Question

Consider an epidemic disease in a country;

The daily emergence of the disease is given in the table below. The first day is the 18th of March, and the last day of the record is the 7th of May. The record is five days apart. There are no records either earlier or later than the dates, or between the dates.

Dates	Daily Emergence of the Epidemic (Ep(t))
18.03.2020	93*x
23.03.2020	293*x
28.03.2020	1704*x
02.04.2020	2456*x
07.04.2020	3892*x
12.04.2020	4789*x
17.04.2020	4353*x
22.04.2020	3083*x
27.04.2020	2131*x
02.05.2020	1983*x
07.05.2020	1380*x

x=0.83.

a) draw a graphic/chart for daily emergence of the epidemic disease, by hand or by using a program,

b) calculate emergence speed of the disease (rate of change (dEp/dt), Ο(h² error level)) for each date and draw a graphic for it.

c) calculate the rate of change of the speed (acceleration of the epidemic (d²Ep/dt²), Ο(h² error level)) for each date and make a graphic, also.

d) what is the total number of infected people during the time period. (use at least two solution techniques and discuss)

e) what is the number of infected people on the 31^st of March.

f) what is going to be the number of infected people on the 9^th of May.

Answer 1

Excel chart made from the given data is shown below

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Graph of daily emergence is given below

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Emergence speed of disease

By Tailor's series , we have, E(x+h) = E(x) + h (dE/dt ) + O(h2 )

hence dE/dt = [ E(x+h) - E(x) ] / h

In the above expressions , h = 5 day time interval , E(x) is number of emergence on x-th day

dE/dt is speed of emergence

On last day , dE/dt = [ E(x) - E(x-h) ] / h

Column in Excel chart with title dE/dt shows the emergence speed of disease

Graph of speed of emergence is shown below

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Rate of change of speed d2E/dt2

E(x+h) = E(x) + h ( dE/dt ) + ( h2 / 2 ) ( d2E/dt2)

E(x-h) = E(x) - h ( dE/dt ) + ( h2 / 2 ) ( d2E/dt2)

By adding above equations , we get,

( d2E/dt2) = [ E(x+h) - 2 E(x) + E(x-h) ] / h2

Rate of change of speed is given in excel chart with title ( d2E / dt2 )

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Total number of infected people is calculated by Integration using trapezoidal rule

As per Sum of term calculated in excel , in column with title , we get

total number of infected people as 1,05,495

The other method is sum of emergence of each day calculated using rate of emergence in between the day of data. This sum shows total number of infected people 84610

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infected people on 31st March = infeected people on 27th march + 4 (dE/dt )

infected people on 31st March = 1769 + 4 (-25) = 1670

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infected people on 9th May = infeected people on 7th May + 2 (dE/dt )

infected people on 9th May = 1145 + 2 (-100) = 945