The model to flatten the curve
You can make you own curve here. We have also made a website to track the daily
As the current covid-19 pandemic sweeping across the global (certainly in New York) and posting dreadful tallies day over day, most of the governments are urging their citizens to stay home, practice the social distancing, in order to “flatten the curve”, so that we can get out from this crisis.
Why we need to flatten the curvePermalink
The curve referenced here usually refers to the number of hospitalizations over time. Being a respiratory disease, the covid-19 patients often require care with medical resources, such as hospital staff and ventilators. The problem is, once the number of patients overwhelms the medical resources in the area, there is almost nothing the doctors can do, but forced to make difficult and agonizing decisions [1].
In order to avoid such meltdown of medical system, the key is to slow the transmission of the disease among the population, such that the rush to the hospitals is spread out over time. Qualitatively, this makes perfect sense, however, how does one calculate the curve quantitatively, namely, put the numbers on the axes?
The model that draws the curvePermalink
The most commonly used framework in modelling infectious disease is the SIR model. The basic construct of the model is pretty simple: the whole population is divided into different “compartments”, and any person at any given time is in one of those buckets. In the most basic setting, there will be three compartments: Susceptible (not yet have the disease), Infectious (have the disease), and Recovered (immuned). A person will move from one compartment to another as time progresses, and the dynamic (rate) of people moving between different compartments dictates the total number of people in each compartment. This dynamics will help us to draw the curve.
From S to IPermalink
At a given time

From I to R (or D)Permalink
Once a person becomes infectious (not necessarily symptomatic), the next compartment(s) for him/her to go is recovered, or unfortunately, dead. Note that in the vanilla version of the SIR model, death is not considered, however, given the unusally high fatality rate of covid-19 [2], we can not ignore the non-zero death rate.
With the above consideration in mind, we will assume the number of people moving from

Putting everything togetherPermalink
Now we have all the ingredients of this SIR(D) model in place, the next thing is to properly describe their dynamics between each other (
This can be easily solved numerically (python code). Below is a simulation with

The most commonly used time step is a day, and with this setting, we can layer in some statistical assumption to bring more meanings to the parameters. Suppose the duration of each infection is exponentially distributed, with an average of
The famous R0Permalink
Another number that is being discussed a lot in the pandemic context, is the basic reproduction number,
where we can define
This simplified form distills the essence of the meaning of
There is an important note we should take about
Parameter estimation and model extensionPermalink
While the construct and the (numerical) solution of the model is relatively straightforward, the real complexity lies in the parameters estimation, e.g., what is the most appropriate values to use for
Aside from the intricacy in parameter estimation, another note worth mentioning here is that, one can extend the current SIR(D) model with more compartments, such as hospitalized, intensive-care-unit’ed (ICU’ed), as long as one can properly model the transition between each compartment, and estimate the parameters with reasonably good accuracy.
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