Question

Measured viral load in plasma of two patients are given in the following table Patient -1 (RNA copies/ml) Patient- 2 (RNA copies/ml) Days post infection 12 14 16 21 23 1353 10081 33612 706640 1097500 79232 798550 1680700 Assuming that the virus grows exponentially, use the given data to calculate initial viral load, growth rate, and doubling time for each patient. Show your best fit model and data in the graph for each patient. Compare viral dynamics in these two patients. a. b. C.

Answer 1

Hi.

Using Excel you can find the models by linearizing de exponential models

y=ax(1 + r)r

Where y is the viral load at a time x, a is the initial vital load and r is the growth rate

This can be calculated using a model like:

and linarizing is you get:

Log(y) = Log(a) + x Log(b)

to find Log a and Log b you use least squares regression.

m = Log(b)

$c=Log(a)=\frac{\sum Log(y)-m\sum x}{N}$

a. We make the following table to find m and c for Patient 1

	x days	P1	Log(y)	x*Log(y)	x^2
	14	10081	4.003504	56.04905	196
	16	33612	4.526494	72.42391	256
	21	706640	5.849198	122.8332	441
	23	1097500	6.040405	138.9293	529
∑	74		20.4196	390.2354	1422

m= Log(b)= 0.235 → b= 100. 235-1.718

c = Log(a) = 0.751 → a = 100.75-5.636

For Patient 1 the model is:

y 5.636 × 1.718'

Meaning that the initial viral load is 5.636

The growth rate is 0.718

And the doubling time can be calculated by:

Log(2) Log(1.718) 1281days

We make the following table to find m and c for Patient 2

	x days	P2	Log(y)	x*Log(y)	x^2
	12	1353	3.131298	37.57557	144
	16	79232	4.898901	78.38241	256
	21	798550	5.902302	123.9483	441
	23	1680700	6.22549	143.1863	529
∑	72		20.15799	383.0926	1370

m = Log(b)= 0.274 → b= 100. 2'4-1.879

cLog(a) 0.114a 10114-1.3

For Patient 2 the model is:

$y=1.3\times1.879^x$

Meaning that the initial viral load is 1.3 approx 6

The growth rate is 0.879

And the doubling time can be calculated by:

Log(2) Log(1.87可:J·099days = 1 ,099days 1.8791-2 → x=

b. Using desmos graphing calculator to graph the models we get

y-5.636x1.718 y = 1.3 x 1.879" 150000 沓 mpulsado po desmos

c. From the graph you can see that even so the Patient 1 started with a higher initial viral load the growth rate is lower that for Patient 2, and with the over time patient 2 will eventually have a higher viral load because his initial viral load was lower but the growth rate was higher.

Note: i personally recomend that you get more meassurements, if you can twice a day, if you can get more data, you can have a more precise model to predict the behavior of the virus.

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