The basic reproduction number, R0, is an essential measure in epidemiology used to quantify the transmission potential of a pathogen. It is a dimensionless quantity that represents the expected number of secondary cases produced by a single infected individual over the course of the individuals infectious period in a fully susceptible population. R0 provides information about the initial dynamics of an emerging disease. Most critically there is a threshold condition that determines whether a pathogen will spread, R0 > 1, or fade out, R0 < 1, because infected hosts do not, on average, replace themselves. Higher values of R0 also increase the likelihood of pathogen establishment given that infected individuals are occasionally entering a naive host population, and hence imply a shorter time to establishment. The magnitude of R0 is also a measure of the risk of an epidemic and indicates the level of effort needed to control or prevent an epidemic. In this talk I present multi-host, multi-pathogen mechanistic models of R0 that explains the zoonotic emergence and persistence of human babesiosis, a tick-borne disease, in the United States. Human babesiosis, spread by the Babesia microti pathogen, is an emerging vector-borne disease transmitted by the Ixodes scapularis tick in the United States. The number of babesiosis cases has been increasing leading to the classification as an “emerging health risk” by the Center for Disease Control. The reasons for emergence remain largely unknown. Given the difficulties in constructing models of tick-borne disease the models presented here are derived by way of ‘direct epidemiological reasoning’ using the assumptions and specifications of the model used. These models are naturally based on the biology of the disease system of interest. The models derived in this work are mechanistic, transparent and almost all parameters can be measured directly by laboratory or field studies.

1. 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

2. J. M. Dunn (QUT) QUT Seminar 08.08.2014 2 / 41

3. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 3 / 41

4. Lyme Disease (Borrelia burgdorferi)
J. M. Dunn (QUT) QUT Seminar 08.08.2014 4 / 41

5. Human Babesiosis (Babesia microti)
Reported cases of Human Babesiosis – United States, 2011
J. M. Dunn (QUT) QUT Seminar 08.08.2014 5 / 41

6. Hosts
White-footed mice (Peromyscus leucopus) Tick (Ixodes scapularis)
J. M. Dunn (QUT) QUT Seminar 08.08.2014 6 / 41

7. 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.
J. M. Dunn (QUT) QUT Seminar 08.08.2014 7 / 41

8. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 8 / 41

9. Tick life cycle
J. M. Dunn (QUT) QUT Seminar 08.08.2014 9 / 41

10. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 10 / 41

11. Tick-borne pathogen transmission routes
J. M. Dunn (QUT) QUT Seminar 08.08.2014 11 / 41

12. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 12 / 41

13. Overview
Model emergence
- Identify the factors driving emergence
- Identify control measures
Model the risk to humans
- Incorporate the identified factors
- Analyse changes in risk
J. M. Dunn (QUT) QUT Seminar 08.08.2014 13 / 41

14. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 14 / 41

15. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 15 / 41

16. Modelling emergence
NGM for tick-borne pathogens
J. M. Dunn (QUT) QUT Seminar 08.08.2014 16 / 41

17. Modelling emergence
Reduction for US Lyme and Human Babesiosis
J. M. Dunn (QUT) QUT Seminar 08.08.2014 17 / 41

18. Modelling emergence
NGM for US Lyme and Human Babesiosis
J. M. Dunn (QUT) QUT Seminar 08.08.2014 18 / 41

19. Modelling emergence
Quantifying R0
J. M. Dunn (QUT) QUT Seminar 08.08.2014 19 / 41

20. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 20 / 41

21. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 21 / 41

22. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 22 / 41

23. Modelling emergence
Internal functions of R0
Efficiency of transmissionInfectivity
Days
H
μ
p(t) = HPe
−1
2
ln t
µP
/σP
2
J. M. Dunn (QUT) QUT Seminar 08.08.2014 23 / 41

24. 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 .
J. M. Dunn (QUT) QUT Seminar 08.08.2014 24 / 41

25. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 25 / 41

26. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 26 / 41

27. Modelling emergence
Implications for control
Given, ¯R0 = 1.57
Vaccination requirements (Roberts, 2003)
V = 1 −
1
R2
0
≈ 60%
J. M. Dunn (QUT) QUT Seminar 08.08.2014 27 / 41

28. 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).
J. M. Dunn (QUT) QUT Seminar 08.08.2014 28 / 41

29. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 29 / 41

30. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 30 / 41

31. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 31 / 41

32. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 32 / 41

33. 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)
J. M. Dunn (QUT) QUT Seminar 08.08.2014 33 / 41

34. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 34 / 41

35. Modelling emergence
Questions?
J. M. Dunn (QUT) QUT Seminar 08.08.2014 35 / 41

36. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 36 / 41

37. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 37 / 41

38. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 38 / 41

39. Modelling risk
Accounting for infectivity of hosts p(t)
J. M. Dunn (QUT) QUT Seminar 08.08.2014 39 / 41

40. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 40 / 41

41. 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
J. M. Dunn (QUT) QUT Seminar 08.08.2014 41 / 41