Biological and Chemical Control of Mosquito Population by Optimal Control Approach

Round 1

Reviewer 1 Report

In this paper, the authors address the question of optimizing the introduction of a predacious species with respect to the use of larvicide and insecticide to reduce a population of mosquitoes. The mathematical model is a system of differential equations which has already already introduced in a former paper. The authors defined their cost functional and prove existence of a minimizers thanks to classical technique. Application of the Pontryagin Maimum Principle allows them to have a short characterization of the optimal solutions. Then several numerical tests are performed to study different scenarios.

The mathematical techniques used in this work are not new. This paper is an example of applications of such strategies in a framework which might be interesting for the fight against mosquitoes, responsible of the transmission of many diseases to human, with an "eco-friendly strategy". Thus, the most interesting part of this work is the comparison and the discussion on the 27 scenarios which are proposed.

Please find below a short list of minor corrections :

• l42, different (with a t).
• l138, 'that that' (please remove an extra 'that').
• l171-172, it is indeed demanding to find a predator that fulfill (5), but also (1). In fact, in this equation, it is supposed that the predator does not need the prey to survive. This is different to standard Lotka-Volterra predator-prey model where the predator species goes to extinction if there is no prey. Maybe a short comment could be added by the authors.
• What is the definition of PC[0,T] in (9) ?
• In equation (13), (T) is missing for the last two equalities.
• Is the characterization of optimal solutions by (15) usefull the the rest of the study ?
• l477, last (with a t).
•  

 

Author Response

*Is the characterization of optimal solutions by (15) usefull the the rest of the study ?    R// Yes, it is useful. In effect, the optimality system is defined by replacing two controls in the direct and adjoint systems (1), (12) with their characterizations according to the formulas (15). The optimality system with two-point boundary conditions (2), (13) is further solved numerically in order to obtain solutions of the optimal control problem under different scenarios.   The manuscript has been adjusted according to the other observations made, which appear in blue, which you can see in the attached document.

Author Response File: Author Response.pdf

Reviewer 2 Report

This paper concerns a cost-benefit analysis of an optimal control approach to a predator-prey model involving the control of mosquitoes. It is a strong contribution to the literature.

• It is more common to use the phrase predator-prey system rather than prey- predator. This occurs in a few places and is easy to fix.

• Page 2, line 42, different is missing a t.

• Page 6, line 188, I recommend giving speci c examples of how the percentage is reduced here. The mathematics indicates that there is a reduction but the magnitude of the reduction may or may not be significant. An example would be helpful.

• Page 7, line 215, functions is missing an s.

• Page 7, line 223, an pair should be a pair.

• Page 9, equation (13), the transversality conditions should be λ2(T) = 0,λ3(T) = 0.

• Page 11, line 324-325, I have no idea what this means.

• Section 4, the difference between ACER and IECR was unclear to me at times. Further detail is needed to clarify this important difference. I suggest adding a short paragraph comparign and contrasting the two concepts.

Comments for author File: Comments.pdf

Author Response

*Page 6, line 188, I recommend giving speci c examples of how the percentage is reduced here. The mathematics indicates that there is a reduction but the magnitude of the reduction may or may not be significant. An example would be helpful.

R// The percentage is calculated using the definition of the normalized forward sensitivity index from [33]. We have made it more precise in the paper text.

* Page 11, line 324-325, I have no idea what this means.

R// We have clarified the meaning of this phrase in the revised version of the paper

*Section 4, the difference between ACER and IECR was unclear to me at times. Further detail is needed to clarify this important difference. I suggest adding a short paragraph comparign and contrasting the two concepts.   R// The ACER estimates the cost to be invested in each intervention strategy for obtaining one unit of the corresponding effect or benefit in comparison to the "no intervention" case and zero cost.  On the other hand, ICER estimates an additional cost per unit of the effect or benefit when two mutually exclusive strategies (bearing different costs and benefits) are compared with each other.  

The paper has been adjusted according to the other observations raised, which appear in green, which you can see in the attached document.

Author Response File: Author Response.pdf