# The Model ## 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](https://app.gitbook.com/@code-for-philly/s/chime/~/drafts/-M2o8KSwkN-ZmcGnQa5n/what-is-chime/parameters) section. ## Additional references Discrete-time SIR modeling: --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://code-for-philly.gitbook.io/chime/what-is-chime/sir-modeling.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.