The authors have declared that no competing interests exist.

In this study, we propose a stochastic SEIQR infectious disease model driven by Lévy noise. Firstly, we study the existence and uniqueness of the global positive solution of the model by using the stop-time. Secondly, the asymptotic behavior of the stochastic system at disease-free equilibrium and endemic equilibrium are discussed. Then, the sufficient condition for persistence under the time mean is studied. Finally, our theoretical results are verified by numerical simulation.

Infectious diseases have always been one of the important threats to human health, and the control of infectious diseases is an important issue in human society. It is well known that Kermack and McKendrick first proposed the SIR Model based on the Indian plague model [

Most of the previous models of infectious diseases were basically considered on the basis of assuming the free movement of individuals in the population, and rarely considered the problem of having isolation chambers. With the onset of COVID-19 in 2020, the prevention and treatment of infectious diseases has become one of the topics of research for governments around the world. After the continuous exploration of prevention, the introduction of isolation chamber Q has an excellent effect on timely controlling of infectious diseases. Therefore compared with the previous SIR, SIRS, SEIR and other models, SEIQR model can more accurately describe the prevention and control of infectious diseases. Liu et al. [

Converting model (^{−1} is called a regenerative matrix. The basic regeneration number is the spectral radius of the regeneration matrix. Basic reproduction number _{0} of system (_{0} ≤ 1, the system (_{0} > 1, system (

Since the infectious disease model is affected by many unpredictable environmental noises, adding random interference to the deterministic model can reflect the transmission law more accurately. In [

However, disease can be affected by a variety of natural mutations, such as volcanic eruptions, chemical pollutants, and sudden climate changes, which are often not accurately described by stochastic models of Brownian motion. Therefore, many studies on natural mutation factors will use Lévy jump to describe. This perturbation can more accurately describe the impact of mutation factors, and more deeply understand and predict the trend of disease spread and development. According to the Lévy-Itô decomposition theorem [_{i}(_{t}}_{t>0} on a complete probability space (Ω, _{i} > 0 (_{i}(_{i}(

^{2,1}(^{d} × [_{0}, ∞]; _{+}), thus the random derivative of

We assume that the jump diffusion coefficient satisfies the following conditions:

(H1): _{1}(_{1}(_{2}(_{2}(_{3}(_{3}(_{4}(_{4}(_{5}(_{5}(

(H2): |_{i}(

_{e}), where _{e} represents the blasting time. To prove the existence of a global solution, just prove _{e} = ∞,

Let _{0} be a sufficiently large positive number such that the initial value (_{0}, the stopping time is defined as:

Obviously, _{k} is monotonically increasing with respect to _{∞} ≤ _{e}. If we could prove that _{∞} = ∞, then _{e} = ∞.

Next, we use the proof by contradiction to prove. Suppose that _{∞} ≠ ∞, then _{∞} ≤

Then there exists an integer _{1} ≥ _{0} such that _{k} ≤ _{1}. Define a ^{2}-function:
_{k} ∧

Let Ω_{k} = {_{k} ≤ _{k}) ≥ _{0} for any _{1}. Notice that for every _{k}, at least one of _{k}, _{k}, _{k}, _{k}, _{k},

Hence
_{k}, letting _{∞} = ∞

_{0} ≤ 1, _{0} is asymptotically stable. Next we will discuss the asymptotic behavior of the solution of the stochastic model at the disease-free equilibrium point.

_{0} ≤ 1, and the following conditions are met:

Thus

Let

According to the Itô formula, then

For the Lyapunov function to be asymptotically stable, then _{1}, _{2}, _{3}, _{4}, _{5} > 0. Thus

Then

_{0}, the wave intensity is related to noise intensity _{i} and _{i}. The bigger the _{i} and _{i}, the bigger the fluctuation. That is, the greater the random disturbance, the farther away the solution of system (_{0} of the deterministic model, at which time the disease will disappear. Next, we will verify the correctness of Theorem 3 through numerical analysis(see _{i} = 0.03(

From the observation of

_{0} > 1,

_{0} > 1, and the following conditions are satisfied:

For any given initial value

According to the Itô formula, then

For the Lyapunov function to be asymptotically stable, then _{1}, _{2}, _{3}, _{4}, _{5} > 0. Where _{1}, _{2}, _{3}, _{4}, _{5},

Thus

This theorem is proved.

Next, we will verify the correctness of Theorem 4 through numerical analysis(see _{i} = 0.01(

It can be seen from

Therefore

Let

Substituting (

Thus

According to

Because Ω is positive invariant set, (

Theorem 7 states that under certain conditions, the disease will continue to spread. This means that the disease persists among the population and is not conducive to further management.

In this work, we have proposed a stochastic SEIQR epidemic model with bilinear incidence rates and Lévy noise based on the randomness of nature and some abrupt fluctuations. By applying the relevant knowledge of stochastic analysis, we have proved the existence and the uniqueness of the global positive solution for the stochastic SEIQR model. Moreover, we showed that the free equilibrium point _{0} and the endemic equilibrium point

PONE-D-24-08895Dynamics of a Stochastic SEIQR Model Driven by L\\'{e}vy Jumps with Bilinear Incidence RatesPLOS ONE

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Reviewer #1: In a revised version the authors should explain this in more details and in simple words and add more physical interpretations, not only refer to theorems and proofs. For more details, please consult my uploaded report. Thank you very much.

Reviewer #2: Review Report

Manuscript ID: PONE-D-24-08895

In this work, a stochastic SEIQR infectious disease model driven by Lévy noise is proposed, considering the impact of discontinuous noise on the transmission process of diseases with latent period. First, the Lyapunov analysis approach is used to demonstrate the existence and uniqueness of the global positive solution of the stochastic SEIQR epidemic model. Then, the asymptotic behavior of stochastic system in the disease-free equilibrium point and endemic equilibrium point is explored by building the Lyapunov function

There is some new contribution and may be consider for publication after addressing the following observations.

1. The abstract is poorly written, it need an improvement, for example, line 2 taking into account may be changed to considering and in line 4, Then, The must be change to Then. the

2. The introduction should also make a compelling case for why the study is useful along with a clear statement of its novelty or originality by providing relevant information and providing answers to basic questions such as:

i. What is already known in the literature?

ii. What was done and how it was done?

3. In introduction section, provide theoretical justification for the choice of Lévy noise over other types of stochastic processes in modeling the abrupt fluctuations inherent in the disease transmission process?

4. In the section on the existence and uniqueness of the global positive solution provide more details on the mathematical intuition and biological implications behind the assumptions (H1) and (H2) related to the jump diffusion coefficient?

5. How did you apply the Ito formula to derive the dynamics of your Lévy-driven stochastic differential equations? A step-by-step mathematical derivation would enhance the clarity. Add detailed mathematical derivations in the subsection discussing the application of the Ito formula.

6. How was the specific form of the Lyapunov function chosen, and what are the mathematical criteria for its selection in proving the asymptotic behavior around the disease-free and endemic equilibrium?

7. Again, in section on the existence and uniqueness of the global positive solution, what mathematical techniques were employed to ensure the solution stays positive and within the biologically feasible region, given the presence of jumps and discontinuities?

8. The conditions for stochastic stability mentioned are quite specific. Provide more insight into how these conditions ensure the stability of the disease-free and endemic equilibrium points? This should be addressed in both the sections on the disease-free equilibrium and the endemic equilibrium analysis.

9. What numerical methods were used for the simulations shown in Figures 1 and 2, and how do these methods accurately capture the effects of Lévy jumps?

10. The references are very few, the author may add some recent related work. They may also add the following recent work, but not mandatory.

11. Author may look for some punctuation, typos and editing issues.

Reviewer #3: This paper deals with the dynamics of a stochastic SEIQR model driven by Lévy jumps. It can be accepted for the publication after some major revisions.See the attached file. For more details, see the attached file.

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At this time, one can see that the quality of this paper has been improved. We express our great thanks again to you. We hope that the corrections and revisions will be satisfactory and that the revised version will be accepted for publication in journal.

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Dynamics of a Stochastic SEIQR Model Driven by L\\'{e}vy Jumps with Bilinear Incidence Rates

PONE-D-24-08895R1

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PONE-D-24-08895R1

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