Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-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
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Overview
1
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Biology of Lyme Disease
Borrelia burgdoferi
Ixodes scapularis
Hosts
2
Base Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
3
Age-Structured Tick Class
Model
Antoine Marc & Carlos Munoz
4
Age-Structured Tick Class &
Seasonality Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Borrelia burgdoferi
Ixodes scapularis
Hosts
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 enhance
transmission intensity and extend the distribution of diseases
such a malaria and dengue as well.
Climate change may open up previously uninhabitable
territory for arthropod vectors as well as increase reproductive
and biting rates, and shorten the pathogen incubation period.
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Borrelia burgdoferi
Ixodes scapularis
Hosts
Borrelia Burgdoferi
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Borrelia burgdoferi
Ixodes scapularis
Hosts
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
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Borrelia burgdoferi
Ixodes scapularis
Hosts
Figure of Life cycle:
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Borrelia burgdoferi
Ixodes scapularis
Hosts
White footed Mouse, Peromyscus leucopus
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Borrelia burgdoferi
Ixodes scapularis
Hosts
Unidentified Alternate Host
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Borrelia burgdoferi
Ixodes scapularis
Hosts
1
Recent data shows that ticks quest at two distinct heights
2
This will help narrow down the search for the other hosts
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Compartmental Model for Base Model
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Base Model
diM
= α β̃ M (1 − iM )iT − µM iM
dt
diT
= β̃ T (1 − iT ) (αiM ) + (1 − α)iA − µT iT
dt
diA
= (1 − α) β̃ A (1 − iA )iT − δ1 µA iA
dt
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Base Model with Seasonality
diM
= α β̃ M (1 − iM )iT − µM iM
dt
diT
= β̃ T (1 − iT ) (αiM ) + (1 − α)iA − µT iT
dt
diA
= δ1 (1 − α) β̃ A (1 − iA )iT − δ1 µA iA
dt
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
middle
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
2
:61-304
3
δ1 =
1 if 61 ≤ t ≤ 304
0
Otherwise
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Parameters
Parameter
ρM
ρT
ρA
βM
βT
βA
α
µM
µT
µA
Definition
Birth rate of the mice into the susceptible class
Birth rate of the ticks into the susceptible class
Birth rate of the alternate host into the susceptible class
Contact Transmission Rate for the mice
Contact Transmission Rate for the Tick
Contact Transmission Rate for the alternate host
Proportion of the the ticks that have a preference for questing at lower heights
Death rate for the Mouse class
Death rate for the Tick class
Death rate for the Alternate Host class
Table: Table of Variables & Parameters for Base Model
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Basic Reproduction Number of a Infection
The nondimensionalized system was reduced and rearranged into
an equation for R0 , which determines whether or not there will be
an epidemic.
If R0 < 1 then the disease will eventually die out of the
population.
If R0 = 1 the disease remains at a constant level in the
population.
If R0 > 1 the level of disease in the population will increase
until there is an epidemic.
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
s
R0 : =
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
α2 β M β T µA + (1 − α )2 β A β T µM
µM µT µA
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Scenarios in which the overall R0 > 1 and an epidemic will
occur in the community.
Base Model for 10 years
Base Model for 10 years
0.6
0.5
Infected Mouse
Infected Tick
Infected Alternate
0.45
Proportion of Total Population
0.5
Proportion of Total Population
Infected Mouse
Infected Tick
Infected Alternate
0.4
0.3
0.2
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.1
0.05
0
0
0
500
1000
1500
2000
2500
3000
3500
Time (Days)
Antoine Marc & Carlos Munoz
0
500
1000
1500
2000
2500
3000
Time (Days)
Lyme Disease: A Mathematical Approach
3500
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Overall R0 > 1 and an epidemic occurs
Base Model for 10 years
Base Model for 10 years
0.3
Infected Mouse
Infected Tick
Infected Alternate
Infected Mouse
Infected Tick
Infected Alternate
0.25
0.3
Proportion of Total Population
Proportion of Total Population
0.35
0.25
0.2
0.15
0.1
0.2
0.15
0.1
0.05
0.05
0
0
0
500
1000
1500
2000
2500
3000
3500
Time (Days)
Antoine Marc & Carlos Munoz
0
500
1000
1500
2000
2500
3000
Time (Days)
Lyme Disease: A Mathematical Approach
3500
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Scenarios in which the overall R0 > 1 and an epidemic will
occur in the community Cont.
Base Model for 10 years
Base Model for 10 years
0.14
0.15
Proportion of Total Population
Proportion of Total Population
0.12
Infected Mouse
Infected Tick
Infected Alternate
0.1
0.05
Infected Mouse
Infected Tick
Infected Alternate
0.1
0.08
0.06
0.04
0.02
0
0
0
500
1000
1500
2000
2500
3000
3500
Time (Days)
Antoine Marc & Carlos Munoz
0
500
1000
1500
2000
2500
3000
Time (Days)
Lyme Disease: A Mathematical Approach
3500
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Overall R0 < 1 and the disease dies out
Base Model for 10 years
Base Model for 10 years
0.1
0.1
Infected Mouse
Infected Tick
Infected Alternate
0.09
0.08
Proportion of Total Population
Proportion of Total Population
0.09
0.07
0.06
0.05
0.04
0.03
0.02
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.01
0
0
0
500
1000
1500
2000
2500
3000
3500
Time (Days)
Antoine Marc & Carlos Munoz
Infected Mouse
Infected Tick
Infected Alternate
0
500
1000
1500
2000
2500
3000
Time (Days)
Lyme Disease: A Mathematical Approach
3500
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Conclusion
Adding an invading species with R0 > 1 can increase the level
of infection for the initial host
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Three Tick Stages
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
System of ODEs Modeling the Spread of Lyme Disease
Diagram that incorporates Criss-Cross Infection
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
System of ODEs Modeling the Spread of Lyme Disease
diM
= αβ M (1 − iM )iTN − µM iM
dt
diTL
= β TL (1 − iTL )iM − ηTL iTL − µTL iTL
dt
diTN E
= β TN (1 − iTN E − iTN ) αiM + (1 − α)iA
dt
(1a)
(1b)
− ηTN iTN
E
− µTN iTN
(1c)
diTN
= ηTL iTL − ηTN iTN − µTN iTN
dt
diTA
= ηTN (iTN + iTN E ) − µTA iTA
dt
diA
= β A (1 − iA ) iTA + (1 − α)iTN − µA iA
dt
Antoine Marc & Carlos Munoz
(1d)
(1e)
(1f)
Lyme Disease: A Mathematical Approach
E
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Parameters
Parameter
ρM
ρT
ρA
βM
β TL
β TN
βA
η TL
η TN
α
µM
µT
µA
Definition
Birth rate of the mice into the susceptible class
Birth rate of the ticks into the susceptible class
Birth rate of the alternate host into the susceptible Larvae class
Contact Transmission Rate for the mice
Contact Transmission Rate for the larval tick
Contact Transmission Rate for the nymphal tick
Contact Transmission Rate for the alternate host
Rate that the larvae molt into nymphs
Rate that the larvae nymphs into adults
Proportion of the the ticks that have a preference for questing at lower heights
Death rate for the Mouse class
Death rate for the Tick class
Death rate for the Alternate Host class
Table: Table of Variables & Parameters for Model with 3 Tick Classes
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Basic Reproduction Number of a Infection
R0 : =
(s
max
αβ M β L ηTL
,
µM (ηTL + µTL )(ηTN + µTN )
Antoine Marc & Carlos Munoz
s
β TN ( 1 − α ) β A η TN
( η TN + µ TN ) µ TA µ A
Lyme Disease: A Mathematical Approach
)
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
To illustrate that the disease will be endemic in the alternate host.
We chose variables in the following manner in the following
manner:
higher β M
lower β TL , β TN , and β A
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Model with 3 Tick Classes α = 1, β M = .21, β TL =
.00041, β TN = .00041, β A = .00041, R0 = 6.2214
Model with 3 Tick Classes for 10 years
0.9
Proportion of Total Population
0.8
0.7
0.6
Infected Mouse
Infected larvae
Exposed Nymph
Infected Nypmh
Infected Adult
Infected Alternate
0.5
0.4
0.3
0.2
0.1
0
0
500
1000
1500
2000
2500
3000
3500
Time (Days)
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Model with 3 Tick Classes α = 1, β M = .01, β TL =
.0041, β TN = .0041, β A = .0041, R0 = 2.9626
Model with 3 Tick Classes for 10 years
0.5
Infected Mouse
Infected larvae
Exposed Nymph
Infected Nypmh
Infected Adult
Infected Alternate
Proportion of Total Population
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
500
1000
1500
2000
2500
3000
3500
Time (Days)
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Compartmental Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Conclusion
It’s very beneficial for the mice to be able to sustain the
disease, but not necessary.
If the the disease is endemic to the mice population, then it’s
very likely that the disease will be endemic for the alternate
host population.
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
System of ODEs
Analysis of R0
Simulations
Conclusion
System of ODEs Modeling the Spread of Lyme Disease
diM
= δ3 αβ M (1 − iM )iTN − µM iM
dt
diTL
= δ1 β TL (1 − iTL )iM − ηTL iTL − µTL iTL
dt
diTN E
= δ3 β TN (1 − iTN E − iTN ) αiM + (1 − α)iA
dt
(2a)
(2b)
− ηTN iTN
E
− µTN iTN
(2c)
diTN
= ηTL iTL − ηTN iTN − µTN iTN
dt
diTA
= ηTN (iTN + iTN E ) − µTA iTA
dt
diA
= β A (1 − iA ) δ5 iTA + δ3 (1 − α)iTN − µA iA
dt
Antoine Marc & Carlos Munoz
(2d)
(2e)
(2f)
Lyme Disease: A Mathematical Approach
E
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
δ1 =
δ3 =
δ5 =
System of ODEs
Analysis of R0
Simulations
Conclusion
1 if active>182 active<283
0
not active
1 : active>119 and active<283
0
not active
1 active>274 active<346 or active>41 active<161
0
not active
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
Antoine Marc & Carlos Munoz
System of ODEs
Analysis of R0
Simulations
Conclusion
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Basic Reproduction Number of a Infection
R0 : =

0s





αβ M β TL ηTL




(ηTL + µTL )(ηTN + ηTN )µM


s
)
(s
αβ M β L ηTL
β TN ( 1 − α ) β A η TN

,
max



µM (ηTL + µTL )(ηTN + µTN )
(ηTN + µTN )µTA µA





0


0
Antoine Marc & Carlos Munoz
0 ≤ t ≤ 121
121 ≤ t ≤ 274
274 ≤ t ≤ 283
283 ≤ t ≤ 346
346 ≤ t ≤ 365
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Model with Seasonality: α = 1, β M = .0011, β TL =
.0011, β TN = .0011, β A = .0011, R0 = .8693
Model with 3 Tick Classes and Seasonality for 10 years
0.1
Infected Mouse
Infected larvae
Exposed Nymph
Infected Nypmh
Infected Adult
Infected Alternate
Proportion of Total Population
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
500
1000
1500
2000
2500
3000
3500
Time (Days)
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Model with Seasonality: α = .5, β M = .011, β TL =
.011, β TN = .011, β A = .011, R0 = .7561
Model with 3 Tick Classes and Seasonality for 10 years
0.1
Infected Mouse
Infected larvae
Exposed Nymph
Infected Nypmh
Infected Adult
Infected Alternate
Proportion of Total Population
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
500
1000
1500
2000
2500
3000
3500
Time (Days)
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Model with Seasonality: α = 0, β M = .0011, β TL =
.0011, β TN = .0011, β A = .0011, R0 = .7036
Model with 3 Tick Classes and Seasonality for 10 years
0.1
Infected Mouse
Infected larvae
Exposed Nymph
Infected Nypmh
Infected Adult
Infected Alternate
Proportion of Total Population
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
500
1000
1500
2000
2500
3000
3500
Time (Days)
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
System of ODEs
Analysis of R0
Simulations
Conclusion
Conclusion
Lyme disease is found in mice in Texas at low levels. The
model indicates that it’s very likely that there is a larger host
that is also an effective carrier.
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
System of ODEs
Analysis of R0
Simulations
Conclusion
References
Branda JA, Rosenberg ES. 2013 Borrelia miyamotoi: A
Lesson in Disease Discovery. Ann Intern Med. 159:61-62.
Fukunaga M. 1995. Genetic and Phenotypic Analysis of
Borrelia miyamotoi sp. nov., Isolated from the Ixodid Tick
Ixodes persulcatus, the Vector for Lyme Disease in Japan. Int.
J. Syst. Bacteriol. 45: 804-810.
Rollend, L. 2013. Transovarial transmission of Borrelia
spirochetes by Ixodes scapularis: a summary of the literature
and recent observations. Ticks Tick-borne Dis. 4(12):46-51.
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach
Biology of Lyme Disease
Base Model
Age-Structured Tick Class Model
Age-Structured Tick Class & Seasonality Model
System of ODEs
Analysis of R0
Simulations
Conclusion
References Cont.
Anderson, J., L. Magnarelli. 1980. Vertebrate host
relationships and distribution of ixodid ticks (Acari: Ixodidae)
in Connecticut, USA. Journal of Medical Entomology,
17:314-323.
Bertrand, M., M. Wilson. 1996. Microclimate-dependent
survival of unfed adult Ixodes scapularis (Acari: Ixodidae) in
nature: life cycle and study design implications. Journal of
Medical Entomology. 33: 619-627.
Antoine Marc & Carlos Munoz
Lyme Disease: A Mathematical Approach