The model consists of individuals who are either Susceptible (S), Infected (I), or Recovered (R).
The epidemic proceeds via a growth and decline process. This is the core model of infectious disease spread and has been in use in epidemiology for many years.
The dynamics are given by the following 3 equations.
To project the expected impact to Penn Medicine, we estimate the terms of the model.
To do this, we use a combination of estimates from other locations, informed estimates based on logical reasoning, and best guesses from the American Hospital Association.
The model's parameters, and , determine the severity of the epidemic.
β can be interpreted as the effective contact rate: β=τ×c
which is the transmissibility τ multiplied by the average number of people exposed c. The transmissibility is the basic virulence of the pathogen. The number of people exposed c is the parameter that can be changed through social distancing.
γ is the inverse of the mean recovery time, in days. i.e.: if γ=1/14 then the average infection will clear in 14 days.
An important descriptive parameter is the basic reproduction number, or . This represents the average number of people who will be infected by any given infected person. When is greater than 1, it means that a disease will grow. A higher implies more rapid transmission and a more rapid growth of the epidemic. It is defined as
is larger when
the pathogen is more infectious
people are infectious for longer periods of time
the number susceptible people is higher
A doubling time of 6 days and a recovery time of 14.0 days imply an of 2.71.
Effect of social distancing
After the beginning of the outbreak, actions to reduce social contact will lower the parameter . If this happens at time , then the effective reproduction rate is , which will be lower than .
For example, in the model, a 50% reduction in social contact would increase the time it takes for the outbreak to double, to 27.5 days from 6.00 days, with a of 1.36.
Using the model
We need to express the two parameters β and γ in terms of quantities we can estimate.
: the CDC recommends 14 days of self-quarantine, we'll use .
To estimate directly, we'd need to know transmissibility and social contact rates. Since we don't know these things, we can extract it from known doubling times. The AHA says to expect a doubling time of 7-10 days. That means an early-phase rate of growth can be computed by using the doubling time formula:
Since the rate of new infections in the SIR model is and we've already computed , becomes a function of the initial population size of susceptible individuals .
The default value for the total size of the susceptible population defaults to the entire catchment area for Penn Medicine entities (HUP, PAH, PMC, CCH)
Delaware = 564696
Chester = 519293
Montgomery = 826075
Bucks = 628341
Philly = 1581000
For other default values, please consult the Data Inputs section.
Discrete-time SIR modeling: https://mathworld.wolfram.com/SIRModel.html