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.

St+1=StβStItS_{t+1} = S_{t} −βS_{t}I_{t}
It+1=It+βStItγItI_{t+1}​ = I_{t} + βS_{t}​I_{t}​−γI_{t}​
Rt+1=Rt+γItR_{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, β\beta and γ\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 R0R_{0} . This represents the average number of people who will be infected by any given infected person. When R0R_{0} is greater than 1, it means that a disease will grow. A higher R0R_{0} implies more rapid transmission and a more rapid growth of the epidemic. It is defined as R0=β/γR_{0}=\beta / \gamma

R0R_{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 R0R_{0}​of 2.71.

Effect of social distancing

After the beginning of the outbreak, actions to reduce social contact will lower the parameter cc . If this happens at time tt , then the effective reproduction rate is RtR_{t} , which will be lower than R0R_{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 RtR_{t}of 1.36.

Using the model

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

  • γ\gamma : the CDC recommends 14 days of self-quarantine, we'll use γ=1/14\gamma = 1/14 .

  • To estimate β\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 TdT_{d} ​ of 7-10 days. That means an early-phase rate of growth can be computed by using the doubling time formula:

g=21/Td1g=2^{1/T_{d}}​−1
  • Since the rate of new infections in the SIR model is g=βSγg=\beta S - \gamma and we've already computed γ\gamma, β\beta becomes a function of the initial population size of susceptible individuals β=(g+γ)\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.

Additional references

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