An Engineering Model of the COVID-19 Trajectory to Predict Success of Isolation Initiatives

Authors’ Response to Dr Ananth Govind Rajan
In this article, King and Striolo have developed a simple engineering model to understand the trajectory of COVID-19 cases. The model is based on three population-related variables and two rate constants. The model also takes into account the effect of intervention strategies applied by the government by scaling down the growth rate of cases. The authors applied this simple model to WHO data from three countries around the world, namely China, Singapore, and South Korea. The authors have also applied the model to the COVID trajectory in the U.K. during the revisions. Although the model is simple and cannot be used to inform complex and far-reaching public health decisions, as the authors rightly caution, the model results in some notable conclusions regarding the pandemic and lockdown strategies to contain it. Over the course of the previous two rounds of review, the authors have improved the manuscript based on the suggestions made. Nevertheless, the manuscript may be improved further by implementing the following revisions, prior to publication:
Authors’ response:
We would like to thank Dr Rajan for their assessment of our work and the useful suggestions. As we discuss below, we have considered all suggestions, and made relevant edits and improvements to the manuscript.

1.    On page 2, the authors introduce SIR-type models and provide a brief overview of such models. The authors should explain the abbreviation SIR and also how such models work. The authors should also briefly describe any alternative or superior strategies for modelling the spread of infectious diseases, which may already have been explored or could be explored in the future.
Authors’ response:
We have now defined the meaning of the SIR abbreviation in the revised manuscript. We have also summarised, briefly, how such models work. As the Reviewer pointed out, the main advantage of these models is their simplicity.
We have already provided references to more complex models, which have been used in the literature. The general trend in the literature with other modelling efforts of the Covid 19 infection trajectory has been to include other variables such as geographical factors, connections between different k factors and demographic indices such as age, population density, mortality rate, etc. Although such models likely represent well the reality, they also require accurate data for parameters fitting. We believe that adding much additional content along these lines is beyond the scope of our paper. One additional perspective, which is considered a merit of the model presented here, is that good fitting of the infection versus time curve have been achieved with minimal parameters.
Nevertheless, we have added a few references to very recent work on this topic. For example, Ref. 15, just published in PNAS, uses analysis of the viral trajectory to identify the containment strategies most likely to be effective.

2.    On page 4, the authors mention that “Fitting is not shown here because abundant analysis is reported on the news as well as on specialist literature.” However, the authors should report the values of k1 and k2 used in their model and how these values were obtained/chosen, along with their 95% confidence intervals. This should be done for all the cases considered in the manuscript.
Authors’ response:
We have added the values for parameters k1 and k2 in notes regarding Figure 1 of the revised manuscript. We respectfully point out that in our analysis the choice of initial constants k1 and k2 to demonstrate the behaviour of the infection with time are arbitrary and thus a 95% CI is not relevant for our purposes. As for the graphs which have used real data, we have used the published values. No 95% CI figures were presented in any case. Some idea of a confidence interval could be estimated from the noise in the advised health figures, but in view of the acknowledged accuracy of data depending as it did on only being able to define confirmed cases in terms of test results, we would such a CI merely speculative.

3.    In Figure 1, the authors mention a R2 value of 1.0 for the exponential fit during the initial stage of the pandemic; however, they do not indicate till what time point the data was included in the fit.
Authors’ response:
Thank you for pointing this out. In our analysis, we find that at Y=2% t= 0.4157 ; the ratio of model to simple exponential growth = 1.0009 (i.e., less than 1/1000 difference in relative terms). Relevant specification commentary has been added to the narrative.

4.    Although the authors provide R2 values for their fits, they do not provide 95% confidence intervals for any of their parameters and error bars for their predictions. These should be added to the manuscript, to quantify the robustness of the model and the predictions.
Authors’ response:
As detailed in the response to Q2 above, R2 where provided are presented to support discussion on the features of the model and not real data. The use of as published case numbers did not enable confidence intervals to be defined.

5.    The authors should clearly mention and indicate that in Figure 1 and all other figures, the vertical axis refers to the percentage of cases, rather than the absolute number of cases. Further, what is the unit of time in the plots shown in the manuscript? This should be mentioned in the manuscript. In Figure 1, the maximum time considered is less than 10, whereas in Figure 2, the maximum time considered is 500. Could the authors explain this choice?
Authors’ response:
We have now amended the labelling of the vertical axes in the figures – thanks for pointing this out. Figures 1 and 2 use adimensional horizontal axis, as these graphs explore the infection trajectory and the effect of intervention – the correspondent graphs and conclusions drawn are independent of the time scale. In Figures 1 and 2, the time units are related to the model parameters used to generate these graphs. All graphs in the manuscript in which specific situations and datasets were discussed have the time units identified.

6.    In Figure 3, it appears that the top two panels plot the absolute number of cases; however, the bottom plot seems to depict the cases as a fraction of the population. The authors must clarify this point.
Authors’ response:
We would like to thank the Reviewer for pointing out this inconsistency, which has been corrected in the revised version of the manuscript.

7.    On page 11, the authors mention that the “fit of the South Korean data is more promising…, as shown by the data in Table 2”. The authors should explain how the ratios A/B and C/D that they have defined can be used to infer the quality of the fit. The authors should also provide concise physical interpretations of these ratios.
Authors’ response:
We believe this question has been addressed on pg 9 of the revised manuscript; the ratios A/B and C/D make the analysis adimensional and enable different infection outbreaks to be compared. No amendment has been made to the manuscript in response to this comment.

8.    Regarding the proposed expression for dX/dt on page 3, could other types of functional expressions (e.g., power laws) in terms of X  and Y  work better in modeling the spread of disease?
Authors’ response:
Because the SIR model considers that the rate change in the numbers of infective cases will be proportional to the frequency of contact, 1st order kinetics are a valid choice. The use of this model is supported by its successful use in modelling other infections (as detailed in the introduction and related references). Of course, other models, including higher order functional forms of the SIR model, could also reproduce the trajectory of Covid-19, as we have already discussed in addressing Q1 above.

9.    On page 4, the authors mention that N=X+Y+Po , but in so doing they have not considered the individuals labeled as REC . Why shouldn’t N = X + Y + REC ?
Authors’ response:
We would like to thank the Reviewer for pointing out this inconsistency, which has been corrected in the revised version of the manuscript.

10.    On pages 4-5, the authors should fix the spellings ‘Hetohcote’ to ‘Hethcote’, ‘mode’ to ‘model’, and ‘out’ to ‘our’.
Authors’ response:
We would like to thank the Reviewer for pointing out this inconsistency, which has been corrected in the revised version of the manuscript.

11.    On page 6, the authors include the effects of government intervention in the simplified model where X, i.e., the number of uninfected individuals, is assumed to be equal to Po. This seems to be a major simplification, which may not be true as the pandemic progresses. What would be the effect of not making this simplification on the model predictions?
Authors’ response:
The model is appropriately considered as being initiated at a time when a very small and the numbers of uninfected persons in essentially P0. The model determines all the classes considered X; uninfected, Y; infected and REC; recovered or deceased. The uninfected persons will not be equal to P0 as the pandemic progresses. No change has been made to the manuscript in response to this comment, as we believe this is explained in the paper.

12.    In Figure 3, the authors refer to the ‘left’ panel, when they should in fact refer to the ‘top’ panel.
Authors’ response:
We would like to thank the Reviewer for pointing out this inconsistency, which has been corrected in the revised version of the manuscript.

13.    On page 11, it would be interesting to see the effect on the model upon addition of another term to Kstep-down that models a reduction in the intervention strength or a reduction in compliance to the intervention measures, which set in after a certain amount of time.
Authors’ response:
We believe that the scenarios modelled in the manuscript have addressed this question in acknowledging that the relaxation in the extent of lockdown above a critical value will prompt a return to exponential growth. No change has been made to the manuscript in response to this question.

Authors’ Response to Dr Kauko Leiviska
This paper considers a topical and important subject, modelling the spreading of Covid-19, and especially using this model in predicting the efficiency of isolation/lockdown activities to stop the virus or at least to control its spreading, and protect the most vulnerable part of the population. These activities are based on the fact, that by decreasing the contacts between the individuals, we can decrease the spreading of the virus. Depending on the extent of these activities, their impacts on the society vary, but they inevitably have negative economic and societal effects. The legal aspects must also be kept in mind; the lockdown activities limit people’s rights of moving and communicating freely. This is clearly visible in the governmental actions in some countries. Now it seems that there are problems in getting vaccines to a wider use, and the virus itself shows no weakening, on the contrary, mutations that are more dangerous are developing, so we will be living in this world of temporary isolation activities for some time.
This paper concludes that a series of strong but short- to - medium term activities are the best way to fight the virus. Timing of these activities is of primary importance: the time to start the measures, and may be even more important, when to lift them. The pressure to officials in both cases is remarkable, and this kind of model could produce more information to support the decision-making. The authors mention in Abstract that the summer 2020 is a good example of the power of isolation activities. The spread of the virus dampened considerably in the countries that took the maximal possible limitations into use. How much of this positive development is due to isolation activities and how much to the “natural” behaviour of the virus is unknown. Similarly, is the negative development towards the autumn 2020 due to too early lifting of the isolation or the virus itself? When more data will be available, the further research will shed more light on this.
As a whole, this paper is a good example of using simple tools to solve complicated problems. As the authors say, it does not solve the whole problem, but to my mind, it could be one tool for decision-makers when planning new measures to fight the virus. The decision on using isolation measures is, of course, political and several aspects should be taken into account, but if the science can provide some impartial facts showing the consequences of the decisions, we can expect better results.
The model:
The model is a simple one consisting of two first order nonlinear differential equations with two variables (the number of uninfected individuals and the number of infected individuals). The model has two parameters (the infection growth rate constant and the constant for recovery/mortality), and it assumes the constant initial value for the population. The infection growth constant was modified so that it could take into account the isolation/lockdown activities. The model includes also an algebraic equation that calculates the number of individuals that are unable to pass on the virus because of the recovery or death of the individual in question. The model is a SIR (Susceptible, Infectious, or Recovered)-type compartmental model used for a long time in describing the propagation of infections in populations. The authors have extended the model so that the isolation/lockdown measures can be simulated.
This is a very simple model and its advantages are the low number of parameters to be tuned and the ease of understanding its operation in spite of its nonlinearity. This kind of model has some weaknesses that must be remembered when applying it. It divides the population in three groups, and cannot handle e.g. age groups or groups with some basic illness separately. As a compartmental model, it assumes both compartments perfectly mixed. This problem could be solved by adding more compartments in the model, but with the cost of more differential equations and parameters. This, of course, would need more detailed data, too. The model is deterministic and many phenomena concerning with the propagation of the disease are stochastic. The model also describes the recovery/mortality with one constant, only. Releasing this assumption means that we have to divide the third compartment in two (SIRD model). In the proposed model, the recovery provides the individual with the immunity. We do not have information enough to know, if this immunity is only temporary. These are the questions needing further research.
The model has been tested using the data from China, South Korea, and Singapore; all highly and densely populated regions. This raises the question of the requirements for the region to be modelled. Is there any minimum number of the initial population Po, or the population density? How does the model work in Europe at the country level, or should we group the countries according to some criteria (economic area, social behaviour, Covid-19 strategy), and model larger regions? The results for UK seem encouraging as for the model performance, but we are still dealing with a densely populated region. The lockdown has in some cases focused on the traffic between countries, and in some cases, it means a severe discontinuity in the working traffic during both the start and stop of the lockdown. A good example is the situation in Finnish and Swedish Lapland, especially in today’s situation when the strategies for fighting the virus are so different in each country. The situation is very much the same in countries that live on tourism. This fits also with the Finnish Lapland where the first case, exactly one year ago, was a tourist. Is there any need to take into account these kind of “exogenous” infection sources?
Another question that the authors might comment with few words is changes in the virus. How does it affect the two constants, if the virus is mutating to the direction of higher transferability (as it happens now) or of higher mortality? Can the model cope with it? What, if we had a mixture of mutated viruses with different properties?
Results:
The model was validated with the WHO data for China, South Korea and Singapore. These countries were chosen because of three reasons: long-time data was available because of the early encounter with the disease (the first study was carried out already in April 2020), high cohort numbers, and cultural similarity. The results together with testing the model with more recent UK data show that the model with a proper parameterisation gives a good fit with the recorded data. There is one exception in the China case, but it seems to be due to problems in data acquisition. Delays in feeding the test results e.g. during the weekend seem to be quite common, and it puts some stress on data pre-processing.
Conclusions:
Section 6 summarises well the results and findings of the study. Adding few lines on the limitations of the model and further research/development directions will answer my earlier comments and questions.

Some corrections:
There are two Sections numbered as Section 4. The easiest way to correct this is to put these Sections together. The authors should also check the caption of Figure 3. Just before Section 6, there is a reference to Figure 2B. Figure 2 has two panels, but they are not numbered, so the reference should be “the right panel of Figure 2”. On the first line of Section 6 there is one “time” too much. There are also some minor typos in the text.

Authors’ response:
We would like to thank Dr Leiviska for their positive assessment of our work. The summary above reflects our motivation in conducting this work and reporting its results. As the Reviewer points out, only future studies will allow us to identify how much the “natural” behaviour of the virus contributes to the success of isolation initiatives.
All the questions raised above are sensible. We have added a few comments regarding the limitations of the model to the conclusions (the last paragraph was added during this second revision), we have added a few recent references, and we have made editorial improvements throughout following the recommendations kindly provided.

Much media and societal attention is today focused on how to best control the spread of Covid-19. Every day brings us new data, and policymakers are implementing different strategies in different countries to manage the impact of Covid-19. To respond to the first ‘wave’ of infection, several countries, including the UK, opted for isolation/lockdown initiatives, with different degrees of rigour. Data showed that these initiatives have yield the expected results in terms of containing the rapid trajectory of the virus. When this manuscript was first prepared (April 2020), the affected societies were wondering when the isolation/lockdown initiatives should be lifted. While detailed epidemiologic, economic as well as social studies would be required to answer this question completely, we employ here a simple engineering model. Albeit simple, the model is capable of reproducing the main features of the data reported in the literature concerning the Covid-19 trajectory in different countries, including the increase in cases in countries following the initially successful isolation/lockdown initiatives. Keeping in mind the simplicity of the model, we attempt to draw some conclusions, which seem to suggest that a decrease in the number of infected individuals after the initiation of isolation/lockdown initiatives does not necessarily guarantee that the virus trajectory is under control. Within the limit of this model, it would appear that rigid isolation/lockdown initiatives for the medium term would lead to achieving the desired control over the spread of the virus. This observation seems consistent with the 2020 summer months, during which the Covid-19 trajectory seemed to be almost under control across most European countries. Consistent with the results from our simple model, winter 2020 data show that the virus trajectory was again on the rise. Because the optimal solution will achieve control over the spread of the virus while minimising negative societal impacts due to isolation/lockdown, which include but are not limited to economic and mental health aspects, the engineering model presented here is not sufficient to provide the desired answer. However, the model seems to suggest that to keep the Covid-19 trajectory under control, a series of short-to-medium term isolation measures should be put in place until one or more of the following scenarios is achieved: a cure has been developed and has become accessible to the population at large; a vaccine has been developed, tested, and distributed to large portions of the population; a sufficiently large portion of the population has developed resistance to the Covid-19 virus; or the virus itself has become less aggressive. It is somewhat remarkable that an engineering model, despite all its approximations, provides suggestions consistent with advanced epidemiologic models developed by several experts in the field. The model proposed here is however not expected to be able to capture the emergence of variants of the virus, which seem to be responsible for significant outbreaks, notably in India, in the spring 2021, it cannot describe the effectiveness of vaccine strategies, as it does not differentiate among different age groups within the population, nor it allows us to consider the duration of the immunity achieved after infection or vaccination.