Parametric model fitting across multiple time periods
Since the vaccine dynamics and effect of mutant variants varied from January to July of 2021, four time periods were considered for fitting, and the 16 parameters (\(K_i,~a_i,~b_i,~c_i,~d_i,~e\) with i = 1-—children, 2—adults, 3—seniors) were obtained for each time period (see Fig. 2a). Three data sets were used for model fitting: COVID-19 Weekly Cases by Age, COVID-19 Weekly Deaths by Age, and COVID-19 Vaccinations by Age. The first time period considered was from January 9, 2021 to March 6, 2021. During this period, the most transmissible strain of COVID-19 present was the alpha variant41. The children were not vaccinated during this time period, but adults and seniors were. In the second period, from March 6 to May 8, 2021, it was assumed that the most transmissible COVID-19 strain present was the Delta strain since the CDC reports the introduction of the Delta variant in the US in early March of 202142. During this period the adults and seniors were vaccinated and the children were not. During the third time period, May 8–June 12, 2021, all three age groups were vaccinated. The alpha and Delta variants were considered the most dominant strains, and the Delta variant was beginning to contribute to a significant proportion of recorded cases43. There was a moderate decrease in new cases and deaths during this period as seen in Fig. 2b, c. During the last period, June 12–July 31, 2021, the proportion of cases due to the Delta variant increased. During this period, the vaccination rates were higher for children as compared to adults and seniors (see Fig. 2e). In summary, 4 sets of 16 parameters were obtained across the four time periods.
The fitted dimensionless parameters have properly reflected the dynamics of COVID-19 transmissibility during these four periods. The value of the relative transmission rate, \(K_i\), shows a decreasing trend for all three age groups during periods 1–3. However, as the delta became dominant, there was a sudden increase in the values of K for all age groups in period 4 (Fig. 2d).
The values of the vaccination parameter, \(a_i\), correctly captured the vaccination dynamics in the US during the fitted time periods. During the first two time periods, \(a_i\) was higher for the senior age group (\(i=3\)), which is consistent with prioritizing senior vaccination. In the last two periods (Fig. 2e), the children and adult age group, on average, have a higher vaccination rate than the senior age group.
The recovery rates for all three age groups show an increasing trend with time (Fig. 2f). This is likely due to the fact that a higher fraction of the population is gaining immunity through vaccinations during the fitted time periods. However, there is an increase in the mortality rate of the senior age group from the third to the fourth time period, indicated by a high value of c in Fig. 2g, which could be due to the Delta variant effect. Relative to the senior age group, the changes in mortality rates of children and adults are negligible, even during the last period when the Delta variant effect is visible in the population. Children contribute most to viral load in the first time period, while seniors contribute most during the second and third time period, and adults contributes most during the last period as indicated by the values of \(d_i\) in Fig. 2h. The dimensionless number e shows a decreasing trend followed by a sharp increase in the last time period as seen in Fig. 2i. This indicates that the time to infect an individual times the viral death rate increases when the Delta variant effect is dominant.
Figure 2 SIRDV-Virulence (Susceptible-Infected-Recovered-Dead-Vaccinated-Virulence) Model Fitting. (a) COVID-19 data are divided over four time periods based on the changing characteristics of the virus and vaccination dynamics. The fitted model is plotted against the data during each time period for (b) cumulative weekly cases, (c) cumulative weekly deaths, and cumulative weekly vaccinations (shown in the supplemental material). Fitted dimensionless parameters are shown for each age group (i=1-children,2-adults,3-seniors) during each of the four time periods: (d) viral transmission \(K_i\), (e) vaccination rate \(a_i\), (f) recovery rate \(b_i\), (g) death rate \(c_i\), (h) viral load \(d_i\), and (i) life span of virus e. In (d–i), the mean is the central point, and the error bars represent the \(25\mathrm{th}\) and \(75\mathrm{th}\) percentile values from Monte Carlo fitting simulations; relative parameters are displayed, where each data point is relative to the average of the respective parameter values of the three age groups for the first time period. Full size image
Figure 3 Effect of mutation on COVID-19 transmission. The mutant variant is modeled by assuming an increase in infection rate K, and an increase in vaccine inefficacy \(\sigma\) and by comparing the (a) original and (b) variant strain, shown pictorially by a higher concentration of virus particles and a higher proportion of particles after vaccination. Future predictions of infected cases and deaths are simulated for (c, f) children, (d, g) adults, and (e, h) seniors at increasing relative infection rates (relative to K from the fourth fitted time period) and a constant \(\sigma\) of 0.05. The worst case scenario (relative K of 2 and \(\sigma\) of 0.2) is simulated, resulting in the predictions of (i) total future active infected cases and (j) total future deaths, where the shaded regions show the error introduced using the \(25\mathrm{th}\) and \(75\mathrm{th}\) percentiles (from Monte Carlo fitting simulations) of the relative K values. Full size image
In the following sections, we will discuss the simulated predictions considering four important factors: (1) effect of the Delta variant, (2) vaccine optimization, (3) effect of Anti/Non-Vaxxers, and (4) effect of reinfection.
Effect of mutation on transmissibility
Mutation of SARS-CoV-2 has been largely responsible for the increased transmissibility due to an increased infection rate and reduced vaccine efficacy44. During the summer of 2021, the Delta variant was responsible for almost all recorded COVID-19 cases45. Therefore, it was important to study the effects of changes in transmissibility, \(K_i\), and vaccine inefficacy, \(\sigma\).
First, we study the variation in \(K_i\) while keeping \(\sigma\) constant. The simulated values of infection rate, \(K_i\), were taken to be 1, 1.2, 1.5 and 2 times the \(K_i\) values from the fourth fitted time period to take into account the effect of original strain and variants (Fig. 3a, b). For all age groups, the number of active cases increases with increasing \(K_i\). Specifically, for a two-fold increase in \(K_i\), the number of active cases at the peak of the pandemic increases by around 1.5–2 times for all age groups (Fig. 3c–e). In addition, a higher infection rate delays the time when the infection reaches its peak value, with differing peak infection times for each age group. Similarly, the total number of deaths increases with an increase in the value of \(K_i\) for all age groups (Fig. 3f–h). The effect is more pronounced in children and adults as compared to seniors, which could be due to a larger fraction of unvaccinated children and adults, relative to the senior population. For a two fold increase in \(K_i\), the total number of deaths increases by approximately 8\(\%\) in children and 10\(\%\) in adults, and only 2\(\%\) in seniors.
For the total US population (excluding ages 12 and under), a comparison was made between two extreme scenarios (Fig. 3i, j): no change in fitted parameters (from the fourth fitted time period) and the worst case scenario. In the worst case scenario, the relative dimensionless infection rate is doubled, and the vaccine inefficacy is increased from 0.05 to 0.2. While the total number of deaths is not significantly affected in the worst case scenario, the peak active infected cases increases by almost 2.5 times. Further simulation scenarios can be found in the supplemental material.
Vaccination optimization strategy
Optimization of vaccine distribution strategies among different age groups remains critical46. Specifically, it is necessary to study the effect of varying the vaccination rates and vaccination prioritization among each of the three age groups. Our study modeled the resulting completed vaccinations, cumulative cases, and cumulative deaths over a future time period as the vaccination parameters were varied.
We first determined the practical range for the dimensionless vaccination parameter, \(a_i\), of each age group. To do this, we assumed that future vaccination rates in the US would not reach the rates that they had reached previously (considering both individual age groups and the entire population under study), given the majority of the senior and adult age groups had already been vaccinated by the initial date of the future simulation time period and peaks had already been reached for the completed vaccination rates in each age group.
To study age group vaccination prioritization, a comparison was done using heat maps (Fig. 4). In general, the minimum infected cases and deaths and the maximum fraction of vaccinated population occur at the highest values of \(a_i\) for these age groups, yet the dependence of vaccination rate in each age group is different. For instance, the future total infections and deaths, as well as total vaccinated fraction are more dependent on \(a_2\) (adult age group) than on \(a_1\) (children age group) as shown in Fig. 4a, b. This is likely because the fraction of the adult population is much higher than that of the children. Comparing the children and senior age groups, it was seen that the total death and infection were more strongly dependent on children than on seniors (\(a_3\)) (Fig. 4d, e). A similar comparison among the senior and adult age group showed that total death and infected cases is more dependent on adults than seniors (Fig. 4g, h).
The dependence of COVID-19 dynamics on adult and children vaccinations are likely due to the differences in completed vaccinations for each age group. A large fraction of seniors (\(\sim\) 81.8%)47 had been fully vaccinated for COVID-19 by the end of July 2021. In comparison, only around 54.4% of the adult population was vaccinated by this time, while the percentage of children was about 34.4%47. Since a large fraction of the children and adult populations had yet to be vaccinated by the end of July 2021, a higher priority was needed to be given to these age groups over the senior age group for the future vaccine distribution, consistent with the strategy in the United States during that time48. For the vaccine distribution strategy, a higher priority to adult and children age groups over the senior age group was predicted to minimize total death and infections as the majority of the population in these two groups were more susceptible to the infection.
Figure 4 Vaccination Allocation Results. The effect of changing the vaccination allocation among each age group and vaccination roll-out speed is studied. (a–c) The senior vaccination parameter, \(a_3\), is held at its maximum practical value, while the children and adult vaccination values, \(a_1\) and \(a_2\), respectively, are varied from 0 to their maximum practical values to predict the proportion of the United States population that will (a) become infected and (b) die as a result of COVID-19 from July 2021 to July 2022. (c) The total number of completed vaccinations from July 2021 to July 2022 as a result of the same parameter changes are shown for reference. (d–f) The adult vaccination parameter is held at its maximum value while the children and senior vaccination parameters are varied. (g–i) The children vaccination parameter is held at its maximum value while the adult and senior vaccination parameters are varied. Full size image
Effect of anti/non-Vaxxer
The long-term effects of individuals unwilling and/or unable to receive the COVID-19 vaccination(s) was studied. Specifically, the effects of varying the proportion of COVID-19 ‘Anti/Non-Vaxxers’ in the susceptible population of each age group were observed to see how changes in the proportion of one age group could affect the number of deaths and cases of that same age group or other age groups.
To simulate this case, a modified compartmental model, PAIRDV-Virulence ( ProVaxxer-AntiVaxxer-Infected-Recovered-Dead-Vaccinated-Virulence), was developed that divided the Susceptible compartment into two new compartments: the COVID-19 ‘Anti/Non-Vaxxer’ compartment and the COVID-19 ‘Pro Vaxxer’ compartment, shown in Fig. 5a. The relationship between the Susceptible compartment of the SIRDV-Virulence ( Susceptible-Infected-Recovered-Dead-Vaccinated-Virulence) model and the COVID-19 ‘Anti/Non-Vaxxer’ and COVID-19 ‘Pro Vaxxer’ compartments of the PAIRDV-Virulence is \(x_{P_i} = (1-\omega _i)x_{S_i}\) and \(x_{A_i} = \omega _i x_{S_i}\), where \(x_{P_i}=P_i/N\) represents the fraction of individuals of age group i in the COVID-19 ‘Pro Vaxxer’ compartment, and \(x_{A_i}\) represents the fraction of individuals of age group i in the COVID-19 ‘Anti/Non-Vaxxer’ compartment. At the beginning of the time period of future simulations, \(x_{S_i}\) is the sum of \(x_{P_i}\) and \(x_{A_i}\), and \(\omega _i\) is the proportion of COVID-19 ‘Anti Vaxxers’ in the susceptible population of age group i.
Dimensionless equations were developed for the PAIRDV-Virulence model in order to conduct the future simulations:
$$\begin{aligned}&\frac{dx_{P,i}}{d\tau } = -x_{P,i}y\kappa _i-a_ix_{P,i}\;, \end{aligned}$$ (1)
$$\begin{aligned}&\frac{dx_{A,i}}{d\tau } = -x_{A,i}y\kappa _i\;, \end{aligned}$$ (2)
$$\begin{aligned}&\frac{dx_{I,i}}{d\tau } = x_{P,i}y\kappa _i+x_{A,i}y\kappa _i+\sigma x_{V,i}y\kappa _i-b_ix_{I,i}-c_ix_{I,i}\;, \end{aligned}$$ (3)
$$\begin{aligned}&\frac{dx_{R,i}}{d\tau } = b_ix_{I,i}\;, \end{aligned}$$ (4)
$$\begin{aligned}&\frac{dx_{D,i}}{d\tau } = c_ix_{I,i}\;, \end{aligned}$$ (5)
$$\begin{aligned}&\frac{dx_{V,i}}{d\tau } = a_ix_{P,i}-\sigma x_{V,i}y\kappa _i\;, \end{aligned}$$ (6)
$$\begin{aligned}&\frac{dy}{d\tau } = \sum ^{3}_{i=1}d_ix_{I,i}-ey\;, \end{aligned}$$ (7)
The PAIRDV-virulence model has the same parameters as the SIRDV-virulence, but one additional compartment. For COVID-19 ‘Anti/Non-Vaxxers’, the only way to exit the \(x_{A,i}\) compartment is by COVID-19 infection whereas the COVID-19 ‘Pro Vaxxers’ can exit the \(x_{P,i}\) compartment by either completing their vaccinations or by becoming infected with COVID-19. It is important to note that when \(\omega _i\) is 0 for all age groups, i, the PAIRDV-Virulence model is identical to the SIRDV-Virulence model, where the COVID-19 ‘Pro Vaxxer’ compartment of the PAIRDV-virulence model acts as the Susceptible compartment of the SIRDV-Virulence model.
As shown in Fig. 5b, an increase in the proportion of Anti/Non-Vaxxers will lead to higher virulence and in turn a higher number of infected cases and deaths. The higher the fraction of vaccinated people, the lesser will be the number of deaths and infected cases due to a lower virulence. The future simulations were run in sets, first varying \(\omega _i\) for each age group while keeping that of the other age groups constant. For these simulations, when \(\omega _i\) was varied for a single age group, i, the dynamics of age group i had significant changes, but negligible changes in the dynamics of other age groups were observed. For the second set of simulations, \(\omega _i\) was varied for all three age groups simultaneously. These results are shown in Fig. 5c–h. Based on these simulations, an increase in \(\omega _i\) will result in an increase in cases (Fig. 5c–e) and deaths (Fig. 5f–h) for all three age groups. The proportion of anti-vaxxers affects the children and adults more than the seniors, the reason being a large fraction of seniors had already been vaccinated by the start of the simulated prediction time period.
Figure 5 COVID-19 Anti/Non-Vaxxer Effect and the PAIRDV-Virulence ( ProVaxxer - AntiVaxxer - Infected - Recovered - Dead - Vaccinated - Virulence) Model. (a) The PAIRDV-Virulence model, a variation of the SIRDV-Virulence ( Susceptible - Infected - Recovered - Dead - Vaccinated - Virulence) model, is introduced, in which the susceptible population, \(S_i\), is divided into a sub-population that will not be vaccinated, \(A_i\), and a sub-population that will be vaccinated, \(P_i\). (b) When the proportion of the susceptible population that will not be vaccinated, \(\omega _i\), increases, the virulence, infections, and deaths increase (c–h). The predicted cases and deaths are shown for (c, f) seniors, (d, g) adults, and (e, h) children as a result of simultaneously changing \(\omega _i\) for each age group. Full size image
Effect of reinfection
We studied the effect reinfection using a modified compartmental model, which has a transition term from the recovered to the infected compartment and a reinfection factor. The modified model was abstracted as its dimensionless form:
$$\begin{aligned}&\frac{dx_{S,i}}{d\tau } = -x_{S,i}y K_i-a_ix_{S,i}\;, \end{aligned}$$ (8)
$$\begin{aligned}&\frac{dx_{I,i}}{d\tau } = x_{S,i}y K_i+\sigma x_{V,i}y K_i-b_ix_{I,i}-c_ix_{I,i}+f_ix_{R,i}y K_i\;, \end{aligned}$$ (9)
$$\begin{aligned}&\frac{dx_{R,i}}{d\tau } = b_ix_{I,i}-f_ix_{R,i}y K_i\;, \end{aligned}$$ (10)
$$\begin{aligned}&\frac{dx_{D,i}}{d\tau } = c_ix_{I,i}\;, \end{aligned}$$ (11)
$$\begin{aligned}&\frac{dx_{V,i}}{d\tau } = a_ix_{S,i}-\sigma x_{V,i}y K_i\;, \end{aligned}$$ (12)
$$\begin{aligned}&\frac{dy}{d\tau } = \sum ^{3}_{i=1}d_ix_{I,i}-ey\;, \end{aligned}$$ (13)
where \(f_i\) is the reinfection parameter accounting for the fraction of recovered people who can be reinfected, and all other dimensionless parameters are defined in Table 1. To test the effect of reinfection on the COVID-19 dynamics, we test varying values of \(f_i\) in our prediction simulations.
Table 1 Definition of variable for the dimensionless SIRDV-Virulence model. Full size table
We simulated the future infections and total deaths for 5 different values of \(f_1=f_2=f_3\) ranging from 0 (no reinfection) to 1 (entire recovered population being susceptible to reinfection). As seen in Fig. 6, as the value of \(f_i\) increases, the number of active infections increases. The increase in deaths with increasing \(f_i\) however is less significant than that for infections.