3.1. Doubling Times, Infection Rates, Reproduction Numbers
First, the doubling time of the pandemic is calculated for different values of the infection rate k11. As mentioned in the Introduction, the worldwide doubling time of COVID-19 outside China was 4 days in the latter half of February and the first half of March 2020. This doubling time was found to correspond with k11 = 0.261 day–1. This value was used as the default in further simulations, unless specified otherwise.
In Europe and North America, doubling times were significantly shorter during that time. For instance, in Italy, the reported number of COVID-19 cases grew approximately exponentially from 150 on February 23 to 10,149 on March 10 (16 days later) (https://www.worldometers.info/coronavirus/country/italy/). An exponential fit to the data leads to a doubling time of 2.66 days (R2 = 0.9841). This is consistent with k11 = 0.344 day–1. Most of the Western world experienced similar growth rates during the same time.
The R0 value was calculated as a function of k11. The relationship between R0 and k11 follows a perfect linear relationship as follows:
where a = 10.0388 days.
The R0 value reaches 1 when k11 = 0.0996 day–1, only 38.1 % of the global average k11 value in late February to early March 2020, and 29.0 % of the value in Italy during that time. As a result, the model-based estimate for R0 in late February to early March is 2.62 worldwide outside China, and 3.45 in Italy and most of the Western world. These are just estimations based on the assumption that the proportion of cases reported remains constant over time.
3.2. Scenarios – Average, Fast, Slow
In this section, a number of scenarios will be run to assess the number of infected and the number of deaths as a function of time, for a population of 100 million, starting with 111 infected (100 incubating, 10 sick and 1 seriously sick) at time 0.
Figures 3 shows the evolution of the epidemic in the base case (doubling time = 4 days, k11 = 0.261 day–1, R0 = 2.62), without intervention. The first deaths are predicted around day 12, when about 1200 people are infected. The number of people showing symptoms at this time is around 460 (250 mild, 20 serious, 190 recovering). This early in the epidemic, it is likely that testing is not yet fully deployed, and the number of reported cases is likely to be on the order of 200 or less.
After one month, the model predicts 30-35 deaths and a total of about 27,000 infected. Of these, about 16,000 show no symptoms, 5,000 show mild symptoms, 500 severe symptoms, and 5000 are recovering. At this point, the official case count is probably a few thousands. Around this time or up to two weeks later, most governments started taking serious precautions to limit the spread of the virus.
After two months without intervention, there are 4.5 million infections and over 6,000 deaths. As a rule of thumb, there is one death per 750 cases in the expansion phase of the disease when the doubling time is 4 days. 2.7 million people are in the incubation phase and 85,000 people are seriously sick.
The peak of seriously sick people is reached on day 95, when over 2.5 million people are seriously sick and over half a million people have died.
After 150 days, the disease is declining but is still overwhelming the health care system, with about 180,000 people seriously sick. The model predicts 1.33 million deaths at this time, 1.33 % of the population. Given the severe lack of care that would occur, the death toll could be underestimated by as much as a factor 2 or 3. About 91.6 million people get infected overall, significantly more than the expected number from “herd immunity” (61.8 million). This is because the disease expands so rapidly that it overshoots and continues to infect people as it winds down past the 62 million mark. This simulation clearly shows that herd immunity is only effective when people are vaccinated before the spread of the disease.
Next, the simulation was repeated for a “fast” scenario where the doubling time is the same as in Italy in late February to early March, 2.66 days (k11 = 0.344 day–1, R0 = 3.45). The results are shown in Figure 4. The main difference with the base case is that the disease spreads faster and peaks sooner. At its peak, 3.2 million people are seriously sick, on day 70. The death burden after 150 days is 1.44 million, or 1.44 % of the population. At this time, the disease has affected 96.4 million people, 96.4 % of the population. Again, this is massively above the number expected from herd immunity (71.0 million people).
During the initial spread of the disease, there is one death every 1800 to 2000 cases, indicating that the epidemic may be underestimated even more when it spreads rapidly. This ratio explains why the case mortality rate of COVID-19 is sometimes incorrectly speculated to be on the order of 0.1 % (https://www.forbes.com/sites/carlieporterfield/2020/04/21/scientists-widely-criticize-studies-that-claim-coronavirus-death-rate-could-be-far-lower-than-believed/#31cde7711517: accessed April 22, 2020).
The next scenario represents a strategy that is popularized as “flattening the curve”: the infection rate is significantly reduced to slow down the spread of the disease in an attempt to avoid overburdening the health care system, but no attempt is made to eradicate the disease, i.e., the R0 remains significantly above 1. The simulation is run with an infection rate k11 = 0.18 day–1 (doubling time 7.65 days, R0 = 1.81). The result is shown in Figure 5.
The peak in the number of seriously sick people is significantly delayed, to day 185, but the number of patients still far exceeds the capacity of any health care system, with 1.4 million seriously sick, half the number of the base case. The death burden in the “flattening the curve” strategy is slightly over 1 million, still over two-thirds of the fast scenario. The total number of people that get infected in a 240-day time span is 73.3 million, again markedly more than the number expected from herd immunity considerations (44.7 million).
Clearly, flattening the curve is an inadequate strategy for fighting the COVID-19 pandemic.
3.3. Scenarios – Social Distancing Intervention
Next, starting from the base case, it is assumed that drastic social distancing measures are taken on day 30 that reduce R0 to below 1. It is assumed that the value of k11 is reduced by 70 % (i.e., from 0.261 day–1 to 0.0783 day–1 i.e., R0 decreases from 2.62 to 0.786). The result is shown in Figure 6. A 70 % effective social distancing intervention with a starting value of k11 = 0.261 day–1, i.e., with respect to the world average, is equivalent with a 77 % effective intervention with a starting value of k11 = 0.344 day–1, i.e., with respect to the situation in Italy and most of the Western world. In other words, in much of the Western world, the results shown in Figure 6 reflect a social distancing initiative that is 77 % effective, not 70 %.
There is a marked decline in the number of infected in this scenario. After 240 days, the number of people who died of COVID-19 is 1420, about three orders of magnitude less than the previous scenarios. Still, this number is 42 times the number people who had died at the onset of the intervention (34).
The number of seriously sick people peaks at a value of 1642 on day 51, again about three orders of magnitude less than in the preceding scenarios.
What is clear from this scenario is that the decline of the epidemic is much slower than its growth. This has important repercussions for any public health policy aiming to save lives. Even seven months into the intervention, the number of infected is comparable to the number of infected two and a half weeks before the intervention. Terminating the intervention would immediately relaunch the epidemic. The reproductive number must be maintained below 1 until the population can be vaccinated on a large scale.
3.4. Scenarios – The Death Burden of Inaction
In this section, the number of deaths will be evaluated as a function of time and effectiveness of the social distancing intervention. The starting point is the base case, with a doubling time of 4 days (k11 = 0.261 day–1, R0 = 2.62).
First the effect of effectiveness of the social distancing intervention is calculated. It is assumed that the intervention starts on day 30 with an effectiveness ranging from 50 % to 80 %. Figure 7 shows the number of deaths after 60, 150, and 300 days.
After 60 days, i.e., 30 days after the start of the intervention, the effect of effectiveness of intervention on mortality is very limited. This is concerning because to observers it appears that the interventions are not working. However, over a 150-day time span, a 5 % decrease of efficiency can triple the mortality. Over a 300-day time span, a 1 % decrease of the efficiency (e.g., from 62 % to 61 %) can cause a 50 % increase in mortality. This explains why some Asian countries treat seemingly trivial violations of the social distancing rules as felonies.
The value of R0 equals 1 at 61.8 % efficiency in this case. The importance of keeping R0 below 1 is immediately obvious from Figure 7. When the initial value of k11 is 0.344 day–1, an efficiency of 71.0 % is needed to lower R0 to 1. This should be the minimum target efficiency of social distancing in Europe and North America.
Next, the effect of timing of introduction of a social distancing intervention on the mortality over 60 days, 150 days, and 300 days is calculated. The results are shown in Figure 14. Probably not surprisingly, the number of deaths doubles with every 4-day delay of the introduction of social distancing. This is an important point, because the number of deaths may seem small at the time of introduction (e.g., from 34 on day 30 to 68 on day 34), the number of deaths after 300 days increases from 1,429 to 2,845 as a result of this delay. Every additional death at the time of intervention represents 42 additional deaths over a 300-day time span.
3.5. Diagnostic Modeling
In this section, modeled deaths versus time will be compared with reported deaths in three countries: Italy, France, and Iran. These countries were chosen because they were hit relatively early so there is more data, the death toll for these countries is relatively high, and they represent three distinct cases. For each country, an analysis was made in early April, and again in late April. The early analyses were presented on YouTube to document and time-stamp the projections (see https://www.youtube.com/watch?v=7Y9fwus0fvQ for Italy, https://www.youtube.com/watch?v=MT4wjniICLY for France, and https://www.youtube.com/watch?v=z1DMM68HHB8 for Iran). The results of the two analyses are compared. The adjustable parameter values obtained in each analysis is compared in Table 2.
Table 2. Adjustable parameters of the COVID-19 spread in Italy, France, and Iran, obtained in early April and late April. Note that tj is the day after the NPI decision whereas tspike is the day of the event leading to the spike.
Country
|
Italy
|
Italy
|
France
|
France
|
Iran
|
Iran
|
Analysis date
|
April 3
|
April 21
|
April 9
|
April 21
|
April 5
|
April 21
|
k11,0 (day–1)
|
0.378
|
0.40
|
0.323
|
0.323
|
0.32
|
0.34
|
correction
|
0.136
|
0.08
|
0.049
|
0.049
|
0.518
|
0.296
|
t1
|
March 9
|
March 2
|
March 24
|
March 24
|
March 5
|
March 5
|
E1
|
0.794
|
0.224
|
0.87
|
0.89
|
0.73
|
0.8
|
t2
|
–
|
March 9
|
–
|
–
|
–
|
–
|
E2
|
–
|
0.46
|
–
|
–
|
–
|
–
|
t3
|
–
|
March 21
|
–
|
–
|
–
|
–
|
E3
|
–
|
0.226
|
–
|
–
|
–
|
–
|
kspike
|
–
|
–
|
–
|
–
|
–
|
0.6
|
tspike
|
–
|
–
|
–
|
–
|
–
|
March 20
|
Population
|
60.5×106
|
60.5×106
|
65.2×106
|
65.2×106
|
83.7×106
|
83.7×106
|
Projected deaths
|
47,620
|
31,323
|
35,156
|
32,499
|
13,676
|
8,061
|
The model fit to the data in Italy is shown in Figure 9. The initial fit was based on a single NPI on March 8, the day a national lockdown was declared. This fit provided a poor prediction of the data after April 3, due to the complexity of the situation. The epidemic started in the region of Lombardy, in the North of Italy, and spread to the rest of the country. The second fit required three NPI phases and still showed some lack of fit. The total mortality projection declined from about 47,000 based on the original fit to about 31,000 based on the second fit.
Figure 10 shows the data for France, with the model fits. The overall efficiency of the NPI is similar to the Italian case, but in France, the lockdown was more sudden in France, and occurred at a later date. The projections of the original model fit is more accurate in the case of France in comparison with Italy, because a single lockdown decision explains the entire data set. The lack of fit in Figure 10 is mainly due to late reporting of some cases, particularly deaths occurring in retirement homes. The mortality projection was 35,156 in the first data fit, and 32,449 in the second data fit. On April 9, the last data point of the first fit, the reported mortality in France was 12,210.
The data for Iran is shown in Figure 11. Prediction of the epidemic in Iran was complicated by the Iranian new year, which occurred on March 20. The quality of fit improved upon adding a spike in the infection rate on that day. Because the spike masked the effectiveness of the NPI in Iran, the mortality projection from the first model fit was a serious overestimate (over 13,000 deaths) in comparison with the second fit (around 8,000 deaths). The relatively low mortality in Iran is thanks to the earlier intervention. As long as the reproduction number of the disease is brought below 1, an early timing of the intervention is more important than the effectiveness.
3.6. Preliminary Mortality Rate Estimation – Netherlands
On April 16, 2020, Reuters reported a study of 10,000 blood donations in The Netherlands, indicating that 3 % of the samples contained antibodies against SARS-CoV-2 (https://www.reuters.com/article/us-health-coronavirus-netherlands-study/dutch-study-suggests-3-of-population-may-have-coronavirus-antibodies-idUSKCN21Y102). The study is non-peer-reviewed and no methodological details were given, so this analysis is preliminary at best. Assuming that the test results are correct and the sample set is representative of the population in The Netherlands, this leads to an estimated 510,000 coronavirus positive people of a population of 17 million. The model was fitted to mortality data in The Netherlands. The obtained values were k11 = 0.34, interventions on March 15 and March 23 with effectiveness 0.34 and 0.58, respectively, and a correction of 0.00118. This leads to a long-term projected mortality of 5,800. On April 8, a week before the report, the number of coronavirus positive cases in The Netherlands is predicted at 360,000 by the model, of the same order of magnitude as the estimate from the blood donation samples. A refit indicates that the model would predict a case number of 510,000 if a case mortality rate of 1.06 % is assumed.
This case mortality rate estimation does not account for inaccuracies in the immunological testing. Assuming a test specificity of 99 % (i.e., a false positive rate of 1 %), the real number of cases would be 343,000, leading to a case mortality rate of 1.57 %. If a test specificity of 98 % is assumed, the real number of cases would be 173,000, and the case mortality rate would be as high as 3.12 %. It follows that the data impose a lower limit of the case mortality on the order of 1 %, and will be higher unless the test used is exceedingly specific.