Comparison to the JHU Data

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UPDATE: 28/04/2020. A Study from Singapore has done exactly what I was going to do in this Section! Check out the link below. Hopefully, you will have enough math background to understand what this study does. As a benefit it allows one to predict the end of the pandemic for various countries.

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When Will COVID-19 End

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UPDATE: 13/05/2020. The web page above has "internalized" their results and are therefore no longer available. That convinced me to finish my work and create my own fits.

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ORIGINAL + FITTING WORK (as of 13/05/2020)

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So far we have just played around with mathematical models. The derivation "makes sense" and follows the phenomenological description of the spread of a virus. But does the math actually match measured data for a real virus outbreak?

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Welcome to the world of parameter-fitting.

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We will first focus on the simple SIR model which has only two parameters: the spread rate "b" and the recovery rate "k". Each pair of values determines the three S, I and R curves as shown in the demo above.

 

We wish to estimate the values of these parameters for a given country from actual data. Of course the SIR model match won't be exact: the data is noisy, the model leaves out many factors, etc. But hopefully it will be "close enough." Making that last statement precise is the art and science of modeling. 

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First we need some real data!

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Since January 22, 2020 the John Hopkin's University has been releasing COVID-19 virus data for every country of the world. Specifically it releases a daily time series for every country of the following three variables:

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• confirmed : # of people who tested positive

• recovered : # of people who recovered

• deaths : # of people who died.

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These variables map easily to the variables of the SIR model. For more information see the JHU web page

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https://systems.jhu.edu/research/public-health/ncov/

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The daily data has been conveniently parsed into a json-format by Rodrigo Pombo. This is how you can access it in your JavaScript (modify it to match your needs, this code snipet just prints out the data to the console).

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fetch("https://pomber.github.io/covid19/timeseries.json")

.then(response => response.json())

.then(data => { data["China"].forEach(

({ date, confirmed, recovered, deaths }) =>

console.log(`${date} C=${confirmed} R=${recovered} D=${deaths}`)

);

});

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In the plots below I am using the number of daily deaths. The reason is that this is a solid number despite the fact that it is probably over-estimated because of co morbidity: a lot of patients die with the virus not because of the virus. In the SIR model the infection rate is assumed to be proportional to the lethality rate.

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To fit the SIR model to the data we use a very simple non-linear least square approach. Our SIR model does not admit an analytical solution. To evaluate the function at a point "x" we therefore have to run the simulation up to the date x. The function also depends on the parameters p=[b, k]:

 

function funSIR(x, p) {

let steps = 10;
let dt = 1.0 / steps;

var S = 1.0;
var y = data[0] / maxData;

for (i = 1; i <= x; i++) {

var S0 = S;
var y0 = y;

for (k = 0; k < steps; k++) {

S = S0 + dt * (-p[0] * S0 * y0);
y = y0 + dt * ( p[0] * S0 * y0 - p[1] * y0);

S0 = S;
y0 = y;

}

}

return y;

}

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The function we are trying to minimize is the difference squared between the SIR function and the "death" data data[i].

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function funCost(p) {

var sum = 0;

for(i = 0; i < nData; i++) {

var diff = funSIR(i, p) - data[i];

sum += diff * diff;

}

return sum;

}

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To fit the parameters we use a very simple multivariate unconstrained optimization technique. The function is initialized with a step for each parameter value. Then separately for each parameter the step is added and the cost function is evaluated. If this cost is smaller than the current one we keep the parameter value and increase the step. We are going in the right direction! If not we decrease the step and reverse its sign. For those readers who know MATLAB, this is a simple version of "fminsearch". We give the full implementation  here in JavaScript.

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function fitFunc(p0, step, maxIter){

let n = p0.length;

var p1 = Array(n);

for(it = 0; it < maxIter; it++) {

for (j = 0; j < n; j++) {

p1 = p0.slice();

p1[j] += step[j];

if(funCost(p1) < funCost(p0)) {

step[j] *= 1.2;

p0 = p1.slice();

}

else {

step[j] *= -0.5;

}

}

}

return p0;

}

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Simple no? ("slice" allows you to copy arrays, don't ask me why).

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Following is a simple JavaScript program that allows you to view the data for a particular country. The data is from the JHU data base. The plot if fitted using the SIR model and the simple optimizer given above. If you do not like the fit try fitting the curve manually to the data by varying the "b" and "k" parameters. We also print the "R0" constant made famous in the movie "Contagion". Instead of a huge scrolling menu you have to manually type in the country's name. Beware! It is case sensitive! *

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The data is updated daily by JHU. So come back daily to increase the number of repeat visitors.

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