Table 1.
Description of variables and parameters from the model in system (5).
Fig 1.
Simulations for a high restoration coefficient ν = 0.8.
The first subplot illustrates the restoration rate f(t, β) (dotted) and the reaction rate g(t, I) (solid). The remaining subplots show: The transmission rate β(t), the effective reproduction number , the new confirmed cases eE(t), and the cumulative cases E(t) + I(t) + R(t). The reaction coefficient in each subplot are chosen to be μ: 0.3 (blue); 0.4 (green); 0.5 (orange) and 0.6 (red), and the remaining parameter values and initial conditions are as in Table 2.
Fig 2.
Simulations for a medium restoration coefficient ν = 0.5.
The first subplot illustrates the restoration rate f(t, β) (dotted) and the reaction rate g(t, I) (solid). The remaining subplots show: The transmission rate β(t), the effective reproduction number , the new confirmed cases eE(t), and the cumulative cases E(t) + I(t) + R(t). The reaction coefficient in each subplot are chosen to be μ: 0.3 (blue); 0.4 (green); 0.5 (orange) and 0.6 (red), and the remaining parameter values and initial conditions are as in Table 2.
Fig 3.
Simulations for a low restoration coefficient ν = 0.2.
The first subplot illustrates the restoration rate f(t, β) (dotted) and the reaction rate g(t, I) (solid). The remaining subplots show: The transmission rate β(t), the effective reproduction number , the new confirmed cases eE(t), and the cumulative cases E(t) + I(t) + R(t). The reaction coefficient in each subplot are chosen to be μ: 0.3 (blue); 0.4 (green); 0.5 (orange) and 0.6 (red), and the remaining parameter values and initial conditions are as in Table 2.
Table 2.
Initial conditions and parameter values and their units, for the simulations in Figs 1–6.
Fig 4.
Simulations for a high restoration coefficient ν = 0.8.
The first subplot depicts the shape of the compliance curve h(t) = (a + 2b)t/(t2 + at + b2), with b = 90, a = 40, and the second plot in the first row illustrates the restoration rate f(t, β) (dotted) and the reaction rate g(t, I) (solid). The remaining subplots show: The transmission rate β(t), the effective reproduction number , the new confirmed cases eE(t), and the cumulative cases E(t) + I(t) + R(t). The maximum value μ0 of the reaction coefficient (μ(t) = μ0 h(t)) used in each subplot is: μ0: 0.3 (blue); 0.4 (green); 0.5 (orange) and 0.6 (red), and the remaining parameter values and initial conditions are as in Table 2.
Fig 5.
Simulations for a medium restoration coefficient ν = 0.5.
The first subplot depicts the shape of the compliance curve h(t) = (a + 2b)t/(t2 + at + b2), with b = 90, a = 40, and the second plot in the first row illustrates the restoration rate f(t, β) (dotted) and the reaction rate g(t, I) (solid). The remaining subplots show: The transmission rate β(t), the effective reproduction number , the new confirmed cases eE(t), and the cumulative cases E(t) + I(t) + R(t). The maximum value μ0 of the reaction coefficient (μ(t) = μ0 h(t)) used in each subplot is: μ0: 0.3 (blue); 0.4 (green); 0.5 (orange) and 0.6 (red), and the remaining parameter values and initial conditions are as in Table 2.
Fig 6.
Simulations for a low restoration coefficient ν = 0.2.
The first subplot depicts the shape of the compliance curve h(t) = (a + 2b)t/(t2 + at + b2), with b = 90, a = 40, and the second plot in the first row illustrates the restoration rate f(t, β) (dotted) and the reaction rate g(t, I) (solid). The remaining subplots show: The transmission rate β(t), the effective reproduction number , the new confirmed cases eE(t), and the cumulative cases E(t) + I(t) + R(t). The maximum value μ0 of the reaction coefficient (μ(t) = μ0 h(t)) used in each subplot is: μ0: 0.3 (blue); 0.4 (green); 0.5 (orange) and 0.6 (red), and the remaining parameter values and initial conditions are as in Table 2.
Fig 7.
Simulation fitting COVID-19 data from Chile.
The first subplot depicts the restoration rate f(t, β) (dotted) and reaction rate g(t, I) (solid); the second plot the transmission rate β(t); and in the third plot the blue dots represent data of daily confirmed new cases of COVID-19 in Chile, from March 16th, 2020, to February 16th, 2021 [60]. The red curve represents the least square fit of the model to the data with parameter values as in Table 3 for the population of Chile, with N = 18 million individuals and initial conditions of the model (5) taken to be E0 = 20, I0 = 81, R0 = 0, S0 = N−E0 − I0 − R0. The fit produces a root-mean-square error (RMSE) of 792.89.
Fig 8.
Simulation fitting COVID-19 data from Italy.
The first subplot depicts the restoration rate f(t, β) (dotted) and reaction rate g(t, I) (solid); the second plot the transmission rate β(t); and in the third plot the blue dots represent data of daily confirmed new cases of COVID-19 in Italy, from February 24th, 2020, to October 31st, 2020 [61]. The red curve represents the least square fit of the model to the data with parameter values as in Table 3 for the population of Italy, with N = 60.5 million individuals and initial conditions of the model (5) taken to be E0 = 81, I0 = 566, R0 = 0, S0 = N − E0 − I0 − R0. The fit produces a root-mean-square error (RMSE) of 1083.49.
Table 3.
Parameters for the simulations in Figs 7 and 8 for the case of Chile and Italy respectively.