Multi-generational SIR modeling: Determination of parameters, epidemiological forecasting and age-dependent vaccination policies

Abstract

We use an age-dependent SIR system of equations to model the evolution of the COVID-19. Parameters that measure the amount of interaction in different locations (home, work, school, other) are approximated from in-sample data using a random optimization scheme, and indicate changes in social distancing along the course of the pandemic. That allows the estimation of the time evolution of classical and age-dependent reproduction numbers. With those parameters we predict the disease dynamics, and compare our results with out-of-sample data from the City of Rio de Janeiro. Finally, we provide a numerical investigation regarding age-based vaccination policies, shedding some light on whether is preferable to vaccinate those at most risk (the elderly) or those who spread the disease the most (the youngest). There is no clear upshot, as the results depend on the age of those immunized, contagious parameters, vaccination schedules and efficiency.

Keywords: COVID-19; Compartmental modeling; Epidemiology; SIR; Vaccination.

Fig. 1

Probability of being clinical among infectives ρ (left) and lethality weight w_d (right).

Fig. 2

In this figure we plot the 7-day moving average data on new daily cases. The dashed red plot corresponds to real data, and the solid blue plot depicts clinical cases modeled by SIR. Above, and in all figures that follow, what we call data is actually a 7-day moving average of the real data.

Fig. 3

This figure is related to fatalities, displaying daily deaths. Again, real data is in dashed red, and computed daily deaths by the model come in solid blue.

Fig. 4

On the left we plot the estimated values of β_w, except for the first one (that has a value roughly ten times bigger than the others). For visualization purposes we interpolated the discontinuous function β_w using cubic splines. On the right we display the values of the lethality strength μ, defined in (12). The blue curve is the least square quadratic curve.

Fig. 5

Reproduction and replacement numbers, ${\mathcal{R}}_0$ and ${\mathcal{R}}_t$. Note that ${\mathcal{R}}_0$ is transient since it depends on the time dependent contact matrix. We also draw a line at y = 1 highlighting the recovering threshold. Comparing with Fig. 2, we see that there is a correspondence between ${\mathcal{R}}_t$ crossing the threshold (up or down) and the number of new cases increasing or decreasing.

Fig. 6

Number of sub-clinical cases in solid blue, corresponding roughly to ten times the number of official cases in dashed red.

Fig. 7

Number of daily new cases. The dashed red plot indicates the data and the solid black plot represents the simulation results for frozen β_w and μ corresponding to the best/intermediary/worst case scenarios. Note that, for most of the 70-day forecast, that data lie within the interval predicted by the model.

Fig. 8

Similarly to Fig. 7, the figure displays the number of daily deaths, dashed red being data and solid black being the best/intermediary/worst cases predicted by the model.

Fig. 9

We present here the accumulated number of cases (left) and deaths (right). In both figures, the dashed red plots correspond to real data and the solid black plots represent the simulation results for frozen β_w and μ, for the best/intermediary/worst case scenarios.

Fig. 10

Population profile of Rio. The last column corresponds to the total population above 75 years old.

Fig. 11

Age-dependent replacement numbers corresponding to the dynamics of the disease in Rio.

Fig. 12

Accumulated deaths for β_w = 0.0064. We assume no vaccination (solid blue), vaccination for group aged 15–34 (solid plots) and over 50 (dashed plots). The vaccination takes place on day 100 (black), 150 (red) or 200 (magenta). Note that vaccinating the elderly population is always optimal in the short term, but not necessarily in the long run.

Fig. 13

Accumulated deaths for β_w = 0.0117, a higher rate of transmission. As before, we assume no vaccination (solid blue), vaccination for group aged 15–34 (solid plots) and over 50 (dashed plots). The vaccination takes place on day 100 (black), 200 (red) or 300 (magenta). Note that vaccinating the elderly population is always optimal in the short term, but not necessarily in the long run.

Fig. 14

Accumulated deaths for β_w = 0.0064, under varying vaccination efficiencies. Assume that at day 150, a vaccination campaign targets either the young (aged 15–34, solid plots) or those over 50 (dashed plots). The vaccination efficiency varies as follows: 100% (blue), 70% (black) or 50% (red).