Executive Summary: The first wave has ended in Brunei on 2020-04-24 (Day 46). We expect a total of 143 cases in the first wave of the COVID-19 pandemic in Brunei, this is an additional 0 cases compared to current case total. New cases have peeked on 2020-03-13 with 9 new cases. Our models assume that after end of the current wave around 0 new cases will appear on an average day, since SARS-CoV-2 is now endemic. Quick, effective, and differentiated mitigation measures need to be in place to prevent a new outbreak.

1 New Cases

According to the Johns Hopkins University global COVID-19 data set [1] a total of 150 COVID-19 cases were reported in Brunei on 2020-11-27. This number of cases increased by 0 percent over the cumulative total cases known one day earlier. At this rate of cases double every days and the basic reproduction rate \(R_0\) in Brunei can be estimated at 2, an estimate of the average number of other people infected by a single infected person. Figure 1.1 shows the time line of new cases reported in the country (black points) together with a five day moving average (purple line) that smoothes daily fluctuations in the reporting that are due to fewer capacities on weekends, holidays and other reporting mishaps.

New cases and five day moving average.

Figure 1.1: New cases and five day moving average.

1.1 Model for new cases (DSM)

New cases peek after a certain time of growth that follows a sigmoidal function and then recedes to a lower level as the epidemic wave fades out and enters an endemic state. This behavior can be modelled with a double-sigmoid function \(f_{base}\), that is obtained by the product of two sigmoidal functions [2].

\[ f_{base}(t) = \frac{1}{ 1+e^{-\alpha_{i} * (t - t_{i})} } * \frac{1}{ 1+e^{-\alpha_{d} * (t - t_{d})} }\]

This function has a maximum \(f_{max} = max(f_{base}(t_{max}))\) at \(t_{max}\).

The model can obtain predictions related to an observed maximum of new cases \(I_{max_{d}}\) and an endemic number of daily cases \(I_{final_{d}}\) as an asymptote (that is ideally 0). Let \(f_{d}(t)\) be a piecewise function that switches between the growth and decay phase at \(t_{max}\), such that

\[\begin{aligned}f_{d}(t) = \left\{ \begin{array}{cc} \frac{I_{max_{d}}}{f_{max}} * f_{base}(t) & \hspace{5mm} t \leq t_{max} \\ \frac{I_{max_{d}} - I_{final_{d} }}{f_{max}} * f_{base}(t) + I_{final_{d}} & \hspace{5mm} t > t_{max} \\ \end{array} \right. \end{aligned}\]

where \(t_{i}\) is the midpoint and \(\alpha_{i}\) the slope of the increase of cases and \(t_{d}\) is the midpoint and \(\alpha_{d}\) is the slope of the decrease in cases. \(I_{max_{d}}\) the peak of new cases and \(I_{final_{d}}\) the asymptotic final value of new cases at the end of the infections wave. Figure 1.2 shows the current double-sigmoidal model ( orange line) that is fitted for the new case time series of Brunei, where the vertical dashed line highlights the midpoint \(t_i\) of the growth phase, the vertical solid line indicates the peak of new infections \(t_max\) and the vertical dotted line the midpoint \(t_d\) of the decay phase. The dashed horizontal line marks the estimated asymptote of endemic new cases \(I_{final_{d}}\) as this infection wave ends. We pragmatically assume that the first wave ends, when 99 percent of the decrease has happened. The dashed green line marks this date.

Double sigmoidal model (DSM) for new cases.

Figure 1.2: Double sigmoidal model (DSM) for new cases.

1.2 Reliability of the DSM

Since the model requires six parameters, it also requires more data points. Hence, it is only feasible to fit this model in later phases of the pandemic (ideally after the peak of daily infections occured) and the model is highly sensitive to new data, particularly with respect to finding the final asymptote\(I_{final_{d}}\)as the following comparison of the model shows. Nevertheless the model renders interesting characteristica of the past developments.

Model for Daily Increases Current Estimate Last Week Estimate Delta (absolute)
\(I_{max_{d}}\) Peak new cases 9 9 0
\(I_{final_{d}}\) Endemic new cases 0 0 0
\(\alpha_{i}\) Slope of growth phase 0.68 0.7 -0.02
\(t_{i}\) Midpoint of growth phase 2020-03-09 (Day 1) 2020-03-09 (Day 1) 0
\(t_{max}\) Peak day 2020-03-13 (Day 5) 2020-03-13 (Day 5) 0
\(\alpha_{d}\) Slope of decay phase 0.15 0.15 0
\(t_{d}\) Midpoint of decay phase 2020-03-24 (Day 16) 2020-03-24 (Day 16) 0
First wave ends 2020-04-24 (Day 46) 2020-04-24 (Day 46) 0

2 Cumulative cases

The cumulative total of COVID-19 cases in Brunei is the number of all COVID-19 cases known up to yesterday and includes all patients already recovered from the infection. This corresponds to the Infected (I) number in the standard epidemiological SIR model. Figure 2.1 shows the data points of the time series together with a Gompertz model for the cumulative cases (red line) as well as the cumulated double-sigmoidal model (orange line). The dashed horizontal lines show the expected maximum number of cases of the respective models.

Cumulative cases with DSM and Gompertz models

Figure 2.1: Cumulative cases with DSM and Gompertz models

2.1 Gompertz model for cumulative cases

The Gompertz model [3] is a sigmoid model that is often fitted to growth data, and often prefered over the standard logistic model (als called Verhulst model) due its asymetry. The Gompertz can be considered as a special case of the four parameter Richards model and is computationally easier to fit as it has one parameter less. The model is based on the cumulative distribution function \(f_{c}(t)\), which is a three parameter logistic function:

\[ f_{c}(t) = I_{max_{c}} * e^{ - \beta_{c} * e^{ - \alpha_{c} * t }} \]

Where the parameter \(I_{max_{c}}\) is the estimated maximum number of cumulative cases (asymptote), \(\beta_{c}\) inflects the curve along time, and \(\alpha_{c}\) is the maximum absolute growth rate of cumulated cases.

2.2 Reliability of the Gompertz model

The Gompertz model change over time as new data is available. Obviously, we are interested whether the models are reliable and in particular show some convergence. When we compare the current model with the model calculated using the information that was available last week, we can again observe how the model parameters have changed.

Parameter Current Estimate Last Week Estimate Delta (absolute)
\(I_{max_{c}}\) Total number of cases 143 143 0
\(\alpha_{c}\) Growth rate 0.145 0.146 -0.001
\(\beta_{c}\) Time shift 3 3 0

3 Projections

3.1 Methodology

Putting in future dates into the current models allows to project future cases. Note: The actual development can differ from model predictions, for example if social distancing norms change or methods in data collection are altered. If such changes happen, a model fitted to the past is no longer valid.