Dynamic Modeling and Analysis of Epidemic Spread Driven by Human Mobility

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The research subject is of, also, practical interest and this work seen as an incremental but solid advance rather than a fundamental breakthrough, compared to the existing literature. The model’s main contribution is the integration of three features often treated separately in the literature: (1) distinct migration matrices for multiple mobile compartments, (2) explicit risk stratification via a low-risk class, and (3) a complete analytical treatment including a block-structured NGM derivation of R₀ and global Lyapunov stability proofs. Compared to standard multi-patch SEIR frameworks, this structure allows richer migration–infection interactions and policy analysis. The inclusion of optimal isolation strategies and sensitivity analysis enhances practical relevance.

There are some corrections and additions needed before considering publication.

1.  In both Eq. (1) and Eq. (2), replace the term φ_iS_i(Ei + I_i)with φI Li(EI + I_i) to match the stated biological assumptions that the low-risk class is infected at rate φi.

2. Clearly define the meaning of aij,bij,cij,dijaij​,bij​,cij​,dij​ (e.g., “rate from region jj to ii”), and use the notation consistently for inflow and outflow terms throughout the equations.

3. Ensure that the corrected infection term appears consistently in the text, all model equations, and in the reproduction number formulas.

4. State explicitly that the Lyapunov weights hI and lI are chosen from the left eigenvector matrix V^{-1}F, ensuring \dot{L} ≤0 when R₀ <1.

5. In theorem 5, derive and explain the role of Eq. (31) in proving \dot{V}≤0.

6. Justify why irreducibility of all migration matrices is required, or state if a weaker condition would suffice.

7. Cite the earlier positivity/boundedness result (Theorem 1) when applying LaSalle’s principle.

8. Related Work Enhancement – Expand the literature review to clearly compare this model’s combination of features (compartment-specific migration + low-risk class + global stability proofs) against representative multi-patch epidemic models.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The approach presented in this paper seems to be technically sound. However, I am not sure how the presented multi-region model is different from meta-population model which has been extensively studied, which is also relevant to the main contribution of this paper. In addition, the paragraphs in section 2 are very long, especially the last paragraph. It would be better if the long paragraphs are divided into a few shorter paragraphs. Furthermore, the authors can improve the conclusion section by describing limitations and possible future work of this paper. 

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The topic of the paper is actual: the study of the spread of infectious diseases by taken into account the human mobility.  The results are interesting and promising..

The conclusions of the analysis of the proposed mathematical model are natural and logical: population migration accelerates the spread of the virus from hight-infected areas to low-infected areas, but the health measures can prevent its transmission. It means that the model is realistic.

However, the paper is not publishable in the present form and some major corrections/additions need to be made. (see the attached file

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript presents an extended SEIR-based multi-regional dynamic model (SLEIQDR) incorporating human mobility. While the extension is mathematically rigorous and includes global stability analysis and parameter sensitivity studies, the novelty is modest compared to prior models that already incorporate mobility and spatial dynamics (e.g., multi-patch SEIR, metapopulation models, reaction-diffusion frameworks).

1. Clarify how this model substantially improves upon or differs from existing spatiotemporal models, particularly those integrating LSTM, Koopman operators, or reaction-diffusion-based approaches.
2. Provide evidence or references supporting the assumptions (e.g., non-mobility of quarantined individuals, no reinfection, and direct migration without infection during travel), or perform a sensitivity analysis to assess how violations of these assumptions affect model outcomes.
3. The mathematical formulation, especially the derivation of the basic reproduction number R0, Lyapunov stability proofs, and eigenvalue matrix decomposition, is dense and spans multiple pages without intermediate explanations.

4. Include a case study or fit the model to actual data to demonstrate applicability, improve credibility, and assess predictive performance.

5. Improve figure clarity, ensure all axes are labeled with units, and explain each subplot concisely in the captions. Consider including a comparative plot showing different scenarios (e.g., with and without migration).
6. The abstract mentions a proposed optimal protective isolation strategy, but the method used for determining this “optimal” strategy is not clearly defined or explained.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have replied to all review comments satisfactorily and amended the manuscript accordingly. Thus, I recommend acceptance of the manuscript

Reviewer 2 Report

Comments and Suggestions for Authors

Thank you for addressing my previous comments. I do not have any further comments.

Reviewer 4 Report

Comments and Suggestions for Authors

All the comments are incorporated in the revised manuscript. The manuscript can be accepted in the present form.