IB Math Standard Level Internal Assessment

Investigation Title: Modeling the Spread of Ebola in Guinea 2014

Candidate Name: Reem Sayed

Candidate Number: 0021

School’s Name: Modern Knowledge Schools

Teacher’s Name: Mrs. Sana Saeed

Exam Session: May 2018

Introduction:

For this Math investigation, I chose to model a virus epidemic. The virus chosen was the Ebola hemorrhagic fever, which is also known as the Ebola virus disease. This disease was initially between wild animals, however, it was eventually transmitted to humans, and the infection was caused through human to human contact. This epidemic disease was a major outbreak especially in the West of Africa, starting in Guinea, my investigating area, and spreading through neighboring countries which are Sierra Leone and Liberia.

This fatal illness is very dangerous if it is left untreated. Eventually, a vaccine was created and a woman named Marie-Paule Kieny which was the Ebola research chief in the World Health Organization talked about it. Although she said that the vaccine was successful, the vaccine can only stop contracting the disease, but it cannot treat it. Although the patients recovered, the disease may still “wake up” once again and will cause another disaster.

Although there are many diseases that have been spread, I chose Ebola, because there needs to be a model to show how the vaccine changed the number of patients diagnosed with the disease. The investigation involves the use of the SIR model to identify the approximate number of infected people in guinea in 2014. Data will be collected through secondary research and will be put through several equations and calculations to finally get the concluding result.

The SIR Model:

Variables:

t = time in days

St = number of susceptible individuals in day t

It = number of infectious individuals in day t

Rt = number of recovered individuals in day t

? = contact rate

? = recovery rate ( 1 )

duration of disease

N = overall population of Guinea

In this mathematical model, there are three main variables. The (S) stands for susceptible individuals, which are the people who do not have the disease yet, but at risk of getting it. The (I) stands for the infected individuals, which are individuals who are currently suffering from the disease. Ebola is a very strong virus that approximately half of the infected individuals die. But, if the individuals survive the six-day infection period, they begin the recovering stage, and (R) stands for the recovered individuals, which are the individuals that passed the infection stage and do not have the disease anymore. However, the virus changes very frequently causing the recovered people to eventually become susceptible of getting the virus again.

The SIR model illustrates the epidemic transfer through the interactions between all three of the susceptible, infected, and recovered individuals of the population. The way this model works is by first moving the people from susceptible to infected at a certain rate known as the infection rate ?, which is the percentage of the product of susceptible and infected individuals. Likewise, the product of infected and recovered is called the recovery rate ?.

Modeling Ebola in Guinea 2014:

The population of Guinea in December 31, 2014 = 1,1810,000 (N0).

Infected individuals = 2707 (I0).

Recovered individuals = 964 (R0).

Data:

– Calculation of ? = people infected = 2707 = 0.00022921

Total population 1,1810,000

– Calculation of rate of recovery ?

? = 1 = 1 = 0.16

duration of disease 6 days

– Initial susceptibility: S0 = N0 – ( I0 + R0)

S0 = 1,1810,000 – (2707 + 964)

S0 = 11,806,329

Data will be found through the calculation of the number of susceptible, infected, and recovered individuals in “Day 0.” The data resulting will be used to calculate those people in the following days till “Day 20.”

Day 0 (t = 0)

S0 + 1 ? S0 = S1 ? S0 = ? ?S0 I0 = ?

N0

S1 ? S0 = – 0.6202

S1 ? 11,806,329 = – 0.6202

S1 = – 0.6202 + 11,806,329

S1 = 11,806,328.4

I0 + 1 ? I0 = I1 ? I0 = ? ?S0 I0 – ?I0 = ? – (0.16)(2707)

N0

I1 ? I0 = – 432.50

I1 ? 2,707 = – 432.50

I1 = – 432.50 + 2,707

I1 = 2,274.5

R0 + 1 ? R0 = R1 ? R0 = ?I0 = (0.16)(2707)

R1 ? R0 = 433.12

R1 ? 964 = 433.12

R1 = 433.12+ 964

R1 = 1,397.12

Day

Susceptible

Infectious

Recovered

0

11,806,329

2707

964

1

11,806,328.4

2,274.5

1,397.12

2

11,806,327.8

1,842

1,830.24

3

11,806,327.1

1,409.5

2,263.36

4

11,806,326.5

977

2,696.48

5

11,806,325.9

544.5

3,129.6

6

11,806,325.3

112

3,562.72

7

11,806,324.7

-320.5

3,995.84

8

11,806,324.03

-753

4,428.96

9

11,806,323.4

-1,185.5

4,862.08

10

11,806,322.8

-1,618

5,295.2

11

11,806,322.2

-2,050.5

5,728.32

12

11,806,321.6

-2,483

6,161.44

13

11,806,320.9

-2,915.5

6,594.56

14

11,806,320.3

-3,348

7,027.68

15

11,806,319.7

-3,780.5

7,460.8

16

11,806,319.1

-4,213

7,893.92

17

11,806,318.5

-4,645.5

8,327.04

18

11,806,317.8

-5,078

8,760.16

19

11,806,317.2

-5,510.5

9,193.28

20

11,806,316.7

-5,943

9,626.4

Table 1: Shows the calculations using the SIR model equations to gain data on the susceptible, infectious, and recovered individuals from Ebola in Guinea 2014.

Graph 1: Displays the number of susceptible individuals from December 31, 2014 and 20 following days.

Overview: This graph shows the reducing number of individuals that are susceptible to the virus after 2014. The individuals’ susceptibility kept decreasing suggesting that less people are at risk of this disease.

Graph 2: The number of infectious individuals from December 31, 2014 and 20 following days.

Overview: This graph shows the reducing number of individuals that are infectious to the virus after 2014. The susceptibility of individuals decreases, and in the seventh day the numbers start to become negative, meaning that there aren’t any infected individuals after they survive the seventh day, since if they pass six days and are still infected, they are more likely to die, because the duration is six days.

Graph 3: The number of recovered individuals from December 31, 2014 and the 20 following days.

Overview: This graph shows the increasing number of individuals that have recovered from the virus through the twenty days. This shows that the Ebola outbreak in Guinea will end soon. The other two graphs confirm that since the number of individuals that are susceptible and infectious is decreasing, so less people are at risk or even infected and recoveries are increasing.

The SIR Model for Differential Equations:

This model is the same as the previous model, except the susceptible, infected, and recovered individuals need to be divided by the number of the total population if Guinea in 2014, so that the result can be a decimal.

This model also has a couple of new variables which are:

t = time in days

? = infection rate = ( 1 ) = 1 = 0.5

duration of contact 2

? = recovery rate = ( 1 ) = 1 = 0.16

Duration of disease 6

Calculations:

The calculations include the differentiation to find the change in the susceptible, infected, and recovered individuals to know whether Ebola will come back again or not.

Step 1:

The above calculations are calculated when t is equal to 0, which is the start of each graph.

To validate the answer, S(0) + I(0) +R(0) = 1

0.9997 + 0.001496 + 0.0005326 ? 1

After that, the answers shown above are replaced in the differential equations:

S’ (0) = – 0.5 x 0.9997 x 0.001496 = -0.00074778

I’ (0) = 0.00074778– (0.16 x 0.0014) = 0.00052378

The answer shows that the individual’s susceptibility will always be negative since there is a negative infection rate.

To find when Ebola will stop recovering, some calculations must be made.

When the susceptibility of the disease is at zero, and no individuals are present, Ebola will not “wake up. ”

S'(t)= -0.5SI=0

S=0 and I=0

I’ (t) = 0.5SI – 0.16I = 0

I(0.5S – 0.16) = 0

I = 0

When the susceptibility of individuals is at zero, the infection will stop occurring.

The differential calculations above reveal that the probability of Ebola spreading again is decreasing since the infection has no individuals to be infected.

Reflection, Evaluation, and Conclusion:

The objective of this investigation was achieved where the SIR model allowed the modeling of Ebola in Guinea. The model shows that the number of susceptible and infectious individuals is reduced, but recovered individuals increased. It seems like the number of susceptible individuals only decreased by 12.3. This can be because there’s a limited number of individuals that are infected and recovered as there are more susceptible individuals in a larger population. It is proved since Day 0, where there were 11,806,329 susceptible individuals, but only 2707 infectious, and 964 recovered. This reduced the disease having an effect on the entire population.

Furthermore, the infectious individual number decreased till it reached zero and even below zero meaning that there are no more infectious individuals at all. However, the number of recovered individuals started off with 964, but in day 20, the number increased by 8,662.4 individuals. This means that the Ebola virus has been treated using the vaccine. Moreover, through the use of differential equations, the percentage of susceptible over infected individuals shows when Ebola will stop coming back. However, this equation did not consider the recovered individuals. This shows that the equation may be adequate, but the number of recovered individuals would have clearly shown the modeling of the spread of Ebola.

Finally, due to lack of secondary data online the data collected was limited. Also, the formulas for the SIR model are used for various infectious diseases, not only Ebola, causing insufficient data and calculations since it isn’t created for the specific location of Guinea. However, sufficient results were acquired by the end of the investigation.

Bibliography:

Press, The Associated. “WHO Chief Praises Guineans for Help with Ebola Vaccine.” CBCnews, CBC/Radio Canada, 4 May 2017, www.cbc.ca/news/health/ebola-vaccine-guinea-1.4099531.

Cenciarelli, Orlando, et al. “Ebola Virus Disease 2013-2014 Outbreak in West Africa: An Analysis of the Epidemic Spread and Response.” International Journal of Microbiology, Hindawi Publishing Corporation, 2015, www.ncbi.nlm.nih.gov/pmc/articles/PMC4380098/.

“Situation SummaryData Published on 31 December 2014.” Ebola Data and Statistics, World Health Organization, apps.who.int/gho/data/view.ebola-sitrep.ebola-summary- 20141231?lang=en.

Nykamp, Duane. “The SIR Infectious Disease Model, Preliminary Analysis”. YouTube. N.p., 2013. Web. 1 Jan. 2016.

https://services.math.duke.edu/education/ccp/materials/diffcalc/sir/sir2.html