term system stability and extinction risk.

Experimental results

Research questions

• RQ1 Under what conditions is malaria stochastically permanent despite random fluctuations in transmission and death rates?
• RQ2 Does a unique stationary distribution exist for the stochastic malaria model over long time periods?
• RQ3 How do different noise intensities—particularly from transmission versus natural death rates—affect the persistence and distribution of malaria?
• RQ4 What is the impact of nonlinear incidence rates and random delays on the long-term behavior of the disease?
• RQ5 Can the stationary distribution of the disease states be numerically approximated when no explicit solution exists?

Key findings

• Higher intensities of white noise processes, especially from natural death rates, drive the system further from the endemic equilibrium and increase the risk of population extinction.
• The stationary distribution for the susceptible population is approximately symmetric with a center (ensemble mean) in the interval [6.5, 8.0] under the given parameter set.
• The stationary distributions for the exposed, infectious, and removed classes are also approximately symmetric, with centers in the intervals [2.45, 2.85], [2.8, 3.6], and [0.95, 1.90], respectively.
• The stationary distribution is unique for a given set of system parameters, though its statistical properties (e.g., mean, variance) vary with parameter values.
• Numerical simulations indicate that noise from natural death rates has a stronger destabilizing effect than noise from transmission rates.
• The model confirms that malaria can persist stochastically under moderate noise, but high noise intensities significantly reduce persistence likelihood.