Mathematical Modelling and Analysis of Stochastic COVID-19 and Hepatitis B Co-Infection Dynamics
Round 1
Reviewer 1 Report
The paper develops a stochastic model for the co-infection of Hepatitis B and COVID-19 in Ghana, examining prevention, vaccination, and environmental effects. It derives reproduction numbers, establishes conditions for extinction and persistence using Lyapunov methods, and illustrates the dynamics of the coupled set of stochastic differential equations through an Euler–Maruyama numerical method.
The paper, at the best of my knowledge is original and new. Nevertheless, I have some questions and remarks:
1) It is not clear why the noise component should be positive. For this reason, I think that the positivity part should be reformulated. The positivity depends on the shape of the function \Psi which weights the noise.
2) Are Levy jumps always positive?
3) Other compartments may affect the noise of a given compartment especially regarding the coinfection dynamics.
4) In the numerical section it is not clear how you calibrate the noise level with respect to data.
5) Related to the previous questions, in recent works, the influence of uncertain quantities have been included in the form of uncertain quantity, see e.g. [https://doi.org/10.1007/s00285-021-01617-y] or heterogeneous populations [https://doi.org/10.1007/s00285-021-01630-1]. It would be beneficial for the reader to understand more closely the interpretation of stochastic quantities in your model.
As above.
Author Response
Comment 1: It is not clear why the noise component should be positive. For this reason, I think that the positivity part should be reformulated. The positivity depends on the shape of the function \Psi which weights the noise.
Response 1: Thank you for pointing this out. We agree with the comment. Therefore in the revised manuscript, we clarify that the positivity of the noise component is not assumed but arises from the structure of the weighting function Ψ. Specifically, we choose Ψ to ensure that the stochastic perturbation respects biological constraints such as non-negative population sizes.
The change can be found on Page 7, paragraph 1, line 8.
Comment 2: Are Levy jumps always positive?
Response 2: Thank you for pointing this out. Lévy jumps are not inherently positive; however, in our model, we restrict the Lévy measure to have support on (0, \infty) to reflect biologically plausible events such as sudden increases in infection. We also assume the Poisson random measure is finite and defined over a càdlàg interval, ensuring mathematical tractability and alignment with standard stochastic epidemic modeling practices. These clarifications have been added to the methodology section.
The change can be found on Page
7, paragraph 1, line 8.
Comment 3: Other compartments may affect the noise of a given compartment especially regarding the coinfection dynamics.
Response 3: Thank you for pointing this out. While we acknowledge that coinfection models often involve interactions across compartments, in this particular case, the transmission mechanisms of COVID-19 and HBV are fundamentally different—COVID-19 is primarily airborne and influenced by super-spreader events, whereas HBV is transmitted through blood and bodily fluids, often via vertical transmission or close contact. Therefore, jumps in COVID-19 due to super spreader activities are unlikely to directly influence HBV dynamics. For this reason, I have treated the noise in each compartment as independent. However, I remain open to exploring indirect pathways or shared healthcare disruptions in future work.
Details can be found on Page 5 [paragraph 4] to Page 6 [paragraph 1]
Comment 4: In the numerical section it is not clear how you calibrate the noise level with respect to data.
Response 4: Thank you for pointing this out. In the current version of the model, the noise level was assumed rather than calibrated directly from data. This choice was made to explore the qualitative impact of stochasticity on epidemic dynamics and to illustrate how random fluctuations can influence trajectory variability. The assumed noise intensity reflects plausible biological variability (e.g., reporting errors, contact heterogeneity) and was selected to preserve positivity and stability of the solution.
Specifically we used \epsilon = 0.05, \lambda = 1
The change can be found in Page 36, Table 2.
Comment 5: Related to the previous questions, in recent works, the influence of uncertain quantities have been included in the form of uncertain quantity, see e.g. [https://doi.org/10.1007/s00285-021-01617-y] or heterogeneous populations [https://doi.org/10.1007/s00285-021-01630-1]. It would be beneficial for the reader to understand more closely the interpretation of stochastic quantities in your model.
Response 5: Thank you for pointing this out. We have expanded the discussion in the manuscript to clarify the biological interpretation of the stochastic terms.
“We include stochastic terms in our model to reflect real-world sources of uncertainty in epidemic dynamics. Firstly, we include small and standard random fluctuation described by Brownian noise to account for continuous fluctuations which may arise from random interactions, environmental factors, or reporting variability. Secondly, we include L\'evy-driven jumps which imprints discrete sudden changes such as super-spreader events as outlined in \cite{Okeahalam2020}.
The change can be found in Page 5, paragraph 2
Further, while recent works see e.g. [https://doi.org/10.1007/s00285-021-01617-y] or heterogeneous populations [https://doi.org/10.1007/s00285-021-01630-1] incorporate uncertainty via random parameters or population heterogeneity, our approach complements these by embedding uncertainty directly into the system dynamics.
Reviewer 2 Report
The manuscript describes a stochastic model of HBV/SARS-CoV-2 population spread. While the mathematical analysis is fine, the study lacks motivation and the model is not complete.
- Why is this combination of diseases of interest?
- The model appears to be missing some compartments. Can people who have recovered from HBV not get COVID? More importantly, people vaccinated against COVID also appear to be immune to HBV --- that is not realistic and likely missing a sizeable proportion of HBV infections.
- There are issues with that/which throughout the manuscript. Please check for correct use.
- I believe assumption A8 should refer to SARS-CoV-2. Vertical transmission is a fairly major mechanism for spread of HBV, so if the authors assume no vertical transmission for HBV, then the model is not correctly capturing disease spread.
Author Response
Comment 1: Why is this combination of diseases of interest?
Response 1: Thank you for pointing this out. The reason for studying the combination of diseases was stated in the manuscript on page 2. It has however been updated in the revised document for clarity as
“Co-infection of HBV and COVID-19 can exacerbate disease severity, leading to adverse outcomes, including increased mortality risk \cite{lin2021patients, sun2024effect}. There is therefore the need for techniques that better predict disease dynamics, and offer recommendations for the prevention, and reduction in disease co-infection spread in the presence of clinical, behavioural and environmental factors.
The change can be found on Page 2,.
Comment 2: The model appears to be missing some compartments. Can people who have recovered from HBV not get COVID? More importantly, people vaccinated against COVID also appear to be immune to HBV --- that is not realistic and likely missing a sizeable proportion of HBV infections.
Response 2: Thank you for pointing this out. In the Revised manuscript we have included a path to indicate that people who have recovered from acute HBV could be infected by COVID-19. Subsequently analyses have been revised to incorporate this path. The change can be found on Page 4.
However, as stated in Assumption 4 (i.e. COVID-19 vaccination wears off after some period of time), people vaccinated against COVID-19 can eventually transit to susceptible state and then to HBV infectious states, hence are not immune to HBV in our model.
Comment 3: There are issues with that/which throughout the manuscript. Please check for correct use.
Response 3: Thank you for pointing this out. Issues with “that/which” have been checked throughout the manuscript. The change can be found Throughout the paper.
Comment 4: I believe assumption A8 should refer to SARS-CoV-2. Vertical transmission is a fairly major mechanism for spread of HBV, so if the authors assume no vertical transmission for HBV, then the model is not correctly capturing disease spread.
Response 4: Thank you for pointing this out. The assumption has been deleted as recommended to correctly capturing HBV disease spread. The change can be found on Page 3.
Reviewer 3 Report
The paper is well- worked and well-organized and it presents original results. The main idea behind is to combine the diseases of Hepatitis B and COVID-19. Therefore the model has many interacting subpopulations. The most general model presented has a stochastic nature driven by Brownian noise together with more general Lévy noise which can include jumps. The simulated examples are also well- worked and discussed and illustrate the theoretical aspects.
The paper is well- worked and well-organized and it presents original results. The main idea is to combine the diseases of Hepatitis B and COVID-19. Therefore the model has many interacting subpopulations. The most general model presented has a stochastic nature driven by Brownian noise together with more general Lévy noise which can include jumps. The simulated examples are also well- worked and discussed and illustrate the theoretical aspects.
Some comments to improve the results in a revised version follow below:
Definition 2: Which is the meaning of “u” and its differential in v(du) in the formula?.
Address more clearly the proof of Theorem 1 which , may be, it needs some corrections and more explanations. Eqn. (5) is based on the fact the neglected four first terms of the former equation are non-negative, or at least, their sum is non-negative. It is not obvious from inspection of (3) that V/C, R/C and R/aC are non-negative. The simplest one to discuss non-negativity which is the last equation of (3) depends, in fact, on the non-negativity or not of S(t) which is being tried to prove.
Proof of Theorem 1, Last separated formula in page 9: why the limit in the second term as time tends to infinity is positive while it involves an exponential with negative exponent?.
Last part of the proof of Theorem 2: Psi/mu is constant so that taking a limit is unneeded in the last right-hand-side term of the penultimate equation. In the last and penultimate equations: Why is it known that the total population has a limit as time tends to infinity?. This is not obvious without some additional explanation or some use of previous result in the article. In case of reasonable doubt limit could be replaced with “limit superior” which always exists and is upper-bounded by psi/mu.
Also, how, or from which previous result ion the paper, is it ensured that the total population does not asymptotically extinguish (N(t) >0 as time tends to infinity, that is, lim inf N(t)>0 as time tends to infinity.
Line 4 after Eqn. (11: There is a typo in the inverse of V since -1 is not written as a superscript.
Comment better on the reason of the equation for the expression “f” next to that of the inverse of V. Why is such a “f” interpreted as a primary infection and why i the subsequent “v” describes the secondary infections.
The expression (12) for the deterministic reproduction number requires technical discussion on the stability ( via largest eigenvalue of the matrix of dynamics) of the stability of the deterministic linearized system around the disease-free equilibrium.
Which is the interpretation of the statement of Theorem 5 concerning I/a(t)?.
Definition 4 and Theorem 12: The definition of persistence is done for a vector while the theorem invokes persistence for an infected subpopulation, as usual. Therefore, it could be defined persistence for the whole state (in the definition) and the to speak about the persistence of its particular components
Author Response
Comment 1: Definition 2: Which is the meaning of “u” and its differential in v(du) in the formula?.
Response 1: Thank you for pointing this out. In the paper we define $v$ as a finite L\'evy measure on a measurable subset $\mathcal{U} \in (0, \infty)$, where $v(\mathcal{U}) < \infty.$ $v(\mathcal{U}) $ gives the expected number of jumps whose size falls within $\mathcal{U}$ in time.
The details can be found on Page 7, paragraph 1.
Comment 2: Address more clearly the proof of Theorem 1 which , may be, it needs some corrections and more explanations. Eqn. (5) is based on the fact the neglected four first terms of the former equation are non-negative, or at least, their sum is non-negative.
Response 2: Thank you for pointing this out. The proof has been revised in the revised paper.
The change can be found on Page 8.
“Eliminating first four positive terms on the right hand side and rearranging terms, we get..."
Comment 3: It is not obvious from inspection of (3) that V/C, R/C and R/aC are non-negative. The simplest one to discuss non-negativity which is the last equation of (3) depends, in fact, on the non-negativity or not of S(t) which is being tried to prove.
Response 3: Thank you for pointing this out. Although not documented in the revised paper. The proof of other states can be proved using similar techniques. For example. The positivity of the solution $V_C(t)$ to \eqref{stochastic system levy} as follows.
We consider the thirteenth equation of \eqref{stochastic system levy}
\begin{equation*}
\begin{split}
\dfrac{dV_C (t)}{dt} &=\theta_V S(t) -(\mu +\sigma_V)V_C (t)+ \epsilon_{13} V_{C}dB_{13}(t).
\end{split}
\end{equation*}
By eliminating first positive term on the right hand side and rearranging terms, we get
\begin{equation}
\begin{split}
\label{positivestocIH3}
\dfrac{d V_{C} (t)}{V_C} & \ge [-(\mu +\sigma_V)V_C (t)) ]dt+ + \epsilon_{13} V_{C}dB_{13}(t). \\
& \ge -rdt+ \epsilon_[13} dB_{13} (t),
\end{split}
\end{equation}
where $r = [-(\mu +\sigma_V)V_C (t)) $. Appying the Itô’sformula
d \ln V_C (t) &=\dfrac {1}{ V_C } dV_C - \dfrac {1}{2} (\epsilon_{13}^ 2 dt),
\end{aligned}$$
A rearrangement yields
\begin{equation}
\label{positivestoc2IH3}
\dfrac {dV_C(t)}{ V_C } = d \ln V_{C} (t) + \dfrac {1}{2} (\epsilon_{13}^ 2 dt).
\end{equation}
Combining \eqref{positivestocIH3} and \eqref{positivestoc2IH3}, and solving for V_C(t) yeilds
$$\begin{aligned}\begin{aligned}
V_C (t)& \ge V_C (0) e^ { - (r + \frac {1}{2} \epsilon_{13}^ 2 ) t + \epsilon_{13} B_{13}(t)}.
\end{aligned}\end{aligned}$$
%\Taking the $\lim_{t\to\infty}$ on both sides we obtain
%\noindent As $$t\to\infty$$
Comment 4: Proof of Theorem 1, Last separated formula in page 9: why the limit in the second term as time tends to infinity is positive while it involves an exponential with negative exponent?.
Response 4: Thank you for pointing this out.
The exponential term in question indeed has a negative linear drift, so its limit as t→∞ is zero almost surely. Our intention was to show positivity for all finite t, not persistence at a positive level in the limit. We have revised the text to clarify that the bound ensures I(t) > 0 for all t, but does not imply \lim_{t\to\infty} I(t) > 0.
The change can be found on Page 9.
Comment 5: [Last part of the proof of Theorem 2: Psi/mu is constant so that taking a limit is unneeded in the last right-hand-side term of the penultimate equation.]
Response 5: [Thank you for pointing this out. We agree with the comment. In the revised manuscript: We have removed the unnecessary limit in the penultimate equation, since \frac{\Psi}{\mu} is constant and does not depend on t.
The change can be found on Page 11.]
Comment 6: [In the last and penultimate equations: Why is it known that the total population has a limit as time tends to infinity?. This is not obvious without some additional explanation or some use of previous result in the article. In case of reasonable doubt limit could be replaced with “limit superior” which always exists and is upper-bounded by psi/mu.]
Response 6: [Thank you for pointing this out. For the total population term, we clarify that convergence as {t \to \infty} was not previously established. Accordingly, we have replaced the limit with the limit superior, which always exists for bounded trajectories and is bounded above by \frac{\Psi}{\mu}. This change ensures the argument remains valid without requiring a proven convergence result.
The change can be found on Page 11.]
Comment 7: [Also, how, or from which previous result in the paper, is it ensured that the total population does not asymptotically extinguish (N(t) >0 as time tends to infinity, that is, lim inf N(t)>0 as time tends to infinity.
Response 7: Thank you for pointing this out. In the proof of invariance of solutions, our goal was not to establish persistence (which is proved later in the paper), but rather to show that the total population remains bounded and nonnegative. For this reason, we have revised the statement to
\frac{\psi}{\mu} \ge \limsup_{t \to \infty} N(t) \ge 0,
where \frac{\psi}{\mu} is a constant upper bound. This formulation reflects the invariance property: solutions remain in a biologically meaningful region, with the population size constrained between zero and a finite bound.
The change can be found on Page 11.]
Comment 8:[ Line 4 after Eqn. (11: There is a typo in the inverse of V since -1 is not written as a superscript.]
Response 8: [Thank you for pointing this out. The typo in the inverse of V in Line 4 after Eqn. (11) as a result of -1 is not written as a superscript has been rectified. The change can be found on Page 12 as
" ...(largest eigenvalue) of $FV^{-1},$ where $F$ and $V$ are the partial derivatives of $f$, and $v$ respectively."]
Comment 9: [Comment better on the reason of the equation for the expression “f” next to that of the inverse of V. Why is such a “f” interpreted as a primary infection and why i the subsequent “v” describes the secondary infections.]
Response 9: [ We appreciate the reviewer’s request for clarification. In our revision we have expanded our explanation of the role of the expression f and its relation to V^{-1}.
The change can be found on Page 12.
Specifically we have revised the section as below:
“When using the NGM method, the basic reproductive number is defined as the spectra radius (largest eigenvalue) of $FV^{-1},$ where $F$ and $V$ are the partial derivatives of $f$, and v respectively...."
Comment 10: The expression (12) for the deterministic reproduction number requires technical discussion on the stability ( via largest eigenvalue of the matrix of dynamics) of the stability of the deterministic linearized system around the disease-free equilibrium.
Response 10: We thank the reviewer for this helpful comment. In the revised manuscript we have expanded the discussion of the deterministic reproduction number in (12) as below: “$\mathcal{R}_{0C}^d$ represents the expected number of secondary infections generated by a single COVID-19 primary infection introduced into a fully susceptible population and $\dfrac {1}{(\rho_C +\mu_C)}$ represents the expected time an individual will spend in the COVID-19 infectious compartment. “ The change can be found on Page 13.
Comment 11:Which is the interpretation of the statement of Theorem 5 concerning I/a(t)?.
Response 11: Thank you for pointing this out.
The statement of Theorem 5 means that when the stochastic reproduction number \mathcal{R}_0^s is less than one, the infected class I_C(t) cannot sustain exponential growth. Biologically, this implies that the infection will eventually die out and the disease‑free equilibrium is locally stable. The change can be found on Page 17.
We have reviewed this in the paper as below: {Local Stability of Stochastic COVID-19 Only Model}. It is crucial to understand the behaviour of a dynamical system near the disease-free equilibrium point. Checking for local stability at this point helps to determine whether the COVID-19 will eventually die out and the disease‑free equilibrium is locally stable.
Comment 12: [Definition 4 and Theorem 12: The definition of persistence is done for a vector while the theorem invokes persistence for an infected sub-population, as usual. Therefore, it could be defined persistence for the whole state (in the definition) and the to speak about the persistence of its particular components]
Response 12: [Thank you for pointing this out. In the revised manuscript, we have clarified the definition of persistence so that it is explicitly tied to the biological compartments of our model. The change can be found on Page 29.
In particular, we now define persistence in the mean for the infected population as :
\begin{Definition}\cite{Anwarud}
\label{anwarud}
Let $\mathcal{X}(t) := I_C (t)+ I_a (t) + I_{aC} (t)+ I_c (t) + I_{cC} (t) $ be the total infected population of \eqref{stochastic system levy}. The persistence for
$\mathcal{X}(t)$ will hold if, there exists $c>0$ such that…"]
Round 2
Reviewer 1 Report
The way the positivity is addressed is somehow unclear since one should assume initial positivity and investigate whether it is propagated in time under some restrictions on the model parameters or scalings.
If a coefficient annihilates the realization if it is not admissible you essentially lose regularity and this is not the standard way to proceed.
I am not satisfied of the answer to my last question. Indeed, when the authors state "our approach complements these by embedding uncertainty directly into the system dynamics" they essentially neglect the role of diffusion in existing literature of epidemic dynamics with uncertain quantities. Superspreaders and deviations from deterministic trajectories are typically computed in terms of quantities of interest. It is not clear the link to the mentioned literature.
comments detailed above
Author Response
Comment 1: The introduction is not sufficient to understand the impact of uncertainties in epidemic dynamics. I suggest a thorough revision.
We have thorough revised the introduction to provide an understanding of the impact of uncertainties in epidemic dynamics. On page 2, paragraph 4
"...
Mathematical models are essential tools that can guide effective interventions for various infectious diseases, e.g., \cite{Allen2010, iboi2020, contreras2020, Diagne2021, Moore2022}. Several deterministic models have been developed to study the transmission dynamics of other infectious disease . Although the relevance of classical deterministic models e.g. \cite{iboi2020, WU2020689, zhao2000mathematical, Shah2024, danane2020mathematical} in explaining disease dynamics cannot be disputed, they often assume that epidemic paths are predictable given specified parameters. In reality, however, epidemic trajectories are rarely deterministic. They are shaped by various layers of uncertainty, including incomplete or noisy data, variability in parameter estimates, and random environmental or demographic fluctuations \cite{Cunha2023}. Ignoring such uncertainties can lead to overconfident forecasts and policy recommendations that fail under real‑world conditions.}
Stochastic models e.g., \cite{ChunyanJiand, iboi2020, HAMA2022105477, pobbi2023} provide a more realistic representation of epidemic dynamics of disease spread . These epidemiological models, unlike classical deterministic counterparts, e.g. \cite{contreras2020, Diagne2021, Moore2022}, account for the effects of random fluctuations in disease transmission dynamics, and allow for the quantification of extinction probabilities and the likelihood of rare but high‑impact events \cite{Cunha2023,Sharan2024}. In the context of COVID‑19, where co‑infections with chronic diseases such as Hepatitis B remain a public health concern, accounting for uncertainty is essential for designing robust and adaptive control measures..."
Comment 2:
The way the positivity is addressed is somehow unclear since one should assume initial positivity and investigate whether it is propagated in time under some restrictions on the model parameters or scalings.
If a coefficient annihilates the realization if it is not admissible you essentially lose regularity and this is not the standard way to proceed.
I am not satisfied of the answer to my last question. Indeed, when the authors state "our approach complements these by embedding uncertainty directly into the system dynamics" they essentially neglect the role of diffusion in existing literature of epidemic dynamics with uncertain quantities. Superspreaders and deviations from deterministic trajectories are typically computed in terms of quantities of interest. It is not clear the link to the mentioned literature.
Response 2: We thank the reviewer for this detailed and insightful comment. We have carefully revised the manuscript to address the three key concerns raised: positivity propagation, regularity of coefficients, and the role of diffusion in the context of uncertainty.On page 8 we have noted " Consequently, we assume strictly positive initial conditions, $X(0)\in \Re_+^d$. To ensure that solutions remain in the admissible domain, the drift and diffusion coefficients are taken to be smooth in the interior of $\Re_+^d$ and to vanish on the boundary \cite{Allen2017}. "
Reviewer 2 Report
The authors did not address all of my previous comments:
- The title specifically references modeling of HBV/SARS-CoV-2 in Ghana. While the authors base some parameter values on information from Ghana, many others are taken from other sources. Simulations are performed using an initial population of 100 susceptible people, which is clearly not the population of Ghana. I don't think the results are really reflective of the dynamics of these diseases in Ghana.
- The conclusions are not put into any context. There seems to be no conclusion reached that would provide guidance to public health officials, particularly as it relates to the coinfected patients that are presumably the focus of this study.
Many of the parameters in table 2 do not have units even though they should.
Author Response
The authors did not address all of my previous comments:
Comment 1:The title specifically references modeling of HBV/SARS-CoV-2 in Ghana. While the authors base some parameter values on information from Ghana, many others are taken from other sources. Simulations are performed using an initial population of 100 susceptible people, which is clearly not the population of Ghana. I don't think the results are really reflective of the dynamics of these diseases in Ghana.
Response 1: Thank you pointing this out. We agree that our original framing may have overstated the Ghana‑specificity of the model. In the revised manuscript we have:
-
Revised the title
as "MATHEMATICAL MODELLING AND ANALYSIS OF STOCHASTIC COVID-19 AND HEPATITIS B CO-INFECTION DYNAMICS" - and the abstract, stating "Finally, motivated by Ghana data, we applied the Euler–Murayama scheme to illustrate the dynamics of the co-infection, COVID-19, HBV and the effect of some parameters on disease transmission dynamics by means of numerical simulations."
- Further, in the numerical analysis section on page 37 we have explained " We emphasise that the initial population of 100 susceptibles and other populations were chosen as a scaled simulation to illustrate the qualitative dynamics of the system and do not reflect the absolute population size."
Comment 2:The conclusions are not put into any context. There seems to be no conclusion reached that would provide guidance to public health officials, particularly as it relates to the coinfected patients that are presumably the focus of this study.
Response 2: Thank you pointing this out. We agree that the original conclusion section did not sufficiently highlight the public health relevance of our findings. In the revised manuscript, we have substantially expanded the Conclusion to provide clearer context and practical implications. on page 42 to 43.
Comment 3: Many of the parameters in table 2 do not have units even though they should.
Response 3: Thank you pointing this out. We agree that many of the parameters in table 2 did not have units. We have provided the units in Table 2 on page 37 in the revised paper.
Reviewer 3 Report
The paper formulates a situation of co-infection Hepatitis-B/COVID-19. The formal results are rigorous and the simulations are illustrative. The current version of the paper has been improved related to the former version and the refereeing critical points have been satisfied successfully. The figures/tables are sufficiently illustrative and clear to interpret.
The paper formulates a situation of co-infection Hepatitis-B/COVID-19. The formal results are rigorous and the simulations are illustrative. The current version of the paper has been improved related to the former version. The figures/tables are sufficiently illustrative and clear to interpret.
Author Response
Comment 1: The paper formulates a situation of co-infection Hepatitis-B/COVID-19. The formal results are rigorous and the simulations are illustrative. The current version of the paper has been improved related to the former version and the refereeing critical points have been satisfied successfully. The figures/tables are sufficiently illustrative and clear to interpret.
Response 1: We are grateful for your comments
