Fig 1.
Real data of vaccination in China [10].
(a) Cumulative vaccination data V(t), (b) Daily vaccination data which approximately gives dV/dt.
Fig 2.
Comparison among the effectiveness of different vaccines of COVID-19.
In China, mainly Sinopharm-Bejing, Sinopharm-Wuhan, Sinovac, CanSinoBio vaccines were administrated (the red squares) [1, 9, 10]. This figure is taken from [11].
Table 1.
Parameter values.
Fig 3.
Compartmental flow diagram of the model (1).
Fig 4.
The red and blue color correspond to the controlled reproduction numbers with under-reporting and without under-reporting respectively.
All other parameter values are same as in the green curves in Fig (17).
Fig 5.
Neural Network architecture for DINNs applied to system (8a)-(8h).
Fig 6.
Proportion of population vaccinated and curve fitting.
Fig 7.
System (8a)-(8h) estimation using synthetic data, dots correspond to training data and lines correspond to DINNs prediction.
Fig 8.
Learning history of parameter estimation (only β) of system (8) using DINNs.
Fig 9.
Box-plot (left) and violin-plot (right) of the relative error of β estimation corresponding to 40 simulations.
Fig 10.
Mean learning history and its 95% confident interval of β estimation corresponding to 40 simulations.
Fig 11.
Learning history of parameter estimation (β and δA) of system (8) using DINNs.
Table 2.
Parameter predictions and relative errors for system (8) using DINNs.
Fig 12.
Learning history of parameter estimation (β, δA and u) of system (16) using DINNs.
Table 3.
Parameter predictions and relative errors for system (16) using DINNs.
Fig 13.
Learning history of parameter estimation (β, δA and θ) of system (16) and Eq (15) using DINNs.
Table 4.
Parameter predictions and relative errors for system (15) and (16) using DINNs.
Fig 14.
Box-plot of the relative error of β estimation corresponding to 30 simulations of each noise level (1%, 5% and 10%).
Fig 15.
Violin-plot of the relative error of β estimation corresponding to 30 simulations of each noise level (1%, 5% and 10%).
Fig 16.
Mean learning history and its 95% confident interval of β estimation corresponding to 30 simulations of each noise level (1%, 5% and 10%).
Table 5.
β = 11 estimation and relative errors for system (8) using DINNs with different percentage of missing data.
Fig 17.
(a) Plot of the controlled reproduction number as a function of the fading rate b and the acquisition rate c as described in the formula (14); (b) Plot of the acquisition-fading kernel type vaccine efficacy function ϕ(t) = A(e−bt − e−ct) (see, formula (14)) corresponding to different choice of b and c as follows: (b, c) = (0.231, 0.0084) (green), (b, c) = (0.061, 0.0033) (red), (b, c) = (0.271, 0.0043) (black), (b, c) = (0.1, 0.004) (blue) and the corresponding values of
are indicated by the dots of corresponding color in the panel (a); (c) The number of daily cases for different choice of (b, c) are shown by corresponding colors.
Fig 18.
Daily cases for θ = 0.05 (blue), θ = 0.6 (green), θ = 0.95 (red).
We assume that pu(θ) = ρ(1 − θ). The parameter values are: αu = 1.5, pr = 0.4, ρ = 0.9, c = 0.8, u(0) = 0.8, and all other parameters are same as before.
Fig 19.
Maximum of daily cases is plotted as a function of pr and θ.
We assume that pu(θ) = ρ(1 − θ). The parameter values are: αu = 1.5, ρ = 0.9, c = 0.8, u(0) = 0.8, and all other parameters are same as before.