Kurt Kreith and Alvin Mendle, University of California, Davis
Covid-19 has left teachers seeking topics that are both engaging and lend themselves to online instruction. As a guiding force for the measures that have reshaped our lives, epidemic modeling stands out as natural. For teachers at the secondary level and those involved in teacher education, this leads to the question: How can an understanding of epidemic modeling be made accessible to students at large?
From the vantage point of evolutionary biology, viewing epidemics as a form of natural selection is a good place to start. An ability to reproduce and mutate rapidly would seem to give the virus a distinct advantage. The tools Humankind can bring to bear include (1) human intellect and (2) a capacity for social organization. Both have figured prominently into efforts to manage Covid-19 and make an appearance in the model to be developed below.
Modern epidemic modeling began with the S.I.R. model created by Kermack and McKendrick in 1927, introducing the use of S, I, and R to designate susceptible, infected, and recovered demographic variables. Here we consider a population P of fixed size (it undergoes neither births nor deaths, and those recovered enjoy total immunity). In set theoretic terms, this can be thought of as
P = S ∪ I ∪ R
where people pass from S to I and from I to R at prescribed rates. The model yields values for all three variables as a function of time.
The fact that S.I.R. was and continues to be described in terms of a system of differential equations poses a barrier to making its insights accessible to students and citizens at large. The calculus-based approach to our version of S.I.R. calls for an application of Euler’s method to this system. However in the digital age, basic algebra combined with spreadsheet apps such as Googlesheets enable us to sidestep such technical challenges.
While a variety of institutions have addressed the issue of difference vs. differential equations in years gone by, computer technology and Covid-19 have changed our world. Not only is it now possible to introduce such models to students at the secondary level, there is a pressing public health reason to do so. Without claiming that the study of epidemic modeling will serve to inoculate society against disinformation, a basic citizen understanding of such tools seems essential for efforts to control the spread of Covid-19.
A broadly inclusive formulation of S.I.R. might begin with a problem such as the following. A hospital has an admissions office and a discharge office, both of which keep daily records. Given a Monday morning population of 43 patients, use the hospital’s admission and discharge records to calculate the patient population for the remainder of the week.
Extending this problem to an entire month sets the stage for introducing functional notation. Here one defines I(n) as the hospital population on (the morning of) the nth day, a(n) as the number admitted that day, and d(n) the number discharged. This can be combined with basic spreadsheet instruction to implement the recursive scheme
I(n+1) = I(n) + a(n) – d(n)
By way of relating this simple exercise to the S.I.R. model, we consider a cruise ship that leaves port with a population of 100 passengers, 90 of which are susceptible to a virus and 10 of which are recovered and thereby immune (the crew are all immune). One morning two of the susceptible passengers are diagnosed as infected and the ship is isolated at sea.
To create a record of the ensuing epidemic, the captain uses S(n), I(n), and R(n) for the number of passengers who are susceptible, infected, and recovered (and thereby immune) on the morning of the nth day. But instead of requiring a head count each morning, she keeps daily records of the numbers admitted to and discharged from the sickbay. This leads to three “hospital problems” and the equations
S(n+1) = S(n) – a(n) S(1) = 88
(*) I(n+1) = I(n) + a(n) – d(n) I(1) = 2
R(n+1) = R(n) + d(n) R(1) = 10
So how does one relate this intuitive set of formulas to S.I.R.? Well, the point of epidemic modeling is to anticipate the course of an epidemic rather than deducing its history. In the context of (), this calls for estimating the values of a(n) and d(n) on the basis of the morning head count S(n), I(n), R(n). With all five of these numbers at hand we can use () to calculate S(n+1), I(n+1), R(n+1), and then to estimate a(n+1) and d(n+1), and then to …
For the captain of our cruise ship, such a tool might be used to take actions aimed at controlling the epidemic to come. For us, the process of anticipating a(n) and d(n) is where important insights are to be gained. Rather than advanced mathematics, it calls for an understanding of human behavior as well as the disease being modeled. Consider the following line of reasoning:
A disease spreads as the result of contacts (meetings) between pairs of infected and susceptible persons. If a population P of (constant) size N averages m meetings/day, then its infected persons will have I(n) × m meetings per day. But only a fraction of these will be with a susceptible person. Of the I(n) × m meetings, I(n) × m × (S(n)/N) will be with a susceptible person. But not all of these meetings will actually transmit the disease. Letting p denote the probability that a meeting between a susceptible and an infected person will actually transmit the disease, we arrive at
a(n) = I(n) × m × (S(n)/N) x p
S.I.R. makes shorter shrift of d(n).
If the disease lasts r days, the number of patients discharged on the nth day can be taken as
d(n) = I(n)/r.
In the resulting model, the constants m, p, r embody human efforts to control an epidemic. Closing schools and businesses serves to reduce “meetings/day”; masks and social distancing serve to reduce p; medical research can reduce the “recovery time” r. These observations breathe life into the implementation of (*) with
(**) a(n) = I(n) x m x (S(n)/N) x p and d(n) = I(n)/r
In the case of our cruise ship, setting m = 6, p = 0.1, r = 12 and implementing a routine spreadsheet simulation for 60 days yields
By way of building on this classical image from epidemic modeling, one can use the inequality a(n) ≤ d(n) and (**) to arrive at S.I.R.’s condition for herd immunity:
S(n)/N ≤ 1/(mpr).
Efforts to temporarily reduce the size of m, p, and/or r serve to ease the requirement for herd immunity. However, a premature easing of these requirements can serve to give the virus free play. In the graph below, our cruise ship captain has imposed rules that reduce m = 6 to m = 1 at n = 5 and returned to m = 6 at n = 30.
In reducing the infection peak from 50 patients to 36, the captain has also deferred the peak’s occurrence from day 16 to day 43. This may allow for the arrival of vaccines that can be used to modify such a scenario.
The question then becomes, is there value in such work? Our answer is “yes.” It allows
K-12 students to begin a journey that uses mathematical modeling to examine issues that affect their daily lives. It puts students in the driver’s seat by allowing them to use data and to draw inferences as they manipulate variables based on different assumptions. And finally, it makes creative use of technology that is free and readily available.
These ideas constituted the first Zoom session of an online special study mathematics course we offered at UC Davis this past Winter quarter. They will also be at the heart of an in-person First Year Seminar to be offered at UC Davis next Winter under the heading “Covid-19, By the Numbers”. We will be glad to share further thoughts and materials with anyone having similar interests.