Mathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD, and SIR Compartment Models
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis article is devoted to an important problem of studies of a five-dimensional compartmental epidemic model, which takes into account the coexisting dynamics between the susceptible-infected-recovered-vaccinated-dead (SIRVD) compartments. It is worth noting that the authors' model also takes into consideration the impact of vaccination campaigns and time-dependent mortality rates on epidemic outbreaks. The authors investigate 1) new classes of exact solutions to the SIRVD and SIRD (susceptible-infected-recovered-dead) equations and 2) apply their recently developed analytical approximations, see [1,2 ], for the SIR (susceptible-infected-recovered) and SIRV (susceptible-infected-recovered-vaccinated) models also to the more general SIRVD model.
The authors' arguments seem correct. The article is very interesting, but too long to check the technical work of the authors in a short time. Thus, I think that it can be recommended for publication in MDPI Mathematics in its current form.
References
1. Schlickeiser, R.; Kröger, M. Analytical solution of the SIR-model for the not too late temporal evolution of epidemics for general time-dependent recovery and infection rates. Covid 2023, 3, 1781--1796. https://doi.org/10.3390/covid3120123.
2. Kröger, M.; Schlickeiser, R. On the analytical solution of the SIRV-model for the temporal evolution of epidemics for general time-dependent recovery, infection and vaccination rates. Mathematics 2024, 12, 326. https://doi.org/10.3390/math12020326.
Author Response
Referee: This article is devoted to an important problem of studies of a five-dimensional compartmental epidemic model, which takes into account the coexisting dynamics between the susceptible-infected-recovered-vaccinated-dead (SIRVD) compartments. It is worth noting that the authors' model also takes into consideration the impact of vaccination campaigns and time-dependent mortality rates on epidemic outbreaks. The authors investigate 1) new classes of exact solutions to the SIRVD and SIRD (susceptible-infected-recovered-dead) equations and 2) apply their recently developed analytical approximations, see [1,2 ], for the SIR (susceptible-infected-recovered) and SIRV (susceptible-infected-recovered-vaccinated) models also to the more general SIRVD model.
1. Schlickeiser, R.; Kröger, M. Analytical solution of the SIR-model for the not too late temporal evolution of epidemics for general time-dependent recovery and infection rates. Covid 2023, 3, 1781--1796. https://doi.org/10.3390/covid3120123.
2. Kröger, M.; Schlickeiser, R. On the analytical solution of the SIRV-model for the temporal evolution of epidemics for general time-dependent recovery, infection and vaccination rates. Mathematics 2024, 12, 326. https://doi.org/10.3390/math12020326.
Response: We thank this referee for a careful reading and positive assessment.
Referee: The authors' arguments seem correct. The article is very interesting, but too long to check the technical work of the authors in a short time. Thus, I think that it can be recommended for publication in MDPI Mathematics in its current form.
Response: Thank you. Note that we have revised the manuscript inline with suggestions by the remaining referees.
Reviewer 2 Report
Comments and Suggestions for AuthorsMathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD and SIR Compartment Models
I have read this paper with great case! In this paper, the authors studied The susceptible-infected-recovered-vaccinated-deceased (SIRVD). The Perov fixed point theorem is adopted to study the established model. Moreover, the authors also apply a recently developed analytical approximation for the SIR and SIRV models to the more general SIRVD model. This approximation offers accurate analytical expressions for epidemic quantities, such as the rate of new infections and the fraction of infected persons, particularly when the cumulative fraction of infections is small. The distinction between recovered and deceased individuals in the SIRVD model affects the calculation of the death rate, which is proportional to the infected fraction in the SIRVD/SIRD cases but often proportional to the rate of new infections in many SIR models using a posteriori approach.. The results are correct and fascinating. The present work is suitable for publication after following revision:
1-Why not divide SI by N in the equations (a1) and (a2)?
2-What are the benefits of your newly discovered exact solutions to the SIRVD and SIRD equations in enhancing our understanding of the spread and control of epidemics compared to previous models?
3-Can you elaborate on the methods used to derive the accurate analytical approximations for epidemic quantities, such as the rate of new infections and cumulative fraction of infections?
4-In what ways does the differentiation between recovered and deceased persons in the SIRVD model impact predictions and potential interventions?
5- What are the limitations or potential improvements of the models you envision that could be addressed in future research? Incorporate that into the conclusion.
6-The paper provides valuable insights into the modeling of epidemic dynamics; however, the practical application of these models could be enhanced by providing case studies or real-world examples where the SIRVD model has been effectively applied to manage an outbreak.
7-The distinction between recovered and deceased individuals is a critical aspect of the SIRVD model. An exploration into how varying fatality rates, influenced by factors such as healthcare capacity or demographic vulnerabilities, affect the model's predictions would be a valuable addition to the study.
Comments on the Quality of English LanguageThe paper is well written. It's important to review it thoroughly and pay attention to the punctuation spots.
Author Response
Referee: I have read this paper with great case! In this paper, the authors studied The susceptible-infected-recovered-vaccinated-deceased (SIRVD). The Perov fixed point theorem is adopted to study the established model. Moreover, the authors also apply a recently developed analytical approximation for the SIR and SIRV models to the more general SIRVD model. This approximation offers accurate analytical expressions for epidemic quantities, such as the rate of new infections and the fraction of infected persons, particularly when the cumulative fraction of infections is small. The distinction between recovered and deceased individuals in the SIRVD model affects the calculation of the death rate, which is proportional to the infected fraction in the SIRVD/SIRD cases but often proportional to the rate of new infections in many SIR models using a posteriori approach.. The results are correct and fascinating. The present work is suitable for publication after following revision:
Response: We thank this referee for a very careful reading, positive assessment, and useful suggestions. We have mentioned the eventual connection with Perov's theorem in the revised conclusions.
Referee: 1-Why not divide SI by N in the equations (a1) and (a2)?
Response: We have chosen to write down the SIRVD equations for population fractions, cf. Eq. (2), as well as the SI equations in (A1) and (A2), i.e. the quantities $S,I,R,V,D$ are normalized to unity. Therefore the total number of the population, $N$, does not appear anywhere. Absolute amounts can be obtained upon multiplying all results (all fractions) by $N$.
Referee: 2-What are the benefits of your newly discovered exact solutions to the SIRVD and SIRD equations in enhancing our understanding of the spread and control of epidemics compared to previous models?
Response: The original SIR model ignores vaccination and is not able to predict the deceased fraction as recovered and deceased fractions are treated together. Using the SIR or SIRV model, the deceased fraction can only be estimated via an a-posteriori approach. We have demonstrated here how the a-posteriori estimate differs from the prediction based on the SIRD or SIRVD models. The SIRVD model moreover considers the effect of vaccination. The qualitative changes in the dynamical evolution of the compartment sizes is highlighted in Figure 1, while the detailed comparison constitutes the main part of our manuscript.
Referee: 3-Can you elaborate on the methods used to derive the accurate analytical approximations for epidemic quantities, such as the rate of new infections and cumulative fraction of infections?
Response: The accurate analytical approximation in two simpler forms, i.e. for the SIR- and SIRV-models, is detailed in our previous article: Kröger, M.; Schlickeiser, R. On the analytical solution of the SIRV-model for the temporal evolution of epidemics for general 895
time-dependent recovery, infection and vaccination rates. Mathematics 2024, 12, 326. https://doi.org/10.3390/math12020326. We refer the reader in the text to this article.
Referee: 4-In what ways does the differentiation between recovered and deceased persons in the SIRVD model impact predictions and potential interventions?
Response: Both the recovered and deceased population does not affect directly the susceptible, infected, or vaccinated fraction. Insofar is the differentiation between these two fraction only of relevance, if the society regards recovered and deceased population as qualitatively different, which is the case. Moreover are recovered and deceased population fractions often recorded separately by health agencies. This allows to determine the rates affecting the recovered and deceased fraction individually. The two rates exhibit very different behavior depending on the type and strength of clinical, social, or political interventions chosen, insofar does it seem appropriate to not merge them into a single rate.
Referee: 5- What are the limitations or potential improvements of the models you envision that could be addressed in future research? Incorporate that into the conclusion.
Response: Potential future improvements include (1) incorporating in the SIRVD model spatially heterogeneous situations by adding spatial diffusion, (2) a detailed testing of the predictions with suitable data from past Covid-19 waves, and (3) derivations of accurate mathematical approximation for more complicated time variations of the ratios $k(\tau )$, $q(\tau )$ and $b(\tau )$. These three issues are now mentioned in the conclusion section.
Referee: 6-The paper provides valuable insights into the modeling of epidemic dynamics; however, the practical application of these models could be enhanced by providing case studies or real-world examples where the SIRVD model has been effectively applied to manage an outbreak.
Response: We agree and therefore have added the new subsection 6.2 where we compare our predictions from Sect. 6.1 in the stationary case with currently available data on past Covid-19 waves. As shown this leads to a new powerful diagnostics method to extract analytically all SIRVD model parameters from measured COVID-19 data of a completed pandemic wave.
Referee: 7-The distinction between recovered and deceased individuals is a critical aspect of the SIRVD model. An exploration into how varying fatality rates, influenced by factors such as healthcare capacity or demographic vulnerabilities, affect the model's predictions would be a valuable addition to the study.
Response: We fully agree. As mentioned in our response to your earlier point 5 time varying fatality rates will be addressed in our future projects. However, our current analysis of the gradually decreasing fatality rate in Sect. 6.3 has shown that such an extension is non-trivial.
Reviewer 3 Report
Comments and Suggestions for AuthorsThis article about SIRVD epidemic compartment model has a theoretical character. The results show analytically” that the temporal dependence of the infected fraction and the 20 rate of new infections differs when considering the effects of vaccinations and when the real-time 21 dependence of fatality and recovery rates diverge. These differences are highlighted for stationary 22 ratios and gradually decreasing fatality rates”. This is an important result in the article.
Expectations when reading this article is that there is a clear connection between results and the practical significance of the results. This is missing in the article.
More comments regarding analyses of the results. For example, all figures (2-7) should have a description of what each graph illustrates.
When reading an article, one drowns in formulas and mathematical conclusions that ought to be discussed more in connection to the main theme of the article that has an applied context.
As a consequence of the shortcomings mentioned above, it difficult to keep focus when reading the article. My assessment is that in this shape, the article does not attract the reader.
The article has uncleared the relationship between scientific - analytical rigor and “mathematics of epidemics” logic concerning to applied meaning of “effects of vaccination”. This makes it difficult to understand meaning of results.
Comments on the Quality of English LanguageGood
Author Response
Referee: This article about SIRVD epidemic compartment model has a theoretical character. The results show analytically” that the temporal dependence of the infected fraction and the rate of new infections differs when considering the effects of vaccinations and when the real-time dependence of fatality and recovery rates diverge. These differences are highlighted for stationary ratios and gradually decreasing fatality rates”. This is an important result in the article.
Response: We thank this referee for a positive assessment and useful suggestions.
Referee: Expectations when reading this article is that there is a clear connection between results and the practical significance of the results. This is missing in the article.
Response: We are bit nonplussed about this statement. We think that our derived results indeed have a practical significance as they allow one to forecast the temporal evolution of pandemic outbursts as well as analyzing monitored data on past COVID-19 waves. We have illustrated this latter point in the newly added subsection 6.2.
Referee: More comments regarding analyses of the results. For example, all figures (2-7) should have a description of what each graph illustrates.
Response: We have improved and extended the captions of several figures.
\referee{When reading an article, one drowns in formulas and mathematical conclusions that ought to be discussed more in connection to the main theme of the article that has an applied context.
Response: Unfortunately a certain number of formulas is unavoidable to ensure mathematical rigorness. As the title says, and in agreement with journal's name, we focus on the mathematical aspects of the SIRVD models. All results derived here can be immediately applied to real data and used to forecast the dynamical evolution of the compartments in an early stage. Depending on the three rates, that can be estimated from day to day using existing data collected during a pandemic, we have for example calculated the peak time and peak heights of the infected and deceased fractions. Accurately estimating such times and amplitudes are of uppermost relevance for handling a pandemic. We have furthermore investigated the effect of changing rates in the course of time. This allows a forecast the strength of interventions on the final numbers, such as the final number of fatalities, or the eventual reduction of required clinical equipment during peak time.
Referee: As a consequence of the shortcomings mentioned above, it difficult to keep focus when reading the article. My assessment is that in this shape, the article does not attract the reader.
Response: Our manuscript deals with solving the complex nonlinear system of partial differential equations describing the temporal evolution of epidemics. We are convinced that we have provided important and original new results. We hope that this will attract the mathematically inclined readers and scientists. Moreover, we hope that the newly-added subsection 6.2, comparing our predictions for the stationary case with currently available monitored data on past Covid-19 waves, will also motivate a broader readership to study our article.
Referee: The article has uncleared the relationship between scientific - analytical rigor and “mathematics of epidemics” logic concerning to applied meaning of “effects of vaccination”. This makes it difficult to understand meaning of results.}
Response: For us analytical rigorship in deriving exact solutions as well as analytical approximations is a must, as is proven by the detailed analysis in this manuscript. We have described how the effects of vaccinations enter the analysis in a rigorous way when we introduced the additional fraction of vaccinated persons in Eqs. (1a)-(1e).
