MODIFIED SIR FOR VECTOR-BORNE DISEASESKatherine Kamis & Jacob Savos

Aims and Objectives To create a universal modified SIR model

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

Introduction A Vector-borne disease is transmitted by

a pathogenic microorganism from an infected host to another organism

We will be creating a model using a tick-borne disease

Literature Review - Ticks Ticks have a two-year life cycle Ticks acquire a vector-borne disease by

feeding on an infected host Once infected, ticks transmit the disease

by feeding on an uninfected host

Lone Star TickDeer Tick

http://www.ent.iastate.edu/imagegal/ticks/iscap/all4.html

http://www.aldf.com/deerTickEcology.shtml

http://www.womenhunters.com/ticks-kim-roberts.html

Literature Review - SIR Susceptible Infected Recovered

Susceptible

RecoveredInfected

Rates of Change 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 S(0 + 1)=S(0)+S’(0)*1

http://www.clas.ucsb.edu/staff/lee/secant,%20tangent,%20and%20derivatives.htm http://www.flatworldknowledge.com/node/30962#web-30962

SIR - Equations

k = Transmittal constantS(1) = B9 + (-$B$4 * B9 * C9)

( ) ( )dS k S t I tdt

The rate of change of the susceptible = The product of the susceptible and infected times the opposite of the transmittal constant

SIR - Equations k = Transmittal constant c = Recovery rate

( ) ( ) ( )dI k S t I t c I tdt

The rate of change of infected = The product of the susceptible, infected, and transmittal constant subtracted by the product of the recovery rate and the infected population

I(1) = C9 + ($B$4*C9*B9)-($B$3*C9)

SIR - Equationsc = Recovery rate

R(1) = D9 + ($B$3 * F9)( )dR c I tdt

The rate of change of recovered = The product of the recovery rate and the infected population

Vector-borne model

Susceptible

Susceptible

Infected

InfectedHosts

Vectors

Death Death

Death Death

Birth

Birth

Gaff Modified SIR Model

( )dN K NB N bNdt K

B – Birth rate of hostsb – Death rate of hostsK – Carrying capacity for hosts per m2

N – Host population

Rate of Change of the Host Population

0 20 40 60 80 100 120 140 1600

0.0005

0.001

0.0015

0.002Population Hosts

Period (Months)H

osts

per

m²

Gaff Modified SIR ModelRate of Change of Tick Population

' ( ) 'dV MN VB V b Vdt MN

V – Tick populationB’ – Birth rate of ticksb’ – Death rate of ticksM – Maximum number of ticks per hostN – Host population

0 20 40 60 80 100 120 140 1600

0.050.1

0.150.2

0.250.3

0.350.4

Population Ticks

Period (Months)Ti

cks

per

m²

Gaff Modified SIR ModelRate of change of the Infected Host

Population( ) ( )dY N Y NYA X B b v Y

dt N K

Y – Infected host populationA – Transmission rate from ticks to hostsN – Host population (total)X – Infected tick populationB – Birth rate for ticksK – Carrying capacity for hosts per m2

b – Death rate of hostsv – Recovery rate of hosts

0 20 40 60 80 100 120 140 1600

0.0002

0.0004

0.0006

0.0008

Infected Hosts

Period (Months)

Hos

ts p

er m

²

Gaff Modified SIR Model Rate of change of the infected tick

population'( )( ) ' 'dX Y VXA V X B b X

dt N MN

X – Infected tick populationA’ – Transmission rate for hosts to ticksY – Infected host populationN – Host population (total)V – Tick population (total)X – Infected tick populationB’ – Tick birth rateM – Maximum number of ticks per hostb’ – Tick death rate

0 20 40 60 80 100 120 140 1600

0.0020.0040.0060.0080.01

0.0120.014

Infected Ticks

Period (Month)

Tick

s pe

r m

²

Change in Host Population = $C$2*(($C$4-B15)/$C$4)*B15 - $C$7*B15

Change in Tick Population = $C$3*C15*((($C$6*B15)-C15)/($C$6*B15)-($C$8*C15)

Change in Infected Host = $C$10*((B15-E15)/B15)*F15-$C$2*((B15*E15)/$C$4)-($C$7+$C$12)*E15Change in Infected Ticks = $C$11*(E15/B15)*(C15-F15) - ($C$3*((C15*F15)/($C$6*B15))) - $C$8*F15

Excel Single Patch Model

Multi-Patch Model1 1 1

1 1 1 1 1, 121

1 1 1 11 1 1 1 1, 1

21 1

1 1 1 1 11 1 1 1 1 1 1, 1

21 1

1 1 1 11 1 1 1 1 1 1, 1

21 1 1

( ) ( )

' ( ) ' ( )

( ) ( ) ( )

' ( )( ) ' ' ( )

x

x x

x

x x

x

x x

x

x x

dN K NB N b N m N Ndt KdV M N VB V b V m V Vdt M NdY N Y N YA X B b v Y m Y Ydt N KdX Y V XA V X B b X m X Xdt N M N

Excel Multi-Patch Model

Methodology Begin with a simple SIR model Develop variables needed to modify the

model Attempt to modify the model to

incorporate all vector-borne diseases

TimelineAOS HCI

Acquire data from external scientists

May-AugFormulate model based on ticks

using Excel Formulate model based on

mosquitoes using Excel

AOS goes to Singapore Finalize model & compare models

Preparation for Finals Presentation Aug

Evaluate and ensure research is validFinalize literature review

Nov-Jan

Set parameters to our model based on characteristics of disease Analyze data & identify vital information required

Collate our data & sort it for proper formation of model Jan-Apr

BibliographyAcademy of Science. Academy of Science Mathematics BC Calculus Text.

Breish, N., & Thorne, B. (n.d.). Lyme disease and the deer tick in maryland. Maryland: The University of Maryland.

Gaff, H. D., & Gross, L. J. (2006). Modeling Tick-Borne Disease: A Metapopulation Model. Mathematical Biology , 69, 265-288.

Neuwirth, E., & Arganbright, D. (2004). The active modeler: mathematical modeling with Microsoft Excel. Belmont, CA: Thomson/Brooks/Cole.

Stafford III, K. (2001). Ticks. New Haven: The Connecticut Agricultural Experiment Station.