ABSTRACT

A. Kinetic models

Kinetic models in biochemistry and biophysics apply to molecules being jiggled around by the molecules surrounding them. Similar models can, and have, been applied to a wide variety of other applications. Diffusion between two compartments can be modeled as a reversible first-order reaction (3). Drug elimination and radioactive decay are processes analogous to an irreversible first-order reaction (1). Population dynamics and epidemiological models can also be formulated using similar mathematical models. The approach we will use is implemented as a simple finite difference (FD) model based on the Euler method, made famous by the 2016 movie Hidden Figures. These methods are ideal for introducing undergraduates to modeling kinetic processes because the computational steps of the FD model are represented by successive rows of the spreadsheet (1, 3). Using Excel's Solver feature (version 2109, Microsoft Corporation, Redmond, WA), the predictions of these FD models can be fitted to experimental data using least squares (LS) techniques (1). As we will discover, the same approach can also be used to model the spread of COVID-19 and compare the model predictions with reported data for confirmed cases of COVID-19 in the United States (2). As students discover, these simple FD models can do a surprisingly good job of modeling the spread of COVID-19 in the United States from February 2020 through August 2021.

B. Learning objectives and pedagogical approach

The educational objectives are for students to learn how to

1. apply FD methods to introductory epidemiological models using the approach presented in (1);

2. apply systematic model development techniques to a complex real-world problem; and

3. use nonlinear LS methods to test the predictions of the various numerical models by fitting them to published data for the United States.

The pedagogical approach is a guided inquiry active learning case study. Students begin by investigating the simplest possible “unlimited growth” epidemiological model. They then validate it by fitting it to published cases per day data for the United States as a whole. That model is then systematically modified to account for finite population size, recovery from COVID-19, changes in the infection rate coefficient due to changes in social distancing (and the delta variant), and finally, to account for vaccinations. The teaching materials use a scaffolded approach that focuses on the impact of each model parameter in a step-by-step manner (2).

The use of a systematic step-by-step approach is important for epidemiological models because they inherently produce exponential growth or decay in the infection rate. As a result, they are mathematically comparable to kinetic models of ion channel permeation that also predict exponential dependence (of electrical current on membrane voltage) and where it was shown that the presence of too many parameters led to fitted parameters of questionable physical significance (4). Hence, in the modeling exercise presented here, students are only asked to add additional model parameters if the data call for them (Occam's razor).

A. Unlimited growth (UG) model

In the simplest models of epidemiology, the population is split into two groups: susceptible and infectious (SI; Fig 1). The simplest of those SI models is the unlimited growth (UG) model, in which the infection rate R_i (arrow in Fig 1) is given by (1) $$R_i\text{ = }{k}_i{N}_i$$ where k_i is the infection rate constant and N_i is the number infectious. The idea behind the UG model and Eq. 1 is that an infectious person wanders randomly throughout the model population, just like a molecule in aqueous solution, infecting others with a rate characterized by an infection rate constant k_i, where k_i = 0.25 d⁻¹ means that an infectious person infects a susceptible person every 4 d, on average, causing them to “jump” from box s → i when they become infectious (usually some days after contact with the infectious person).

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I chose to use the symbol N_i for the number infectious to make the notation match standard biochemical practice for molecular systems. However, epidemiologists prefer using the single uppercase letter I instead of N_i. They also prefer to use Greek letters for the rate constants so that the infection rate is written as βI (5). We will stick with using the traditional chemistry k with a descriptive subscript for rate constants and N with a descriptive subscript for numbers in the boxes of the models.

The UG model of Figure 1 is unlimited because the model population is assumed to be infinite, and an infectious person stays infectious forever. Both of those assumptions are clearly incorrect, but they make for the simplest model. We will discuss making the population finite and modeling recovery in the next sections. However, the UG model gives students important insights into the initial spread of the virus: when the general population did not know that SARS-CoV-2 was in their local area.

Equation 1 is different from most first-order reactions because the rate of jumps into box i is proportional to the number N_i already in box i. This can be contrasted with other processes such as first-order drug elimination, where the rate of jumps out of the body (and into the bladder) is proportional to the number in the body (1).

According to the UG model of Figure 1, the FD equation for the small change δN_i in the number infectious N_i during a short time δt (the timestep) is given by (2) $$\text{δ}{N}_i=R_i\text{δ}t$$ where R_i is given by Eq. 1. Hence, the model can be implemented in a spreadsheet using the following condensed FD instructions (2): (3) $$t^{\text{new}}=t^{\text{old}}+\text{δ}t$$ (4) $$R_i^{\text{new}}=k_i*N_i^{\text{old}}$$ and (5) $$N_i^{\text{new}}=N_i^{\text{old}}+R_i^{\text{new}}*\text{δ}t$$ where the superscript “old” refers to the previous row in the spreadsheet (previous computational step) and “new” refers to the current row of the spreadsheet (current computational step). Hence, $N_i^{\text{old}}$ is the old number infectious (previous row at time t^old) and $N_i^{\text{new}}$ is the new number infectious (current row at time t^new = t^old + δt); see the glossary of symbols in the appendix. The asterisk symbol is included in Eqs. 4, 5, and others to remind students that it is required in Excel formulas. The first activity asks students to write out a complete FD algorithm based on Eqs. 3–5 to calculate the infection rate R_i(t) and number infectious N_i(t). Students implement the algorithm in the rows of a preformatted spreadsheet, and by plotting the model predictions on semi-log graphs, they discover that the UG model predicts an exponential growth in both the number infectious N_i and the infection rate R_i (2, 6).

In a “show-that” exercise, students solve the elementary differential equation implied by Eqs. 1 and 2 to give the following analytical solution (6) $$N_i=N_0{e}^{k_{i}t}$$ which predicts an exponential growth in the number infectious from an initial number infectious N₀ (at t = 0). Substituting Eq. 6 into Eq. 1 yields (7) $$R_i=k_i{N}_0{e}^{k_{i}t}$$ for the infection rate R_i(t). Equation 7 is important because R_i corresponds to the publicly available number of new confirmed COVID-19 cases reported per day. Because both N_i and R_i are exponential functions of time, students are able to show that the exponential growth can be characterized by a doubling time (8) $$t_d=\frac{\ln 2}{k_i}$$

Students then compare the predictions of Eq. 7 with published cases per day data using a preformatted spreadsheet: first using Excel's “exponential trendline” and then using the LS techniques implemented using Excel's Solver. Figure 2 shows the resulting LS fit to Eq. 7, with data reported (7) for the United States during the 19 d after 26 February 2020. As students discover, the model does a surprisingly good job of explaining the reported data, validating the UG model's prediction of exponential growth for the initial uncontrolled spread of the contagion (2).

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B. Finite population (FP) model

The FD diagram in Figure 3 shows a simple modification of the UG model that accounts for the finite size of the population. In this finite population (FP) model, the infection rate is given by (9) $$R_i\text{ = }{k}_i{N}_{i}s$$ where s is the susceptible fraction of the population that is defined by (10) $$s\equiv \frac{N_s}{N}$$ where N_s is the number susceptible and N (with no subscript) is the total number of people in the model population, where (11) $$N\text{ = }{N}_s\text{ + }{N}_i$$

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Hence, we are assuming that the model population size does not change during the modeling time (no births or deaths). As a result, we can update N_s using the instruction (12) $$N_s^{\text{new}}=N-N_i^{\text{new}}$$

The idea behind Eq. 9 is that people behave like molecules in solution. They randomly bump into each other at a constant rate, on average. Infections occur with a fixed probability when people get close together. If we assume that encounters occur at a constant rate, then the probability that an infectious person interacts with a susceptible person (as opposed to another infectious person) is simply s, the fraction of the population that is susceptible. In other words, s is the fraction of people an infectious person encounters that are still susceptible to the virus. These simple assumptions are easy to understand, but it is important to remember that people are not molecules.

By substituting the definition of s (Eq. 10), into Eq. 9 and solving Eq. 11 for N_s, students show that the FP model can be calculated using (13) $$R_i^{\text{new}}=k_i*N_i^{\text{old}}*N_s^{\text{old}}/N$$ and the instructions in Eqs. 5 and 12.

Students write out a complete algorithm for the FP model and implement it in a preformatted spreadsheet. They are then able to investigate how social distancing can “flatten the curve” by halving the infection rate constant from k_i = 0.3 d⁻¹ to k_i = 0.15 d⁻¹, as shown in Figure 4. The effect of having a finite population is that the infection rate no longer increases exponentially without limit, and there is a peak in the R_i(t) curve that can be flattened by social distancing. Interestingly, the N_i(t) curve (not shown) exhibits the classic logistic growth first reported by Verhulst (8) and later by McKendrick (9). According to the FP model, social distancing merely delays the inevitable. Eventually, everyone in the model population becomes infectious despite the reduced infection rate constant. As we will discuss in the next section, the SIR model is qualitatively different, because social distancing can reduce the ultimate number infected, and it can even prevent the outbreak from occurring at all.

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C. SIR model

The main problem with the FP model is that people do not recover—ever. Clearly, that is not realistic. People do recover from COVID-19 and hence stop being infectious after a period of time. Figure 5 shows an FD diagram for the SIR epidemiological model. Named after the letters used for the three boxes, it is an important base model in epidemiology developed by Kermack and McKendrick in 1927 (10).

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The three boxes in Figure 5 represent the possible states of people in the model population. In the SIR model, N_s, s, and N_i have the same meaning as the FP model. The new state variable is N_r, which represents the number recovered. It is the number of individuals in the model population that have been infected but have now recovered and are no longer infectious and are further assumed to be immune to the disease forever. The symbol N_r more correctly stands for the number removed from the susceptible or infectious boxes. In addition to recovering, individuals can be removed from the number infectious by being isolated or quarantined from the susceptible portion of the population, and they are also removed by death. All those individuals are represented by box r. We also have a relationship with the total number N in the model population, and it spells out the initials of the SIR model in the subscripts of the bookkeeping equation (14) $$N\text{ = }{N}_s\text{ + }{N}_i\text{ + }{N}_r$$

Now that we have discussed the three boxes, let us talk about the arrows between boxes in Figure 5. The first arrow from box s → i represents the rate of infection R_i = k_iN_is (Eq. 9), the same equation that we used for the FP model of Figure 3. The second arrow from box i → r represents the rate of recovery. That recovery rate is given by (15) $$R_r\text{ = }{k}_r{N}_i$$ where k_r is the recovery rate constant and the mean residence time in box i (1) is predicted to be (16) $${\tau }_i=\frac{1}{k_r}$$

We will call τ_i the mean infectious time. It can be approximated by a quantity that can be measured clinically, the mean recovery time.

Using the information above, students write out FD instructions using Eqs. 13 and 17–20. (17) $$R_r^{\text{new}}=k_r*N_i{}^{\text{old}}$$ (18) $$N_i{}^{\text{new}}=N_i{}^{\text{old}}+\left(R_i^{\text{new}}-R_r^{\text{new}}\right)*\text{δ}t$$ (19) $$N_r^{\text{new}}=N_r^{\text{old}}+R_r^{\text{new}}*\text{δ}t$$ (20) $$N_s^{\text{new}}=N-N_i{}^{\text{new}}-N_r^{\text{new}}$$

They then write out a complete algorithm for the SIR model, implement it in a preformatted spreadsheet, and then they answer a series of questions comparing the properties of the SIR model with the previous models.

Subsequently, students investigate the effects of social distancing by reducing the infection rate constant k_i. Figure 6 shows the kind of graphical information students observe in the spreadsheets. As shown in Figure 6, they discover that not everyone in the model population needs to be infected by the end of the pandemic if social distancing is implemented and maintained until the pandemic has subsided, i.e., the number susceptible N_s does not reach zero at the end of the pandemic, meaning that some susceptible individuals were never infected (see week 50 in Fig 6a).

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A universal feature of the SIR model is that it predicts an “exponential dragon” in the infection rate R_i(t) (fig 6b and fig 12.15 of (2)), whose duration is determined by the value of the infection rate constant k_i. The peak of the exponential dragon exhibits a characteristic inverted vee shape on a semi-log plot (2). The idea of using a dragon analogy for explosive exponential growth was inspired by the expression “tickling the dragon's tail” that is based on a remark by Richard Feynman about the dangers of some ill-advised early nuclear experiments in which exponential growth had the potential for similar catastrophic consequences. See section 12.4 of (2).

In a guided inquiry exercise, students investigate the effect of the model population size N on the predictions of the model. They discover that model populations of all sizes in the SIR model behave in a similar manner and produce the same shape curves for N_s, N_i, N_r, R_i, and R_r, independent of the size of the model population. In a later show-that problem, they discover that the SIR model can be reformulated in terms of fractional variables s, i, and r, where the fraction susceptible s is defined by Eq. 10, the fraction infectious i is defined by (21) $$i\equiv \frac{N_i}{N}$$ and the fraction recovered r is defined by (22) $$r\equiv \frac{N_r}{N}$$

The fact that the SIR model predicts the same behavior independent of model population size is an important property of the SIR model.

D. Herd immunity

After the peak in N_i(t), the SIR model predicts that the number infectious N_i will steadily decline because the rate of infection R_i is less than the rate of recovery R_r. This can be related to the epidemiological concept of herd immunity (11). One way to see how the two concepts are related is to consider the quantity 1 − s_p, which is the cumulative fraction that have been infected at the time t_p of the peak in N_i(t). We can then define i_p as the fraction infectious and r_p as the fraction recovered at time t_p, respectively. Hence, from the bookkeeping Eq. 14, we have (23) $$s_p\text{ + }{i}_p\text{ + }{r}_p\text{ = 1}$$ so that 1 − s_p = i_p + r_p. Hence, the quantity 1 − s_p is the sum i_p + r_p, which is the cumulative total number of people that have been infected at time t_p.

Our SIR model assumes that individuals who have been infected cannot be infected again. Hence, anyone who has already been infected is permanently immune in our SIR model. As a result, the fraction of the model population that are immune at any time can be written as (24) $$h\text{ = }i\text{ + }r\text{ = 1 }-s$$ where h is the fraction immune or the immune fraction of the model population. Once the immune fraction h reaches (25) $$h_p\text{ = 1}-s_p$$ the recovery rate R_r is larger than the infection rate R_i, and the model predicts that the disease will be in decline and eventually die out. The fraction h_p is the herd immunity threshold. If the fraction immune is greater than or equal to h_p, i.e., if (26) $$h\ge h_p$$ then the disease will be in decline rather than growing (as indicated by whether N_i(t) decreases or increases, respectively).

E. Finding the peak in the curve and ${ℛ}_0$

In a guided inquiry exercise, students are asked to consider why the infection rate curve R_i(t) always cuts through the peak in the recovery rate curve R_r(t); see, for example, Figure 6b. Students discover that the peak in N_i(t) and R_r(t) occurs when (27) $$R_i\text{ = }{R}_r$$

By substituting Eqs. 9 and 15 into Eq. 27, students show that the value of the fraction susceptible s at the peak in N_i is given by (28) $$s_p=\frac{k_r}{k_i}=\frac{1}{{ℛ}_0}$$ where ${ℛ}_0$ is the basic reproduction number that is given by (29) $$R_0\equiv k_i{\tau }_i=\frac{k_i}{k_r}=\frac{1}{s_p}$$

The basic reproduction number ${ℛ}_0$ was made famous in the 2011 movie Contagion, and it is arguably the most important widely discussed parameter of epidemiological models (12). The first part of Eq. 29, ${ℛ}_0\equiv k_i{\tau }_i$ ≡ k_iτ_i defines $s\approx 1;k_i$ as the number infected by a single individual at the beginning of the outbreak when s ≈ 1; k_i is the average number of people infected by a single infectious individual per day; and τ_i is the mean number of days that they are infectious.

Because ${ℛ}_0$ = 1/s_p, we can write the herd immunity threshold h_p in Eq. 25 in terms of the basic reproduction number ${ℛ}_0$ (30) $$h_p=1-\frac{1}{{ℛ}_0}$$

The best fit value of ${ℛ}_0$ students obtain by fitting the US data with τ_i = 8 d is ${ℛ}_0$ ≈ 4.1 so that the herd immunity threshold for the original variants of COVID-19 is h_p ≈ 0.76 or about 76%, in the absence of any social distancing measures. Recall that in the SIR model, the value of the infection rate constant k_i depends on the level of social distancing. Hence, whenever the infection rate R_i(t) is decreasing, we have technically passed the herd immunity threshold for the current value of k_i.

A. Initial transition to social distancing

When students come to this section, they already know that the SIR model can successfully model the pandemic in the United States during the initial period of exponential growth (epoch 1 in Fig 7) using an infection rate constant of k_i = k₁ = 0.505 d⁻¹. In a guided inquiry activity, students discover that the SIR model can successfully model the exponential decay during the second epoch (epoch 2 of social distancing) up to Memorial Day, 25 May 2020. The fit is excellent, but the initial conditions for the fit are arbitrary and meaningless. My first attempt at modeling the transition using a step change in the infection rate coefficient k_i was an abject failure. As shown in Figure 7, the model can successfully model both epoch 1 (exponential growth) and epoch 2 (social distancing), but the step change in k_i between epochs 1 and 2 produced a spike in the fitted model of R_i(t) that is clearly inconsistent with the reported data during the transition period.

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As the title of this section states, the solution to this conundrum was to change the transition function for k_i(t) from a step change to a Gaussian transition function. Once again, it turns out that Excel is a convenient platform for modeling COVID-19 because it includes a function NORM.DIST that implements the Gaussian distribution, both as a probability density function and as a cumulative probability. The latter is what we need for our Gaussian transition function. The justification for a Gaussian transition function is that not all states, communities, or individuals took up social distancing at the same time (or to the same extent). The simplest assumption is that those transition times are normally distributed and hence can be represented by a Gaussian transition function. From a modeling perspective, a Gaussian transition function is appealing because it introduces only one additional parameter for the standard deviation of the Gaussian that accounts for the statistical spread in the times when individual people changed their level of social distancing. The implementation that students use in Excel has two parameters: t₁₂ is the mean transition time between epochs 1 and 2; and σ₁₂ is the standard deviation of the distribution of transition times between epochs 1 and 2. The Excel function for calculating the cumulative probability F₁₂(t) of the normal (Gaussian) distribution for the transition times between epochs 1 and 2 is (31) $$F_{12}^{\text{new}}=\text{NORM}.\text{DIST}\left(t^{\text{new}},t_{12},{\sigma }_{12},\text{TRUE}\right)$$ where $F_{12}^{\text{new}}$ is the value of the Gaussian cumulative probability at time t^new and TRUE is the value of the “cumulative” parameter of the NORM.DIST Excel function (2, 13). The corresponding probability density p₁₂(t) can be calculated using cumulative = FALSE, i.e. (32) $$p_{12}^{\text{new}}=\text{NORM}.\text{DIST}\left(t^{\text{new}},t_{12},{\sigma }_{12},\text{FALSE}\right)$$

The time-dependent infection rate coefficient k_i(t) can then be calculated using (33) $$k_i^{\text{new}}=k_1+F_{12}^{\text{new}}*\left(k_2-k_1\right)$$

Students implement the model in a preformatted spreadsheet and fit the model to the published data using LS techniques and Excel's Solver (Fig 8) (2, 13).

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Using this transition function, students estimate the number of lives that were lost because the rest of America did not follow New York City's lead with mask wearing and social distancing. The estimate they obtain is 60,000+ lives lost by Memorial Day (25 May 2020) based on the observed crude mortality ratio of m_c = 0.0595 in the European Centre for Disease Prevention and Control (ECDC) data (7). This estimate is based on a simple empirical correlation students discover between the observed mortality rate and the observed infection rate R_i (2, 13).

B. The summer surge

Using similar techniques, students are also able to model the summer surge resulting in the fit to the US data up to Labor Day (7 September 2020) shown in Figure 9. Students add parameters for epoch 3 with an infection rate constant k₃ for the relaxed social distancing at the beginning of the summer surge, a transition time t₂₃, and standard deviation σ₂₃. The decline in the summer surge is modeled with an infection rate constant k₄ for the stricter social distancing during the decline in the summer surge and a corresponding transition time t₃₄ and standard deviation σ₃₄.

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C. The fall surge and the effect of population size on the SIR model

According to a Centers for Disease Control and Prevention (CDC) report dated 19 January 2021, only 1 in 4.6 (95% uncertainty interval (UI), 4.0−5.4) of total COVID-19 infections were reported in the period from February to December 2020 (14, 15). Rounding up to one significant figure, that means that only about 1 in 5 of actual COVID-19 infections are represented in the data to which the model is fitted in Figure 10, i.e., q ≈ 20%, where q is the fraction of the actual US population that is represented in the reported cases per day data, defined by (34) $$q\equiv \frac{N}{N^{\star }}$$ and N^⋆ = 3.3 × 10⁸ is the estimated actual population of the United States. There are two ways to account for this shortfall in reported cases per day data. One way is to multiply the cases per day by a factor of 1/q = 5, and the other way is to reduce the model population size to N = qN^★, or 20% of the actual US population. We chose the latter, because it makes the model predictions directly comparable to the reported cases per day data.

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Figure 10 shows the predictions of the SIR model when students fit a fifth epoch with infection rate constant k₅, a transition time t₄₅, and standard deviation σ₄₅. The subsequent model then predicts a fall exponential dragon that depends on the model population size. Figure 10 shows the range of predictions of the fitted model for model population sizes of q = 10%, 20%, and 40% of the actual US population, i.e., N = 3.3 × 10⁷, 6.6 × 10⁷, and 1.32 × 10⁸, respectively.

As shown in Figure 10, changing the model population size and refitting has no visible effect on the fitted model up to Thanksgiving Day (26 November 2020). However, the model predictions for the fall exponential dragon almost immediately diverge, depending on the size of the model population. In a guided inquiry exercise, students discover that the fitted infection rate constants (k₁ − k₅) are different for the three fitted models and that the difference in the predicted behavior after Thanksgiving is primarily due to differences in the values of the susceptible fraction s at Thanksgiving and partially due to differences in the fitted infection rate constants. The values of s at Thanksgiving are s = 0.61, 0.80, and 0.90, and the fitted values of the infection rate constant are k₅ = 0.23, 0.18, and 0.17 d⁻¹ for models with q = 10%, 20%, and 40%, respectively. The fitted value of k₅ = 0.23 d⁻¹ for q = 10% is clearly higher than the other two fits, and students discover that q = 10% of the actual US population is just about the smallest model population size that is consistent with the data up to 26 November 2020 (Thanksgiving).

For the two higher fits in Figure 10, the infection rate constants are approximately the same (k₅ ≈ 0.18 d⁻¹) so that the difference between them is primarily due to the difference in the susceptible fraction at Thanksgiving (s = 0.80 and 0.90), respectively. Recall that the susceptible fraction s is a monotonically decreasing function in the model, and the q = 20% model starts closer to the peak value of s_p = 0.70 for epoch 5. In other words, the primary difference between the fits with q = 20% and 40% is that there are more susceptible people left in the model population at Thanksgiving if q = 40% rather than q = 20%.

A. SIRV model

In late December 2020, the US Federal Drug Administration (FDA) approved COVID-19 vaccines for use in the United States (emergency use authorization). Figure 11 shows a simple model of how vaccination can be added to the SIR model. The new feature is box v for fully vaccinated individuals. The arrows entering box v indicate the rates at which individuals are effectively vaccinated. They are then assumed to be permanently immune to COVID-19. The three rates leading to box v are labeled R_v,s, R_v,i, and R_v,r where the subscripts “s,” “i,” and “r” indicate the originating box. These three vaccination rates are related to the total rate of vaccination R_v in the model population by (35) $$R_v\text{ = }{R}_{v,s}\text{ + }{R}_{v,i}\text{ + }{R}_{v,r}$$

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The bookkeeping equation for the susceptible-infected-recovered-vaccinated (SIRV) model is (36) $$N\text{ = }{N}_s\text{ + }{N}_i\text{ + }{N}_r\text{ + }{N}_v$$ where the subscripts once again spell out the letters of the model.

The number of vaccinated individuals in the model population is calculated from the number fully vaccinated $N_v^{*}$ reported by Our World in Data (OWID) (16) using $N_v^{\text{new}}=q{N}_v^{\text{*new}}$. However, because the OWID spreadsheet data contain some blank cells, the following Excel instruction is used (37) $$N_v^{\text{new}}=\text{IF}\left(N_v{}^{\star \text{new}}=0,N_v^{\text{old}},q*N_v{}^{\star \text{new}}\right)$$ and the FD instruction for the vaccination rate in the model population is (38) $$R_v^{\text{new}}=\left(N_v^{\text{new}}-N_v^{\text{old}}\right)/\text{δ}t$$

The rate of vaccination of susceptible individuals in the model population can be calculated using (39) $$R_{v,s}^{\text{new}}=N_s^{\text{old}}*R_v^{\text{new}}/\left(N_s^{\text{old}}+N_i^{\text{old}}+N_r^{\text{old}}\right)$$ and similarly for $R_{v,i}^{\text{new}}$, and $R_{v,r}^{\text{new}}$. Eq. 39 and the corresponding equations for $R_{v,i}^{\text{new}}$ and $R_{v,r}^{\text{new}}$ assume that individuals in each of the three boxes s, i, and r are equally likely to be vaccinated. Hence, in the SIRV model, the numbers in boxes i and r can be calculated using (40) $$N_i{}^{\text{new}}=N_i{}^{\text{old}}+\left(R_i^{\text{new}}-R_r^{\text{new}}-R_{v,i}^{\text{new}}\right)*\text{δ}t$$ (41) $$N_r^{\text{new}}=N_r^{\text{old}}+\left(R_r^{\text{new}}-R_{v,r}^{\text{new}}\right)*\text{δ}t$$

Combining Eqs. 39–41 with bookkeeping Eq. 36 yields the following FD instructions for the numbers in boxes i, r, and s. (42) $$N_i{}^{\text{new}}=N_i{}^{\text{old}}+\left(R_i^{\text{new}}-R_r^{\text{new}}-N_i^{\text{old}}*R_v^{\text{new}}/\left(N-N_v^{\text{old}}\right)\right)*\text{δ}t$$ (43) $$N_r^{\text{new}}=N_r^{\text{old}}+\left(R_r^{\text{new}}-N_r^{\text{old}}*R_v^{\text{new}}/\left(N-N_v^{\text{old}}\right)\right)*\text{δ}t$$ and (44) $$N_s^{\text{new}}=N-N_i{}^{\text{new}}-N_r^{\text{new}}-N_v^{\text{new}}$$

Using a preformatted spreadsheet, students use Eqs. 37, 38, and 42–44 to implement the SIRV model and compare its predictions with reported data (see below). Note that when comparing the model variables with the published data, it is important to recall that all vaccinations are reported, but only about 1 in 5 infections are reported.

B. Modeling epoch 5—the fall dragon

Figure 12 shows the SIRV model fitted to the US data up to Thanksgiving Day (26 November 2020), with a model population of q = 21.7% of the actual US population, i.e., N = 7.17 × 10⁷ based on the CDC estimate (15). The predictions of the SIRV model in Figure 12 were made using the number fully vaccinated $N_v^{\text{*new}}$ that was reported daily by OWID (16). Figure 12a includes a plot of the infection rate coefficient k_i(t), showing the changes in social distancing in the fitted model. An important feature of the prediction is that none of the model parameters were changed after Thanksgiving Day. Specifically, the infection rate coefficient k_i remains constant at the same value k_i = k₅ = 0.18 d⁻¹ that started the fall surge, throughout the entire holiday period and beyond.

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Figure 12b shows the same fitted SIRV model as in Figure 12a, but Figure 12b now includes additional US data from Thanksgiving (26 November 2020) through 14 February 2021. Those additional data (grey diamonds) were not used in the fit and hence test the predictions of the model after Thanksgiving, throughout the 2020 holiday period, and the first month and a half of 2021. Because of the large fluctuations in data reported over the holiday period, students are introduced to the idea of plotting a 7-d moving average of the US data. Because we are interested in the fit to the data, students do not use Excel's built-in moving average that averages the 7 d up to the current day (you may have seen graphs of this same type widely reported in the popular press). Instead, students use a centered moving average that does not produce a systematic 3-d delay in the average curve because it is centered on the current day. The centered moving average can easily be implemented using Excel's AVERAGE() function (2) and provides a good visual comparison with the model predictions. Day 400 corresponds to 1 April 2021.

C. Epoch 6 and epoch 7 (the delta variant)

Figure 13 shows the SIRV model fitted to US data up to today (27 August 2021). Two additional epochs, 6 and 7, have been added to the model, and the parameters were fitted in a similar manner to that described above. Unlike Figure 12, the fit shown in Figure 13 includes the model population size as a fitted parameter. The fitted value is N = 6.64 × 10⁷ or q = 20.1% of the actual US population. This number is consistent with the value of q = 21.7% that was assumed for the fit in Fig 12. The LS fit parameters for the 7 epochs are recorded in Table 1. The other model parameters used in the fit were δt = 1 d, τ_r = 8.0 d, and N₀ = 5. The calculated parameters corresponding to this fit are ${ℛ}_0$ = 4.1 and t_d = 1.8 d at the beginning of the pandemic.

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Table 1 SIRV model parameters for the 7 epochs of the COVID-19 pandemic in the United States.

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A. Model assumptions

The models presented here can be criticized because the simplifying assumptions are clearly not 100% accurate. However, there is a long tradition of oversimplified models providing important insights into the behavior of real systems. As an example, consider the ideal gas. The assumption that gas molecules do not interact with each other is clearly wrong; gas molecules cannot pass through each other, just like they do not pass through the walls of their container. However, students of thermodynamics know that the ideal gas reference state is central to the formulation of classical and statistical thermodynamics. The ideal gas model is ultimately justified a posteriori, by comparing its behavior with real data and by the utility of the insights it provides. In a similar manner, the simplifying assumptions of the models presented here are ultimately justified by successful fits to the reported data and by the utility of the insights provided by the simplified model.

B. Educational objectives and scope

The teaching materials discussed here are designed as a case study in modeling a complex data set. They are not meant to directly inform public policy. However, simple models often provide useful insights into complex phenomena, not just by what they model successfully, but also by what they cannot explain. These insights are not usually provided by forecasting models (18).

C. Model extension—the SEIR model

An obvious extension to the work presented here is to change the base SIR model to the SEIR model. The main new feature of the SEIR model is the explicit inclusion of a box $e$ for exposed individuals who have been infected but are not yet infectious. SEIR model box $e$ is inserted between boxes $s$ and $i$ of the original SIR model. Modeling using the SEIR model is not included in this case study because it would add another adjustable parameter to the base model, and following the principle of Occam's razor, it has been omitted. Further investigation of the SEIR model and its SEIRV variant is left as a research exercise for students. However, a preliminary investigation has shown that the SEIR model also needs a Gaussian transition function to successfully model the transition from epoch 1 to epoch 2.