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.

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

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.

the pathogen is more infectious

people are infectious for longer periods of time

the number susceptible people is higher

**Effect of social distancing**

**Using the model**

We need to express the two parameters β and γ in terms of quantities we can estimate.

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.

Additional references

Discrete-time SIR modeling: https://mathworld.wolfram.com/SIRModel.html

$S_{t+1} = S_{t} −βS_{t}I_{t}$

$I_{t+1} = I_{t} + βS_{t}I_{t}−γI_{t}$

$R_{t+1} = R_{t} + γI_{t}$

The model's parameters, $\beta$ and $\gamma$ , determine the severity of the epidemic.

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

A doubling time of 6 days and a recovery time of 14.0 days imply an $R_{0}$of 2.71.

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.

$\gamma$ : 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 $\beta = (g+\gamma)$.