Dynamical Analysis of a Modified Epidemic Model with Saturated Incidence Rate and Incomplete Treatment

Round 1

Reviewer 1 Report

The manuscript presents a mathematical model of TB that incorporates a saturated incidence function and incomplete home treatment of TB. The inclusion of the saturated incidence function is not well justified, but the model is otherwise sound. I have concerns about other parts of the manuscript:

1. Section 3.2: Are the equilibrium points not found by setting derivatives to 0? This needs to be stated.
2. Lines 108-109: This sentence is unclear does incomplete treatment increase or decrease R0?
3. Lines 121-123: It is not immediately clear that these conditions hold in general. I think an appendix showing that they hold is needed. The current appendices that calculate eigenvalues numerically for specific values of the parameters is insufficient.
4. Parameter values of table 2 do not have units.
5. More generally, I have an issue with "assumed" parameters. This is even more problematic in this manuscript than it looks because some parameters are cited as being taken from a particular previously published manuscript. Unfortunately, this manuscript also "assumed" the vast majority of its parameters. You cannot claim to be modeling a particular disease if you are using parameters pulled out of thin air. The introduction mentions a few papers that fit similar models to actual data --- parameters derived in this way can be said to describe a specific disease. If you are guessing at parameters, then your model describes an infection disease that follows saturated incidence and incomplete treatment. It DOES NOT necessarily describe tuberculosis.
6. Why is figure 6 before figure 5?
7. What is on the x-axes of figures 7 and 8?
8. There are problems with grammar throughout the manuscript.

Author Response

 We thank the experts for their comments/suggestions, which greatly improve the manuscript. This is a detailed response to the reviewers’ comments and suggestions. We have agreed with the experts’ comments/suggestions and made all appropriate changes. The experts’ comments are written first and followed by our response written in italics. Changes included in the manuscript are written in blue. We hope that the revised manuscript is suitable for publication. 

Author Response File: Author Response.pdf

Reviewer 2 Report

• Overview of manuscript

The authors presented the Tuberculosis (TB) model considering saturated incidence and incomplete treatment. They analyzed the existence of all equilibrium points. They investigated the local and global stability of the non-endemic and endemic points. Some numerical simulations are presented to support the analytical results.

• Comments on text

1. New contribution

The authors need to talk about all the new contributions of their modeling approach compared to the previous TB models.

2. English

The English in this paper is good.

Comments

• Please compare your model with other models with the saturated infection rate. Is there any novelty or what is the advantages of your model compared to others? Did anyone test the models with the saturated infection rate using data? You may need to discuss more on this part of your paper.
• Latin Hypercube Sampling and Partial Rating Correlation Coefficients methods are two global stabilities however, you may need to describe these methods analytically as well. Could you provide the bar chart for PRCC to explore the most influential parameters since it would be easier to track the results.
• The authors may need to improve the quality of figures 7 and 8.
• That would be better if authors can explain the reason to use LHS and PRCC methods and no other sampling methods.
• Please include the y-label in figure 9.
• The authors may need to add more explanations related to figure 10 and optimal control problem.

Author Response

 We thank the experts for their comments/suggestions, which greatly improve the manuscript. This is a detailed response to the reviewers’ comments and suggestions. We have agreed with the experts’ comments/suggestions and made all appropriate changes. The experts’ comments are written first and followed by our response written in italics. Changes included in the manuscript are written in blue. We hope that the revised manuscript is suitable for publication. 

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments to authors:

1. The application of the formula for calculating the model of the tuberculosis epidemic process is possible only after clarifying the procedure for the formation of coefficents for patients who are treated at home and in a hospital.
2. From the point of view of the organization of medical care, there can be no equal mortality rates from tuberculosis during treatment at home and in hospital.
3. Also, there can be no identical control parameters in the two groups of influence on these groops. If there are identical control parameters? an additional destription of the process of providing medical care to tuberculosis patients in the hospital is required.

Author Response

 We thank the experts for their comments/suggestions, which greatly improve the manuscript. This is a detailed response to the reviewers’ comments and suggestions. We have agreed with the experts’ comments/suggestions and made all appropriate changes. The experts’ comments are written first and followed by our response written in italics. Changes included in the manuscript are written in blue. We hope that the revised manuscript is suitable for publication. 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have not addressed my biggest concerns about their work here. As I indicated in my previous report, the model itself is structurally fine, and can be called a model of infectious disease, but it is NOT a model of tuberculosis. What converts a generic infectious disease model to a model for a specific disease is the values of the parameters. The parameter values set the time scale and population scale to reproduce a specific disease. The authors have NOT included units in Table 2, instead they included "Time" and "Humans" in Table 1. "Time" is NOT a unit. It does not set a scale; "minutes", "days", etc. are units as they set the time scale for the model.

While there are no units in Table 2, I will assume that the time scale is in days since the subsequent graphs are in days. If we look carefully at the values presented in Table 2, we can see why I don't think the authors can call this a model of TB. According to the parameters in the table (assuming they are in days): it takes, on average, 5 days (delta) to die from TB after hospitalization; it takes, on average, 1 day (epsilon) to move from initial infection to the hospital and 11 days (theta) to go from initial infection to symptomatic; it takes, on average, 11 days (k1) of treatment at home to recover and 1.4 days (k2) of treatment at the hospital to recover. In actual reality, TB treatment takes weeks or months, the latent period is also on a much longer time scale than a few days. Additionally, all the graphs presented here run less than a year when TB epidemics are typically on a multi-year time scale. The time scale for the parameter values used here is NOT representative of TB, so the authors cannot claim that they are modeling TB.

As the authors mentioned in their response, there are previous publications with similar "assumed" parameters. To me, this is all the more reason to not allow this practice to continue. It is irresponsible to continue to propagate misinformation in scientific journals. As scientists, we have a responsibility to base our work on facts and correct information.

While that is my major concern with this manuscript. The authors have also not actually proved theorems 3 and 4. Just because they are true for the specific parameter values they have "assumed", does not make them generally true. A true mathematical proof requires the authors to prove that the Routh-Hurwitz criteria hold for ALL values of the parameters.

Author Response

We thank the experts for their comments/suggestions, which greatly improve the manuscript. This is a detailed response to the reviewers’ comments and suggestions. We have agreed with the experts’ comments/suggestions and made all appropriate changes. The experts’ comments are written first and followed by our response written in italics. Changes included in the manuscript are written in blue. We hope that the revised manuscript is suitable for publication.

Author Response File: Author Response.pdf

Reviewer 3 Report

I agree with the changes made to the article. The article can be published in the submitted amended version.

Author Response

 We thank the experts for their comments/suggestions, which greatly improve the manuscript. This is a detailed response to the reviewers’ comments and suggestions. We have agreed with the experts’ comments/suggestions and made all appropriate changes. The experts’ comments are written first and followed by our response written in italics. Changes included in the manuscript are written in blue. We hope that the revised manuscript is suitable for publication 

Author Response File: Author Response.pdf

Round 3

Reviewer 1 Report

The authors have addressed my major concerns. English language and grammar need correction throughout.