Simulation results and comparison of 12 hypothetical disease transmission scenarios
In this study, we first examined the model theoretical performances using a few reasonable but hypothetic infection rate patterns. Three such patterns expressed in r code format were given below.
rr = c(rep(3,30), 2,2,1.5,1.5, 1.2,1.2,rep(0.8,4),rep(0.5,10),rep(0.1,50))
rr = c(rep(3,25), 2,2,1.5,1.5, 1.2,1.2,rep(0.8,4),rep(0.5,10),rep(0.1,55))
rr = c(rep(2.5,30), 2,2,1.5,1.5, 1.2,1.2,rep(0.8,4),rep(0.5,10),rep(0.1,50))
Literature reported that the typical range for Rt values was between 2 to 4 for COVID–19 spread without intervention [1, 3, 9, 14]. Assuming the simulation period started before any government interventions, therefore, Rt values were set to 3 for the first 30 days in the pattern 1 and then decreasing gradually over the subsequent stages of the 100 days presumably due to the effect of government interventions. In pattern 2, we would like to check how the outbreak outcomes would change if the government’s intervention enforced five days earlier; further in pattern 3, we examined how different it could be if the starting Rt values were lower than 3 (e.g., 2.5) given other things equal. As indicated in Table 1, there were two parameters (muT and sizeV) for defining a negative binomial distribution [10, 11] which was used for describing the time interval for a susceptible person gets infected. In the study, we also compared different scenarios by examining different muT and sizeV values. The three different simulation settings that examined were: TransSimu(nd = 100, muT = 4, sizeV = 1); TransSimu(nd = 100, muT = 4, sizeV = 0.9); and TransSimu(nd = 100, muT = 3.6, sizeV = 1).
Therefore, we finally ended up with a total of 12 hypothetical disease transmission scenarios for us to investigate/compare using the proposed simulation model and the results were presented in Figure 2.
The interpretations of the graphical outcomes of Figure 2 were as follows. The three panels in the top row showed that by only five days earlier of enforcing intervention, the number of the active cases would peak earlier accordingly with a much lower peak value (black curves versus blue curves). The three panels in the bottom row showed that by reducing Rt from 3 to 2.5 for the first 30 days, the number of the active cases would reach the peak in the same day but with a much lower peak value (black curves versus red curves). The comparison of the panels in the second column versus the first column showed that reducing the muT values (3.6 versus 4 days) would increase the peak level quite substantially. The comparison of the panels in the third column versus the first column showed that reducing the sizeV values (0.9 versus 1, a smaller sizeV value implies a larger variance) would increase the peak level even further. By comparing panels in columns three and two seemed indicating the impact on the peak level due to the change (10%) in muT was somehow smaller than that due to change (10%) in sizeV. Therefore, in summary, this part of the simulation study showed that, as expected/implied by the underlying theory, a higher Rt or a smaller muT would result in a higher number of infected people keeping other things equal.
Simulation study results for Australia and UK data
The simulation model was then tested with two real COVID–19 data sets: the confirmed COVID–19 cases reported for Australia and United Kingdom (UK) over the period 1 March to 18 April 2020 [2]. The infection rate pattern for Australia was specified as
rr = c(rep(2.5,5), rep(2.3,5),rep(2.9,5), rep(3,5),rep(2.1,5), rep(0.9,5), rep(0.3,5), rep(0.2,15), rep(0.1,50)) and the infection rate pattern for UK was
rr = c(rep(3.4,10), rep(3.1,10),rep(2,5),rep(1.8,5),rep(1.6,5),rep(1.5,5),rep(0.7,5), rep(0.6,5),rep(0.3,10),rep(0.1,40)).
Different from the model default setting for the initial cases (n0 = 1), it was decided to use the newly confirmed cases over the period of two weeks prior to 1 March as the initial infectious case number. Therefore, it was n0 = 10 for Australia data and n0 = 9 for UK data in running the simulation model. The parameter muT was also adjusted in the simulation setting to give more flexibility in fitting the observed data patterns. The simulation settings were:
TransSimu(nd = 100, muT = 4.3,sizeV = 1,n0 = 10) for Australia and TransSimu(nd = 100, muT = 3.95,sizeV = 1,n0 = 9) for UK. The muT values decided here implied that the average time needed for getting a susceptible people infected was shorter in UK than in Australia.
The simulation study results based on the Australian confirmed COVID–19 cases data were presented in Figure 3 (the model estimated number of the active cases) and Figure 4 (the cumulative number of the confirmed cases). Figure 3 indicated that, according to the model prediction, the number of the active cases in Australia would peak around 30 March. Practically, this would be the time the Australian health care system having the highest pressure. Decisions might then be made for preparation for the possible worst scenario situation (how much and when) as predicted by the model. Figure 4 presented the model predicted/estimated pattern for the cumulative number of cases and the actually observed/recorded number of confirmed COVID–19 cases were superimposed as those dot points. The 49 observed data points (period over 1 March to 18 April) fitted nicely to the predicted median curve. By examining the resulting Rt values, it was found that the infection rate in Australia was actually higher in later March than in early March which might be explained by the fact that a substantial large number of new cases were identified related to several cruise ships arrived in March (e.g., the Ruby Princess cruise ship case https://www.abc.net.au/news/2020–04–05/ruby-princess-cruise-coronavirus-deaths-investigated-nsw-police/12123212). As we have argued in the previous sections the determination of Rt values involved different types of influential factors and these factors were very much time dependent. Nevertheless, the simulation model predicted that at the peak time, the number of active cases could reach 1629 with the interquartile range of (1245, 1970); by the end of the COVID–19 outbreak, it would have a total number of 6606 confirmed cases in Australia with the interquartile range of (5049, 8126). Both Figures 3 and 4 indicated that the COVID–19 outbreak in Australia should be over by early May should the assumed Rt pattern in this model proved to be true.
The simulation study results based on the United Kingdom’s confirmed COVID–19 cases data were presented in Figure 5 (the model estimated number of the active cases) and Figure 6 (the cumulative number of the confirmed cases). Figure 5 indicated that, according to the model prediction, the number of the active cases in UK would peak around 11 April. The simulation model predicted that at the peak time, the number of active cases could reach 24603 with the interquartile range of (20061, 29454). Practically, this would be the time the UK’s health care system having the highest pressure. For example, assuming 5% of the active case individuals would require ICU beds, this meant UK health care system should prepare at least 1230 ICU beds before 11 April.
Same as the Australian case in Figure 4, the UK case in Figure 6 showed a near perfect fit between the model predicted/estimated median curve and the observed data points. The simulation model predicted that by the end of the COVID–19 outbreak, UK could have a total number of 136333 confirmed cases with the interquartile range of (110847, 163006). Due to the uncertain and complex nature in determination of the Rt values, one should not be too confident about the model estimation results for both the Australia case and the UK case in terms of ultimate true cumulative number of confirmed COVID–19 cases because, as it was said “Models are only as good as the assumptions on which they are based” [15]. However, in the process of determining the Rt values for the simulation model more insights were gained on the dynamics of the COVID–19 spread over time in both countries.
The feasible interpretations of these two infection rate patterns were as follows. Overall, Australia had a lower infection rate pattern than UK had. The details of the differences was summarised by presenting a descriptive summary of the first 50 Rt values (which mainly were determined by the observed number of confirmed cases) as in Table 2. The results should explain the reason why both countries started with almost the same confirmed case numbers in the beginning of March (26 versus 23) but now ended with UK had as 16 times higher of the confirmed cases than Australia had (6533 versus 108692).
Table 2: A numeric summary of the estimated infection rate patterns over the period 1 March to 19 April
Rt values
|
minimum
|
25th percentile
|
median
|
mean
|
75th percentile
|
maximum
|
Australia
|
0.2
|
0.2
|
1.5
|
1.46
|
2.5
|
3
|
UK
|
0.6
|
1.5
|
1.9
|
2.12
|
3.1
|
3.4
|
The infection rate patterns could also be examined by different stages over the simulation period. Beginning from 1 March, the first 10 days both countries were under pre-intervention stage and the estimated/assumed infection rate was between 2.3 and 2.5 for Australia and as high as 3.4 for UK. Then the unusual changes happened for the Australian infection rates over the second 10-day in March that it increased to as high as 3. On the other hand, the UK infection rates steadily decreased over the same 10-day period. From 20 March onwards, the infection rates kept constantly decreasing with a sharper drop in Australia than in UK. The latter gradually decreasing pattern of Rt values might be interpreted as a reflection of how intervention measures were enforces and intensified over time. Of the total 100 Rt values in each pattern, the first 49 Rt values were largely determined by fitting the model to the observed data. The remaining 51 Rt values would then primarily be a result of subjective judgement or some wishful assumptions. For example, by assuming the last 40 Rt values to be 0.1, we practically assume that, for both countries after 1 May the interventions would keep in effect, the general public would respond accordingly, and the confirmed cases keep recovering so that the overall effect was to such an extent that the infection rate would drop to a very low level of 0.1. However, what the future reality this might finally roll out could only be “what will be will be”.