Lyme Disease: A Mathematical Approach · 2015. 7. 25. · Antoine Marc & Carlos Munoz Lyme Disease: A Mathematical Approach. Biology of Lyme Disease Base Model Age-Structured Tick

Biology of Lyme DiseaseBase Model

Age-Structured Tick Class ModelAge-Structured Tick Class & Seasonality Model

Lyme Disease: A Mathematical Approach

Antoine Marc & Carlos Munoz

July 21, 2015

Antoine Marc & Carlos Munoz Lyme Disease: A Mathematical Approach



Overview

1 Biology of Lyme DiseaseBorrelia burgdoferiIxodes scapularisHosts

2 Base ModelCompartmental ModelSystem of ODEsAnalysis of R0SimulationsConclusion

3 Age-Structured Tick ClassModel

Compartmental ModelSystem of ODEsAnalysis of R0SimulationsConclusion

4 Age-Structured Tick Class &Seasonality Model

System of ODEsAnalysis of R0SimulationsConclusion




Borrelia burgdoferiIxodes scapularisHosts

Why is Lyme Disease Important to Study?

The shortening of Winter in the North led to the following:

Warmer temperatures have been predicted to both enhancetransmission intensity and extend the distribution of diseasessuch a malaria and dengue as well.

Climate change may open up previously uninhabitableterritory for arthropod vectors as well as increase reproductiveand biting rates, and shorten the pathogen incubation period.





Borrelia Burgdoferi





Ixodes scapularis

Ixodes scapularis, the black-legged tick

Can be found throughout the country including Texas

Take a blood meal every time they molt

Once infected, they are infected for life

Must attach for 36 hours to transmit the bacteria

No vertical transmission

Questing season is changing due to climate change





Figure of Life cycle:





White footed Mouse, Peromyscus leucopus





Unidentified Alternate Host





1 Recent data shows that ticks quest at two distinct heights

2 This will help narrow down the search for the other hosts





Compartmental Model for Base Model





Base Model

diMdt

= αβ̃M(1 − iM)iT − µM iMdiTdt

= β̃T (1 − iT )((αiM) + (1 − α)iA

)− µT iT

diAdt

= (1 − α)β̃A(1 − iA)iT − δ1µAiA





Base Model with Seasonality

diMdt

= αβ̃M(1 − iM)iT − µM iMdiTdt

= β̃T (1 − iT )((αiM) + (1 − α)iA

)− µT iT

diAdt

= δ1(1 − α)β̃A(1 − iA)iT − δ1µAiA





middle2

3:61-304

δ1 =

{1 if 61 ≤ t ≤ 3040 Otherwise





Parameters

Parameter Definition

ρM Birth rate of the mice into the susceptible classρT Birth rate of the ticks into the susceptible classρA Birth rate of the alternate host into the susceptible classβM Contact Transmission Rate for the miceβT Contact Transmission Rate for the TickβA Contact Transmission Rate for the alternate hostα Proportion of the the ticks that have a preference for questing at lower heightsµM Death rate for the Mouse classµT Death rate for the Tick classµA Death rate for the Alternate Host class

Table: Table of Variables & Parameters for Base Model





Basic Reproduction Number of a Infection

The nondimensionalized system was reduced and rearranged intoan equation for R0, which determines whether or not there will bean epidemic.

If R0 < 1 then the disease will eventually die out of thepopulation.

If R0 = 1 the disease remains at a constant level in thepopulation.

If R0 > 1 the level of disease in the population will increaseuntil there is an epidemic.





R0 :=

√α2βMβT µA + (1 − α)2βAβT µM

µMµT µA





Scenarios in which the overall R0 > 1 and an epidemic willoccur in the community.

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

po

rtio

n o

f T

ota

l P

op

ula

tio

n

0

0.1

0.2

0.3

0.4

0.5

0.6

Base Model for 10 years

Infected Mouse

Infected Tick

Infected Alternate

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

po

rtio

n o

f T

ota

l P

op

ula

tio

n

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Infected Mouse

Infected Tick

Infected Alternate





Overall R0 > 1 and an epidemic occurs

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

po

rtio

n o

f T

ota

l P

op

ula

tio

n

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Infected Mouse

Infected Tick

Infected Alternate

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

po

rtio

n o

f T

ota

l P

op

ula

tio

n

0

0.05

0.1

0.15

0.2

0.25

0.3


Infected Mouse

Infected Tick

Infected Alternate





Scenarios in which the overall R0 > 1 and an epidemic willoccur in the community Cont.

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

po

rtio

n o

f T

ota

l P

op

ula

tio

n

0

0.05

0.1

0.15


Infected Mouse

Infected Tick

Infected Alternate

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

po

rtio

n o

f T

ota

l P

op

ula

tio

n

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14


Infected Mouse

Infected Tick

Infected Alternate





Overall R0 < 1 and the disease dies out

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

po

rtio

n o

f T

ota

l P

op

ula

tio

n

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Base Model for 10 years

Infected Mouse

Infected Tick

Infected Alternate

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

po

rtio

n o

f T

ota

l P

op

ula

tio

n

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1


Infected Mouse

Infected Tick

Infected Alternate





Conclusion

Adding an invading species with R0 > 1 can increase the levelof infection for the initial host





Three Tick Stages





System of ODEs Modeling the Spread of Lyme DiseaseDiagram that incorporates Criss-Cross Infection





System of ODEs Modeling the Spread of Lyme Disease

diMdt

= αβM(1 − iM)iTN − µM iM (1a)diTLdt

= βTL(1 − iTL)iM − ηTL iTL − µTL iTL (1b)

diTN Edt

= βTN (1 − iTN E − iTN )(

αiM + (1 − α)iA)− ηTN iTN E − µTN iTN E

(1c)

diTNdt

= ηTL iTL − ηTN iTN − µTN iTN (1d)diTAdt

= ηTN (iTN + iTN E )− µTA iTA (1e)

diAdt

= βA(1 − iA)(iTA + (1 − α)iTN

)− µAiA (1f)





Parameters

Parameter Definition

ρM Birth rate of the mice into the susceptible classρT Birth rate of the ticks into the susceptible classρA Birth rate of the alternate host into the susceptible Larvae classβM Contact Transmission Rate for the miceβTL Contact Transmission Rate for the larval tickβTN Contact Transmission Rate for the nymphal tickβA Contact Transmission Rate for the alternate hostηTL Rate that the larvae molt into nymphsηTN Rate that the larvae nymphs into adultsα Proportion of the the ticks that have a preference for questing at lower heightsµM Death rate for the Mouse classµT Death rate for the Tick classµA Death rate for the Alternate Host class

Table: Table of Variables & Parameters for Model with 3 Tick Classes






R0 :=

max

{√αβMβLηTL

µM(ηTL + µTL)(ηTN + µTN ),

√βTN (1 − α)βAηTN(ηTN + µTN )µTAµA

}





To illustrate that the disease will be endemic in the alternate host.We chose variables in the following manner in the followingmanner:

higher βM

lower βTL , βTN , and βA





Model with 3 Tick Classes α = 1, βM = .21, βTL =.00041, βTN = .00041, βA = .00041, R0 = 6.2214

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

port

ion o

f T

ota

l P

opula

tion

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Model with 3 Tick Classes for 10 years

Infected Mouse

Infected larvae

Exposed Nymph

Infected Nypmh

Infected Adult

Infected Alternate





Model with 3 Tick Classes α = 1, βM = .01, βTL =.0041, βTN = .0041, βA = .0041, R0 = 2.9626

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

port

ion o

f T

ota

l P

opula

tion

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Model with 3 Tick Classes for 10 years

Infected Mouse

Infected larvae

Exposed Nymph

Infected Nypmh

Infected Adult

Infected Alternate





Conclusion

It’s very beneficial for the mice to be able to sustain thedisease, but not necessary.

If the the disease is endemic to the mice population, then it’svery likely that the disease will be endemic for the alternatehost population.





System of ODEs Modeling the Spread of Lyme Disease

diMdt

= δ3αβM(1 − iM)iTN − µM iM (2a)diTLdt

= δ1βTL(1 − iTL)iM − ηTL iTL − µTL iTL (2b)

diTN Edt

= δ3βTN (1 − iTN E − iTN )(

αiM + (1 − α)iA)− ηTN iTN E − µTN iTN E

(2c)

diTNdt

= ηTL iTL − ηTN iTN − µTN iTN (2d)diTAdt

= ηTN (iTN + iTN E )− µTA iTA (2e)

diAdt

= βA(1 − iA)(

δ5iTA + δ3(1 − α)iTN)− µAiA (2f)





δ1 =

{1 if active>182 active119 and active274 active41 active





R0 :=

0 0 ≤ t ≤ 121√αβMβTLηTL

(ηTL + µTL)(ηTN + ηTN )µM121 ≤ t ≤ 274

max

{√αβMβLηTL

µM(ηTL + µTL)(ηTN + µTN ),

√βTN (1 − α)βAηTN(ηTN + µTN )µTAµA

}274 ≤ t ≤ 283

0 283 ≤ t ≤ 3460 346 ≤ t ≤ 365





Model with Seasonality: α = 1, βM = .0011, βTL =.0011, βTN = .0011, βA = .0011, R0 = .8693

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

port

ion o

f T

ota

l P

opula

tion

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Model with 3 Tick Classes and Seasonality for 10 years

Infected Mouse

Infected larvae

Exposed Nymph

Infected Nypmh

Infected Adult

Infected Alternate





Model with Seasonality: α = .5, βM = .011, βTL =.011, βTN = .011, βA = .011, R0 = .7561

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

port

ion o

f T

ota

l P

opula

tion

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09


Infected Mouse

Infected larvae

Exposed Nymph

Infected Nypmh

Infected Adult

Infected Alternate





Model with Seasonality: α = 0, βM = .0011, βTL =.0011, βTN = .0011, βA = .0011, R0 = .7036

Time (Days)

0 500 1000 1500 2000 2500 3000 3500

Pro

port

ion o

f T

ota

l P

opula

tion

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09


Infected Mouse

Infected larvae

Exposed Nymph

Infected Nypmh

Infected Adult

Infected Alternate





Conclusion

Lyme disease is found in mice in Texas at low levels. Themodel indicates that it’s very likely that there is a larger hostthat is also an effective carrier.





References

Branda JA, Rosenberg ES. 2013 Borrelia miyamotoi: ALesson in Disease Discovery. Ann Intern Med. 159:61-62.

Fukunaga M. 1995. Genetic and Phenotypic Analysis ofBorrelia miyamotoi sp. nov., Isolated from the Ixodid TickIxodes persulcatus, the Vector for Lyme Disease in Japan. Int.J. Syst. Bacteriol. 45: 804-810.

Rollend, L. 2013. Transovarial transmission of Borreliaspirochetes by Ixodes scapularis: a summary of the literatureand recent observations. Ticks Tick-borne Dis. 4(12):46-51.





References Cont.

Anderson, J., L. Magnarelli. 1980. Vertebrate hostrelationships and distribution of ixodid ticks (Acari: Ixodidae)in Connecticut, USA. Journal of Medical Entomology,17:314-323.

Bertrand, M., M. Wilson. 1996. Microclimate-dependentsurvival of unfed adult Ixodes scapularis (Acari: Ixodidae) innature: life cycle and study design implications. Journal ofMedical Entomology. 33: 619-627.


Biology of Lyme DiseaseBorrelia burgdoferiIxodes scapularisHosts

Base ModelCompartmental ModelSystem of ODEsAnalysis of R0SimulationsConclusion

Age-Structured Tick Class ModelCompartmental ModelSystem of ODEsAnalysis of R0SimulationsConclusion

Age-Structured Tick Class & Seasonality ModelSystem of ODEsAnalysis of R0SimulationsConclusion

Documents

Lyme Disease: A Mathematical Approach · 2015. 7. 25. · Antoine Marc & Carlos Munoz Lyme Disease: A Mathematical Approach. Biology of Lyme Disease Base Model Age-Structured Tick