COVID-19 models including reopening and second wave

Reopening of many countries has begun and testing rates have also ramped up.  Some countries are now better able to detect infections and the age profile of infected people has in many cases shifted towards younger people.  Some countries have the outbreak under good apparent control while others have not yet seen even the first peak.  

We have updated our models for Ireland, the US and Italy to include the latest ECDC data.  On the basis of our modeling assumptions, and in the absence of further closures/ measures, Italy may not see a second wave for a considerable period.  Ireland may see this second wave sooner (and authorities will act to control it).  

A population (whole country) based model of any country is an approximation, as there are always localized effects, especially when applied to large countries like the US.  For example, cities in the North East had large first waves and are having relatively small second waves on reopening.  Cities in other regions had small first waves that never really fell away and are now seeing the effects of those waves continuing to build after somewhat earlier reopening.  Viewed as a whole, the US clearly has a second wave, but for the above reasons that terminology is not necessarily accurate locally.

Model fit to death rates in the US as a whole, up to 23 July 2020.  Model predictions indicate that a second peak in death rate has started and tightening of measures is required to limit its severity.

We have included in the latest models parameters to reflect increased detection by testing (as positive test rates are reducing in many nations) and also reduced mortality as the age profile shifts.  Those extra parameters add uncertainty but are necessary in order to continue to describe the evolving situation well.  On the basis of the model and its prior versions referenced in previous posts, parts of the US will need to continue to take all possible measures to limit the spread of COVID-19 in order to prevent a significantly larger death toll.

COVID-19 second wave risk: models for Ireland, Italy, US and Singapore

We've taken another look at ECDC data to date and updated the models for the US, Italy and Ireland with data to 13 June.  We've also added a model fitted to Singapore data.

To date, the same model structure has fitted every region and inferred rates of physical contact between those infected with the disease and the broader community have tallied qualitatively with mobility data from Apple and Google.  That continues to be true for Ireland, Italy and the United States.  

Now that many countries are easing restrictions, the risk of a second wave of infections is growing.  In the case of the US, model parameters are tending to suggest that a second wave could be nearer than in the other regions.  Every feasible way to reduce transmission should be considered and it is possible that relatively small measures such as widespread wearing of masks could make the difference between a manageable and unmanageable second wave.

We've added some factors to the scenarios tabs of the models that define how restrictions are lifted.  That makes it easier for you to change those (in Simulator>Set Parameters) and you can use them to make response surfaces.  We took Ireland as an example and estimated the number of reported deaths at end 2020 for a variety of 'new normal' levels of contact (m_normal) and periods over which we adjust from lockdown to those levels (t_normal).  

Response surface (contour plot) for projected Year End 2020 reported deaths in Ireland from COVID-19, as a function of 'new normal' levels of movement / transmission (m_normal) relative to baseline and the length of time over which we move from lockdown to new normal (t_normal).  While there is some sensitivity to t_normal, the main sensitivity is to m_normal (vertical axis).  Every reasonable step should be taken to keep this value as low as possible.

Here's hoping that our greater knowledge about COVID-19 puts us in a position to avoid the less favourable regions on this diagram.

We added the Singapore model at the request of a customer there.  Singapore was highly praised in the early stages of the outbreak, with a small number of cases, extensive testing and vigilance and a very low mortality rate from COVID-19.  However an outbreak in dormitories used by migrant workers led to a spike in cases and a period of lockdown referred to as the 'circuit breaker'.  Mortality rates remain low and this is attributed to both compliance and the young age profile of the majority of cases / migrant workers.  To fit the data, we had to allow for a burst of increased contact (the breakout in dormitories) and the model now fits the data well.

All models are available here as usual.  All you need to run them is the Excel file and Dynochem.

COVID-19 exit strategies from lockdown

We have been using a 'mixing' variable to represent the effects of non-pharmaceutical measures / interventions on the rate of exposure to COVID-19:

The rate of change of the number of people currently exposed depends on i) the relative contact rate between infectious and susceptible and ii) the probability of transmission during contact; these are reflected together in the 'mixing' term

This parameter reflects the relative contact rate between infectious and susceptible and also the probability of transmission during contact.  Baseline mixing = 1.0, i.e. what we did before the outbreak.  To date, this variable has been reduced by lockdowns and closures (reducing our contact rate) and physical distancing, handwashing, mask wearing (reducing the probability of transmission).  When we are exiting lockdowns, our contact rate will return gradually towards the baseline but if we maintain or improve current standards of physical distancing, handwashing and mask wearing, the mixing term in the model will not return to 1.0.  This will be very important in preventing large subsequent waves of infection.

As noted in last week's update, Ireland, Italy and the United States have reduced infection rates during lockdown by at least two thirds.  Some of the benefit came from reduced contact and some from reduced probability of infection; at this time it is difficult to split the effects accurately; however it is likely that the reduced probability accounts for a significant portion of the effect.

In addition to including the latest data in our models this week, we have updated the lines (on the process worksheet) that allow simulation of the effect of relaxing the measures that reduced contact.  Most countries will relax measures over a period of several months starting soon, while maintaining focus on hygiene and distancing and in many cases encouraging the wearing of masks.  Our models currently simulate a linear relaxation of measures, though in practice these will occur in 'steps' or 'phases'.  We also include the option of a return to current lockdown for a short period at any time.  Users can define the ultimate 'relaxed' value of the mixing variable and to begin with we have set this to 45% of baseline.  We chose this value in the hope that measures affecting probability of transmission will be maintained and that they are highly effective; and that as many people as possible will be able to continue to work from home.

Example projections are shown below for the US as a whole, with linear relaxation of lockdown measures to mixing = 45% of baseline (inset) over a period of months.  Even with these potentially optimistic parameter values, a very large second wave of infections could occur before the end of the year.  It seems likely therefore that in order to control the level of infection, it will be necessary to have periods during which we return to current levels of lockdown.

Model fit to known cases and reported deaths to date for the US (symbols and curves of the same colour).  Then projecting to end 2020 with a linear relaxation of restrictions on movement [inset], to a new stable mixing level of 45%.

COVID-19 models at 3 May Ireland Italy and US

Our models have been updated to include the data from Ireland, Italy and the US for the week just ended.  Trends and parameters are stable in all regions.  We do not adjust R0 in any region (this was fitted to the early 'exponential' data) but we refit the 'mixing' profiles each week and values have been steady for some time.  Since 11 April, the models also predict death rates and the related fitted parameters are also steady.

Of the three regions, R0 may have been highest in the US (3.92) and lowest in Italy (2.94), with Ireland in between (3.52).  We estimate that R_effective (or Rt) after adoption of restrictions is now 1.1 in the US, 0.98 in Italy and 0.84 in Ireland.  These figures translate to mixing levels of 28%, 33% and 24% of baseline respectively in the periods of most severe 'lockdown'.

In order to predict the safest and most beneficial ways to release restrictions, data on the incremental effectiveness of each separate measure (e.g. social distancing, wearing masks, cocooning the vulnerable, staying at home, closing shops and places of work) would be extremely useful but are not yet available.  We may learn these parameters as the outbreak progresses and be better able to predict exit strategies.

In the meantime, Google and Apple are providing anonymized data related to movement, recorded when our devices talk to their servers.  In general, these data trend in line with the mixing percentages we estimate from the predictive model.  Results are shown below for the United States as an example. Though the mobility data are scattered, the trends align with the predictive model and indicate that mobility data are helpful indicators of progress.

Mobility data from Apple devices compared to the mixing variable fitted to COVID-19 case data for the US.  Though the data are scattered, the trends align with the predictive model.

COVID-19 models at 26 April Ireland Italy and US

Our predictive models for Ireland, Italy and the US are stable and identical for each region, differing only in fitted parameter values (e.g. R0 and people movement versus time).  As the progress of COVID-19 is slowed by restrictions, the fitted parameter values for people movement have become more consistent from week to week.  Models with data to 26 April 2020 are available here as usual.
Good progress is being made and restrictions will be gradually loosened and occasionally re-tightened over the coming weeks and months, depending on trends in case numbers. 

Data for Ireland remains harder to interpret because of delayed testing results and some changes in the basis for reporting deaths this week.  A graph showing backdating of rest results was shared by the government on 23 April 2020 and we have used this to update the Ireland model; however there are spurious peaks (see below) and in addition, focused testing in care/nursing homes generated high case numbers this week that probably do not reflect a trend.  We therefore did not do a tight fit to the Ireland data.

Known case data for Ireland reported by the ECDC and a backdated version of the same information published on Thursday this week.  Both versions contain spurious trends that make direct use for parameter estimation difficult.

Approximate model fitted to backdated Ireland data indicates that the number of infectious in the first wave probably peaked around 1 April 2020.

Over the next weeks we will explore developing a more granular predictive model for the loosening of restrictions, especially focusing on the degree to which communities can return towards normality while those especially vulnerable remain protected. 

COVID-19 at 19 April Ireland Italy and US

Each of the regions we have been modeling continues to make good progress against COVID-19 and some relaxation of the restrictions on people movement are being discussed or applied carefully with a view to avoiding second and later waves.

The numbers of infectious appear to have peaked clearly in Italy and the US.  The Ireland data has been harder to read because of around 3000 positive test results delivered three weeks late;  without a definitive restatement of the correct dates for those case numbers, it is difficult to be confident of the Ireland parameters at this time.

Models with current data are available as usual here.  Sample results are shown below.  As usual in the graphs below, discrete symbols are measured data (cases and deaths) and curves of the same colour are model predictions of those data.

Italy:

Predictive model fit to known cases and reported deaths from Italy indicates a peak in the number of infectious in mid March.

US:

Predictive model fit to known cases and reported deaths from Italy indicates a peak in the number of infectious at end March.

COVID-19 models for Ireland, Italy and US - 11 April

Thanks again for all the positive feedback on this work.

In this week's update we have included data to 11 April for each region.  The structure of the predictive model is the same as before, with the additional calculation of deaths from the outbreak now included.  We have also simplified and we hope improved the workflow for application to other regions, by replacing the manual adjustment of an 'imposed' population mixing profile with use of the Dynochem Fitting window to fit a 'piecewise-linear' mixing profile to all case data (second scenario in the model).

The models (available here) fit very well to both cases and deaths in each region (first scenario for initial period and second for the outbreak to date).  The now usual caveats apply about known coronavirus case numbers: they lag and obscure real case numbers and their meaning varies depending on the testing criteria, volume and delay in each region and over time. Case data for Ireland has been muddled in recent days by the addition of results from swabs taken over the last month that took several weeks to test; we have attempted to reconcile this unhelpful number over the relevant period and refitted parameters for Ireland on that basis.

Each region's degree of control over the outbreak may be assessed by the current effective 'R', taking account of its original value (R0) and the restrictions.  R_effective should be near or preferably below 1 before restrictions can be safely relaxed for a short period (when it will rise).

The are many 'peaks' in such an outbreak and to say 'it has peaked' requires a more specific definition of the type of peak.  While infection rate may already have peaked (in the first wave), detection lags that, so peaks later; death rates also lag and peak later again.

Various peaks that occur in an outbreak like COVID-19, using model data for Ireland as an example.  Natural time lags as well as testing delays may cause peaks in infection, detection, death (on secondary y-axis) and ICU bed occupation rates to occur at different times.

As usual in the graphs below, discrete symbols are measured data (cases and deaths) and curves of the same colour are model predictions of those data.  'Mixing' reflects the reduced interaction of the population with each other.

Ireland:

Ireland: Including the burst of old case data results received this week, Ireland looks to have a little further to go before the first wave of the outbreak places peak demand on health services.  In order to bring that peak into April, the public will need to observe restrictions more tightly than at present.  Estimated current R_effective=1.1.  The peak infection rate may have occurred around 3 April.

Italy:

Italy: The first wave of the outbreak has recently passed the point of peak demand on health services. Peak infection rate was probably around 19 March. Estimated current R_effective=0.97.

United States (as a whole):

US: Peak infection rate may have occurred in early April.  Peak demand on the health services could be in late April or early May.  R_effective appears similar to Ireland at approximately 1.1.