The Mathematical Epidemiology of Human Babesiosis in the North-Eastern United States - Jessica Dunn, QUT

The Mathematical Epidemiology of Human Babesiosis in
the North-Eastern United States
Jessica Margaret Dunn, Dr. Stephen Davis (RMIT), Dr. Andrew
Stacey (RMIT), Assoc. Prof. Maria Diuk-Wasser (Yale/Columbia)
J. M. Dunn (QUT) QUT Seminar 08.08.2014 1 / 41

Tick-borne disease in the USA
The geographical range of tick-borne diseases are expanding. There are
seven emerging tick diseases:
Lyme disease
Human babesiosis
Human anaplasmosis
Powassan
Deer tick encephalitis
B. miyamotoi borreliosis
Deer tick ehrlichiosis

Lyme Disease (Borrelia burgdorferi)

Human Babesiosis (Babesia microti)
Reported cases of Human Babesiosis – United States, 2011

Hosts
White-footed mice (Peromyscus leucopus) Tick (Ixodes scapularis)

Research Objective
To identify the key factors driving human babesiosis (B. microti) and
Lyme disease (B. burgdorferi) in endemic sites, and their expansion
into new areas in the north-eastern United States.

Mathematical Modelling Challenges
Deriving mathematical models of tick-borne disease transmission is
notoriously diﬃcult!
Multiple hosts (competent and non-competent)
Tick life-cycle (biting rate)
Multiple tranmission routes
Multiple pathogens

Tick life cycle

Tick-phenology
Densities-Northeast
Weeks
Density
0 5 10 15 20 25 30 35 40 45 50
0
50
100
150
200
250
300
350
400
450
500 Larvae
Nymphs
Adults

Tick-borne pathogen transmission routes

Modelling challenges
The modelling challenge then becomes to one of incorporating these
complexities whilst maintaining a model that:
1 is representative of the transmission cycle
2 can be used with ﬁeld data which will provide meaningful estimates of
the parameters
3 has a minimal number of parameters to ensure the model can be
adequately analysed

Overview
Model emergence
- Identify the factors driving emergence
- Identify control measures
Model the risk to humans
- Incorporate the identiﬁed factors
- Analyse changes in risk

Modelling emergence
Modelling emergence
The basic Reproduction number, R0
In single host systems, R0 is the expected number of secondary cases
produced by one infectious individual in a fully susceptible population.
R0 = 1 provides a threshold condition:
pathogen will spread R0 > 1
pathogen will fade out R0 < 1

Modelling emergence
R0 for multiple hosts
Next generation Matrix (NGM) (Diekmann and Heasterbeek)
Deﬁne kij as the expected number of new cases that have state at
infection i caused by one individual at state at infection j, during its whole
infectious period.
For example given 2 host types i and j there are four possibilities:
K = (kij ) =
k11 k12
k21 k22
R0 is the dominant eigenvalue of the NGM such that
vk+1 = Kvk

Modelling emergence
NGM for tick-borne pathogens

Modelling emergence
Reduction for US Lyme and Human Babesiosis

Modelling emergence
NGM for US Lyme and Human Babesiosis

Modelling emergence
Quantifying R0

Modelling emergence
Internal functions of R0
Tick Phenology
0 50 100 150 200 250 300 350
Day
Mean nymph burden
Mean larvae burden
Representativemeantick
countpermouse
52050
μ
H
τ

Modelling emergence
Block Island
Connecticut
100 250150 200 100 150 200 250
100 150 200 250100 150 200 250
0
1
5
20
50
150
0
1
5
20
50
150150
50
20
5
1
0
150
50
20
5
1
0
Day of year Day of year
Day of year Day of year
LarvaltickburdenLarvaltickburden
NymphaltickburdenNymphaltickburden

Modelling emergence
Brunner and Ostfeld (2008)
¯ZN(t) = HNe
−1
2
ln
(t−τN )
µN
/σN
2
if t ≥ τN;
0 otherwise
¯ZL(t) =



HE e
−1
2
t−τE
µE
2
if t ≤ τL;
HLe
−1
2
ln
(t−τL)
µL
2
+ HE e
−1
2
t−τE
µE
2
otherwise

Modelling emergence
Internal functions of R0
Eﬃciency of transmissionInfectivity
Days
H
μ
p(t) = HPe
−1
2
ln t
µP
/σP
2

Modelling emergence
Global Sensitivity Analysis of R0
Ranks the parameters by their contribution to the variation of R0 using
Sobol’s indices:
Main effect: calculates the effect of parameter xi on R0 fixing all
other variables
Total effect: includes the main effect for xi plus all other interaction
involving xi .

Modelling emergence
Global Sensitivity Results
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Parameter
Sobol’sfIndices
MainfEffect
TotalfEffect
H τ μ σ τ H τ μ H μ σ H Dq ρ σμ s cN N N N L L L L P P PLE E E NN

Modelling emergence
Implications for emergence
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
Proportion of fed larval ticks that survive to become unfed nymphs (S
N
)
R
0
Threshold R
0
=1
Fixed point estimate

Modelling emergence
Implications for control
Given, ¯R0 = 1.57
Vaccination requirements (Roberts, 2003)
V = 1 −
1
R2
0
≈ 60%

Modelling emergence
The Coinfection Story
J.M. Dunn et al. Borrelia burgdorferi enhances the enzootic establishment of
Babesia microti in the northeastern United States, PLOS ONE(2014).

Modelling emergence
Modiﬁcation of R0
k13
k31
k13
k31 k32
k23
k32
k23
White-footedm1:
White-footedm2:
Tickainfectedaw3:
Ka= 0 0
0 0
0
1 2
3
R0 = k13k31 + k23k32
. . .
t=365
t=0
. . . ψ
t =365−t
t =0
p1(t ) . . . dt + (1 − ψ)
t =365−t
t =0
p2(t ) . . . dt dt

Modelling emergence
Implications of coinfection on emergence
0.6 0.8 1
c
0.4 0.6 0.8 1
0.3
0.4
0.5
c
0.6 0.8 1
c
0.4 0.6 0.8 1
0.3
0.4
0.5
0.6
0.7
c
sN
B. microti
B. microti C8B. Burgdorferi BL2068
fade8out
fade8out
emergence emergence
80w8B. burgdorferi8BL2068prevalence
in8mice

Modelling emergence
Timing is everything!120 140 160 180 200 220 240 260 280 300
0
5
120 140 160 180 200 220 240 260 280 300
0
5
10
15
Re
ouseProportion3of3infected3larval3ticks3per3mouse
Representative3mean3tick3count3per3mouse
Connecticut
3330.233333330.1
Mean nymph burden
Mean larvae burden
Babesia3+3Borrelia
Babesia
3333333333333333333333333333330.93333330.83333330.73333330.63333330.53333330.43333330.333333330.233333330.1

Modelling emergence
Coinfection is not the whole story!
Accounting for aggregation on hosts
k13
k13
32
1
5
4
2
k51
k15
k12
k21
k14
k41
2: High aggregation white footed mouse
- infected with Bb
4: Low aggregation white footed mouse
- infected with Bb
3: High aggregation white footed mouse
- infected with Bm
5: Low aggregation white footed mouse
- infected with Bm
R0 = k12k21 + k12k21 + k13k31 + k14k41 + k15k51

Modelling emergence
Scenario Estimated R0
No co-aggregation; no coinfection 0.70 (0.62,0.78)
Low co-aggregation; no coinfection 0.80 (0.71,0.86)
Moderate co-aggregation; no coinfection 0.97 (0.81,1.04)
High co-aggregation; no coinfection 1.13 (1.00, 1.21)
High co-aggregation; coinfection 1.78 (1.64, 1.91)

Modelling emergence
Conclusions
Epidemiological:
Values of R0 are consistently low 1 < R0 < 3
Transmission eﬃciency drives emergence
Timing is everything!
Mathematical:
Models are mechanistic, transparent, linked directly with ﬁeld data
Step towards a model for more complicated tick-borne pathogens
First such model that that assesses the importance of (i) coinfection
and (ii) aggregation

Modelling emergence
Questions?

Modelling risk
Modelling risk to humans
Risk is directly proportional to the infection prevalence in nymphal ticks.
Compartment type SIR Model: (S)usceptibles to (I)nfectives to
(R)ecovered

Modelling risk
Three generation based compartments:
Sk(t), Ik(t) and Ck(t)
dSk
dt
= −βk(t)Sk
dIk
dt
= βk(t)Sk − γI
dCk
dt
= γI

Modelling risk
Force of Infection
The force of infection is related to the unfed nymphs from the previous
year k − 1
βk(t) =
1
DN
νk
¯ZN(t)qN.
with the proportion of infected unfed nymphs, νk, in year k is given by
νk =
365
0
aL(t)¯p
Ik−1
Nk−1
dt

Modelling risk
Accounting for infectivity of hosts p(t)

Modelling risk
dSk
dt
= −βk(t)Sk + b(t)Nk − (µ +
Nk
K
)Sk
dIk,1
dt
= βk(t)Sk − (µ +
Nk
K
)Ik,1 − γI1
dIk,2
dt
= γI1 − (µ +
Nk
K
)Ik,2 − γI2
dIk,3
dt
= γI2 − (µ +
Nk
K
)Ik,3 − γI3
dIk,4
dt
= γI3 − (µ +
Nk
K
)Ik,4 − γI4
dIk,5
dt
= γI4 − (µ +
Nk
K
)Ik,5 − γI5
dIk,6
dt
= γI5 − (µ +
Nk
K
)Ik,6 − γI6
dCk
dt
= γI6 − (µ +
Nk
K
)C

Modelling risk
νk =
365
0
aL(t) ¯p1
Ik−1,1
Nk−1
+ ¯p2
Ik−1,2
Nk−1
+ ¯p3
Ik−1,3
Nk−1
+ ¯p4
Ik−1,4
Nk−1
+ ¯p5
Ik−1,5
Nk−1
+ ¯p6
Ik−1,6
Nk−1
dt
0 5 10 15 20 25 30
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
time (years)
Proportionofinfectedfedlarvae

The Mathematical Epidemiology of Human Babesiosis in the North-Eastern United States - Jessica Dunn, QUT

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