HIV Dynamics with a Trilinear Antibody Growth Function and Saturated Infection Rate

Round 1

Reviewer 1 Report

This manuscript presents a modification of an existing mathematical model for HIV dynamics and a detailed analysis of its positivity, boundedness and equilibria. The model and its contexts is relevant to the journal. The manuscript is well organized and consistent, but the writing could be improved (see some suggestions below). The presented mathematical results are interesting and consistent. The computer simulations needs a careful revision. General and specific comments to improve the manuscript's impact and quality are provided next.  
General and major comments

    \item The introduction is way too short, even for a mathematical paper. Other works in the field should be mentioned and detailed. At the very beginning, in the first paragraph many technical details of the complex biological infection are given. Please improve the introduction in terms of: (i) motivation and presentation of the context, (ii) related works, (iii) contributions of this work.
    %
    \item The authors should better emphasize the novelties of this work. Is the model given in equation (1) entirely new? Please make it clear. Also, why the modification of the previous model was required? What is the main motivation and application of this modification?
   
    \item Although well known, the authors could add in the manuscript how one can obtain the given reproductive numbers $R_0$, $R_1$, and so on. At least, indicating what steps are taken to obtain them. This can ease the reading of newcomers.
   
    \item The authors must include details about the numerical method and time-step size used for the simulations.
   
    \item One very important point that should be addressed in the review. The numerical simulations for each case converge to the equilibria found using the mathematical analysis. This indicate somehow that the simulations are correct. However, one can clearly see many numerical oscillations in the simulation results (see Figs 2-5). This should be discussed. Are they biological reasonable? Why? Please explain.
   
    \item If the numerical oscillations can not be explained with biological arguments, the review suspects it is a numerical artifacts from the numerical solution. Perhaps the authors are using a explicit numerical scheme with a very large time step size for the simulation. This point should be fixed, otherwise it weakens the entire analysis made by the authors.

 

Specific comments and suggestions

\item Please add equations number to more equations in the manuscript.

\item Please consider changing the constant $d$, since when multiplied by $x$, it looks like $dx$, which might be confused with the differential $dx$.

\item Line 31: and constants?

\item Line 60: Fix the beginning of the sentece. Something is wrong or missing. \textit{We now steady the ...}. Maybe \textit{We now show...}?

Author Response

Authors responses to referees remarks to the paper

 

 

HIV Dynamics With A Trilinear Antibody Growth Function And Saturated Infection Rate

 

We are very grateful to the referees for their valuable remarks. All of them are taken into account in the revised version of the paper (text in blue). Detailed responses are given below.

Reviewer #3:

• The introduction is way too short, even for a mathematical paper. Other works in the field should be mentioned and detailed. Please improve the introduction in terms. 

Authors’ response: We have added our explanation in the text.

 

• The authors should better emphasize the novelties of this work.

Authors’ response: We have added our motivation in the text.

 

• The authors could add in the manuscript how one can obtain the given reproductive numbers $R_0$, $R_1$, and so on. At least, indicating what steps are taken to obtain them. 

Authors’ response: We have added our explanation and some references in the introduction.

 

• The authors must include details about the numerical method and time-step size used for the simulations.

Authors’ response: We have taken into consideration references [13,17,18,19] to prove our method.

 

• One can clearly see many numerical oscillations in the simulation results.

Authors’ response: You are right. We can clearly see many numerical oscillations in the simulation results, but our biological goal of having stability in the equilibrium points already calculated theoretically.

 

• The authors are using a explicit numerical scheme with a very large time step size for the simulation.

Authors’ response: We have taken into consideration this remark.

 

• Specific comments and suggestions.

Authors’ response:    

• We have only numbered the equations which are usable other times in our work.
• Thank you for your remark. The differential $dx$ can only be used in the system of equation (1), otherwise $d$ can be considered as a constant in the multiplication with other variables.
• We have added your proposal in the text.

Author Response File: Author Response.pdf

Reviewer 2 Report

A mathematical model describing the human immunodeficiency virus (HIV) is presented in this paper. CTL immunity and antibodies with trilinear growth functions are incorporated into the model. Biological studies support the boundedness and positivity of solutions for non-negative initial data. Local stability of the equilibrium is established. To support our theoretical findings, numerical simulations are performed.

 

The manuscript is interesting, but the following comments should be considered before publication. 

1-  It needs proofreading to correct the typos and improve the presentation. 

2- The motivation for using model (1) is not clear and convincing in the manuscript. There are many suggested models in this direction and the authors should show the advantages of using the underlying model; See "Dynamical Analysis of Post-Treatment HIV-1 infection model, Complexity, 2022, Article ID 9752628".

3- More discussions about the numerical simulations and Figures are required. 

4- The following references may be useful: Hopf Bifurcation and Stability of Periodic Solutions for Delay Differential Model of HIV Infection of CD4+ T- cells, Abstract and Applied Analysis, Volume 2014 (2014), Article ID 838396; Delay Differential Model for Tumor-Immune Dynamics with HIV Infection of CD4+ T-cells, International Journal of Computer Mathematics, 90(3) (2013) 594–614.

Author Response

Authors responses to referees remarks to the paper

 

 

HIV Dynamics With A Trilinear Antibody Gowth Function And Saturated Infection Rate

 

We are very grateful to the referees for their valuable remarks. All of them are taken into account in the revised version of the paper (text in blue). Detailed responses are given below.

 

 Reviewer #1:

• It needs proofreading to correct the typos and improve the presentation. 

Authors’ response: Thank you very much, we have done our best to correct them.

 

• The authors should show the advantages of using the underlying model.

Authors’ response: We have add our motivation in the text.

 

• More discussions about the numerical simulations and Figures are required. 

Authors’ response: We have add our motivation in the text.

 

• Some following references may be useful.

Authors’ response: The references are taken into consideration.

 

 

Reviewer #2:

 

Major comments:

1. The authors should give a more detailed description of the biology behind each term in the model. 

Authors’ response: We have added our explanation in the text.

 

2. Why is the non-linear function needed to improve the model?

Authors’ response: We have added our explanation in the text.

 

3. What are the useful biological insights obtained from this study?

Authors’ response: We have taken into consideration this question.

 

 

Minor comments:

 

4. The title is misspelled. 

Authors’ response: We have taken into consideration this remark.

 

 

5. Ref (4), is unrelated to the statement. 

Authors’ response: The references are taken into consideration.

 

 

6. Did the authors check the conditions on lines 57-58 in Mathematica or Maple?

Authors’ response: We checked the conditions manually; we base ourselves on the conditions of the Routh-Hurwitz theorem.

 

 

 

7. Why are the notations for E0 and E1 dependent on R_0 but not E2, E3 and E4?

Authors’ response: All points depend on R0 only to simplify calculations.

 

 

8. The biological interpretations of R1, R2, R3 and R4.

Authors’ response: The number R1 represents the basic defense rate and R3 is the half harmonic mean of R0 and R2.

 

 

9. I suggest adding additional simulations to explore this aspect (see 8).

Authors’ response: We have taken into consideration this remark.

 

 

10. Test what happen if the respective R crosses over the threshold. Is the stability lost? Does a bifurcation occur, etc..

Authors’ response: Indeed, stability will be lost.

 

 

 

 

 

 

 

Reviewer #3:

• The introduction is way too short, even for a mathematical paper. Other works in the field should be mentioned and detailed. Please improve the introduction in terms. 

Authors’ response: We have added our explanation in the text.

 

• The authors should better emphasize the novelties of this work.

Authors’ response: We have added our motivation in the text.

 

• The authors could add in the manuscript how one can obtain the given reproductive numbers $R_0$, $R_1$, and so on. At least, indicating what steps are taken to obtain them. 

Authors’ response: We have added our explanation and some references in the introduction.

 

• The authors must include details about the numerical method and time-step size used for the simulations.

Authors’ response: We have taken into consideration references [13,17,18,19] to prove our method.

 

• One can clearly see many numerical oscillations in the simulation results.

Authors’ response: You are right. We can clearly see many numerical oscillations in the simulation results, but our biological goal of having stability in the equilibrium points already calculated theoretically.

 

• The authors are using a explicit numerical scheme with a very large time step size for the simulation.

Authors’ response: We have taken into consideration this remark.

 

• Specific comments and suggestions.

Authors’ response:    

• We have only numbered the equations which are usable other times in our work.
• Thank you for your remark. The differential $dx$ can only be used in the system of equation (1), otherwise $d$ can be considered as a constant in the multiplication with other variables.
• We have added your proposal in the text.

Author Response File: Author Response.pdf

Reviewer 3 Report

The manuscript studies the dynamics of a within-host HIV infection under the action of antibodies and cytotoxic T lymphocytes. I have several comments regarding the current manuscript.

Major comments:

1. The model formulation is not well-motivated. Given that it is a variation of a previous model, the authors should give a more detailed description of the biology behind each term in the model. For example, it does not make sense why antibody production depends on healthy cells, viruses, and the current number of antibodies (e.g., gxvw). I have a similar concern with the production of cytotoxic T Lymphocytes, which depends on healthy cells, infected cells, and the current number of cytotoxic T Lymphocytes (e.g., cxyz).

2. Why is the non-linear function needed to improve the model? Are there experimental evidences that support this new model?

3. What are the useful biological insights obtained from this study? How are they relevant to the ongoing HIV research?

The mathematical proofs are generally correct; however, their presentation can be improved. There are some minor comments regarding the manuscript and the proofs.

Minor comments:

1. The title is misspelled. 

2. Ref (4), cited on line 13, is unrelated to the statement.

3. The positivity portion of proposition 1 is not a proof. I suggest the authors cite relevant theorem, explicitly prove it, or state that it is trivial.

4. Did the authors check the conditions on lines 57-58 in Mathematica or Maple? If so, I suggest including it in a Supplementary or Appendix.

5. Why are the notations for E0 and E1 dependent on R_0, but not E2, E3, and E4?

6. R1, R2, R3, and R4 are just conditions for local stability. Why did the authors consider them reproduction numbers? What are their biological interpretations?

7. While propositions 3 to 6 are fine, these conditions are only sufficient. Due to the difficulty in studying the stability, I suggest adding additional simulations to explore this aspect (see 8).

8. For all simulations to confirm the theoretical results, please adjust the values so that the respective R is closer to 1 (either slightly larger or slightly less than 1). Also, for exploratory purposes, do test what happen if the respective R crosses over the threshold. Is the stability lost? Does a bifurcation occur, etc.?

Author Response

Authors responses to referees remarks to the paper

 

 

HIV Dynamics With A Trilinear Antibody Growth Function And Saturated Infection Rate

 

We are very grateful to the referees for their valuable remarks. All of them are taken into account in the revised version of the paper (text in blue). Detailed responses are given below.

Reviewer #2:

 

Major comments:

1. The authors should give a more detailed description of the biology behind each term in the model. 

Authors’ response: We have added our explanation in the text.

 

2. Why is the non-linear function needed to improve the model?

Authors’ response: We have added our explanation in the text.

 

3. What are the useful biological insights obtained from this study?

Authors’ response: We have taken into consideration this question.

 

Minor comments:

 

4. The title is misspelled. 

Authors’ response: We have taken into consideration this remark.

 

 

5. Ref (4), is unrelated to the statement. 

Authors’ response: The references are taken into consideration.

 

 

6. Did the authors check the conditions on lines 57-58 in Mathematica or Maple?

Authors’ response: We checked the conditions manually; we base ourselves on the conditions of the Routh-Hurwitz theorem.

 

 

 

7. Why are the notations for E0 and E1 dependent on R_0 but not E2, E3 and E4?

Authors’ response: All points depend on R0 only to simplify calculations.

 

 

8. The biological interpretations of R1, R2, R3 and R4.

Authors’ response: The number R1 represents the basic defense rate and R3 is the half harmonic mean of R0 and R2.

 

 

9. I suggest adding additional simulations to explore this aspect (see 8).

Authors’ response: We have taken into consideration this remark.

 

 

10. Test what happen if the respective R crosses over the threshold. Is the stability lost? Does a bifurcation occur, etc..

Authors’ response: Indeed, stability will be lost.

 

 

 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

It is clear that the authors have addressed many issues and suggestions made by the reviewer.

With respect to the oscillations, now it is clear that it is a part of the model itself. Perhaps adding reference near the new sentences about this behavior could make it even clear.

We have studied numerically the stability of the problem equilibria. We have found that the numerical tests are consistent with the theoretical results [4,5,14,15,16]. (this is a suggestion, the authors need to find the appropriate references to cite here).

The title contains a typo "gowth" instead of "growth". I forgot to mention that in the first review.

Please add details of the numerical method and software (MATLAB? Python? in-house code?) used to carry out the numerical simulations. This is important for the reproducibility of your research.

 

Author Response

Authors responses to referees remarks to the paper

 

 

HIV Dynamics With A Trilinear Antibody Gowth Function And Saturated Infection Rate

 

We are very grateful to the referees for their valuable remarks. All of them are taken into account in the revised version of the paper (text in red). Detailed responses are given below.

 

 Reviewer #1:

• It needs proofreading to correct the typos and improve the presentation. 

Authors’ response: Thank you very much, we have done our best to correct them.

 

• The authors should show the advantages of using the underlying model.

Authors’ response: We have add our motivation in the text.

 

• More discussions about the numerical simulations and Figures are required. 

Authors’ response: We have add our motivation in the text.

 

• Some following references may be useful.

Authors’ response: The references are taken into consideration.

 

• More discussions about the oscillations are required.

Authors’ response: We have added some references in the text.

 

• The title contains a typo "gowth" instead of "growth". 

Authors’ response: Thank you very much, we have taken into consideration this remark.

 

• Add details of the numerical method and software (MATLAB? Python? in-house code?) used to carry out the numerical simulations.

Authors’ response: We have added our explanation in the text.

 

 

 

 

 

Reviewer #2:

 

Major comments:

1. The authors should give a more detailed description of the biology behind each term in the model. 

Authors’ response: We have added our explanation in the text.

 

2. Why is the non-linear function needed to improve the model?

Authors’ response: We have added our explanation in the text.

 

3. What are the useful biological insights obtained from this study?

Authors’ response: We have taken into consideration this question.

 

 

Minor comments:

 

4. The title is misspelled. 

Authors’ response: We have taken into consideration this remark.

 

 

5. Ref (4), is unrelated to the statement. 

Authors’ response: The references are taken into consideration.

 

 

6. Did the authors check the conditions on lines 57-58 in Mathematica or Maple?

Authors’ response: We checked the conditions manually; we base ourselves on the conditions of the Routh-Hurwitz theorem.

 

 

 

7. Why are the notations for E0 and E1 dependent on R_0 but not E2, E3 and E4?

Authors’ response: All points depend on R0 only to simplify calculations.

 

 

8. The biological interpretations of R1, R2, R3 and R4.

Authors’ response: The number R1 represents the basic defense rate and R3 is the half harmonic mean of R0 and R2.

 

 

9. I suggest adding additional simulations to explore this aspect (see 8).

Authors’ response: We have taken into consideration this remark.

 

 

10. Test what happen if the respective R crosses over the threshold. Is the stability lost? Does a bifurcation occur, etc..

Authors’ response: Indeed, stability will be lost.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Reviewer #3:

• The introduction is way too short, even for a mathematical paper. Other works in the field should be mentioned and detailed. Please improve the introduction in terms. 

Authors’ response: We have added our explanation in the text.

 

• The authors should better emphasize the novelties of this work.

Authors’ response: We have added our motivation in the text.

 

• The authors could add in the manuscript how one can obtain the given reproductive numbers $R_0$, $R_1$, and so on. At least, indicating what steps are taken to obtain them. 

Authors’ response: We have added our explanation and some references in the introduction.

 

• The authors must include details about the numerical method and time-step size used for the simulations.

Authors’ response: We have taken into consideration references [13,17,18,19] to prove our method.

 

• One can clearly see many numerical oscillations in the simulation results.

Authors’ response: You are right. We can clearly see many numerical oscillations in the simulation results, but our biological goal of having stability in the equilibrium points already calculated theoretically.

 

• The authors are using a explicit numerical scheme with a very large time step size for the simulation.

Authors’ response: We have taken into consideration this remark.

 

• Specific comments and suggestions.

Authors’ response:    

• We have only numbered the equations which are usable other times in our work.
• Thank you for your remark. The differential $dx$ can only be used in the system of equation (1), otherwise $d$ can be considered as a constant in the multiplication with other variables.
• We have added your proposal in the text.

 

• The authors provide additional references on similar forms of the model and how they were applied to study viral dynamics in real patients.

Authors’ response: We have taken into consideration this remark.

 

Author Response File: Author Response.docx

Reviewer 3 Report

The authors have addressed my comments satisfactorily. However, it would be nice if the authors provide additional references on similar forms of the model and how they were applied to study viral dynamics in real patients. This will enhance the readability and relevance of the manuscript.

Author Response

Authors responses to referees remarks to the paper

 

 

HIV Dynamics With A Trilinear Antibody Gowth Function And Saturated Infection Rate

 

We are very grateful to the referees for their valuable remarks. All of them are taken into account in the revised version of the paper (text in red). Detailed responses are given below.

 

 Reviewer #1:

• It needs proofreading to correct the typos and improve the presentation. 

Authors’ response: Thank you very much, we have done our best to correct them.

 

• The authors should show the advantages of using the underlying model.

Authors’ response: We have add our motivation in the text.

 

• More discussions about the numerical simulations and Figures are required. 

Authors’ response: We have add our motivation in the text.

 

• Some following references may be useful.

Authors’ response: The references are taken into consideration.

 

• More discussions about the oscillations are required.

Authors’ response: We have added some references in the text.

 

• The title contains a typo "gowth" instead of "growth". 

Authors’ response: Thank you very much, we have taken into consideration this remark.

 

• Add details of the numerical method and software (MATLAB? Python? in-house code?) used to carry out the numerical simulations.

Authors’ response: We have added our explanation in the text.

 

 

 

 

 

Reviewer #2:

 

Major comments:

1. The authors should give a more detailed description of the biology behind each term in the model. 

Authors’ response: We have added our explanation in the text.

 

2. Why is the non-linear function needed to improve the model?

Authors’ response: We have added our explanation in the text.

 

3. What are the useful biological insights obtained from this study?

Authors’ response: We have taken into consideration this question.

 

 

Minor comments:

 

4. The title is misspelled. 

Authors’ response: We have taken into consideration this remark.

 

 

5. Ref (4), is unrelated to the statement. 

Authors’ response: The references are taken into consideration.

 

 

6. Did the authors check the conditions on lines 57-58 in Mathematica or Maple?

Authors’ response: We checked the conditions manually; we base ourselves on the conditions of the Routh-Hurwitz theorem.

 

 

 

7. Why are the notations for E0 and E1 dependent on R_0 but not E2, E3 and E4?

Authors’ response: All points depend on R0 only to simplify calculations.

 

 

8. The biological interpretations of R1, R2, R3 and R4.

Authors’ response: The number R1 represents the basic defense rate and R3 is the half harmonic mean of R0 and R2.

 

 

9. I suggest adding additional simulations to explore this aspect (see 8).

Authors’ response: We have taken into consideration this remark.

 

 

10. Test what happen if the respective R crosses over the threshold. Is the stability lost? Does a bifurcation occur, etc..

Authors’ response: Indeed, stability will be lost.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Reviewer #3:

• The introduction is way too short, even for a mathematical paper. Other works in the field should be mentioned and detailed. Please improve the introduction in terms. 

Authors’ response: We have added our explanation in the text.

 

• The authors should better emphasize the novelties of this work.

Authors’ response: We have added our motivation in the text.

 

• The authors could add in the manuscript how one can obtain the given reproductive numbers $R_0$, $R_1$, and so on. At least, indicating what steps are taken to obtain them. 

Authors’ response: We have added our explanation and some references in the introduction.

 

• The authors must include details about the numerical method and time-step size used for the simulations.

Authors’ response: We have taken into consideration references [13,17,18,19] to prove our method.

 

• One can clearly see many numerical oscillations in the simulation results.

Authors’ response: You are right. We can clearly see many numerical oscillations in the simulation results, but our biological goal of having stability in the equilibrium points already calculated theoretically.

 

• The authors are using a explicit numerical scheme with a very large time step size for the simulation.

Authors’ response: We have taken into consideration this remark.

 

• Specific comments and suggestions.

Authors’ response:    

• We have only numbered the equations which are usable other times in our work.
• Thank you for your remark. The differential $dx$ can only be used in the system of equation (1), otherwise $d$ can be considered as a constant in the multiplication with other variables.
• We have added your proposal in the text.

 

• The authors provide additional references on similar forms of the model and how they were applied to study viral dynamics in real patients.

Authors’ response: We have taken into consideration this remark.

 

Author Response File: Author Response.docx