# The Model

How does the CHIME model work?

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.

The model's parameters,

$\beta$

and $\gamma$

, determine the severity of the epidemic.β can be interpreted as the

*effective contact rate*: β=τ×cwhich 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$\beta = (g+\gamma)$.

- 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

Last modified 2yr ago