bdocs rambling

Introduction

There has been some debate about the timing and effectiveness of the lockdown measures adopted by the UK in its response to the Coronavirus pandemic. I’m sceptical about the sense of trying to rewrite the past, but I was intrigued as to what my model’s findings might have been. But, we are where we are, and most energy should be focused on the here and now, and the future. But, no doubt, there are lessons to be learned eventually.

NB Government health warning – this is pure speculative modelling, and it is more about the sensitivities in the setting of modelling parameters, and not a statement of fact! Don’t quote me!

Background

Some commentators and academics (Professor Rowland Kao at Edinburgh University tweeted this BBC coverage of his modelling work for Scotland) feel that the UK might well have applied the lockdown measure 2 weeks earlier, on the 9th March, as Italy did, instead of on the 23rd March. UK Government states that our first cases were later than for other countries, and so too, therefore, was our lockdown. Time will tell how significant this is.

I was interested to see what my model (the Prof Alex de Visscher code, with my parameters, modifications and published data for the UK situation) might have made of this for the UK, and I have run the model for some options.

A major problem with running any model from 9th March for the UK is that the first UK deaths, specifically attributed to Covid-19, on the original basis, prior to the inclusion of Care Home figures, were on the 10th March, although, of course, there had been many more cases of infection reported, with people being admitted to hospital and ICU departments in the hundreds. I’m sure that, as discussed above, there will have been deaths associated with the pandemic, but not officially recorded as such.

My current model forecast

It’s important, first of all, to state my baseline, which is my model’s current forecasts for the pandemic in the UK on the basis of the March 23rd lockdown, which embraced working from home, social distancing, school, restaurant and pub closures and several other measures. UK lockdown did allow for excursions for exercise once a day (not allowed at that time in Spain and Italy, for example), and travel for essential medical supplies and food (possibly to help others with those aspects).

In that baseline work, current as of May 14th, these are the graphical representations of the outlook, which forecasts a little over 42,000 deaths by the late summer 2020, stabilising at about that number into the future, assuming no changes to the 84.1% lockdown intervention effectiveness, and no pharmaceutical intervention yet (either vaccine or radically more effective treatment of symptoms). The orange curve always represents the reported numbers.

My model projection is broadly in agreement with the following projection from the IHME, the Institute for Health Metrics and Evaluation, an independent global health research center at the University of Washington, which forecasts between 43,000 and 44,000 deaths in the UK by the summer.

The number of cases, below, is predicted at 2.8 million in the UK, assuming that only 8% of those with the virus are actually tested and diagnosed, a percentage that is informed by the ratio of deaths to real cases derived from other countries’ experience. The fit of the model, therefore, isn’t as good for cases, as the data is fuzzy. The 12.5 multiple on reported numbers mentioned on the chart takes account of that 8%.

It is likely, therefore, that we are not yet aware of anything like the real number of cases, and even death data is now felt to have been understated, currently attributed by confirmed test diagnoses only, not by a simple mention of Covid-19 on death certificates. These latter figures are now being analysed by Government. My model will be updated (as it has been recently to include Care Homes) when those figures are adjusted, as they surely will be when the data is cleansed. The UK Office for National Statistics are working on this, and have already published some numbers (but with some lag, and so are not up to date).

The UK Government does now report for Care Homes in the “all settings” figures it reports, as of nearly two weeks ago, and it retroactively adjusted all historic reported numbers to take those into account, which my model does too; but the death certificate reference to Covid-19 is still not a factor in assigning deaths to Covid-19 in the official published UK daily data.

It is fair to say that policies on this vary hugely from country to country, and from region to region within countries, so it isn’t easy to see where to draw the line.

Options for parametric sensitivities for the March 9th lockdown scenario

It is hard to be sure what the effectiveness of lockdown measures might have been in early March. Some other countries, such as China, were seen to enforce very stringent measures at (and before) that time (having encountered the Coronavirus earlier) and in his own work, Alex de Visscher has used 90% effectiveness for China. This figure relates to the extent of the reduction in infection rate in the model, 90% effectiveness reducing the rate to 10% of its original value. See my blog article for a description of the model and its variables.

At the other end of the scale, the USA model was set at 71% initially, and Canada at 75%; Italy at 79.1% and Spain at 85% were some other choices. Lots of sensitivities were run around these settings, but this gives an idea of the possible range.

The best fit of my current model, as above, for the March 23rd lockdown is currently for an effectiveness of 84.1%, and it is continuing to match published data at that setting, and therefore remains the basis for my current March 23rd lockdown forecasting.

Without any changes to the underlying infection rates, I ran my 9th March UK lockdown model for three values of interv_success, the model variable for lockdown (intervention) effectiveness; for the current 84.1%, and then for 80% and 75%.

I should say that in the University of Edinburgh work mentioned above, which was for Scotland, information on commuting movements was input to their models, using Google mobility data, to try to get an understanding of how social restrictions affect spread of the virus. They found that “Not moving so far away from one’s home is one of the big impacts of what lockdown is doing”.

I haven’t been able to do such an activity analysis, but the % effectiveness in my model is a reflection of assumptions about that. My feeling is that in the UK, we were, and are, unlikely to be able to enforce lockdown as strongly as China did. In the UK, for the last 6 or 7 weeks, we have, as I mentioned, had more freedoms than, for example, Italy and Spain to go outside for exercise (running or cycling, say).

Our recent relatively high public compliance rate with UK Government directives for lockdown measures (working from home where possible, social distancing etc) might not have been as good earlier on, when the impact of the pandemic here in the UK wasn’t confronting people so directly, or to such an extent, as two weeks later, on the 23rd March, by when 359 people had died, and published case numbers were 6650. These numbers compare with no deaths as of 9th March (on the original basis, without inclusion of Care Homes), and just 321 cases.

I should again emphasise that the reported case numbers might well be understated by a factor of 10 to 15 (12.5 is the figure Alex has worked with), judging by the experience in other countries further down the pandemic track.

Taking all this into account, my other options for interv_success are for lower % impact on the infection rate than works best currently, not higher – the current 84.1%, and then for 80% and 75% as mentioned above.

Results

I present the outcomes in graphical form. The first sets of graphs, for lockdown starting on 9th March, are for these three options for intervention effectiveness.

84.1 % effectiveness (as per my current, well fitted model)

75% effectiveness

Finally, on the larger chart, presenting the situation up to May 13th, instead of the current 33186 deaths, the model would forecast 1123 deaths by 13th May if lockdown had happened on March 9th, assuming 75% intervention effectiveness. The long term deaths would be nearly 6000, but with still a few deaths per day even a year later, because of the continuing lower intervention effectiveness (assumed by the model), and cases would have reached a little under 400,000 in a year’s time, with negligible growth by then.

Discussion

These results show a marked reduction in deaths forecast for May 13th by my model for the March 9th scenario lockdown, compared with the actual March 23rd lockdown results. This is only a model and will have deficiencies in a) the reduced data available for calibration of the model before March 9th (no deaths, for example) for model fitting, and b) the lack, therefore, of a firm basis for setting transmission rates at that time, and testing them.

But it does show the high dependency the later results have on early data.

There is another feature in the numbers, however, that is more noticeable in the charts for the assumed 75% lockdown effectiveness. This is is clearer when we look more closely at the long term outcome for deaths, starting with that 75% case, as above,

where we can see that even in April 2021 (in the absence of pharmaceutical measures, or any change to the intervention effectiveness) the deaths are still increasing (if there were no further increase, the log chart would be flat, i.e. horizontal, at that point). The 5738 deaths at that time are not the maximum.

This outcome is even more stark if I choose 70% intervention effectiveness, and in this case I present both the linear y-axis and the log y-axis charts, since on the linear chart, the numbers are significant enough to be perceived visually. The orange curve is always the actual reported numbers, whatever the model scenario.

70% effectiveness analysis

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The linear chart makes clear that by the end of April 2021, there continues to be a high rate of increase of deaths, and, equivalently, we can also see that the log chart is much further from flattening than in the 75% case, even after more than a year of lockdown, at this level (70%) of effectiveness. The modelled number of deaths by April 2021 is over 250,000, with 17 million cases.

Visually, the reported deaths look very unlikely to get anywhere near that, and in my base March 23rd lockdown model (which has a good fit with reported numbers at the moment, May 13th, the orange curve here), deaths are projected to flatten at 42,000, with cases at 2.8 million.

I totally accept that the model will have its deficiencies – the calibration phase to March 9th is short, and the infection rate parameter might not be realistic as a result.

But from a comparative model behaviour perspective, this is another example, in an exponential growth situation (as a pandemic is) that early numbers are not a good indicator of outcomes. Small early differences make very large differences down the track. It’s a non-linear relationship, with high sensitivity of the pandemic growth rate to the % effectiveness of intervention measures (most noticeably at the lower % effectiveness).

It does seem that if the effectiveness of interventions is not high enough, while there can appear to be good early results (the number of deaths by May 13th 2020, in even the 70%, least effective model, is 1,980, as compared with 540 for the 84.1% case above (starting March 9th)) the pandemic eventually overcomes the measures.

By the end of 2020, in this 70% scenario, the modelled deaths are likely already to exceed the reported deaths, and would be growing quite fast by then, as can be seen from the charts.

This tells me that if pandemic intervention measures are to work effectively, and strategically (i.e. long term), they need not only to be early, but also at least at 80% effectiveness in terms of reducing the infection transmission rate for this Coronavirus.

Long term 80% effectiveness outlook

We can confirm this from the 80% effectiveness longer term outlook for the March 9th lockdown scenario (assuming no changes in the lockdown measures, and no available pharmaceutical measures).

At this level of intervention effectiveness (80%), the modelled deaths curve peaks at about 1000, reaching that point by autumn 2020 as stated in the earlier section, with about 67,000 cases by that time, stabilising at 68,000 before a year’s time, as we see in the chart below. Again, the orange line is the current status of the reported cases (x12.5 as before).

Summary

This article simply addresses the theoretical, earlier lockdown scenario that has been much discussed. The modelling I have done is probably not adequate in terms of absolute numbers, but it is clear, from comparisons of my scenarios, that on similar assumptions to the current modelling for the March 23rd lockdown initiation, the number of deaths at May 13th would have been less – probably far less, as asserted by the University of Edinburgh study for Scotland.

Whether those assumptions are valid (i.e. that the intervention effectiveness would have been the same) is questionable.

Furthermore, it requires just a 10% to 15% diminution of that effectiveness percentage to lead to a worse outcome in the long term.

I see this as an indicator that, in the absence of pharmaceutical measures (a vaccine, ideally (of lasting effect), or medicines that handle symptoms effectively, and save lives) the intervention measures have to continue to be carefully monitored, including an understanding of which are the most effective at reducing infection transmission rates.

It may well be that as the researchers at Edinburgh stated, the “stay-at-home” policy is the most effective measure, and that trips away from home should continue to be minimised even into the long term, by working from home where possible, and otherwise travelling only for medical and food purchases.

In my research on appropriate (available and feasible (for me)) modelling tools for epidemics, I discovered this paper by Prof. Alex de Visscher (to be called Alex hereafter). He has been incredibly helpful.

https://arxiv.org/pdf/2003.08824.pdf

Thanks to Alex including the MatLab code (a maths package I have used before) in the paper, and also a detailed description of how the model works, it qualifies as both available and feasible. There is also a GNU (public and free) version of MatLab (called Octave)) which is mostly equivalent in terms of functionality for these purposes, which makes it practical to use the mathematical tools.

In my communications with Alex, he has previously helped me with working out the relationship between the R0 Reproductive Number (now being quoted every time on Government presentations) and the doubling period for the case number of disease outbreak. I published my work on this at:

https://docs.google.com/document/d/1b9lB4RszIHJipS27raXxN6ECTFbihJ_fQFhdy5oCEwA/edit?usp=sharing

since mathematical expression are too cumbersome here on WordPress.

I would still say the Government is being a little coy about these numbers, saying that R0 is below 1 (a crucial dividing point between epidemic increase at R0>1 and epidemic slow down and dying out for R0<1). They now say (April 14th) it’s less than one outside hospitals and care homes, but not inside these places, as if (as Alex says) Covid-19 is not lethal, except for those people who die from it.

At least the Government have now stopped using graphs with the misleading appearance of y-axis log charts.

These following charts (unlike the original Government ones) make it pretty clear what kind of chart is being shown (log or linear) and also illustrate how different an impression might be given to a member of the public looking on a TV screen a) if the labelling isn’t there or obscurely referenced, and b) how the log charts to some extent mask the perception of steep growth in the numbers up until late March.

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This chart from 14th April show the Government doing a little better over recent days in presenting a chart that visually represents the situation more clearly. Maybe the lower growth makes them more comfortable showing the data on linear axes.

Alex de Visscher model description

I’ll try to summarixe Alex’s approach in the model I have begun to use to work with the UK numbers (Alex had focused on other countries (US and Canada nearer to home, plus China, Italy, Spain, Iran with more developed outbreak numbers since the epidemic took hold).

I have previously described the SIR (Susceptible, Infected and Recovered) 3-compartment model on this blog at https://www.briansutton.uk/?p=1296, but in this new model we have as many as 7 compartments: Uninfected (U), Infected (I), Sick (S), Seriously Sick (SS), Deceased (D), Better but not recovered (B) and Recovered (R). They are represented by this diagram:

Firstly, an overall infection rate r₁ above can be calculated as a weighted sum of these four sub-rates;

r₁ = (k₁₁I+ k₁₂S+ k₁₃SS+k₁₄B)U/P

where the infection rates k₁₁, k₁₂, k₁₃, k₁₄, are set separately, towards I, S, SS and B respectively, from healthy people, with the rates proportional to the fraction of uninfected people, compared to the total population, U/P.

All other transitions between compartments are assumed as first order derivatives with their own rate constants, with time t measured in days:

dU/dt = -r1; the rate of movement from Uninfected to Infected; and similarly, with in-/out-puts added and subtracted for other compartments:

Transition rates between compartments	Differential equation
r1 = (k11I+ k12S+ k13SS+k14B)U/P	dU/dt = -r1
r2=k2I for I to S transitions	with dI/dt = r1-r2
r3=k3S for S to SS transitions	with dS/dt = r2-r3-r5
r4=k4SS for SS to D transitions	with dD/dt = r4
r5=k5S for S to B transitions	with dB/dt = r5+r6-r7
r6=k6SS for SS to B transitions	with dSS/dt = r3-r4-r6
r7=k7B for B to R transitions	with dR/dt = r7

Some simplifying assumptions are made: k₁₂ = k₁₁/2, k₁₃ = k₁₁/3 and k₁₄ = k₁₁/4, in other words scaling the infection rates to I from the other compartments S, SS and B, respectively, to the principal rate from U to I. (Subsequently k₁₄ = 0*k₁₁/4 was chosen to compensate for long contagious times once the SS median had been changed from to 10 days from 3.5 days. See below).

So we can recognise that the bones of the previous SIR model are here, but there is compartment refinement to allow an individual’s status to be more precisely defined, along with the related transitions.

The remaining parameters that can be set in the model are:

Parameter (units)	Value
k11 (day-1)	variable
k2 (day-1)	loge(2)/5.1
k3 (day-1)	k5/9
k4 (day-1)	k6.(15/85)
k5 (day-1)	loge(2)/3.5
k6 (day-1)	loge(2)/10
k7 (day-1)	loge(2)/10

k₁₁ was initially chosen by inspection of the observed doubling rate for the disease, and can be modified. k₂ is based on the 5.1 day incubation rate observed in the paper by Lauer et al at https://annals.org/aim/fullarticle/2762808/incubation-period-coronavirus-disease-2019-covid-19-from-publicly-reported

k₃/k₅ is based on the assumption that 10% of the infected (I) become seriously sick (SS), whereas 90% get better without developing serious symptoms; less than the 80:20 proportion observed normally, because many with mild symptoms go unreported.

The k₄/k₆ ratio is set assuming a 15% non-survival rate for hospitalised cases.

k₅ and k₇ were set in the original model assuming a median rate of 3.5 days in each of the S and B stages before moving to the Recovered R stage (ie a median 7 days to recover from mild symptoms). k₇ has now been set to a median of 10 days (implying a 13.5 day median for recovery from mild symptoms), and to compensate for the resulting long contagious times, k₁₄ = 0*k₁₁/4 has been chosen as stated above.

k₆ is based on an SS stage patient remaining in that stage for 10 days before transitioning to B. Because the other pathway is to the Deceased D stage, the median is actually less, 8.5 days. This would be consistent with a mean duration of 12.26 days for getting better (B), which has been observed clinically.

Implementation of the model and further parameter setting

I have been operating the model with some UK data using Octave, but Alex was using the equivalent (and original licensed software) MatLab. He used the Ordinary Differential Equation Solver module in that software, ODE45, which I have used before for solving the astronomical Restricted Three Body Problem, so I know it quite well.

Just as all of the various Governmental, Johns Hopkins University and Worldometer published graphs and data are presented with a non-zero case baseline to start the simulations, as the lead-in time is quite long, so does this model. These initial parameters for the UK model so far are chosen as:

Population P	67.8 million
Initial infected number I	100
Initial Sick number S	10
Initial Seriously Sick number SS	1
Deaths D	0
Better B	0
Recovered R	0

Thus there are (in the model) 111 infected people at the start, among the UK population of 67.8 million. This is still quite an early start for the simulation, with less than 10 known infections; the doubling time is estimated from the total of people in any infected stages at days 29 and 30 using a similar approach to the one I explained in that previous paper at https://docs.google.com/document/d/1xg4XY2wZPZFM2O5E76Y6zBPc_42l7PMPE4NF6XipDPQ/edit?usp=sharing

Thus the doubling time I called T_D there is given by the equation:

T_D log_e2 = log_e (C₃₀ /C₂₉)

where C_d is the reported number of cases on a given day d. Known cases are assumed to be 5% of infected cases I, 1/3 of sick cases S, 90% of seriously sick SS, 12% of recovering R and 90% of deceased cases D (not all are known but we assume, for example, that reporting is more accurate for SS and D). The calculated doubling time in the model is not sensitive to the choice of these fractions.

The critical factor R₀, the Reproduction Number, can be calculated by a separate program module, for one infected patient, where the number of newly infected starting with that case can be calculated over time.

Solving the equations

The model, with the many interdependent parameter choices that can be made, solves the Differential Equations described earlier, using the initial conditions mentioned in Table 3. Alex’s paper describes runs of the model for different assumptions around interventions (and non-interventions), and he has presented some outcomes of National models (for example for China) on his YouTube channel at:

https://www.youtube.com/channel/UCSH5Bzb8Oa0BojCVMdXBW_g/

and Alex’s introduction to the model on that channel, for both China, and the world outside China which describes the operation of the model, and the post processing fitting to real data charts can be seen at

Alex’s early April update on Italy is at:

UK modelling

I’m at an early stage of working on a UK model, using Alex’s model and techniques, and the most recent UK data is presented at the UK Government site at:

https://www.gov.uk/government/publications/covid-19-track-coronavirus-cases

using the link to the total data that has been presented to date on that site. Unfortunately, each day I look, the presentation format changes (there used to be a spreadsheet which was the most convenient) and now the cumulative case count for the UK has increased suddenly by 3000 on top of today’s count. Death numbers are still in step. It isn’t hard to see why journalists might be frustrated with reporting by the UK Government.

I have, as a first cut, run the numbers through the model to get an initial feel as to outcomes. The charts produced are not yet annotated, but I can see the shape of the predictions, and I will just include one here before I go on to refine the inputs, the assumptions, the model parameters and the presentation.

In this chart, days are along the x-axis, and the y-axis in a log scale to base 10 (equal intervals are a multiple of 10 of the previous y-gridline) of numbers of individuals.

The red curve represents active cases, the purple curve the number of deaths D, and the yellow curve the number of seriously sick patients, SS. This chart is the simpler of the two that the model produces, but serves to illustrate what is going on.

I have not yet made any attempt to match the curves against actual UK outcomes for the first 50 days (say) as Alex has done for the other countries he has worked on.

The input choices for this version for the UK are:

interv_time = 52;    % intervention time: Day 51 = March 23 – 1 = March 22

    interv_success = 0.85; % fractional reduction of k₁₁ during intervention

    k₁₁ = 0.39;          % infection rate (per day)

    k₅ = log(2)/3.5;

    k₆ = log(2)/10;

    k₇ = log(2)/10;

    f_SS = 0.1;     % fraction seriously sick

    f_CR = 0.04;    % fraction critical

    f_mort = 0.015; % mortality rate

    correction = 0.0012;

    P = 67.8e6;       % total population

The correction number is set to adjust for the correct starting number of cases. On Alex’s advice I set k₁₁ = 0.39 which seems to provide a good initial match to death data.

In terms of numerical outputs, this is just a model, and hasn’t even been tuned, let alone fine tuned. The charts above reflect the output numbers in the model, which were:

doubling_time = -7511748644 – not relevant here for a declining epidemic;
final_deaths = 39,319 reached after a little over 100 days;
total_infections = 2,621,289 peaking at about 60 days

and I’m still making sense of these and working out some possible adjustments. Firstly I want to match with real data up to to date, so there is some spreadsheet and plotting work to do.

Now the model has been run, there can be a combinatorial number of changes that could be made to get the model to fit the first 50 days’ data.

The work has only just started, but I feel that I have a worthwhile tool to begin to parallel what the Government advisers might be doing to prepare their advice to Government. Watch this space!

In the news today (coincident with my current work on this post) we can see how important the measures taken through social distancing, self-isolation and (partial) lockdown are to reducing the rate of infections. So far, the SHTM researchers say “…the R0 could be cut…” and “likely lead to a substantial impact and decline…” – i.e. we are on track theoretically, but we are yet to see the full benefits. It is still my view, expressed in my last post, that in the UK, we are still at an R0 figure of 2.5-3.5, with a doubling period of 2-3 days (2 and 3 are VERY different figures in this exponential context, as we see from my earlier posts).

While it’s a small study, it’s interesting to note the intention to reduce the R0 Reproduction Number (that I talked about in my last post), equivalent to the average number of people a person infects in the “susceptible” part of the population (those not yet infected, or recovered). The SIR model I have been working on reflects this concept in generic terms, as described below.

The model is, in effect, a Markov Chain solution to the infection activity, where each state of the SIR (Susceptible-Infected-Recovered compartmentalised) population at time t+δt depends only on the previous state at time t, solved through Monte-Carlo sampling for the statistical behaviour (ie not analytically), using an iterative procedure called the Gillespie algorithm, described last time, to solve the differential equations of the SIR model.

Thanks to some great work by Tomasz Knapik (Sutton family friend) I can show show a couple of new charts relating to variations in the data for the Gillespie algorithm for solving the SIR equations I talked about in my previous post.

You’ll recall this chart from last time, where for a population of 350, with infection rate α, and recovery rate β set, the progress of this simple SIR model shows the peak in infections from 1 index case, with a corresponding reduction in susceptible individuals, and an increase in recoveries until the model terminates with all individuals recovered, and no susceptible individuals left.

I drew this with a log scale (base 10) for time, because in this model, with the default input data, not much happens for quite a while from time t=0, until the infections take off rapidly at t=0.1 (on the generic time scale) and reach a peak at t=0.3. The model is fully played out at t=10. The peak for infections, out of a population of 350, is 325, for this input data:

Key parameters for the basic SIR model are:

The total population, N; the rate of infection α; and the rate of cure β.

Base default data for the simulation is for 1 index infected case at t=0, and the “Spatial Parameter” V, set at 100 in the generic units used by the model. I’ll talk more about this parameter in a later post.

A larger V reduces the infection growth through “distancing” the infected cases from the susceptible compartment of the population, and a smaller V increases the pace of infection by reducing the distance in general terms.

The equations for the simple SIR model being analysed, for a limited total population N, where:

With linear scales for both the x-axis (time) and the y-axis (cases) we can see, in the following two charts, that a change in the spatial parameter (V) in the differential equations above does precisely what the Government is trying to achieve with its measures regarding self-isolation, social distancing and reduction in opportunities for people to meet other than in their household units.

Firstly, in the the modified Python Gillespie model (the Knapik, Sutton & Sutton model!) for the case illustrated above, but with linear axes:

Second, I change the spatial parameter, V, to 1000, for illustrative purposes:

We see above that the spatial parameter, representing increasing “social distancing” in this SIR application, does what is intended by the social distancing behaviour encouraged by most Governments – it pushes the peak of the infections from t=0.3 further out, towards t=2, and also flattens the peak, down to about 140 cases, from 325 before.

My task now is to relate the following key parameters of this situation:

First of all, the Reproduction Number R0 (the number of people one person might infect).

Secondly, the generation interval (how long it takes that infection to take effect (a vital number, and the subject of some difference in the literature between medical practitioners on the one hand, and demographic / biological practitioners on the other). An epidemiological formula, and a demographic/ecological/evolutionary biology formula can both be used to assess the generation interval, from R0, but can lead to very different results, to summarise a point made in paper on exactly this topic (ref. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1766383/).

Third, the doubling period – what is the right logarithmic multiplier in the Johns Hopkins University charts, given R0?

And lastly, the time taken, relatively, to see the R0 come down to a value near 1, when the epidemic stabilises and would tend to die out if R0<1 can be achieved.

No account is taken by my SIR model of recovery and re-infection; the SEIS model mentioned last time, or a variation of it, would be needed for that.

No-one is clear, furthermore, as to the seasonality of this Coronavirus, Covid-19, with regard to winter and summer, ambient humidity and temperatures, or the likelihood of mutation.