# 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: