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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.