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The Model
How does the CHIME model work?

# Compartmental Modeling

## ​Discrete-time SIR modeling of infections/recovery

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.
$S_{t+1} = S_{t} −βS_{t}I_{t}$
$I_{t+1}​ = I_{t} + βS_{t}​I_{t}​−γI_{t}​$
$R_{t+1} ​= R_{t} + γI_{t}​$
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.

## Parameters

The model's parameters,
$\beta$
and
$\gamma$
, 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
$R_{0}$
. This represents the average number of people who will be infected by any given infected person. When
$R_{0}$
is greater than 1, it means that a disease will grow. A higher
$R_{0}$
implies more rapid transmission and a more rapid growth of the epidemic. It is defined as
$R_{0}=\beta / \gamma$
$R_{0}$
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
$R_{0}$
​of 2.71.
Effect of social distancing
After the beginning of the outbreak, actions to reduce social contact will lower the parameter
$c$
. If this happens at time
$t$
, then the effective reproduction rate is
$R_{t}$
, which will be lower than
$R_{0}$
.
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
$R_{t}$
of 1.36.
Using the model
We need to express the two parameters β and γ in terms of quantities we can estimate.
• $\gamma$
: the CDC recommends 14 days of self-quarantine, we'll use
$\gamma = 1/14$
.
• To estimate
$\beta$
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
$T_{d}$
​ of 7-10 days. That means an early-phase rate of growth can be computed by using the doubling time formula:
$g=2^{1/T_{d}}​−1$
• Since the rate of new infections in the SIR model is
$g=\beta S - \gamma$
$\gamma$
,
$\beta$
becomes a function of the initial population size of susceptible individuals
$\beta = (g+\gamma)$
.

### Initial Conditions

• 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.