You signed in with another tab or window. Reload to refresh your session.You signed out in another tab or window. Reload to refresh your session.You switched accounts on another tab or window. Reload to refresh your session.Dismiss alert
"[Discrete-time SIR modeling](https://mathworld.wolfram.com/SIRModel.html) of infections/recovery"
381
-
)
382
-
st.markdown(
383
-
"""The model consists of individuals who are either _Susceptible_ ($S$), _Infected_ ($I$), or _Recovered_ ($R$).
384
-
385
-
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."""
386
-
)
387
-
st.markdown("""The dynamics are given by the following 3 equations.""")
"""To project the expected impact to Penn Medicine, we estimate the terms of the model.
395
-
396
-
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.
397
-
398
-
399
-
### Parameters
400
-
401
-
The model's parameters, $\\beta$ and $\\gamma$, determine the virulence of the epidemic.
402
-
403
-
$$\\beta$$ can be interpreted as the _effective contact rate_:
404
-
"""
405
-
)
406
-
st.latex("\\beta = \\tau \\times c")
407
-
408
-
st.markdown(
409
-
"""which is the transmissibility ($\\tau$) 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.
410
-
411
-
412
-
$\\gamma$ is the inverse of the mean recovery time, in days. I.e.: if $\\gamma = 1/{recovery_days}$, then the average infection will clear in {recovery_days} days.
413
-
414
-
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. Higher $R_0$'s imply more rapid growth. It is defined as """.format(
415
-
recovery_days=int(parameters.infectious_days)
416
-
)
417
-
)
418
-
st.latex("R_0 = \\beta /\\gamma")
419
-
420
-
st.markdown(
421
-
"""
422
-
423
-
$R_0$ gets bigger when
424
-
425
-
- there are more contacts between people
426
-
- when the pathogen is more virulent
427
-
- when people have the pathogen for longer periods of time
428
-
429
-
A doubling time of {doubling_time} days and a recovery time of {recovery_days} days imply an $R_0$ of {r_naught:.2f}.
430
-
431
-
#### Effect of social distancing
432
-
433
-
After the beginning of the outbreak, actions to reduce social contact will lower the parameter $c$. If this happens at
434
-
time $t$, then the number of people infected by any given infected person is $R_t$, which will be lower than $R_0$.
435
-
436
-
A {relative_contact_rate:.0%} reduction in social contact would increase the time it takes for the outbreak to double,
437
-
to {doubling_time_t:.2f} days from {doubling_time:.2f} days, with a $R_t$ of {r_t:.2f}.
438
-
439
-
#### Using the model
440
-
441
-
We need to express the two parameters $\\beta$ and $\\gamma$ in terms of quantities we can estimate.
442
-
443
-
- $\\gamma$: the CDC is recommending 14 days of self-quarantine, we'll use $\\gamma = 1/{recovery_days}$.
444
-
- 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:
- 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.
0 commit comments