Modified SIR for Vector-Borne DiseasesGroup 9-019Gay Wei En Colin 4i310Chua Zhi Ming 4i307Katherine Kamis

AOSJacob Savos AOS

Aims & Objectives To create a universal modified SIR model

for vector-borne diseases to make predictions of the spread of these diseases.

Motivation In 2005, there was an

epidemic of Dengue in Singapore.

Since then the number of cases has been at an increased level.

The number of cases of Lyme disease has been increasing in Loudoun County, Virginia, an area previously devoid of Lyme disease.

Hwa Chong Institution

Academy of Science

A model will help to predict if the current trend will continue.

MotivationHwa Chong Institution

Academy of Science

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 20100

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Lyme Disease Cases

20002001200220032004200520062007200820090

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Dengue Fever Cases

Simple SIR The SIR model is used to predict outbreaks of diseases.

Considers 3 compartments: Susceptible Infected Recovered

Two directions of change, namely from Susceptible to Infected or from Infected to Recovered

Susceptible Infected Recovered

Simple SIR S’(t) = -k * S(t) * I(t) I’(t) = -S’(t) – R’(t) R’(t) = c * I(t)

k – Transmittal constant

c – Recovery rate

0 5 10 15 20 25 30 35 40 45 500

50001000015000200002500030000350004000045000 SIR

SuspectibleInfectedRecovered

Time (Days)

Popu

latio

n

Euler’s Method Tangent line – slope at a certain point Tangent lines are estimates of the rates of

change Rates of change can be used to estimate

actual points S(t + h) = S(t) + S’(t)*h

Vector-Borne Disease Model

Susceptible

SusceptibleInfected

InfectedHosts (N)

Vectors (V)

Death Death

Death Death

Birth

Birth

Net Migration Net Migration

Z. Qiu Equations (2008) S – Susceptible host

population I – Infected host population T – Susceptible vector

population X – Infected vector population B – Birth rate of hosts µ – Death rate of hosts N – Total Host population b – Contact rate β – Disease transmission

probability (vectors to host) γ – Recovery Rate m – Migration Rate

Rate of Change of Susceptible Hosts

Rate of Change of Infected Hosts

Z. Qiu Equations (2008) S – Susceptible host

population I – Infected host population T – Susceptible vector

population X – Infected vector population N – Total Host population b – Contact rate V – Total Vector population B’ – Birth rate of vectors ε– Death rate of vectors M – Maximum number of

vectors per host α – Disease transmission

probability

Rate of Change of Susceptible Vectors

Rate of Change of Infected Vectors

Assumptions A person recovers from Dengue after 2

weeks 5% of the mosquito population is

infected with dengue fever Birth, death, and migration rates stay

the same after 2004 Vector birth and death rates stay

constant

Data Collection Dengue Fever Statistics Singapore Human Population Statistics Singapore Climate Data

Temperature Precipitation

Dengue Fever Statistics

Human Population Statistics

http://www.indexmundi.com/

Climate Data

Data Analysis - Climate Spearman’s Rank Correlation Coefficient

Used to determine to what extent temperature and precipitation are correlated to the number of cases of dengue fever.

Spearman’s Rank Correlation Coefficient

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24-0.050

0.050.1

0.150.2

0.250.3

0.350.4

|SCC Prep||SCC Temp|

Time Lag (Weeks)

Spea

rman

's R

ank

Corr

elat

ion

Coeffi

cien

t

Data Analysis - Climate

0 10 20 30 40 50 600

0.00000002

0.00000004

0.00000006

0.00000008

0.0000001

0.00000012

0.00000014

0.00000016

200420052006200720082009

Period (Weeks)

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smitt

al C

onst

ant

k

Data Analysis - Climate Using the 14 week lag and the statistics

for weekly cases, we can determine when to extract our transmittal constant.

Data Analysis - Seasonality Tick populations are affected by

different seasons in the United States Summer & Winter:

Birth rate is lower Death rate is higher

Spring & Fall: Birth rate is higher Death rate is lower

Dengue Fever Data

Dengue Fever Data

0 50 100 150 200 250 300 3500

5000100001500020000250003000035000

Period (Weeks)

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cept

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Ve

ctor

s

Susceptible Vectors Population

0 50 100 150 200 250 300 3500

200400600800

100012001400160018002000

Period (Weeks)Popu

latio

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Infe

ted

Vect

ors

Infected Vectors Population

0 50 100 150 200 250 300 3504,100,0004,200,0004,300,0004,400,0004,500,0004,600,0004,700,0004,800,000

Period (Weeks)

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osts

Susceptible Hosts Population

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200400600800

1000120014001600

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Hos

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Infected Hosts Population

Dengue Fever Data

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5000100001500020000250003000035000

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ctor

s

Susceptible Vectors Population

0 50 100 150 200 250 300 3500

200400600800

100012001400160018002000

Period (Weeks)Popu

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Infe

ted

Vect

ors

Infected Vectors Population

2000 2001 2002 2003 2004 2005 2006 2007 2008 20090

2000400060008000

10000120001400016000

Dengue

Z. Qiu Model –Estimates from Initial Data

0 100 200 300 400 500 6000

20406080

100120140160180200

Infected Population(Hosts)

Period (Weeks)Popu

latio

n of

Infe

cted

Hos

ts

0 100 200 300 400 500 600302304306308310312314316

Susceptible Population(Vector)

Period (Weeks)

Popu

latio

n of

Sus

cept

ible

Ve

ctor

s

0 100 200 300 400 500 60013.5

1414.5

1515.5

1616.5

1717.5

Infected Population(Vector)

Period (Weeks)

Popu

latio

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Infe

cted

Vec

tors

0 100 200 300 400 500 6004400000450000046000004700000480000049000005000000

Susceptible Population (Hosts)

Period (Weeks)Popu

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Sus

cept

ible

H

osts

Z. Qiu Model –Theoretical Population

0 100 200 300 400 500 600440000045000004600000470000048000004900000

Susceptible Population (Hosts)

Period (Weeks)Popu

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cept

ible

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osts

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Infected Population(Hosts)

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Susceptible Population(Vector)

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cept

ible

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ctor

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0 100 200 300 400 500 6000

2000400060008000

1000012000

Infected Population(Vector)

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cted

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tors

Lyme Disease

0 50 100 150 200 250 3000

200400600800

10001200

Susceptible Population (Hosts)

Period

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cept

ible

H

osts

0 50 100 150 200 250 3000

200400600800

10001200

Infected Population(Hosts)

Period

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Infe

cted

Hos

ts

0 50 100 150 200 250 3000

50000

100000

150000

200000

250000Susceptible Population

(Vector)

Period

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cept

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ctor

s

0 50 100 150 200 250 3000

100020003000400050006000700080009000

Infected Population(Vector)

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ctor

s

Problems & Limitations Data was not accessible for use in either

country Collection of data for models was difficult If this data was available we would be

able to create models to help predict infected populations and determine: Health Care Costs Wellness of the Population Control Methods

Future Work We have created a model that we can

use to predict trends in populations With field work, the necessary data such

as mosquito population count and transmission rates along with other parameters that can be measured in the field can used with the model to predict the spread of Dengue and Lyme disease

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Dengue

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Thank You!Any Questions?