# 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 $g=\beta S - \gamma$ and we've already computed $\gamma$, $\beta$ 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.