Students develop and test simple kinetic models of the spread of coronavirus disease 2019 (COVID-19) caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus. Microsoft Excel is used as the modeling platform because it is nonthreatening to students and it is widely available. Students develop finite difference models and implement them in the cells of preformatted spreadsheets following a guided inquiry pedagogy that introduces new model parameters in a scaffolded step-by-step manner. That approach allows students to investigate the implications of new model parameters in a systematic way. Students fit the resulting models to reported cases per day data for the United States using least squares techniques with Excel's Solver. Using their own spreadsheets, students discover for themselves that the initial exponential growth of COVID-19 can be explained by a simplified unlimited growth model and by the susceptible-infected-recovered (SIR) model. They also discover that the effects of social distancing can be modeled using a Gaussian transition function for the infection rate coefficient and that the summer surge was caused by prematurely relaxing social distancing and then reimposing stricter social distancing. Students then model the effect of vaccinations and validate the resulting susceptible-infected-recovered-vaccinated (SIRV) model by showing that it successfully predicts the reported cases per day data from Thanksgiving through the holiday period up to 14 February 2021. The same SIRV model is then extended and successfully fits the fourth peak up to 1 June 2021, caused by further relaxation of social distancing measures. Finally, students extend the model up to the present day (27 August 2021) and successfully account for the appearance of the delta variant of the SARS-CoV-2 virus. The fitted model also predicts that the delta variant peak will be comparatively short, and the cases per day data should begin to fall off in early September 2021, counter to current expectations. This case study makes an excellent capstone experience for students interested in scientific modeling.

### 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

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

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

• 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 Ri (arrow in Fig 1) is given by
$$\def\upalpha{\unicode[Times]{x3B1}}$$$$\def\upbeta{\unicode[Times]{x3B2}}$$$$\def\upgamma{\unicode[Times]{x3B3}}$$$$\def\updelta{\unicode[Times]{x3B4}}$$$$\def\upvarepsilon{\unicode[Times]{x3B5}}$$$$\def\upzeta{\unicode[Times]{x3B6}}$$$$\def\upeta{\unicode[Times]{x3B7}}$$$$\def\uptheta{\unicode[Times]{x3B8}}$$$$\def\upiota{\unicode[Times]{x3B9}}$$$$\def\upkappa{\unicode[Times]{x3BA}}$$$$\def\uplambda{\unicode[Times]{x3BB}}$$$$\def\upmu{\unicode[Times]{x3BC}}$$$$\def\upnu{\unicode[Times]{x3BD}}$$$$\def\upxi{\unicode[Times]{x3BE}}$$$$\def\upomicron{\unicode[Times]{x3BF}}$$$$\def\uppi{\unicode[Times]{x3C0}}$$$$\def\uprho{\unicode[Times]{x3C1}}$$$$\def\upsigma{\unicode[Times]{x3C3}}$$$$\def\uptau{\unicode[Times]{x3C4}}$$$$\def\upupsilon{\unicode[Times]{x3C5}}$$$$\def\upphi{\unicode[Times]{x3C6}}$$$$\def\upchi{\unicode[Times]{x3C7}}$$$$\def\uppsy{\unicode[Times]{x3C8}}$$$$\def\upomega{\unicode[Times]{x3C9}}$$$$\def\bialpha{\boldsymbol{\alpha}}$$$$\def\bibeta{\boldsymbol{\beta}}$$$$\def\bigamma{\boldsymbol{\gamma}}$$$$\def\bidelta{\boldsymbol{\delta}}$$$$\def\bivarepsilon{\boldsymbol{\varepsilon}}$$$$\def\bizeta{\boldsymbol{\zeta}}$$$$\def\bieta{\boldsymbol{\eta}}$$$$\def\bitheta{\boldsymbol{\theta}}$$$$\def\biiota{\boldsymbol{\iota}}$$$$\def\bikappa{\boldsymbol{\kappa}}$$$$\def\bilambda{\boldsymbol{\lambda}}$$$$\def\bimu{\boldsymbol{\mu}}$$$$\def\binu{\boldsymbol{\nu}}$$$$\def\bixi{\boldsymbol{\xi}}$$$$\def\biomicron{\boldsymbol{\micron}}$$$$\def\bipi{\boldsymbol{\pi}}$$$$\def\birho{\boldsymbol{\rho}}$$$$\def\bisigma{\boldsymbol{\sigma}}$$$$\def\bitau{\boldsymbol{\tau}}$$$$\def\biupsilon{\boldsymbol{\upsilon}}$$$$\def\biphi{\boldsymbol{\phi}}$$$$\def\bichi{\boldsymbol{\chi}}$$$$\def\bipsy{\boldsymbol{\psy}}$$$$\def\biomega{\boldsymbol{\omega}}$$$$\def\bupalpha{\bf{\alpha}}$$$$\def\bupbeta{\bf{\beta}}$$$$\def\bupgamma{\bf{\gamma}}$$$$\def\bupdelta{\bf{\delta}}$$$$\def\bupvarepsilon{\bf{\varepsilon}}$$$$\def\bupzeta{\bf{\zeta}}$$$$\def\bupeta{\bf{\eta}}$$$$\def\buptheta{\bf{\theta}}$$$$\def\bupiota{\bf{\iota}}$$$$\def\bupkappa{\bf{\kappa}}$$$$\def\buplambda{\bf{\lambda}}$$$$\def\bupmu{\bf{\mu}}$$$$\def\bupnu{\bf{\nu}}$$$$\def\bupxi{\bf{\xi}}$$$$\def\bupomicron{\bf{\micron}}$$$$\def\buppi{\bf{\pi}}$$$$\def\buprho{\bf{\rho}}$$$$\def\bupsigma{\bf{\sigma}}$$$$\def\buptau{\bf{\tau}}$$$$\def\bupupsilon{\bf{\upsilon}}$$$$\def\bupphi{\bf{\phi}}$$$$\def\bupchi{\bf{\chi}}$$$$\def\buppsy{\bf{\psy}}$$$$\def\bupomega{\bf{\omega}}$$$$\def\bGamma{\bf{\Gamma}}$$$$\def\bDelta{\bf{\Delta}}$$$$\def\bTheta{\bf{\Theta}}$$$$\def\bLambda{\bf{\Lambda}}$$$$\def\bXi{\bf{\Xi}}$$$$\def\bPi{\bf{\Pi}}$$$$\def\bSigma{\bf{\Sigma}}$$$$\def\bPhi{\bf{\Phi}}$$$$\def\bPsi{\bf{\Psi}}$$$$\def\bOmega{\bf{\Omega}}$$$$\tag{1}{R_i} = {k_i}{N_i}$$
where ki is the infection rate constant and Ni 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 ki, where ki = 0.25 d−1 means that an infectious person infects a susceptible person every 4 d, on average, causing them to “jump” from box si when they become infectious (usually some days after contact with the infectious person).
Fig 1

FD diagram of the UG epidemiological model. The two boxes represent the two parts of the model population. Box $$s$$ represents people that are susceptible to the disease. Box $$i$$ represents people that are infectious.

Fig 1

FD diagram of the UG epidemiological model. The two boxes represent the two parts of the model population. Box $$s$$ represents people that are susceptible to the disease. Box $$i$$ represents people that are infectious.

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I chose to use the symbol Ni 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 Ni. 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 Ni 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 δNi in the number infectious Ni during a short time δt (the timestep) is given by
$$\tag{2}{\rm{\updelta }}{N_i} = {R_i}{\rm{\updelta }}t$$
where Ri is given by Eq. 1. Hence, the model can be implemented in a spreadsheet using the following condensed FD instructions (2):
$$\tag{3}{t^{{\rm{new}}}} = {t^{{\rm{old}}}} + {\rm{\ \updelta }}t$$
$$\tag{4}R_i^{{\rm{new}}} = {k_i}*N_i^{{\rm{old}}}$$
and
$$\tag{5}N_i^{{\rm{new}}} = N_i^{{\rm{old}}} + R_i^{{\rm{new}}}*{\rm{\updelta }}t$$
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, $$N_i^{{\rm{old}}}$$ is the old number infectious (previous row at time told) and $$N_i^{{\rm{new}}}$$ is the new number infectious (current row at time tnew = told + 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. 35 to calculate the infection rate Ri(t) and number infectious Ni(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 Ni and the infection rate Ri (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
$$\tag{6}{N_i} = {N_0}{e^{{k_i}t}}$$
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
$$\tag{7}{R_i} = {k_i}{N_0}{e^{{k_i}t}}$$
for the infection rate Ri(t). Equation 7 is important because Ri corresponds to the publicly available number of new confirmed COVID-19 cases reported per day. Because both Ni and Ri are exponential functions of time, students are able to show that the exponential growth can be characterized by a doubling time
$$\tag{8}{t_d} = {{\ln 2} \over {k_i}}$$

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).

Fig 2

Excel chart comparing the exponential growth model of Eq. 7 for Ri(t) with reported data for the United States in the 19 d after 26 February 2020. The solid line is a LS fit to the US data. The fitted model parameters are $${N_0} = 23$$ and $${k_i} = 0.22 \ {{\rm{d}}^{ - 1}}$$⁠. Data source ECDC (7).

Fig 2

Excel chart comparing the exponential growth model of Eq. 7 for Ri(t) with reported data for the United States in the 19 d after 26 February 2020. The solid line is a LS fit to the US data. The fitted model parameters are $${N_0} = 23$$ and $${k_i} = 0.22 \ {{\rm{d}}^{ - 1}}$$⁠. Data source ECDC (7).

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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
$$\tag{9}{R_i} = {k_i}{N_i}s$$
where s is the susceptible fraction of the population that is defined by
$$\tag{10}s \equiv {{N_s} \over N}$$
where Ns is the number susceptible and N (with no subscript) is the total number of people in the model population, where
$$\tag{11}N = {N_s} + {N_i}$$
Fig 3

FD diagram of a two-box epidemiological model exhibiting limited growth. The two boxes in this finite population (FP) model represent the two parts of the population that can be affected by the disease. Box $$s$$ represents the portion of the population susceptible to the disease. Box $$i$$ represents the portion infectious. Lowercase $$s$$ is the fraction of the population that are still susceptible to infection.

Fig 3

FD diagram of a two-box epidemiological model exhibiting limited growth. The two boxes in this finite population (FP) model represent the two parts of the population that can be affected by the disease. Box $$s$$ represents the portion of the population susceptible to the disease. Box $$i$$ represents the portion infectious. Lowercase $$s$$ is the fraction of the population that are still susceptible to infection.

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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 Ns using the instruction
$$\tag{12}N_s^{{\rm{new}}} = N - N_i^{{\rm{new}}}$$

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 Ns, students show that the FP model can be calculated using
$$\tag{13}R_i^{{\rm{new}}} = {k_i}*N_i^{{\rm{old}}}*N_s^{{\rm{old}}}/N$$
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 ki = 0.3 d−1 to ki = 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 Ri(t) curve that can be flattened by social distancing. Interestingly, the Ni(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.

Fig 4

Excel area chart showing the predictions of the FP model for Ri(t) (Eq. 9) for a model population of $$N = 1,\!000$$⁠, an infection rate constant of $${k_i} = 0.3\ {{\rm{d}}^{ - 1}}$$⁠, an initial number infectious of $${N_0} = 1$$⁠, and a timestep of $${\rm{\updelta }}t = 1\;{\rm{d}}$$⁠. The “with social distancing” curve shows the effect of reducing the infection rate constant by a factor of $$2$$ on day $$0$$ to $${k_i} = 0.15\ {{\rm{d}}^{ - 1}}$$ by implementing social distancing and mask wearing.

Fig 4

Excel area chart showing the predictions of the FP model for Ri(t) (Eq. 9) for a model population of $$N = 1,\!000$$⁠, an infection rate constant of $${k_i} = 0.3\ {{\rm{d}}^{ - 1}}$$⁠, an initial number infectious of $${N_0} = 1$$⁠, and a timestep of $${\rm{\updelta }}t = 1\;{\rm{d}}$$⁠. The “with social distancing” curve shows the effect of reducing the infection rate constant by a factor of $$2$$ on day $$0$$ to $${k_i} = 0.15\ {{\rm{d}}^{ - 1}}$$ by implementing social distancing and mask wearing.

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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).

Fig 5

FD diagram of the SIR epidemiological model. The three boxes represent the three parts of the model population that can be affected by the disease. Box $$s$$ represents the portion that is susceptible to the disease. Box $$i$$ represents the portion infectious. Box $$r$$ represents the portion that is recovered from the infection (or died). Sometimes this box is labeled removed, as in removed from consideration.

Fig 5

FD diagram of the SIR epidemiological model. The three boxes represent the three parts of the model population that can be affected by the disease. Box $$s$$ represents the portion that is susceptible to the disease. Box $$i$$ represents the portion infectious. Box $$r$$ represents the portion that is recovered from the infection (or died). Sometimes this box is labeled removed, as in removed from consideration.

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The three boxes in Figure 5 represent the possible states of people in the model population. In the SIR model, Ns, s, and Ni have the same meaning as the FP model. The new state variable is Nr, 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 Nr 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
$$\tag{14}N = {N_s} + {N_i} + {N_r}$$
Now that we have discussed the three boxes, let us talk about the arrows between boxes in Figure 5. The first arrow from box si represents the rate of infection Ri = kiNis (Eq. 9), the same equation that we used for the FP model of Figure 3. The second arrow from box ir represents the rate of recovery. That recovery rate is given by
$$\tag{15}{R_r} = {k_r}{N_i}$$
where kr is the recovery rate constant and the mean residence time in box i (1) is predicted to be
$$\tag{16}{\tau _i} = {1 \over {k_r}}$$

We will call τi 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 1720.
$$\tag{17}R_r^{{\rm{new}}} = {k_r}*{N_i}^{{\rm{old}}}$$
$$\tag{18}{N_i}^{{\rm{new}}} = {N_i}^{{\rm{old}}} + \left( {R_i^{{\rm{new}}} - R_r^{{\rm{new}}}} \right)*{\rm{\updelta }}t$$
$$\tag{19}N_r^{{\rm{new}}} = N_r^{{\rm{old}}} + R_r^{{\rm{new}}}*{\rm{\updelta }}t$$
$$\tag{20}N_s^{{\rm{new}}} = N - {N_i}^{{\rm{new}}} - N_r^{{\rm{new}}}$$

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 ki. 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 Ns does not reach zero at the end of the pandemic, meaning that some susceptible individuals were never infected (see week 50 in Fig 6a).

Fig 6

Excel charts showing the predictions of the SIR model for a model population of $$N = 1,\!000$$⁠, an infection rate constant of $${k_i} = 0.12\ {{\rm{d}}^{ - 1}}$$⁠, a mean infectious time of $${\tau _i} = 16\ {\rm{d}}$$⁠, an initial number infectious of $${N_0} = 1$$⁠, and a timestep of $${\rm{\updelta }}t = 0.01\;{\rm{d}}$$⁠. Chart (a) shows the numbers in the three boxes $$s$$⁠, $$i$$⁠, and $$r$$ of the SIR model. Chart (b) shows the exponential dragon predicted for the infection rate Ri (solid blue line) and the recovery rate Rr. Note that because of Eq. 15, the recovery rate $${R_r}$$ is directly proportional to the number infectious Ni.

Fig 6

Excel charts showing the predictions of the SIR model for a model population of $$N = 1,\!000$$⁠, an infection rate constant of $${k_i} = 0.12\ {{\rm{d}}^{ - 1}}$$⁠, a mean infectious time of $${\tau _i} = 16\ {\rm{d}}$$⁠, an initial number infectious of $${N_0} = 1$$⁠, and a timestep of $${\rm{\updelta }}t = 0.01\;{\rm{d}}$$⁠. Chart (a) shows the numbers in the three boxes $$s$$⁠, $$i$$⁠, and $$r$$ of the SIR model. Chart (b) shows the exponential dragon predicted for the infection rate Ri (solid blue line) and the recovery rate Rr. Note that because of Eq. 15, the recovery rate $${R_r}$$ is directly proportional to the number infectious Ni.

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A universal feature of the SIR model is that it predicts an “exponential dragon” in the infection rate Ri(t) (fig 6b and fig 12.15 of (2)), whose duration is determined by the value of the infection rate constant ki. 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 Ns, Ni, Nr, Ri, and Rr, 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
$$\tag{21}i \equiv {{N_i} \over N}$$
and the fraction recovered r is defined by
$$\tag{22}r \equiv {{N_r} \over N}$$

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 Ni(t), the SIR model predicts that the number infectious Ni will steadily decline because the rate of infection Ri is less than the rate of recovery Rr. 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 − sp, which is the cumulative fraction that have been infected at the time tp of the peak in Ni(t). We can then define ip as the fraction infectious and rp as the fraction recovered at time tp, respectively. Hence, from the bookkeeping Eq. 14, we have
$$\tag{23}{s_p} + {i_p} + {r_p} = 1$$
so that 1 − sp = ip + rp. Hence, the quantity 1 − sp is the sum ip + rp, which is the cumulative total number of people that have been infected at time tp.
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
$$\tag{24}h = i + r = 1 - s$$
where h is the fraction immune or the immune fraction of the model population. Once the immune fraction h reaches
$$\tag{25}{h_p} = 1 - {s_p}$$
the recovery rate Rr is larger than the infection rate Ri, and the model predicts that the disease will be in decline and eventually die out. The fraction hp is the herd immunity threshold. If the fraction immune is greater than or equal to hp, i.e., if
$$\tag{26}h \ge {h_p}$$
then the disease will be in decline rather than growing (as indicated by whether Ni(t) decreases or increases, respectively).

### E. Finding the peak in the curve and $${\boldsymbol {\cal R}_{\bf 0}}$$

In a guided inquiry exercise, students are asked to consider why the infection rate curve Ri(t) always cuts through the peak in the recovery rate curve Rr(t); see, for example, Figure 6b. Students discover that the peak in Ni(t) and Rr(t) occurs when
$$\tag{27}{R_i} = {R_r}$$
By substituting Eqs. 9 and 15 into Eq. 27, students show that the value of the fraction susceptible s at the peak in Ni is given by
$$\tag{28}{s_p} = {{k_r} \over {k_i}} = {1 \over {{{\cal R}_0}}}$$
where $${{\cal R}_0}$$ is the basic reproduction number that is given by
$$\tag{29}{{\cal R}_0} \equiv {k_i}{\tau _i} = {{k_i} \over {k_r}} = {1 \over {s_p}}$$

The basic reproduction number $${{\cal R}_0}$$ 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, $${{\cal R}_0}$$kiτi defines $${{\cal R}_0}$$ as the number infected by a single individual at the beginning of the outbreak when s ≈ 1; ki is the average number of people infected by a single infectious individual per day; and τi is the mean number of days that they are infectious.

Because $${{\cal R}_0}$$ = 1/sp, we can write the herd immunity threshold hp in Eq. 25 in terms of the basic reproduction number $${{\cal R}_0}$$
$$\tag{30}{h_p} = 1 - {1 \over {{{\cal R}_0}}}$$

The best fit value of $${{\cal R}_0}$$ students obtain by fitting the US data with τi = 8 d is $${{\cal R}_0}$$ ≈ 4.1 so that the herd immunity threshold for the original variants of COVID-19 is hp ≈ 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 ki depends on the level of social distancing. Hence, whenever the infection rate Ri(t) is decreasing, we have technically passed the herd immunity threshold for the current value of ki.

The ultimate goal of this section is to model transitions between different levels of social distancing in a straightforward manner by making the infection rate coefficient a function of time ki(t) in the SIR model (personal communication, R. Hilborn, American Association of Physics Teachers, 5 June 2020). The title of this section is a spoiler because it really was not obvious, at least to me, that Gaussian transition functions between the different epochs in the pandemic were a good way to go, even though, in hindsight it seems rather obvious.

### 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 ki = 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 ki 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 ki between epochs 1 and 2 produced a spike in the fitted model of Ri(t) that is clearly inconsistent with the reported data during the transition period.

Fig 7

Excel chart showing the fitted SIR model with a step change in the infection rate coefficient from ki = k1 = 0.505 d−1 to ki = k2 = 0.118 d−1), where ki is calculated from the initial epoch 1 of exponential growth (orange dots) and k2 is the value students calculated for the epoch 2 of social distancing (green dots). The only adjustable parameter in the fit is the transition time t12 = 35 d between k1 and k2. The other parameters in the model are N = 8.25 × 107, δt = 1 d, τi = 8 d, and N0 = 5.54 (2). As shown, the fit produces a transition spike (grey line) that does not match the reported transition data. Data source ECDC (7).

Fig 7

Excel chart showing the fitted SIR model with a step change in the infection rate coefficient from ki = k1 = 0.505 d−1 to ki = k2 = 0.118 d−1), where ki is calculated from the initial epoch 1 of exponential growth (orange dots) and k2 is the value students calculated for the epoch 2 of social distancing (green dots). The only adjustable parameter in the fit is the transition time t12 = 35 d between k1 and k2. The other parameters in the model are N = 8.25 × 107, δt = 1 d, τi = 8 d, and N0 = 5.54 (2). As shown, the fit produces a transition spike (grey line) that does not match the reported transition data. Data source ECDC (7).

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As the title of this section states, the solution to this conundrum was to change the transition function for ki(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
$$\tag{31}F_{12}^{{\rm{new}}} = {\rm{NORM}}.{\rm{DIST}}\left( {{t^{{\rm{new}}}},{t_{12}},{\sigma _{12}},{\rm{TRUE}}} \right)$$
where $$F_{12}^{{\rm{new}}}$$ is the value of the Gaussian cumulative probability at time tnew 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.
$$\tag{32}p_{12}^{{\rm{new}}} = {\rm{NORM}}.{\rm{DIST}}\left( {{t^{{\rm{new}}}},{t_{12}},{\sigma _{12}},{\rm{FALSE}}} \right)$$
The time-dependent infection rate coefficient ki(t) can then be calculated using
$$\tag{33}k_i^{{\rm{new}}} = {k_1} + F_{12}^{{\rm{new}}}*\left( {{k_2} - {k_1}} \right)$$

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).

Fig 8

Excel chart showing the SIR model (solid orange line) fitted to the US data (blue circles). SIR model parameters, $${k_1} = 0.60\ {{\rm{d}}^{ - 1}}$$⁠, $${k_2} = 0.12\ {{\rm{d}}^{ - 1}}$$⁠, $${t_{12}} = 27\;{\rm{d}}$$⁠, and $${\sigma _{12}} = 12.3\;{\rm{d}}$$⁠, were fit simultaneously using the LS method. The remaining SIR model parameters were set to $$N = 8.25 \times {10^7}$$⁠, $${\rm{\updelta }}t = 1\ {\rm{d}}$$⁠, $${\tau _i} = 8\;{\rm{d}}$$⁠, and $${N_0} = 5$$⁠. The transition from $${k_1} \to {k_2}$$ is modeled using a Gaussian (normal) distribution with mean $${t_{12}}$$ and standard deviation $${\sigma _{12}}$$⁠. The dotted line shows $${p_{12}}\left( t \right)$$⁠, the probability density function (Eq. 32) of the Gaussian transition function $${F_{12}}\left( t \right)$$ (Eq. 31). Circled numbers indicate epochs 1 and 2 of the pandemic. Data source ECDC (7).

Fig 8

Excel chart showing the SIR model (solid orange line) fitted to the US data (blue circles). SIR model parameters, $${k_1} = 0.60\ {{\rm{d}}^{ - 1}}$$⁠, $${k_2} = 0.12\ {{\rm{d}}^{ - 1}}$$⁠, $${t_{12}} = 27\;{\rm{d}}$$⁠, and $${\sigma _{12}} = 12.3\;{\rm{d}}$$⁠, were fit simultaneously using the LS method. The remaining SIR model parameters were set to $$N = 8.25 \times {10^7}$$⁠, $${\rm{\updelta }}t = 1\ {\rm{d}}$$⁠, $${\tau _i} = 8\;{\rm{d}}$$⁠, and $${N_0} = 5$$⁠. The transition from $${k_1} \to {k_2}$$ is modeled using a Gaussian (normal) distribution with mean $${t_{12}}$$ and standard deviation $${\sigma _{12}}$$⁠. The dotted line shows $${p_{12}}\left( t \right)$$⁠, the probability density function (Eq. 32) of the Gaussian transition function $${F_{12}}\left( t \right)$$ (Eq. 31). Circled numbers indicate epochs 1 and 2 of the pandemic. Data source ECDC (7).

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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 mc = 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 Ri (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.

Fig 9

Excel chart showing the predictions of the SIR model (solid orange line) when fitted to US data reported as confirmed cases per day by the ECDC (blue circles) up to Labor Day (7 September 2020) for four epochs of the pandemic (circled numbers): epoch 1, the initial exponential growth; epoch 2, the epoch of social distancing; epoch 3, the relaxation of social distancing following Memorial Day; and epoch 4, the return to social distancing following the Fourth of July. The vertical dashed lines indicate Memorial Day and the Fourth of July. The graph also includes the infection rate coefficient $${k_i}\left( t \right)$$ (green dotted line) on the secondary vertical axis. Data source ECDC (7).

Fig 9

Excel chart showing the predictions of the SIR model (solid orange line) when fitted to US data reported as confirmed cases per day by the ECDC (blue circles) up to Labor Day (7 September 2020) for four epochs of the pandemic (circled numbers): epoch 1, the initial exponential growth; epoch 2, the epoch of social distancing; epoch 3, the relaxation of social distancing following Memorial Day; and epoch 4, the return to social distancing following the Fourth of July. The vertical dashed lines indicate Memorial Day and the Fourth of July. The graph also includes the infection rate coefficient $${k_i}\left( t \right)$$ (green dotted line) on the secondary vertical axis. Data source ECDC (7).

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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
$$\tag{34}q \equiv {N \over {{N^ \star }}}$$
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.
Fig 10

Excel chart showing the predictions of the SIR model when fitted to 5 different epochs: epoch 1, the initial exponential growth; epoch 2, the initial period of social distancing; epoch 3, the relaxation of social distancing following Memorial Day; epoch 4, the return to social distancing following the Fourth of July; and epoch 5, the fall surge following Labor Day (7 September 2020). The blue circles show the US data reported as confirmed cases per day up to Thanksgiving (26 November 2020). The solid orange line represents a model population of 20% of the actual US population. The dashed orange lines represent predictions of the fitted model for model populations of 10% and 40% of the actual US population. Data source OWID (16).

Fig 10

Excel chart showing the predictions of the SIR model when fitted to 5 different epochs: epoch 1, the initial exponential growth; epoch 2, the initial period of social distancing; epoch 3, the relaxation of social distancing following Memorial Day; epoch 4, the return to social distancing following the Fourth of July; and epoch 5, the fall surge following Labor Day (7 September 2020). The blue circles show the US data reported as confirmed cases per day up to Thanksgiving (26 November 2020). The solid orange line represents a model population of 20% of the actual US population. The dashed orange lines represent predictions of the fitted model for model populations of 10% and 40% of the actual US population. Data source OWID (16).

Close modal

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 (k1k5) 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 sp = 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 Rv,s, Rv,i, and Rv,r where the subscripts “s,” “i,” and “r” indicate the originating box. These three vaccination rates are related to the total rate of vaccination Rv in the model population by
$$\tag{35}{R_v} = {R_{v,s}} + {R_{v,i}} + {R_{v,r}}$$
Fig 11

FD diagram of a simple modification of the SIR model that accounts for vaccinations, the SIRV model. The four boxes represent the four parts of the model population that can be affected by the disease. Box $$s$$ represents the portion of the population that is susceptible to the disease. Box $$i$$ represents the portion of the population that is infectious. Box $$r$$ represents the portion of the population that has recovered from the infection (or died). Box $$v$$ represents the fully vaccinated portion of the population.

Fig 11

FD diagram of a simple modification of the SIR model that accounts for vaccinations, the SIRV model. The four boxes represent the four parts of the model population that can be affected by the disease. Box $$s$$ represents the portion of the population that is susceptible to the disease. Box $$i$$ represents the portion of the population that is infectious. Box $$r$$ represents the portion of the population that has recovered from the infection (or died). Box $$v$$ represents the fully vaccinated portion of the population.

Close modal
The bookkeeping equation for the susceptible-infected-recovered-vaccinated (SIRV) model is
$$\tag{36}N = {N_s} + {N_i} + {N_r} + {N_v}$$
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 $$N_v^ \star$$ reported by Our World in Data (OWID) (16) using $$N_v^{{\rm{new}}} = qN_v^{\star{\rm{new}}}$$⁠. However, because the OWID spreadsheet data contain some blank cells, the following Excel instruction is used
$$\tag{37}N_v^{{\rm{new}}} = {\rm{IF}}\left( {N{{_v^ \star }^{{\rm{new}}}} = 0,N_v^{{\rm{old}}},q*N{{_v^ \star }^{{\rm{new}}}}} \right)$$
and the FD instruction for the vaccination rate in the model population is
$$\tag{38}R_v^{{\rm{new}}} = \left( {N_v^{{\rm{new}}} - N_v^{{\rm{old}}}} \right)/{\rm{\updelta }}t$$
The rate of vaccination of susceptible individuals in the model population can be calculated using
$$\tag{39}R_{v,s}^{{\rm{new}}} = N_s^{{\rm{old}}}*R_v^{{\rm{new}}}/\left( {N_s^{{\rm{old}}} + N_i^{{\rm{old}}} + N_r^{{\rm{old}}}} \right)$$
and similarly for $$R_{v,i}^{{\rm{new}}}$$⁠, and $$R_{v,r}^{{\rm{new}}}$$⁠. Eq. 39 and the corresponding equations for $$R_{v,i}^{{\rm{new}}}$$ and $$R_{v,r}^{{\rm{new}}}$$ 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
$$\tag{40}{N_i}^{{\rm{new}}} = {N_i}^{{\rm{old}}} + \left( {R_i^{{\rm{new}}} - R_r^{{\rm{new}}} - R_{v,i}^{{\rm{new}}}} \right)*{\rm{\updelta }}t$$
$$\tag{41}N_r^{{\rm{new}}} = N_r^{{\rm{old}}} + \left( {R_r^{{\rm{new}}} - R_{v,r}^{{\rm{new}}}} \right)*{\rm{\updelta }}t$$
Combining Eqs. 3941 with bookkeeping Eq. 36 yields the following FD instructions for the numbers in boxes i, r, and s.
$$\tag{42}{N_i}^{{\rm{new}}} = {N_i}^{{\rm{old}}} + \left( {R_i^{{\rm{new}}} - R_r^{{\rm{new}}} - N_i^{{\rm{old}}}*R_v^{{\rm{new}}}/\left( {N - N_v^{{\rm{old}}}} \right)} \right)*{\rm{\updelta }}t$$
$$\tag{43}N_r^{{\rm{new}}} = N_r^{{\rm{old}}} + \left( {R_r^{{\rm{new}}} - N_r^{{\rm{old}}}*R_v^{{\rm{new}}}/\left( {N - N_v^{{\rm{old}}}} \right)} \right)*{\rm{\updelta }}t$$
and
$$\tag{44}N_s^{{\rm{new}}} = N - {N_i}^{{\rm{new}}} - N_r^{{\rm{new}}} - N_v^{{\rm{new}}}$$

Using a preformatted spreadsheet, students use Eqs. 37, 38, and 4244 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 $$N_v^{\star{\rm{new}}}$$ that was reported daily by OWID (16). Figure 12a includes a plot of the infection rate coefficient ki(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 ki remains constant at the same value ki = k5 = 0.18 d−1 that started the fall surge, throughout the entire holiday period and beyond.

Fig 12

Excel charts showing the USA data and the predictions of the SIRV model. The blue circles show US data reported as confirmed cases per day up to Thanksgiving Day (26 November 2020). The jagged black line shows the centered 7-d moving average of the US data. The solid orange line shows the predictions of the SIRV model, assuming that the model population is $$q = 21.7\%$$ of the actual population (15). Chart (a) shows the infection rate coefficient $${k_i}\left( t \right)$$ as a function of time on the secondary vertical axis. Circled numbers indicate the epochs of the pandemic. Chart (b) shows additional US data (grey diamonds) up to 14 February 2021 that were not used in the fit and the corresponding 7-d average (jagged black line). These unfitted data validate the predictions of the SIRV model with a constant infection rate coefficient of $${k_5} = 0.18\ {{\rm{d}}^{ - 1}}$$ in epoch 5 of the pandemic. Data source OWID (16).

Fig 12

Excel charts showing the USA data and the predictions of the SIRV model. The blue circles show US data reported as confirmed cases per day up to Thanksgiving Day (26 November 2020). The jagged black line shows the centered 7-d moving average of the US data. The solid orange line shows the predictions of the SIRV model, assuming that the model population is $$q = 21.7\%$$ of the actual population (15). Chart (a) shows the infection rate coefficient $${k_i}\left( t \right)$$ as a function of time on the secondary vertical axis. Circled numbers indicate the epochs of the pandemic. Chart (b) shows additional US data (grey diamonds) up to 14 February 2021 that were not used in the fit and the corresponding 7-d average (jagged black line). These unfitted data validate the predictions of the SIRV model with a constant infection rate coefficient of $${k_5} = 0.18\ {{\rm{d}}^{ - 1}}$$ in epoch 5 of the pandemic. Data source OWID (16).

Close modal

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, τr = 8.0 d, and N0 = 5. The calculated parameters corresponding to this fit are $${{\cal R}_0}$$ = 4.1 and td = 1.8 d at the beginning of the pandemic.

Fig 13

Excel chart showing the predictions of the SIRV model. The blue circles show US data reported as confirmed cases per day up to 27 August 2021. The jagged black line shows the centered 7-d moving average of the reported US data. The smooth orange line shows the SIRV model fitted to the 7 epochs of the pandemic in the US, with a fitted value of $$N = 6.64 \times {10^7}$$ or $$q = 20.1\%$$ of the actual US population. Data source OWID (16).

Fig 13

Excel chart showing the predictions of the SIRV model. The blue circles show US data reported as confirmed cases per day up to 27 August 2021. The jagged black line shows the centered 7-d moving average of the reported US data. The smooth orange line shows the SIRV model fitted to the 7 epochs of the pandemic in the US, with a fitted value of $$N = 6.64 \times {10^7}$$ or $$q = 20.1\%$$ of the actual US population. Data source OWID (16).

Close modal
Table 1

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

“All models are wrong, but some are useful” is a common aphorism in epidemiology (17). In sympathy with that statement, the modeling approach presented here follows the principle of Occam's razor so that the models students investigate are made as simple as possible and include no unnecessary parameters. This allows our attention to be focused on the essential model assumptions and their qualitative and quantitative consequences.

According to Holmdahl and Buckee (18), COVID-19 models generally fall into one of two general categories that they call “forecasting models” and “mechanistic models.” The phenomenological models discussed here are mechanistic models whose purpose is to gain insights into the real system being modeled. This can be contrasted with forecasting models, such as the original versions of the model from the Institute for Health Metrics and Evaluation that received a great deal of attention in the popular press in 2020 (18).

### 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.

#### 1. Infection is a Poisson process

All the models presented here include a transition from box si that is characterized by an infection rate coefficient ki. In the UG model, the infection rate per infectious person is ki, a constant that is independent of everything. This assumes that infectious people only interact with susceptible people and that there is an unlimited supply of susceptible people with which they can interact. In the FP model, and later SIR models, the number of interactions of infectious people with others is still assumed to be constant, but now it is assumed that the probability that the interaction is with a susceptible person, rather than another infectious person (or recovered or vaccinated individual in the SIR or SIRV models), is given by s, the susceptible fraction of the entire model population. This might be a reasonable assumption for a small model population limited to a small geographic region where the entire population interacts with each other directly or indirectly on the timescale of infection and recovery, but there are clearly problems with applying this assumption to the United States as a whole. Most people stay local and do not interact with others outside of their hometown or county. Also, there are regional differences in levels of social distancing.

In comparing the models with reported cases per day data, students must interpret the meaning of the jump from box si carefully. The model jumps do not occur at the time of infection but rather at the time that the individual becomes infectious. As with models of diffusion that include similar jumps (1), it is important for students to realize that these model jumps are not instantaneous, and they occur with a distribution of inter-event times (19).

#### 2. Recovery is a not a Poisson process

The arrows in Figures 1, 3, 5, and 11 represent rates that depend only on the current state of the system. For example, for jumps from box ir in the SIR-based models, the rate of recovery (or removal) is Rr = krNi (Eq. 15), which means that the probability of any infectious person recovering is a constant, independent of everything, including how long they have been infectious (and hence been in box i). Because the probability is constant, the model intrinsically assumes that recovery is a Poisson process with an exponential distribution of residence times that has a peak at zero time (1, 19). Students are reminded that is a good approximation for drug elimination and radioactive decay, but it is clearly not correct for recovery from COVID-19 because infectious people take a week or so to recover from the disease (even if they have mild or asymptomatic cases). However, if we consider the ensemble average number of people in box i as a whole, then it seems reasonable that the ensemble average recovery rate depends directly on the number infectious Ni, if we consider times significantly longer than the mean infectious time τi. The main advantage of Eq. 15 is that it is easy to understand, and it gives us a simple way to predict the recovery rate Rr based solely on the current number infectious Ni. Hence, we do not have to keep track of each individual and how long they have been infected in the model, which would be difficult to do in Excel. Students are reminded that we are only trying to understand the basics of epidemiology with our SIR model (2).

#### 3. Infection rate data

For simplicity, published cases per day data are compared directly with the model infection rate Ri in these materials. This correspondence is only approximate. There is usually a delay of about a week or so between exposure to the SARS-CoV-2 virus and when an individual becomes infectious and transitions to box i. Usually, there is a further delay before someone in the model population tests positive and appears in the published data. Hence, the time when an individual first becomes infectious (the jump from box si) is usually somewhere between the time of exposure and the time a positive test is reported. One delay is biological; it takes time for the SARS-CoV-2 virus to be reproduced in the body to a level that makes the person infectious and that is detectable in a COVID-19 test. The other delay is sociological; most people were not tested every day so that the time of their positive test depends on when they were tested and when that positive test was reported and tabulated in the data used here. For example, early in the pandemic, testing resources were scarce, and only symptomatic patients were tested. This uncertainty in the timing of an individual's COVID test adds to the uncertainty in the fitted transition times between epochs. Hence, the delay between exposure and becoming infectious and the delay between becoming infectious and a positive test result both need to be accounted for if one wants to interpret the correlation between public health policy changes and the fitted transition times. Further discussion of these fascinating public health policy issues is beyond the scope of this article and is left as a research question for students.

#### 4. Gaussian transition functions

Within the model, the infection rate coefficient ki is the only model parameter that changes between epochs of the pandemic. The use of Gaussian transition functions is an empirical choice that is supported by the central limit theorem, namely, if you combine enough random events, the distribution of outcomes will approach the Gaussian (normal) distribution. However, recalling that the data being fitted is the total number of reported cases across the whole United States, there are potentially many independent or correlated distributions that should be combined. Additionally, the model does not explicitly include the delay between exposure to the virus and becoming infectious, and the model does not include the subsequent delay before a positive test appears in the reported data. As a result, the fitted transition times tij are likely delayed from changes in public health guidelines or mandates, and it is likely that the standard deviations σij of the Gaussian transition functions are too large. Both issues may be partially addressed by including a box for individuals who have been infected (exposed) but are not yet infectious, as is done in the susceptible-exposed-infected-recovered (SEIR) model, see the discussion below.

#### 5. Model population size

It is a bold assertion that the underreporting of positive cases can be accounted for by a single parameter qN/N, where N is the model population size and N ≈ 3.3 × 108 is the estimated actual population of the United States. The use of a single q throughout the pandemic cannot be supported by direct measurement of actual infections; those data are simply not available. The best estimate I have been able to find was published by the CDC (15), but their estimates have changed over time. The most recent CDC estimate (27 July 2021) is 1 in 4.2 (95% UI, 3.6−4.9) COVID-19 infections were reported from February 2020−May 2021 (20), i.e., q = 24%. From a modeling perspective, the assumption that q is a constant throughout the pandemic can only be justified a posteriori, if it is consistent with the published cases per day data over the entire course of the pandemic. In addition, modeling of the pandemic at the present time (late August 2021) is becoming increasing problematic, as basic assumptions of the SIRV model are being shown to be incorrect, at least for some reported cases. For example, some fully vaccinated individuals are now testing positive, and additional vaccination shots are planned for those previously considered to be “fully vaccinated.”

The simplest way to interpret q is that it is the percentage of actual infections that appear in the reported data. Using that as a measure of what is happening in the actual population assumes that those outside of the model population, spread COVID-19, are infected by COVID-19, recover from COVID-19, and are vaccinated, in a similar manner to those in the model population. This implies that individuals in the model population are mixed in with the rest of the actual population and that the rate of reported cases is proportional to the rate of actual cases.

In the SIR models, the first time that the model population size affects the qualitative behavior of the fitted model is after Thanksgiving Day (26 November 2020), during the fall exponential dragon (Fig 10). The reason is that the third peak in the pandemic is the first model peak that corresponds to the exponential dragon predicted by the SIR model (Fig 6b). The peak in the exponential dragon occurs when s = kr/ki, i.e., when the fitted model first reaches s = sp (the herd immunity threshold). For epoch 5, ki = k5 = 0.18 d−1, and the peak occurs when s = sp = kr/k5 = 0.70 or hp = 30% (Table 1). Recall that sNs/N so that it depends on the model population size N. Also, recall that s is a monotonically decreasing function of time in all the models presented here (reinfection is not possible in all the models discussed here).

The fourth peak in the fitted SIRV model occurs in epoch 6 when $$s = {s_p} = {k_r}/{k_6} = 0.39$$⁠, or $${h_p} = 61\%$$ (Table 1). The fact that the fit shown in Figure 13 has essentially the same value of $$q$$ as the fit in Figure 12 provides strong support for the SIRV model and the hypothesis that $$q$$ is approximately constant (at least up to the beginning of epoch 7). The fitted value of $${k_6} = 0.32\;{{\rm{d}}^{ - 1}}$$ is nearly double $${k_3}$$ and $${k_5}$$⁠, reflecting a significant further reduction in social distancing during epoch 6, although the infection rate constant is still nearly half of what it was during the uncontrolled spread in epoch 1 with $${k_1} = 0.59\ {{\rm{d}}^{ - 1}}$$ at the beginning of the pandemic.

The much larger fitted infection rate constant of $${k_7} = 0.89\ {{\rm{d}}^{ - 1}}$$ during epoch 7 can be attributed to the emergence of the delta variant of the SARS-CoV-2 virus in the United States (and the low level of social distancing). In an analogous manner to the third and fourth peaks, the model predicts that the fifth peak in the SIRV model will occur when $$s = {s_p} = {k_r}/{k_7} = 0.14$$ or $${h_p} = 86\%$$ (Table 1), assuming the infection rate coefficient remains constant. The fitted infection rate constant for the delta variant (⁠$${k_i} = {k_7} = 0.89\ {{\rm{d}}^{ - 1}}$$⁠) is $$\sim 2.8$$ times higher than the previous variants of the virus during epoch 6, which, if the level of social distancing is the same, would imply that the basic reproduction number for the delta variant could be as high as $${{\cal R}_0} \approx 11$$ in the absence of any social distancing measures.

After the fitted period (up to 27 August 2021), the SIRV model makes a rather bold prediction that the curve will peak near the end of August so that the cases per day data are predicted to fall off during September and October 2021 (Fig 13, day 600 corresponds to 18 October 2021). This prediction relies on the infection rate coefficient remaining constant at $${k_7} = 0.89\ {{\rm{d}}^{ - 1}}$$ and that the model population size is constant at $$q \approx 20\%$$ throughout the entire pandemic.

Of all the assumptions made in the SIRV model, the assumption that $$q$$ did not and will not change with time (and is thus a single constant throughout the pandemic) is probably the most questionable. COVID-19 tests were scarce at the beginning of the pandemic, and estimates were made that fewer than 1 in 10 cases were reported, i.e., $$q \lt 10\%$$⁠. According to the CDC, this number increased to an average of over $$q = 20\%$$ by January 2021 and $$q = 24\%$$ from February 2020 to May 2021 in the report dated 27 July 2021 (20). Clearly, this assumption must be kept in mind when considering the model parameters, particularly at the beginning and the end of the pandemic. For example, one implication of the CDC-estimated increase in $$q$$ is that the current value of $$s$$ (on 27 August 2021) in the fitted SIRV model is probably too low so that the predicted delta variant peak (epoch 7) likely occurs too soon because there are more susceptible individuals $$s$$ left in the actual population than the SIRV model predicts. Figure 10 illustrates this same delay and increase in the size of the predicted exponential dragon as $$q$$ is increased from $$q = 10\%$$ to $$q = 20\%$$⁠. In addition, the fitted value of $${k_7}$$ is probably too high, in a similar manner to the fitted value of $${k_5} = 0.23{{\rm{d}}^{ - 1}}$$ being too high in the fit with $$q = 10\%$$ (Fig 10).

#### 6. Vaccinations

The form of the SIRV model shown in Figure 11 was chosen to match the vaccination program in the United States. COVID-19 tests were not a prerequisite for vaccination. Hence, the status of individuals receiving vaccinations is not included in the data. As a result, the SIRV model assumes that vaccinations were administered to anyone in the population that was asymptomatic at the time. That assumption is reflected in Eq. 39 and the corresponding equations for $$R_{v,i}^{{\rm{new}}}$$ and $$R_{v,r}^{{\rm{new}}}$$ so that the rate of vaccination of individuals in each of boxes $$s$$⁠, $$i$$⁠, and $$r$$ is directly proportional to the numbers currently in each respective box. This assumption overestimates the vaccination rate of people in box $$i$$ (because symptomatic individuals were not supposed to be vaccinated) and underestimates vaccinations of individuals in boxes $$s$$ and $$r$$⁠. The model does not consider partially vaccinated individuals. Recall, $$N_v^{\star^{\rm{new}}}$$ is the reported number of fully vaccinated individuals.

People are considered fully vaccinated two weeks after their second dose of the Pfizer-BioNTech or Moderna COVID-19 vaccines or two weeks after a single dose of Johnson & Johnson's Janssen COVID-19 vaccine (21). Just like the other jumps in the SIR models, this extended process is approximated by a single jump transition of variable duration. As a result, students should once again be reminded that we are only trying to understand the basics of epidemiology with our SIRV model (2).

Finally, the initial vaccination rollout in the United States was targeted at specific groups, health care workers, nursing home residents, and the elderly, with progressively lower age restrictions until the vaccine was released for everyone 12 or older. As of late August 2021, COVID-19 vaccines have not been FDA-approved for children under 12. The models presented here do not take age into account, treating everyone in the model population in the same manner, irrespective of their age. Clearly, this is another assumption that is questionable, as different age groups behaved in different ways during the pandemic and had different susceptibilities to the disease.

#### 7. Immunity is permanent?

A basic assumption of the SIR model is that recovery from COVID-19 imparts permanent immunity. There is no mechanism in the model for an individual becoming infectious a second time. That assumption is not universally correct. If immunity granted by past infection were permanent, then immunizing those who have recovered from COVID-19 would be pointless because they would not gain any benefit from vaccination.

A basic assumption of the SIRV model is that being fully vaccinated always imparts permanent immunity. In late August 2021, that assumption was known to be not accurate. There have been reported cases of COVID-19 in individuals who had been previously fully vaccinated.

The assumption that immunity is permanent provides a basic constraint on all the SIR-based models presented here. At a constant infection rate (of any fixed value), the pandemic is predicted to peter out when enough people become immune, either by being infected (and recovering) or by being vaccinated. According to the model, the third and fourth peaks in epochs 5 and 6 of the pandemic were caused by this affect, and the predicted peak near the end of August (epoch 7) is also reliant on that assumption. Hence, if $${k_7} = 0.89\ {{\rm{d}}^{ - 1}}$$ represents the infection rate constant for the delta variant of COVID-19 (at the current level of social distancing) and delta is the last variant to appear in the data, then in late August 2021, we are closing in on a final test of the permanent immunity hypothesis in our fitted SIRV model and the related assumption that $$q$$ is a constant throughout the pandemic.

### 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).

#### 1. Finite difference methods

Students discover that the simple FD methods that they first learned in the context of molecular biophysics can also be applied to epidemiological models of COVID-19 in the United States. All the models that students investigate predict exponential growth at the beginning of the pandemic. Exponential growth is qualitatively different from most models in molecular biophysics. Comparison of exponential growth with exponential decay provides students with further insights into the properties of all models that predict proportional change (1) and into challenges with the accuracy of the FD method during rapid exponential growth (2).

#### 2. Systematic model development and LS fits

The teaching materials discussed here can be used as an introduction to performing LS fits. Students are guided through the process of calculating the residuals between observed data and the predictions of the model, calculating $$Q$$⁠, the sum of the squares of the residuals, and then using Excel's Solver to find the minimum in $$Q$$ (2). If needed, students can also be directed to Chapter 6 of (1) for a more detailed introduction to LS fits in the context of O2 binding to myoglobin.

As mentioned in the introduction to this article, performing LS fits to complex data is more of an art than a science, particularly when the model predictions depend exponentially on the fitted parameters (4). Utilizing the principle of Occam's razor is central to the approach presented here. Students are guided through the modeling process starting with fitting the UG model to epoch 1 with Excel's exponential trendline feature and then an LS fit. They then fit the SIR model to epoch 2 (after the transition is complete) to reproduce Figure 7. In producing the LS fit, shown in Figure 8, students first estimate $${t_{12}}$$ and $${\sigma _{12}}$$ by hand, adjusting the values in the spreadsheet and observing the effect on the model predictions. Only once they have an approximate fit, do students use Excel's Solver to find the minimum in $$Q$$⁠. Once students have a fit as shown in Figure 8, they systematically investigate how changing model parameters $$N$$ and τi affects the model and its fitted parameters. They discover that the model, up to the end of epoch 2, can be successfully fitted with any reasonable values of $$N$$ and τi. The remainder of the fits to the SIR and SIRV models are done in a similar manner by systematically adding one epoch at a time. That approach ensures that students understand how each added parameter affects the model and helps them avoid lack of convergence problems that can plague complex LS fits.

LS fits using the SIR model are not like arbitrary polynomial fits. Polynomials can be fitted to almost any shape curve, but the SIR model (with constant ki) always predicts a characteristic exponential dragon shape for the infection rate Ri(t); see Figure 6b. After a transition to a new epoch, the shape of the Ri(t) curve is determined by the current value of the susceptible fraction $$s$$ (and the infectious fraction $$i$$⁠) and the new value of the infection rate coefficient ki because all the other model parameters are held constant throughout the pandemic. As students discover, epoch 1 corresponds to exponential growth at the beginning of the dragon; epoch 2 corresponds to a gradual exponential decay during the dragon's tail caused by s$$\lt {k_r}/{k_2}$$⁠, (⁠$$h \gt {h_p}$$⁠) with low ki; epoch 3 corresponds to exponential growth with $$s \gt {k_r}/{k_3}$$⁠, (⁠$$h \lt {h_p}$$⁠); epoch 4, similar to epoch 2, corresponds to a gradual exponential decay caused by $$s \lt {k_r}/{k_4}$$⁠, (⁠$$h \gt {h_p}$$⁠) with low ki; epoch 5 corresponds to an exponential dragon, where the susceptible fraction starts with $$s \gt {k_r}/{k_5}$$⁠, (⁠$$h \lt {h_p}$$⁠) and transitions to $$s \lt {k_r}/{k_5}$$⁠, (⁠$$h \lt {h_p}$$⁠), as the susceptible fraction decreases and passes through $$s = {k_r}/{k_5} = {s_p}$$ (herd immunity for $${k_i} = {k_5}$$⁠); epoch 6 corresponds to an exponential dragon in which the susceptible fraction decreases and passes through the value of $$s = {k_r}/{k_6} = {s_p}$$ (herd immunity for $${k_i} = {k_6}$$⁠); and finally, epoch 7 (fitted up to 27 August 2021), corresponds to another exponential dragon that is predicted to peak around the present time (27 August 2021) when the susceptible fraction passes through the value of $$s = {{k_r} / {k_7}} = {s_p}$$ (herd immunity for $${k_i} = {k_7}$$⁠); see Figure 13 and Table 1.

The fit to epoch 1 is a foundational confirmation that the UG model (and the subsequent SIR models) are reasonable, as they successfully predict exponential growth at the beginning of the pandemic. The fits to epochs 2, 3, and 4, are not particularly impressive from a modeling perspective, as the fitted model does not appear significantly different from a straight line (outside of the Gaussian transition periods). Almost any model can predict linear behavior. However, the fact that the fitted model predicts a peak in epoch 5 that has the correct approximate timing, height, and width, with a single constant value of $${k_i} = {k_5} = 0.18\ {{\rm{d}}^{ - 1}}$$⁠, is a strong validation of the SIR model during that time. Recall that the SIR model always predicts an exponential dragon peak for Ri(t) of the form shown in Figure 6b (if ki is constant). Similarly, the fact that the SIRV model successfully models the peak in epoch 6 with a fitted value of $$q \approx 20\%$$ (in agreement with Fig 12) and a constant infection rate constant $${k_6}$$ through the peak is another strong validation of the SIRV model. Finally, the SIRV model also successfully explains the emergence of the delta variant in epoch 7 and makes a rather bold prediction that the peak in Ri(t) will be reached near late August 2021, assuming the infection rate coefficient does not increase beyond the fitted value of $${k_7} = 0.89\ {{\rm{d}}^{ - 1}}$$ and that the fraction of the actual US population that is represented in the reported cases per day data remains constant at $$q \approx 20\%$$⁠.

The model prediction that the delta variant peak is upon us at the end of August 2021 and that the cases per day data should fall off rapidly during September and October 2021 is based on the increasing questionable assumption that the model population is constant at $$q \approx 20\%$$ throughout the pandemic (and that the infection rate coefficient will remain constant at $${k_i} = {k_7} = 0.89\;{{\rm{d}}^{ - 1}}$$⁠). As discussed above, the CDC report dated 27 July 2021, estimates that $$q$$ is increasing as a larger fraction of actual cases are reported (20). Hence, it is likely that the height of the predicted delta variant peak occurs too soon because there are more susceptible individuals left in the actual population than a value of $$q \approx 20\%$$ predicts. In addition, it is likely that the value of $${k_7} = 0.89{{\rm{d}}^{ - 1}}$$ is an overestimate in a similar manner to the fitted value of $${k_5} = 0.23\;{d^{ - 1}}$$ being too high in the fit shown in Figure 10 with $$q = 10\%$$⁠. The validity of the assumption that $$q \approx 20\%$$ throughout the pandemic will be tested in the next few weeks or so (2).

In summary, the systematic least squares approach enables students to appreciate that the SIR model and its SIRV variant do a surprisingly good job of modeling the pandemic in the United States from 26 February 2020 to 27 August 2021. The most important question is not what is wrong with the oversimplified model, but rather, why does it work so well?

### 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.

This project was motivated by a question. Can undergraduates use spreadsheets to successfully model the spread of COVID-19? The answer is a resounding yes! In fact, this topic makes an excellent capstone experience for students interested in scientific modeling. The FD methods used are accessible to students at the level of introductory physics, and they reinforce the universal applicability of computational methods in scientific modeling. Along the way, students gain a different perspective on kinetic models and rate constants by applying them to the behavior of people. Although people do not jiggle around like molecules in solution, they do have interactions with others at a rate that can be successfully modeled using familiar biophysical techniques.

Excel is an often underrated platform for computational modeling. It has numerous advantages for undergraduate students and their instructors that facilitate the learning objectives of this case study. Excel is familiar and nonthreatening to students; most undergraduates have already used it to plot data in science labs. It also has many features that make it ideal for modeling the spread of COVID-19. The most obvious feature is that calculations are laid out spatially, which makes it easier for students without programming experience to follow the logic of the computational approach. Another advantage is the ease of graphing.

FD methods can be easily implemented in spreadsheets (1), allowing students to understand and calculate solutions to differential models that have no analytical solution. In addition, there is an extremely simple procedure for performing LS fits to computational models using Excel's Solver (1). Hence, Excel is an excellent platform for practical reasons and because it lets students and their instructors focus on the learning objectives of (a) developing FD methods, (b) using systematic model development techniques, and (c) validating the models by fitting them to real data using least squares. A scaffolded guided inquiry approach is used so that students are actively engaged in investigating the consequences of the model assumptions in a systematic step-by-step manner. That approach facilitates student understanding of the FD models as they develop them, and it enables students to see how the model parameters affect the qualitative and quantitative predictions of these introductory models, as they are systematically developing them, while simultaneously validating them using LS fits to reported data. Although the approach is aimed at students without formal programming experience, it can be easily adapted for students with programming languages, such as Python (22).

Even though the approach uses only introductory methods, the modeling approach is surprisingly successful in modeling the spread of COVID-19 in the United States. Because of its simplicity, the model also provides unexpected insights into the spread of the virus. Notably (after the initial exponential outbreak), the behavior of the US population up to 14 February 2021 can be separated into two categories: “stricter social distancing” and “relaxed social distancing.” Epochs 2 and 4 of the model in Figure 12a correspond to stricter social distancing with $${k_2} \approx {k_4} = 0.122 \pm 0.001\ {{\rm{d}}^{ - 1}}$$⁠, and epochs 3 and 5 of the model correspond to more relaxed social distancing with $${k_3} \approx {k_5} = 0.176 \pm 0.006\ {{\rm{d}}^{ - 1}}$$⁠. Hence, the inception of both the summer and fall surges can be explained by a modest $$30\%$$ increase in the infection rate coefficient ki.

A significant feature of the modeling approach is that it uses as few parameters as possible to model the published data. It is easy for students (and instructors) to be seduced into the notion that the model can do better (and it can), but every time an additional parameter is added, the question that should be asked, is will we learn anything new from it (4, 23)? As the data came in day by day, it seemed clear to me that there were changes in the infection rate coefficient occurring during the beginning of the fall surge. However, it turned out that on a longer timescale, a single transition function could fit the data almost as well and with a simpler and more insightful observation that $${k_5} \approx {k_3}$$⁠.

As the model is extended beyond Thanksgiving (26 November 2020), students discover that the size of the model population $$N$$ becomes an important parameter in the fitted model. As shown in Figure 12b, the projected model, with a model population of $$q \approx 20\%$$ of the actual US population, appears to match the US data quite well. Once vaccinations began, students added vaccination to the SIR model, resulting in an SIRV model that explicitly includes a separate box $$v$$ for the fully vaccinated. The success of the models in predicting the basic shape, height, and timing of the third peak (Fig 12) is a significant validation of the predictions of the SIR model and its SIRV variant because they have no wiggle room in the form of the predicted exponential dragon assuming constant $${k_i} = {k_5}$$ and a constant value of $$q \approx 20\%$$⁠.

The success of the SIRV model in explaining the fourth smaller peak in the pandemic (Fig 13) with $${k_i} = {k_6}$$ and a fitted value of $$q \approx 20\%$$ (the same as the rest of the pandemic) is a significant additional validation of the SIRV model. A final test of the SIRV model is whether it can explain the increase in the cases per day data during the beginning of epoch 7. As shown in Figure 13, the SIRV model can not only fit the data with the same value of $$q \approx 20\% ,$$ but it also provides insights into just how infectious the delta variant is compared with the original variants of the SARS-CoV-2 virus. Only time will tell if the SIRV model's predictions for the head and tail of the delta variant exponential dragon are correct.

Although the 5 peaks in Figure 13 have a similar appearance, it is important to note that the third, fourth, and fifth peaks in Figure 13 are qualitatively different from the first two peaks. The first peak is caused by the transition from uncontrolled spread (epoch 1) to the first period of stricter social distancing (epoch 2). The second peak is similarly caused by a transition from relaxed social distancing (epoch 3) to a second epoch 4 of stricter social distancing. In contrast, the third, fourth, and fifth peaks, during the middle of epochs 5, 6, and 7 of Figure 13, are simply exponential dragons (Fig 6b) that are intrinsic to the SIR model with a constant infection rate coefficient. The fitted infection rate constant in epoch 6 is nearly twice that of the earlier epochs 3 and 5, indicating a further substantial reduction in social distancing measures.

The fitted infection rate constant in epoch 7 is $$\sim 2.8$$ times higher than epoch 6, consistent with the delta variant being $$\sim 2.8$$ times more transmissible than the original variants of the SARS-CoV-2 virus. The SIRV model makes a bold prediction that the peak caused by the delta variant is upon us at the end of August 2021 and that the cases per day data should fall off rapidly during September and October 2021. Those predictions are based on the increasingly questionable assumption that the model population is constant at $$q \approx 20\%$$ throughout the pandemic (and that the infection rate coefficient remains constant at $${k_i} = {k_7} = 0.89\;{{\rm{d}}^{ - 1}}$$ and the infection rate amongst vaccinated and recovered individuals is negligible). Only time will tell if those assumptions remain applicable.

The success of the SIRV model in explaining and predicting the quantitative behavior of the spread of COVID-19 from 26 February 2021, through 27 August 2021, is a significant validation of the basic SIR model and its SIRV variant. As students discover, people are not molecules, but sometimes they behave like them.

Thanks to Paul Ginsparg, Robert Hilborn, and Jaqueline Lynch for helpful comments on earlier drafts of the manuscript. Support from the National Institutes of Health (Fellowship GM20584) and the National Science Foundation (grants 0836833 and 1817282) is gratefully acknowledged.

### APPENDIX. SYMBOL GLOSSARY

Table A1

Nonletter symbols.

Table A2

Prefixes, suffixes, subscripts, and superscripts.

Table A3

Letter and letter like symbols.

1.
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P. H.
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Accessed 20 June 2021.
3.
Nelson,
P. H.
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https://arxiv.org/abs/1210.3641 (preprint posted October 22, 2012)
4.
Nelson,
P. H.
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J Chem Phys
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5.
Jones,
J. H.
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Notes On $${{\cal R}_0}$$
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Accessed 20 February 2021.
6.
Nelson,
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The coronavirus outbreak—exponential growth
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Accessed 20 February 2021.
7.
European Centre for Disease Prevention and Control.
2020
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Download historical data (to 14 December 2020) on the daily number of new reported COVID-19 cases and deaths worldwide
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Accessed 15 December 2020.
8.
Mathematical Association of America.
2021
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Logistic Growth Model
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Accessed 7 April 2021.
9.
McKendrick,
A.,
and
Pai
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The rate of multiplication of micro-organisms: A mathematical study
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10.
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W. O.,
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McKendrick
A. G.
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A contribution to the mathematical theory of epidemics
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Proc R Soc Lond A
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721
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11.
Metcalf,
C. J. E.,
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and
Grenfell
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Understanding herd immunity
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753
755
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12.
Heesterbeek,
J. A. P.
2002
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A brief history of Ro and a recipe for its calculation
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Acta Biotheor
50
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189
204
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13.
Nelson,
P. H.
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Lives lost because people didn't wear masks
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Accessed 20 February 2021.
14.
Reese,
H.,
Iuliano
A. D.,
Patel
N. N.,
Garg
S.,
Kim
L.,
Silk
B. J.,
Hall
A. J.,
Fry
A.,
and
Reed
C.
2020
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Estimated incidence of coronavirus disease 2019 (COVID-19) illness and hospitalization—United States, February–September 2020
.
Clin Infect Dis
72
:
e1010
e1017
.
15.
Centers for Disease Control and Prevention.
2021
.
Estimated disease burden of COVID-19
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Accessed 1 April 2021.
16.
Our World in Data.
2021
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United States: coronavirus pandemic country profile
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Accessed 27 August 2021.
17.
Box,
G. E.
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Launer
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18.
Holmdahl,
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Buckee
C.
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N Engl J Med
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303
305
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19.
Nelson,
P. H.,
Kaiser
A. B.,
and
Bibby
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1991
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J Catal
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101
112
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20.
Centers for Disease Control and Prevention.
2021
.
Estimated disease burden of COVID-19
.
Accessed 19 August 2021.
21.
Centers for Disease Control and Prevention.
2021
.
Key things to know about COVID-19 vaccines
.
Accessed 19 August 2021.
22.
Ginsparg,
P.
2021
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Info 3950 problem set “9.” Accessed 11 June 2021
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23.
Nelson,
P. C.
2004
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W. H. Freeman and Company
,
New York
.