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Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Modeling the spread of a virus like COVID-19 using recursion.

Let

N = total population (assumed constant, disregarding deaths, births, immigration, and emigration).

= number who are susceptible to the disease at time n (n is in weeks).

= number who are infected (and contagious) at time n.

= number who are recovered (and not contagiuous) at time n.

The total population is divided between these three groups:

N = S

+ I

+ R

There are several hidden assumptions here that may or may not apply to COVID-19, such as a recovered

person is assumed to not be able to get the disease a second time, at least within the time window being

examined.

On week 0 (the start), you assume a certain small number of people have the infection (just to get things

going). Everyone else is initially susceptible, and no one is recovered.

There are two constants of interest:

Let period = time period that it takes for an infected person to recover (recover meaning they become

not infectious to someone else).

Let ratio = infection ratio (an infected person can infect this many others in period weeks),

assuming the

entire population was susceptible)

We can deduce three equations that track how S, I, and R change from time period n to time period n+1:

First it may be helpful to define three intermediate variables.

The infection ratio is for the entire infection period, so the weekly infection ratio would be a fraction of

that, ratio / period. Also, since the infection ratio is the ratio if all the population was susceptible, the

effective ratio is really this ratio scaled down for the percent of the population that is susceptible, S

/ N:

(A) effectiveWeeklyRatio = ratio/period * S

The number of new infections will be the number currently infected times the effectiveWeeklyRatio:

(B) newInfections = I

* effectiveWeeklyRatio

On average 1/period of the infected people will recover in a single time period:

/ period

Using these intermediate variables you calculate new values for R, S, and I:

(1) R

n+1

= R

+ newRecovered

(2) S

n+1

= S

– newInfections

(3) I

n+1

= I

+ newInfections − newRecovered

Instead of using this equation, perhaps better:

(3) I

n+1

= N − R

n+1

− S

n+1

(keeps the total number N constant even after rounding errors)

You will also maintain a “total” variable to add together those that ever got the virus at any point.

total

+= newInfections

These equations lend themselves naturally to recursion. The recursive function is passed values of the

variables for the previous week and calculates new values for the variables for the current week.

For the this project, you will input a value for ratio.

You can assume N = 1000000 (one million) and period=1.0 (one week)

Use initial values of I=100, R=0, S=999900, week=0, and total=100 just to get things going.

Call a function named NextWeek recursively for each new week. Show in three columns the week, the

number of new infections, and the running total of new infections. Right justify each column in a field of

appropriate width. Stop when the number of new infections equals zero.

Note that most values are integers, except that period, ratio, and weeklyEffectiveRatio are floating-point

(doubles). You may need to force a division to be floating point, and you will need to round some

answers off to the nearest integer. To round a double to int, use #include <math.h>

(#include <cmath> for c++) and use the round() function. The function will return the total (all the

way back up the recursive tree so main can get it).

int NextWeek(I, R, S, N, total, ratio, period, week)

{

declare variables of appropiate type:

weeklyEffectiveRatio, newRecovered, newInfections

increment week

calc weeklyEffectiveRatio

calc newRecovered (remember to round to int)

calc newInfections (remember to round to int)

add newInfections to total

calc new values for I, R, S

print week, newInfections, total in asked for format

if (base case)

return total

return NextWeek(..)

}

COVID NOTES:

The estimates for ratio range from about 2.5 to about 5.

The estimates for the period that one is infectious vary widely, it is probably more than the 1.0 week

given here. Assuming you keep ratio the same, the period doesn’t affect the total that eventually get the

disease much, it just shortens or lengthens the total time for the course of the disease,.

Example output/input (assuming a value for r of 2.5),

showing the first few lines:

Enter r:

2.5

WEEK NEW CASES TOTAL CASES

---- ----------- -----------

1 250 350

2 625 975

3 1561 2536

4 3893 6429

5 9670 16099

(only first five weeks shown here, but program goes until zero new cases)

Note that there are two lines of header that are printed out once as literal strings (in main):

WEEK NEW CASES TOTAL CASES

"---- ----------- -----------"

Then for each week,

Right justified width 4 for the week

Right justified width 12 for new cases

Right justified width 12 for total cases

Submission

Submit one file in D2L under Assignments:

Submit your source code in one file (named with an extension .c for C, .cpp for C++).

I will run the program, so you don't need to supply the output

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odeling the spread of a virus like COVID-19 using recursion. Let N = total population (assumed constant, disregarding deaths, births, immigration, and emigration). S n = number who are susceptible to the disease at time n (n is in weeks). I n = number who are infected (and contagious) at time n.