[Paper Review] A novel analysis approach of uniform persistence for a COVID-19 model with quarantine and standard incidence rate

Songbai Guo, Yuling Xue+2 more|arXiv (Cornell University)|May 31, 2022

COVID-19 epidemiological studies4 citations

TL;DR

This paper proposes a novel analytical approach to estimate the ultimate lower bound of infected individuals in a COVID-19 model with standard incidence rate and quarantine measures. By refining persistence analysis, it proves that the disease is uniformly persistent if the control reproduction number $\mathcal{R}_c > 1$, offering a more precise method applicable to broader biomathematical models and improving existing persistence techniques.

ABSTRACT

A coronavirus disease 2019 (COVID-19) model with quarantine and standard incidence rate is first developed, then a novel analysis approach for finding the ultimate lower bound of COVID-19 infectious individuals is proposed, which means that the COVID-19 pandemic is uniformly persistent if the control reproduction number $\mathcal{R}_{c}>1$. This approach can be applied to other related biomathematical models, and some existing works can be improved by using it. In addition, the COVID-19-free equilibrium $V^0$ is locally asymptotically stable (LAS) if $\mathcal{R}_{c}<1$ and linearly stable if $\mathcal{R}_{c}=1$, respectively; while $V^0$ is unstable if $\mathcal{R}_{c}>1$.

Motivation & Objective

• To develop a more refined analysis method for estimating the long-term lower bound of infected individuals in a COVID-19 model with standard incidence and quarantine.
• To establish conditions under which the disease is uniformly persistent in the population.
• To improve existing persistence analysis techniques in epidemiological models using a new analytical framework.
• To demonstrate the practical applicability of the method through numerical simulations and parameter estimation.

Proposed method

• The authors formulate a six-compartment COVID-19 model (S, E, I, A, Q, R) with standard incidence rate, replacing bilinear incidence with density-dependent transmission.
• The model incorporates distinct quarantined rates and recovery rates for symptomatic and asymptomatic individuals, enhancing realism.
• A novel persistence analysis technique is developed to derive an explicit lower bound $\nu$ for $\liminf_{t\to\infty} E(t)$, the long-term number of exposed individuals.
• The method relies on constructing a Lyapunov-like function and using inequalities involving parameters such as $\theta$, $\eta$, $\varrho$, and $\xi$ to bound the infimum of the exposed population.
• Theoretical results are validated using MATLAB simulations with realistic parameter values from the literature.
• The approach is shown to improve upon existing methods in [3, 9, 14, 15, 19, 13, 16, 18, 17, 24, 30, 31, 37] by providing tighter and more explicit estimates.

Experimental results

Research questions

• RQ1 Under what conditions is the COVID-19 model with standard incidence and quarantine uniformly persistent?
• RQ2 Can a more precise and explicit lower bound be derived for the long-term number of exposed individuals when $\mathcal{R}_c > 1$?
• RQ3 How does the proposed analytical method improve upon existing persistence analysis techniques in epidemic models?
• RQ4 What is the impact of different quarantine and recovery rates on the persistence threshold and lower bound of infection?
• RQ5 Can the method be generalized to other biomathematical models beyond the current COVID-19 framework?

Key findings

• The model exhibits uniform persistence if and only if the control reproduction number $\mathcal{R}_c > 1$, ensuring the disease persists in the population.
• The COVID-19-free equilibrium $V^0$ is locally asymptotically stable when $\mathcal{R}_c < 1$, indicating disease extinction.
• When $\mathcal{R}_c = 1$, the linearized system at $V^0$ is stable, but the equilibrium is not asymptotically stable.
• The novel method provides a computable lower bound $\nu \approx 6.725 \times 10^6$ for $\liminf_{t\to\infty} E(t)$, which is approximately 72% of the estimated equilibrium value $E^* \approx 9.396 \times 10^6$.
• Numerical simulations confirm the validity of the theoretical lower bound estimate under realistic parameter settings.
• The method improves upon prior persistence analysis in multiple cited works by offering sharper, more explicit estimates of the ultimate lower bound of infected individuals.