Teaching undergraduate physical chemistry lab with kinetic analysis of COVID-19 in the United States

Abstract

A physical chemistry lab for undergraduate students described in this report is about applying kinetic models to analyze the spread of COVID-19 in the United States and obtain the reproduction numbers. The susceptible-infectious-recovery (SIR) model and the SIR-vaccinated (SIRV) model are explained to the students and are used to analyze the COVID-19 spread data from U.S. Centers for Disease Control and Prevention (CDC). The basic reproduction number R₀ and the real-time reproduction number R_t of COVID-19 are extracted by fitting the data with the models, which explains the spreading kinetics and provides a prediction of the spreading trend in a given state. The procedure outlined here shows the differences between the SIR model and the SIRV model. The SIRV model considers the effect of vaccination which helps explain the later stages of the ongoing pandemic. The predictive power of the models is also shown giving the students some certainty in the predictions they made for the following months.

GRAPHICAL ABSTRACT

INTRODUCTION

The unprecedented times we live in have brought about a call for chemical professionals to provide solutions for instructing laboratory techniques in our rapidly changing world. The onset of the novel coronavirus or SARS-CoV-2 in 2019 (COVID-19) changed the way chemistry is taught and brought many hurdles to conquer. In the past three years, we have seen a shift from traditional learning spaces to remote solutions allowing students to be instructed in their studies without risking contact during the pandemic.^1–4 This is difficult as an important part of chemical education is physically taking part in chemical experimentation.⁵ Brilliant and dedicated educators developed and employed new procedures to provide that critical part of a well-rounded education in chemical science, for example, Silverberg published an article on an approach to laboratory instruction that allows great flexibility for students who may be in or out of quarantine.⁶ It is this goal that needs meet and, it needs to be kept in mind that not every laboratory or classroom has equal access to equipment nor does every student have equal access to a stable internet connection. This switch from in-person instruction to remote across the world has made that fact front and center to programs with a diverse range of students in regards to their race, ethnicity, and socio-economic conditions.^1,2,7,8 The laboratory procedure outlined here was built on a well-understood model in epidemiology, the susceptible-infectious-recovered (SIR) model.^9–11 Students only need access to a computer capable of running Microsoft Excel, Microsoft Word, and access to the world wide web. This procedure seeks to be very accessible with nearly any chemistry laboratory being able to conduct it with their students.

Modeling techniques have been in use in the field of epidemiology for centuries now.^9,11–14 One of the earliest examples found in its use to track the life expectancy in patients who were inoculated for smallpox.¹⁵ The foundation is then there to take the raw data derived from public health authorities and create models to understand and predict the direction of an ongoing health crisis.¹⁶ One such model is the SIR model that has its roots in a simple relationship between three possible states of being susceptible, infectious, and recovered.^9,17,18 The transitions between these three states can be treated in the same manner of chemical kinetics where a concentration of the states exists that change in time depending on various factors.^10,19,20 Three of those factors that will be focused heavily on in the procedure will be replacement number R_t, effective replacement number R_e, and the basic replacement number R₀, whose physical meanings have been explained in the literature and will be explained again in the next section when the equations are introduced.^21,22 The SIR model correlates the replacement number with the relative spreading speed of the virus. By adding additional parameters such as vaccination rates, SIR model can be extended to simulate the real-world data with a reasonable replacement number, especially at key points of public policy or social events.¹⁶

We have reported a laboratory procedure last year to analyze COVID-19 spreading in a state at the early stage without taking into account vaccinations.²³ This year we added the vaccination data into the analysis and updated the SIR model to SIR-vaccinated (SIRV) model to analyze and predict the trends of COVID-spreading in the U.S. The skills taught and utilized in this procedure are vital in a physical chemistry classroom alongside many other places in the educational journey of an undergraduate student.²⁴ The ability for students to see in real time the impact of public policy, and the statistical parameters they will derive from the model leaves a lasting impression about the usefulness and limitations of such techniques.

PROTOCOL

This laboratory procedure can be completed in two weeks (4 three-hour labs). The students are instructed to collect literature, process data, write reports, and given time to receive additional instruction outside of class time if needed. The SIR model and a modified version The SIRV model which considers vaccination rates are introduced and practiced. Pre-laboratory instruction about the SIR model along with information about the history of mathematical modeling in the study of pandemics is necessary to explain the practical usefulness to the students.^3,20,25 Microsoft Excel is the only computational program required to conduct this laboratory. While it could be conducted just the same with other programs that allow for data modeling we used Excel since the vast majority of students at our institution are far more familiar with it than similar programs. The loan mortgage calculations are practiced in Excel at the beginning to warm up the students about model simulations in Excel, which has been explained in detail in our previous publication.²³ Students will also need to source and download data of COVID case numbers and vaccination rates of the region or regions they are designated to. This could be a state in the United States, it could be just the city of New York, or it could be an entire nation. The only real requirement is that the region being examined has reliable and easily accessed data on COVID case rates per day and vaccination rates. We instructed students to use a state in the United States as their region of interest for this procedure. Both COVID-19 cases and vaccination data in the U.S. are originated from U.S. Centers for Disease Control and Prevention (CDC). The COVID cases are summarized and downloaded from the Github site of the New York Times.²⁶ The vaccination data is downloaded from Our World in Data’s State by State data on Covid-19.²⁷ Both are accessed and downloaded around Jan. 27, 2022. The data from the CDC is treated in the form of a sliding window average smooth for a 7-day period by averaging data from three days before to three days after a given day. Five days are added before the first available data point with an infected patient zero to account for the time required for the first case to become infectious and then get tested. Five days are chosen due to an incubation time of 3–7 days^16,28,29, allowing a day past the medium time of 4 days for an infected carrier to get tested¹⁶. We assume at this point the infected carrier will become isolated and although they may be physically contiguous for a longer period they cannot infect others while in active isolation. While all of this this is very unusual when modeling pandemics for the purpose of research in epidemiology since the patient zero might have been missed or there might very well be multiple patient zeros in a community, we found it useful to imagine an initial patient zero for the educational purpose of this laboratory showing the timeline of the infection and testing. The fittings show the same results with or without these additional five days for the rest of the dates (see supporting information SI). Data is checked and downloaded again on April 14, 2022 after the analysis has been done to check the predictions.

SIR Model:

The SIR model follows a flow quite like the kinds of simple diagrams you may see drawn out in a modern chemistry textbook showing you the simplest model of chemical reaction kinetics (Figure 1).¹⁰ In the model, given a set of data with S, I, R in the unit number of people in a community, the two rate constants β/N and γ are the key parameters to predict the spreading trend of the disease. β/N is the second-order rate constant with the unit number of people per day when using the number of people for the units of S, I, and R. k’=βS/N is the pseudo-first-order rate constant referring to the average number of people infected by a carrier during a period (day), where S is the total population of the susceptible group, γ is the frequency of removing an infected carrier and, N refers to the total population of a region.¹⁶ γ is the first order rate constant of recovering/isolation. All listed rates will include units of days⁻¹ as the data provided by public and private sources on the spread of the virus will be most often in days.

Fig. 1.

Scheme of the SIR model. S, susceptible; I, infectious; R, recovered population; β/N, infecting rate where N is the total population of interest; and γ, recovery/isolation rate. The sizes of the boxes are chosen to represent their relative values at a random point of time. Merging of arrows indicates a “bimolecular” reaction.

Historically, the two rate constants in the SIR model are combined into a new parameter called the reproduction number. Three different forms of reproduction numbers are examined in this lab, namely the time-dependent reproduction number R_t, the time-dependent effective reproduction number R_e, and the basic reproduction number, R₀ (Equations 1–3). The time-dependent reproduction number, R_t, reflects the infectivity of the virus and the social interaction frequency of the society and is defined:

$$R_t=\frac{\beta \left(t\right)}{\gamma \left(t\right)}$$ (1)

which is the ratio between the second-order reaction rate constant of infecting and the first-order rate constant of removing an active carrier, thus, it is unitless. For a given variant of the virus, assuming a fixed γ value that is related to the average incubation time, the R_t value should be mainly proportional to the frequency of social interactions and virus transportation rate which are affected by various conditions such as public awareness, government regulations, social distance, air circulation, hand washing, and wearing masks. Its value is proportional to the virus transmission probability of an interaction between an infectious person and a susceptible person and is not affected by the decrease in the concentration of the susceptible population due to recovery or vaccination.

An effective reproduction number, R_e, has been defined in the literature the quasi-first order reaction rate over the recovering rate:

$$R_e=\frac{\beta \left(t\right)}{\gamma \left(t\right)}\frac{S\left(t\right)}{N\left(t\right)}$$ (2)

Comparing to R_t, R_e is reduced by the decrease of the concentration of the susceptible population along the epidemic. R_e>1 reflects an exponential-like growth in the adjacent days but will no longer be exponential as a significant number of people in the susceptible population move groups.⁹ R_e=1 a steady state, and R_e<1 sees a decay of the number of infectious people,^9,21 corresponding to the upside and the downside of the waves of infected cases each day.

The basic reproduction rate, R₀, is the initial reproduction number which is the same as the initial R_t and R_e at the beginning, t = 0, when the susceptible population is assumed to be the same as the total population and no action is yet taken to mitigate the spreading,

$$R_0=\frac{\beta \left(t=0\right)}{\gamma \left(t=0\right)}$$ (3)

The rate equations of the SIR model are listed below. The derivative of the susceptible population is the negative rate constant times the product of the number of susceptible people S and the number of infectious people I:

$$\frac{dS}{dt}=-\frac{\beta }{N}SI$$ (4)

The changing rate of the infectious population is a tug of war between the newly infected population and the recovered population each day. The latter being a rate constant of removing an infectious carrier, γ, times the total number of infectious people. The value γ is the one over the number of days before each infectious person is removed from the infectious category mainly by isolation once they are aware of being infected, we used a fixed average value of 0.2 day⁻¹ according to the literature for simplicity,^9,23,28–30 even though it should slightly vary over different population groups and time.¹⁷ Normally γ would reflect the rate of recovery by acquired immunity. In most cases acquiring natural immunity from becoming infected will take longer than the period between being exposed and becoming isolated. People who are isolated and still contiguous are effectively “recovered” in this model as they cannot act as a member of the infectious population that can infect the susceptible population.

$$\frac{dI}{dt}=\frac{\beta }{N}SI-\gamma I$$ (5)

$$\frac{dR}{dt}=\gamma I$$ (6)

SIRV Model:

The SIRV model is nearly identical to the SIR model but includes the effect of vaccination on reducing the number of susceptible people (Figure 2). The vaccination rate is time-dependent for COVID-19 as vaccines become more available and the number of people who are willing to take the vaccine varies over time. Equations (7–11) show the relationship between the variables in the SIRV model. The vaccinated population V is kept separate from the recovered population in our calculations. While the immune population is made up of those who either in the recovered group or/and were vaccinated, we kept them distinct to emphasize the different ways of obtaining immunity.

Fig. 2.

Scheme of the SIRV model. S, susceptible; I, infectious; R, recovered population; β/N, infecting rate where N is the total population; γ, recovery/isolation rate; V, vaccinated population; r_V, the vaccination rate. The immune population M is assumed to be recovered or/and vaccinated people.

In this model, the rate at which the susceptible population is decreasing is a combination of the infected rate with the vaccinated rate. The vaccination rate is the increase of the fully vaccinated population over the total population in a period. People are assumed to be immune the day after receiving the one-dose vaccine and the second dose of the two-dose vaccine.

$$r_V=\frac{1}{N}\frac{dV}{dt}$$ (7)

Thus, assuming the vaccination is evenly distributed across the total population, the vaccination rate in the susceptible population is the same as r_V,

$$\frac{dS}{dt}=-\frac{\beta }{N}SI-r_{V}S$$ (8)

$$\frac{dI}{dt}=\frac{\beta }{N}SI-\gamma I$$ (9)

$$\frac{dR}{dt}=\gamma I$$ (10)

The immune population M is made up of both the recovered population and the vaccinated population.

$$\frac{dM}{dt}=r_{V}S+\gamma I$$ (11)

All the data analysis in the lab is carried out using discrete numerical methods instead of the continuous analytical model described above. These equations are converted to discrete equations with the time interval dt ⇒ Δt = t_i+1 – t_i ≡ 1 day, where the subscript i+1 and i meaning two adjacent days, e.g. i+1 is today then i is yesterday. Choosing 1 day instead of a much finer period is compatible with the raw data and does not affect the fitting. However, it will introduce an error in the accumulated simulations if one wants to predict a long period after a reference point (SI convergence tests). Examples of these recursive equations are provided in the following for the SIRV model. Equations for the SIR model use the same principle.

$$r_{V\left(i+1\right)}=\frac{V_{i+1}-V_i}{N\Delta t}$$ (12)

$$S_{i+1}=S_i-\left(\frac{{\beta }_i{I}_i{S}_i}{N}+r_{Vi}{S}_i\right)\Delta t$$ (13)

$$I_{i+1}=I_i+\left(\frac{{\beta }_i{I}_i{S}_i}{N}-{\gamma }_i{I}_i\right)\Delta t$$ (14)

$$R_{i+1}=R_i+\left({\gamma }_i{I}_i\right)\Delta t$$ (15)

$$V_{i+1}=V_i+\left(r_{Vi}{S}_i\right)\Delta t$$ (16)

$$M_{i+1}=M_i+\left({\gamma }_i{I}_i+r_{Vi}{S}_i\right)\Delta t$$ (17)

Where S_i, I_i, R_i, V_i, and M_i are the number of susceptible people, the number of active infectious people, the number of recovered people, the number of vaccinated people, and the number of immune people on day i, respectively. The vaccinated population was calculated from the CDC data using Equation 12 and Equation 16. The model predicted daily new cases number is

$$new{I}_{i+1}= \frac{{\beta }_i{I}_i{S}_i}{N}\Delta t$$ (18)

The students start the numerical calculation by guessing the first date of the first infectious person in a state, e.g., five days before the first available data. These days are added just for an educational purpose about reaction time and can be removed. The student can practice manual fitting by changing R_t values piecewise as explained previously by comparing the simulated new cases per day with the raw data.²³ They can also directly calculate the real-time R_t and R_e values by estimating β from Equation 18 and using Equation 1 by replacing the simulated new case with the smoothed raw data of daily case from CDC, newI_raw:

$$R_{t_{\left(i+1\right)}} =\frac{new{I}_{ra{w}_{i+1}}}{\Delta t}*\frac{N}{S_i{I}_i{\gamma }_i}$$ (19)

Δt is held at 1 day and γ is held at 1/5 day⁻¹ throughout the fitting,²⁵ consistent with literature values and CDC suggestion of 3–7 days of infectious period.^4,29,30 From R_t(i+1), one can calculate β and R_e

$${\beta }_{i+1}={\gamma }_{i+1}{R}_{t_{\left(i+1\right)}}$$ (20)

Or one can calculate β first and then use β to calculate R_e,

$${\beta }_{i+1}=\frac{new{I}_{ra{w}_{i+1}}}{\Delta t}*\frac{N{\gamma }_{i+1}}{S_i{I}_i{\gamma }_i}$$ (21)

$$R_{e_{\left(i+1\right)}} =\frac{{\beta }_{i+1}{S}_{i+1}}{N{\gamma }_{i+1}}$$ (22)

These values can be used to calculate S, I, M, and daily new cases (Equation 12–18) that day. Please see SI Excel sheets for the detailed calculations on both models.

RESULTS

Covid Data Acquisition:

Each student was assigned a state and instructed to use a source whose data can be verified by the CDC.^26,27 The daily case count can be found by subtracting subsequent values of the cumulative case count. We assume that there are no cases of natural immunity and that all infectious people are tested without serious human error. A smoothed daily case count can then be generated with a moving average of the daily change in cases with 3 days before and after (Figure 3).

Fig. 3.

(a) Example of daily cases produced by subtracting a day’s accumulated positive cases from the previous number overlaid with the seven day smooth for the state of Ohio from 3/4/2020 to 1/25/2022. (b) Number of accumulated fully vaccinated people in the state of Ohio till 1/25/2022.

Data Analysis:

Once students have collected data for their state, they were instructed to find the base reproduction factor R_o by manually fitting the SIR model to data from the first two weeks. This was done by taking the calculated cases per day generated from the piecewise SIR model and fitting it to the cases per day from their smoothed data when minimizing the sum of the residuals.

$$\mathit{\text{Residual}}=I_{t_i}-I_i$$ (23)

where I_t,i is the data from CDC and I_i is simulated active cases on day i. Typical R₀ value of a U.S. state is found to be at ~3.^23,30

Then R_t is manually adjusted in a short period of choice to minimize the sum of absolute residuals or directly calculated from the daily cases in a state. An example is shown with the data in the state of Ohio (Figure 4). Both the R_t and R_e values are found to be oscillating around 1 along the waves of outbreaks.

Fig. 4.

Fitting of R_t and R_e over time in the state of Ohio from 3/9/2020 to 1/25/2022 using (a) piecewise averaging fitting, and (b) direct calculation of R_t with the SIR model.

When considering vaccination, assuming immunity is gained after being fully vaccinated, and refitting the same data with the SIRV model, the R_t value gradually bends up after March 2021 and reaches somewhat close to the original basic reproduction value R₀~3 and the R_e values remains similar to those in the SIR model (Figure 5).

Fig. 5.

Fitting of R_t and R_e over time in Ohio calculation from 3/9/2020 to 2/22/2022 using (a) piecewise averaging fitting, and (b) direct calculation of R_t with the SIRV model.

The R_t value of several randomly chosen states are listed in Fig. 6 including Ohio shown above. The real-time fitting at the beginning of the outbreak is very noisy. The states investigated show a similar trend. Some large noises in summer 2021 are from missing data points and readjustment on the number of the positive cases in some states.

Fig. 6.

Comparison of real-time R_t value obtained from SIRV model fitting of several randomly chosen states.

Students were instructed to continue their calculated daily cases well past 1/25/2022 after then available data using the average reproduction factor for the last five days of available data. Their predictions overlap well with data obtained at a later date giving the students some predictive power (Figure 7).

Fig. 7.

Prediction of the trend in Ohio after 1/25/2022 with different models overlaid with the data obtained 4/14/2022 from U.S. CDC (red dashed line). An R_t value averaged from 1/20–1/25 is used for all days after 1/25.

DISCUSSION

The basic reproduction number of COVID is better extracted from the piecewise method of the first two weeks of data due to the relatively high sensitivity to the noise at the beginning when case numbers are small. Two weeks of data give consistent results from state to state with R₀ ~ 3. The piecewise method can be applied to the full data set but is time-consuming. Thus, with proper smoothing, the real-time method is better to determine the R_t values over time for the middle and later stages.

R_t is directly related to the second-order reaction rate constants which reflect the collision frequency of individuals and the energy barriers of the disease infection. R_e is related to the quasi-first-order reaction rate constant that has been normalized to the concentration of the susceptible population in the community. The R_e value as designed directly correlated with the growth and decline of the daily cases and its magnitude away from 1 reflects the exponential-like growth in the adjacent days or decline rate like the interest rate or payback rate in a mortgage. However, it does not reflect the effect of social regulations as well as R_t, especially in the later stage of the spread. In the later stage, the concentration of the susceptible population has been significantly reduced due to immunity either from recovery or vaccination. Thus, R_t is a more valuable value than R_e to trace the pandemic during these spreading stages. The resulting R_t from the SIRV model (Figure 6) is consistent with our experience that travel and social activities are almost fully restored around the 2022 new year which brings the R_t value back to near R₀ in all the randomly chosen states. Both SIR (Figure 4) and SIRV model (Figure 5) give the same R_e values but SIR model significantly lowers the R_t estimated values when vaccination is significant.

Breakthrough, natural immunity, and partial immunity are all ignored during the analysis to simplify the model. Variants of the virus such as delta and omicron, social density, transportations, cultural differences, changing γ values, and variations of these values among different locations are ignored but may have been reflected in the variations of the β values. These parameters could be added to create more reaction pathways for varying subpopulations, which is beyond the scope of this data analysis lab course. With information on what number of daily cases represent separate variants in the total number of daily new cases and the number of reinfections, R→S→I, the model can be expanded to account for the impact of reported higher rates of reinfection by the Omicron variant.^30,31

The model provides the students some prediction power on the spread trend of COVID-19 (Fig. 7). For example, if a mask can block 50% of the virus from spreading, then we can reduce the R_t value by 50% if everybody has put on a mask; reducing 50% of social activity further halves the R_t value as seen in the first several months of the pandemic (Figure 4–5 piecewise R_t). In general, the R_t value is unlikely to exceed the R₀ value and must be larger or equal to zero for the same virus which set the limits on the upper and lower boundary of the spreading rate. The effect of the vaccination on the spreading rate is readily visible by comparing the simulations with and without vaccination and assuming no change in the social behaviors. In both SIR model and SIRV model, either R_t or R_e provides some prediction ability to the near future (Figure 7), but it makes more sense to use R_t of SIRV to predict the trend in a longer period. There is no obvious evidence observed for the different spreading rates among the different variants such as delta and omicron, which will need data on the variants among the positive cases to be analyzed.

CONCLUSION

Extending the kinetic model from SIR to SIRV helps the students to understand the later stages of disease prevention and experience kinetic model selection. Comparing the R_t and R_e values help the students to distinguish the second-order reaction and the quasi-first order reaction models for the same reaction mechanism, which echoes what they have learned in the class as well. Comparison between the mortgage model, the COVID-19 kinetics, and the conventional chemical reaction kinetics allow the students to compare the units in different systems, especially on the rate constants, thus helping them to understand the purpose of such analysis and the prediction task better. The real-time R_t values correlate with social events the best thus provides more prediction abilities, but their value is dependent on various conditions such as models, assumptions, data availability, and data accuracy. Overall, applying what they have learned in chemistry classes to a real-world problem motivates the students to learn and practice various skills in chemistry.