Bifurcation and Optimal Control Analysis of an HIV/AIDS Model with Saturated Incidence Rate

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This is a good model, however I have the following suggestions

1.  Please edit the abstract so that it reflects the aim and objectives of the study. In simpler terms it must outline the research questions being answered by the work.
2. Besides HIV treatment there is HIV prevention in the form of PrEP and PEP. I think it is relevant to add these to the model as it really is the current topic of discussion and would add to the body of knowledge. Please refer to these two works and probably highlight it in your future work if it is late to incorperate it. https://www.sciencedirect.com/science/article/pii/S2666720723001571   
3. Please correct this eta_1,eta_2 \in (0,1) and also mention that eta_1>eta_2 and it meaning on the dynamics.
4. Elaborate further on this "The parameter that represents 174
   the psychological or inhibitory effect is denoted by 𝜔." what is the impact of 𝜔. and what values does it assume i.e what happens when 𝜔=0?
5. Please correct grammar in the paper...for instance "It is further reduced by a natural death 177 at a rate 𝜇." who or what is it?Line 177.
6.  I think it best to use S_1 and S_2 for the E_1 and E_2 classes to avoid them being taken as Exposed classes (commonly used classes).
7. I would suggest the inclusion of a group of individuals whose treatment fails and they go back to I_a. This can be due to drug resistance or adherence issues.
8. Why have you left out death due to infections in the infected classes?
9. Line 308, please make the fractions bigger (typesetting required for fractions in the doument).
10. I think this is a typing error"Strategi". Do you mean "Strategy"?
11. What are the values of your weights w__1.

Comments on the Quality of English Language

The paper needs a grammar and spell check

Author Response

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Reviewer 2 Report

Comments and Suggestions for Authors

The author proposed an epidemic model to describe the HIV transmission in a population. Based on the dynamic analysis, they also formulate an optimal control problem. Then, they do some cost-effectiveness analysis for the optimal control strategies. This topic seems interesting. However, the theoretical results and numerical results obtained in this manuscript are trivially. There are some comments:

In page 7, the proof of Lemma 1 is not correct. It should be proved that every variable is nonnegative.

In Page 10, line 306, “the polynomial (13) has one positive real root” maybe a mistake. The expression of equilibrium (13) cannot have root.

In Theorem 4, the condition “B_0<0” is strange and it makes the theorem 4 confused.

In section 4, the authors consider the parameters \xi, \theta, \tau as continuous variables that depend on time to formulate an optimal control problem. I do not think it is reasonable to formulate the optimal control problem without any infection data case.  

The current version should be undergo major revisions further progress.

Author Response

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Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The comments I made have been addressed. Well done! However, I have the following minor issues

1. Line 190 still contains terms in E if E is a parameter; please replace it with a Greek letter instead.
2. The same is observed in lines 209, Equations 2, 3, 4, etc., and throughout the document as well. Please check and replace accordingly.
3. Figure 1 and Table 1 also need correction, there are E terms (change to \zeta or \xi.... Greek letters)

Author Response

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Reviewer 2 Report

Comments and Suggestions for Authors

Comments on "Bifurcation and optimal control analysis of an HIV/AIDS 2 
model with a saturated incidence rate "

The authors proposed an epidemic model to describe the transmission of HIV. The dynamics analysis of the model was proved and the optimal control problem was formulated and studied. However, the method and the theoretical results in the manuscript are common. There is a point should be addressed.

i)In page 13, line 407, “To establish the global stability of the disease-free equilibrium, we apply the method used in [33].” The method is unknown and not clearly.

Author Response

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