Fig 1.
Example of the piecewise function f(t) in Eq (1).
Here, we present the shape of the function when the vaccination process is completed after 240 days (blue) or 360 days (orange) after the beginning of vaccination (T0).
Fig 2.
The vaccine process is defined with green line, while the waning immunity process is indicated with red line.
Table 1.
Model parameters.
Table 2.
Model variables and assumptions.
Fig 3.
Percentage positive tests, tests turnaround times (24h-48h) and cumulative cases are reported from March 17, 2020 to November 28, 2020 in Toronto.
The percentage positive tests decrease from May 31 to August 24, when it starts increasing, showing, however a magnitude smaller than the first wave in March. Until May 31 about 50% of the tests were returned within 24h, while until the end of August, the trend shows an initial decreasing followed by an increasing trend. After this, the curve decreases again, raising up in October and remaining below 50%. However, it slightly decreases again after November 10. Similar pattern for 48h tests turnaround time. The cumulative cases increase until May 31, and then remain stable until the end of August when the increase starts again and continue until Dec 2, 2020, with a slight decrease after the beginning of November.
Fig 4.
R0 4D contour plot in the βh, r24, ρ parameters space.
The values of the reproduction number are presented under Phase 1-2 and 3-4 (a, b, respectively) considering 0%, 30%,90% vaccine coverage, lifelong immunity and efficacy 70%. In all cases, we observe that as the vaccine coverage increases, the reproduction number decreases visibly. Also, the yellow region (indicating the highest values of the reproduction number) increases as the probability of the transmission in the household increases. In the phases with higher transmission (1, 3 and 4), we observe that the reproduction number is always greater than 1, except for 90% coverage. We also observe that if more people receive their test back within 24 hours, the reproduction number decreases. Similarly, in general, to maintain the reproduction number below 2, more than 20% of the infectious individuals need to be tested.
Fig 5.
Cumulative cases under Phase 1: Efficacy 70% and vaccination completed in 120 days.
Cumulative cases when the vaccine efficacy is 70%, vaccination is completed in 120 days under Phase 1 conditions, reopening in December 2021 (a-c) or May 2021 (b-d) and immunity does not wane (a-b) or wanes over 180 days (c-d).
Fig 6.
Cumulative cases under Phase 3: Efficacy 70% and vaccination completed in 120 days.
Cumulative cases when the vaccine efficacy is 70%, vaccination is completed in 120 days under Phase 3 conditions, reopening in December 2021 (a-c) or May 2021 (b-d) and immunity does not wane (a-b) or wanes over 180 days (c-d).
Fig 7.
Cumulative cases under Phase 2: Efficacy 70% and vaccination completed in 120 days.
Cumulative cases when the vaccine efficacy is 70%, vaccination is completed in 120 days under Phase 2 conditions, reopening in December 2021 (a-c) or May 2021 (b-d) and immunity does not wane (a-b) or wanes over 180 days (c-d). sub-figures (e-f) are zoomed images of sub-figures (a) and (c).