Models of Infectious Diseases

4858 words (19 pages) Essay

5th Sep 2017 Health And Social Care Reference this

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1.1 Introduction

As an aspiring doctor, I am always on the lookout for medical news and keeping up to date with the current affairs is something that has always interested me. Another thing which has sparked my attention over the recent years is the ability of mathematicians and scientists to model infectious disease, in order for the medical professionals to make decisions on how best to act.

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For instance, prior to the 2016 Rio Olympic Games there was a lot of controversy due to competitors pulling out of their events because of the scare of contracting and spreading the Zika Virus. It is times like these that processes, such as modelling infectious disease, are so important; in order to understand whether an infectious disease will spread, how it will spread and what the dangers of the spread of the diseases are.

These models allow decisions to be made such as; whether the border control procedures need to change, whether there should be research into developing new drugs or whether they need to take further action and produce a vaccination.

The SIR model

One method of modelling infectious disease is through using the SIR model. This model starts from the premise that the whole population can be divided into three distinct classes; susceptible, infectious and recovered. ‘Susceptible’ refers to anybody that is currently healthy, but could potentially contract the disease. ‘Infectious’ refers to anybody that is carrying or suffering the disease, meaning that they could spread the disease to anybody that is susceptible. Finally, the ‘recovered’ class refers to anybody that is assumed to be immune to the disease; this could be naturally or through vaccination.

The SIR model is used in epidemiology in order to understand how the disease has evolved over time, and more importantly how it will evolve over time. It is therefore known as a dynamical system; because it tells us how state variables change over time. In this case, the state variables are the population of the three classes; ‘susceptible’, ‘infectious’ or ‘recovered’.

Notation to understand:

In order to model infectious disease, two parameters must be given:

a = the recovery rate

b = the infectious rate

The following notation represents the state variables:

St= the number of susceptible individuals at time t

It = the number of infective individuals at time t

Rt = the number of recovered individuals at time t

T = the time, in days

The changing state variables:

If St is the number of susceptible individuals at time t then…

(1)

And…

  (2)

It can therefore be deduced that the change in the number of susceptible individuals is equal to the negative number of individuals that become infected in day t. The number is negative because in order for someone to become infected they move out of the susceptible state and therefore the population of susceptible individuals decreases.

(3)

The change in the number of infective individuals is equal to the number of individuals that are infected in day t minus the number of individuals that recover in day t.

(4)

Thirdly, the change in the number of recovered individuals is equal to equation (2), the recovery rate multiplied by the number of infective individuals at time t.  

(5)

A theoretical example:

If we specify some values for the recovery rate and the rate of infection, we can begin to understand how individuals move between these classes over time.

Parameters for an infectious disease:

a = 0.05

b = 0.0001

Initial state variable values:

S0 = 10000

I0 = 1000

R0 = 0

Applying equation (3) tells us the number of individuals that will become infected on day 1:

Therefore, from day 0 to day 1 the number of susceptible individuals will change to:

The change in the number of recovered people, given in equation (5) is as follows:

(5)

So the number of recovered people after day 1 will be:

Finally, the number of infectious individuals will be:

(4)

Therefore, we can see that from day 0 to day 1 the populations have changed to:

S1 = 9000

I1 = 1950

R1 = 50

If we repeat the process for the next 3 days, we can begin to see how the disease could spread over time:

Day 2:

(3)

(5)

(4)

Day 3:

(3)

(5)

(4)

Table to show the populations after four days:

Day

Number of susceptible individuals

To the nearest whole number

0

10000

10000

1

9000

9000

2

7245

7245

3

4631.36625

4631

Day

Number of infectious individuals

To the nearest whole number

0

1000

1000

1

1950

1950

2

3607.5

3608

3

6040.75875

6041

Day

Number of recovered individuals

To the nearest whole number

0

0

0

1

50

50

2

147.5

148

3

327.875

328

If we plot these on a graph against time we can see how this disease has spread through the population:

Continuing this process for the next fifteen days gives the following results:   

Day:

Number of susceptible people

Number of infectious people

Number of recovered people

0

9000

1950

50

1

7245

3607.5

147.5

2

4631.36625

6040.75875

327.875

3

1833.66963

8536.417432

629.9129375

4

268.3726905

9674.8935

1056.733809

5

8.724970602

9450.796545

1540.478484

6

0.4791784

8986.50251

2013.018311

7

0.048564611

8537.607998

2462.343437

8

0.00710205

8110.769061

2889.223837

9

0.001341741

7705.236368

3294.76229

10

0.000307898

7319.975584

3680.024108

11

8.25174E-05

6953.97703

4046.022887

12

2.5135E-05

6606.278236

4393.721739

13

8.53011E-06

6275.964341

4724.035651

14

3.17665E-06

5962.166129

5037.833868

15

3.17665E-06

5962.166129

5037.833868

Rounding these numbers to the nearest whole numbers, and excluding negative numbers gives the following table:

Day:

Number of susceptible people

Number of infectious people

Number of recovered people

0

9000

1950

50

1

7245

3608

148

2

4631

6041

328

3

1834

8536

630

4

268

9675

1057

5

9

9451

1540

6

0

8987

2013

7

0

8538

2462

8

0

8111

2889

9

0

7705

3295

10

0

7320

3680

11

0

6954

4046

12

0

6606

4394

13

0

6276

4724

14

0

5962

5038

15

0

5962

5038

From the graph we can see that there is a point at which all of the groups level off. For example, by day 14 the recovered and infectious group level off, and be day 5 the susceptible group levels off at 0. The SIR model is therefore useful at this basic level in order to see how the three state variables interact with one another.

Using the SIR model for Ebola:

However, in order to model a disease such as Ebola, a few changes to the equations must be made. Two parameters must be introduced; the mortality rate and the duration of the disease.

The SIR model will be used to model the spread of Ebola in Guinea, according to the data provided by WHO, in 2014.

Calculating the rate of infection (b) and the rate of recovery (a) requires the mortality rate and the duration of the disease. The infection rate is equal to the mortality rate divided by the number of susceptible people. The recovery rate is equal to 1 divided by the duration of the disease.

In June 2014, there were 390 total cases of Ebola, 273 of these resulting in death. The duration of the disease is between 2 and 21 days. The duration of the disease used for the SIR model will be the midpoint; 11.5 days. The mortality rate of Ebola at this point was (273/390) 0.7. The population of susceptible people in Guinea in 2014 was 11,474,383

Therefore, the infection rate and recovery rate are as follows:

Bringing back the equations from before, the following values are going to be used, having rounded the recovery and infection rate to 1 significant figure.

a =

S0 = (11474383 – 117) 11,474,266

I0 = (390 – 273) 117

R0 = 273

T0 = the start of June 2014

The following tables show the process of calculating the changes in populations:

a

b

s

i

r

t

0.1

0.00000006

11,474,266

117

273

0

0.1

0.00000006

80.54934732

68.84935

11.7

1

0.1

0.00000006

127.9481926

109.3633

18.58493

2

0.1

0.00000006

122.6893008

104.868

17.82126

3

0.1

0.00000006

147.484557

126.0614

21.42313

4

0.1

0.00000006

158.9781358

135.8852

23.09295

5

0.1

0.00000006

180.3286874

154.134

26.19466

6

0.1

0.00000006

199.6512232

170.6493

29.00192

7

0.1

0.00000006

223.579187

191.1009

32.47833

8

0.1

0.00000006

249.0221104

212.8471

36.17502

9

0.1

0.00000006

278.0642545

237.6695

40.39479

10

0.1

0.00000006

310.1130082

265.0614

45.05166

11

0.1

0.00000006

346.0453313

295.7723

50.27308

12

0.1

0.00000006

386.0276536

329.9443

56.08336

13

0.1

0.00000006

430.6727697

368.1011

62.57165

14

0.1

0.00000006

480.4377735

410.6332

69.80454

15

0.1

0.00000006

535.9504153

458.077

77.87343

16

0.1

0.00000006

597.8468096

510.9758

86.87102

17

0.1

0.00000006

666.8678465

569.9626

96.90528

18

0.1

0.00000006

743.8203017

635.7265

108.0938

19

0.1

0.00000006

829.610506

709.0416

120.5689

20

0.1

0.00000006

925.2410683

790.7643

134.4768

21

0.1

0.00000006

1031.828496

881.8479

149.9806

22

0.1

0.00000006

1150.611313

983.3501

167.2612

23

0.1

0.00000006

1282.964805

1096.445

186.5198

24

0.1

0.00000006

1430.413979

1222.434

207.9795

25

0.1

0.00000006

1594.649251

1362.761

231.8879

26

0.1

0.00000006

1777.54255

1519.023

258.5196

27

0.1

0.00000006

1981.165088

1692.987

288.1784

28

To the nearest whole number, the following table shows the number of people in each population over 28 days.

Day:

Number of susceptible people after day t

Number of infectious people after day t

Number of recovered people after day t

0

11474185

186

285

1

11474058

178

30

2

11473935

214

36

3

11473787

231

39

4

11473628

262

45

5

11473448

290

49

6

11473248

325

55

7

11473025

362

61

8

11472776

404

69

9

11472498

451

77

10

11472188

503

85

11

11471842

561

95

12

11471456

626

106

13

11471025

698

119

14

11470544

779

132

15

11470008

869

148

16

11469411

969

165

17

11468744

1081

184

18

11468000

1206

205

19

11467170

1345

229

20

11466245

1500

255

21

11465213

1673

284

22

11464063

1865

317

23

11462780

2080

354

24

11461349

2319

394

25

11459755

2585

440

26

11457977

2882

490

27

11455996

3212

547

28

11455996

1693

288

Plotting these on graphs…

To calculate the equation of graph 1…

The graph appears to be exponential, so using the form y = Aekt , I will try to calculate the equation.

Taking days 3 and 24, I can solve the equation simultaneously.

11473787 = Ae3k

11461349 = Aekt

Ln (11473787) = ln A x 3k

Ln (11461349) = ln A x 24k

K = -0.000051648769

Substituting into the original equations to find A…

Ln (11433787) = ln A + 3k

Ln A = ln (11473787) – 3k

Ln A = 16.25568…

A = 11475064.9

So…

N (of susceptible) = 11475064.9e-0.000051648769t

Due to the massive range of numbers, the three series cannot be plotted on the same graph, and be of any use. Therefore, three separate graphs are necessary. The graphs show that with the parameters calculated for Ebola in 2014, the population of infectious people seems appears to be increasing exponentially, with no sign of a slowing rate. A similar pattern is shown in the recovered population, whilst the number of susceptible people is appearing to decrease exponentially.

The graphs over just 28 days did not show much, so the equations were used over a longer period of time. These graphs are shown below:

As expected, and shown in the previous graphs, the number of recovered and infected people increases rapidly over approximately the first 75 days, after which point the populations of both of these populations reduce in size. The number of susceptible people decreases after approximately 45 days, because more people have become infected or recovered. The number of susceptible people is important in predicting the way that the disease will spread, as the more susceptible people there are, along with a high infection rate, the more likely it is that another epidemic will occur.

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The study of models of the Ebola virus are particularly important at the moment, as there are worries that the disease will spread in a different way, as it adapts to become airborne. Therefore, it is very important to look at how the disease has spread in the past, not just how it could spread in the future.

The graph below shows the number of cases of Ebola in Guinea, Liberia and Sierra Leone.

(1)

And…

  (2)

It can therefore be deduced that the change in the number of susceptible individuals is equal to the negative number of individuals that become infected in day t. The number is negative because in order for someone to become infected they move out of the susceptible state and therefore the population of susceptible individuals decreases.

(3)

The change in the number of infective individuals is equal to the number of individuals that are infected in day t minus the number of individuals that recover in day t.

(4)

Thirdly, the change in the number of recovered individuals is equal to equation (2), the recovery rate multiplied by the number of infective individuals at time t.  

(5)

A theoretical example:

If we specify some values for the recovery rate and the rate of infection, we can begin to understand how individuals move between these classes over time.

Parameters for an infectious disease:

a = 0.05

b = 0.0001

Initial state variable values:

S0 = 10000

I0 = 1000

R0 = 0

Applying equation (3) tells us the number of individuals that will become infected on day 1:

Therefore, from day 0 to day 1 the number of susceptible individuals will change to:

The change in the number of recovered people, given in equation (5) is as follows:

(5)

So the number of recovered people after day 1 will be:

Finally, the number of infectious individuals will be:

(4)

Therefore, we can see that from day 0 to day 1 the populations have changed to:

S1 = 9000

I1 = 1950

R1 = 50

If we repeat the process for the next 3 days, we can begin to see how the disease could spread over time:

Day 2:

(3)

(5)

(4)

Day 3:

(3)

(5)

(4)

Table to show the populations after four days:

Day

Number of susceptible individuals

To the nearest whole number

0

10000

10000

1

9000

9000

2

7245

7245

3

4631.36625

4631

Day

Number of infectious individuals

To the nearest whole number

0

1000

1000

1

1950

1950

2

3607.5

3608

3

6040.75875

6041

Day

Number of recovered individuals

To the nearest whole number

0

0

0

1

50

50

2

147.5

148

3

327.875

328

If we plot these on a graph against time we can see how this disease has spread through the population:

Continuing this process for the next fifteen days gives the following results:   

Day:

Number of susceptible people

Number of infectious people

Number of recovered people

0

9000

1950

50

1

7245

3607.5

147.5

2

4631.36625

6040.75875

327.875

3

1833.66963

8536.417432

629.9129375

4

268.3726905

9674.8935

1056.733809

5

8.724970602

9450.796545

1540.478484

6

0.4791784

8986.50251

2013.018311

7

0.048564611

8537.607998

2462.343437

8

0.00710205

8110.769061

2889.223837

9

0.001341741

7705.236368

3294.76229

10

0.000307898

7319.975584

3680.024108

11

8.25174E-05

6953.97703

4046.022887

12

2.5135E-05

6606.278236

4393.721739

13

8.53011E-06

6275.964341

4724.035651

14

3.17665E-06

5962.166129

5037.833868

15

3.17665E-06

5962.166129

5037.833868

Rounding these numbers to the nearest whole numbers, and excluding negative numbers gives the following table:

Day:

Number of susceptible people

Number of infectious people

Number of recovered people

0

9000

1950

50

1

7245

3608

148

2

4631

6041

328

3

1834

8536

630

4

268

9675

1057

5

9

9451

1540

6

0

8987

2013

7

0

8538

2462

8

0

8111

2889

9

0

7705

3295

10

0

7320

3680

11

0

6954

4046

12

0

6606

4394

13

0

6276

4724

14

0

5962

5038

15

0

5962

5038

From the graph we can see that there is a point at which all of the groups level off. For example, by day 14 the recovered and infectious group level off, and be day 5 the susceptible group levels off at 0. The SIR model is therefore useful at this basic level in order to see how the three state variables interact with one another.

Using the SIR model for Ebola:

However, in order to model a disease such as Ebola, a few changes to the equations must be made. Two parameters must be introduced; the mortality rate and the duration of the disease.

The SIR model will be used to model the spread of Ebola in Guinea, according to the data provided by WHO, in 2014.

Calculating the rate of infection (b) and the rate of recovery (a) requires the mortality rate and the duration of the disease. The infection rate is equal to the mortality rate divided by the number of susceptible people. The recovery rate is equal to 1 divided by the duration of the disease.

In June 2014, there were 390 total cases of Ebola, 273 of these resulting in death. The duration of the disease is between 2 and 21 days. The duration of the disease used for the SIR model will be the midpoint; 11.5 days. The mortality rate of Ebola at this point was (273/390) 0.7. The population of susceptible people in Guinea in 2014 was 11,474,383

Therefore, the infection rate and recovery rate are as follows:

Bringing back the equations from before, the following values are going to be used, having rounded the recovery and infection rate to 1 significant figure.

a =

S0 = (11474383 – 117) 11,474,266

I0 = (390 – 273) 117

R0 = 273

T0 = the start of June 2014

The following tables show the process of calculating the changes in populations:

a

b

s

i

r

t

0.1

0.00000006

11,474,266

117

273

0

0.1

0.00000006

80.54934732

68.84935

11.7

1

0.1

0.00000006

127.9481926

109.3633

18.58493

2

0.1

0.00000006

122.6893008

104.868

17.82126

3

0.1

0.00000006

147.484557

126.0614

21.42313

4

0.1

0.00000006

158.9781358

135.8852

23.09295

5

0.1

0.00000006

180.3286874

154.134

26.19466

6

0.1

0.00000006

199.6512232

170.6493

29.00192

7

0.1

0.00000006

223.579187

191.1009

32.47833

8

0.1

0.00000006

249.0221104

212.8471

36.17502

9

0.1

0.00000006

278.0642545

237.6695

40.39479

10

0.1

0.00000006

310.1130082

265.0614

45.05166

11

0.1

0.00000006

346.0453313

295.7723

50.27308

12

0.1

0.00000006

386.0276536

329.9443

56.08336

13

0.1

0.00000006

430.6727697

368.1011

62.57165

14

0.1

0.00000006

480.4377735

410.6332

69.80454

15

0.1

0.00000006

535.9504153

458.077

77.87343

16

0.1

0.00000006

597.8468096

510.9758

86.87102

17

0.1

0.00000006

666.8678465

569.9626

96.90528

18

0.1

0.00000006

743.8203017

635.7265

108.0938

19

0.1

0.00000006

829.610506

709.0416

120.5689

20

0.1

0.00000006

925.2410683

790.7643

134.4768

21

0.1

0.00000006

1031.828496

881.8479

149.9806

22

0.1

0.00000006

1150.611313

983.3501

167.2612

23

0.1

0.00000006

1282.964805

1096.445

186.5198

24

0.1

0.00000006

1430.413979

1222.434

207.9795

25

0.1

0.00000006

1594.649251

1362.761

231.8879

26

0.1

0.00000006

1777.54255

1519.023

258.5196

27

0.1

0.00000006

1981.165088

1692.987

288.1784

28

To the nearest whole number, the following table shows the number of people in each population over 28 days.

Day:

Number of susceptible people after day t

Number of infectious people after day t

Number of recovered people after day t

0

11474185

186

285

1

11474058

178

30

2

11473935

214

36

3

11473787

231

39

4

11473628

262

45

5

11473448

290

49

6

11473248

325

55

7

11473025

362

61

8

11472776

404

69

9

11472498

451

77

10

11472188

503

85

11

11471842

561

95

12

11471456

626

106

13

11471025

698

119

14

11470544

779

132

15

11470008

869

148

16

11469411

969

165

17

11468744

1081

184

18

11468000

1206

205

19

11467170

1345

229

20

11466245

1500

255

21

11465213

1673

284

22

11464063

1865

317

23

11462780

2080

354

24

11461349

2319

394

25

11459755

2585

440

26

11457977

2882

490

27

11455996

3212

547

28

11455996

1693

288

Plotting these on graphs…

To calculate the equation of graph 1…

The graph appears to be exponential, so using the form y = Aekt , I will try to calculate the equation.

Taking days 3 and 24, I can solve the equation simultaneously.

11473787 = Ae3k

11461349 = Aekt

Ln (11473787) = ln A x 3k

Ln (11461349) = ln A x 24k

K = -0.000051648769

Substituting into the original equations to find A…

Ln (11433787) = ln A + 3k

Ln A = ln (11473787) – 3k

Ln A = 16.25568…

A = 11475064.9

So…

N (of susceptible) = 11475064.9e-0.000051648769t

Due to the massive range of numbers, the three series cannot be plotted on the same graph, and be of any use. Therefore, three separate graphs are necessary. The graphs show that with the parameters calculated for Ebola in 2014, the population of infectious people seems appears to be increasing exponentially, with no sign of a slowing rate. A similar pattern is shown in the recovered population, whilst the number of susceptible people is appearing to decrease exponentially.

The graphs over just 28 days did not show much, so the equations were used over a longer period of time. These graphs are shown below:

As expected, and shown in the previous graphs, the number of recovered and infected people increases rapidly over approximately the first 75 days, after which point the populations of both of these populations reduce in size. The number of susceptible people decreases after approximately 45 days, because more people have become infected or recovered. The number of susceptible people is important in predicting the way that the disease will spread, as the more susceptible people there are, along with a high infection rate, the more likely it is that another epidemic will occur.

The study of models of the Ebola virus are particularly important at the moment, as there are worries that the disease will spread in a different way, as it adapts to become airborne. Therefore, it is very important to look at how the disease has spread in the past, not just how it could spread in the future.

The graph below shows the number of cases of Ebola in Guinea, Liberia and Sierra Leone.

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