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

The Mathematical Epidemiology of Human Babesiosis inthe North-Eastern United States

Jessica Margaret Dunn, Dr. Stephen Davis (RMIT), Dr. AndrewStacey (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 areseven 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) andLyme disease (B. burgdorferi) in endemic sites, and their expansioninto new areas in the north-eastern United States.


Mathematical Modelling Challenges

Deriving mathematical models of tick-borne disease transmission isnotoriously 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 45 500

50

100

150

200

250

300

350

400

450

500 LarvaeNymphsAdults


Tick-borne pathogen transmission routes


Modelling challenges

The modelling challenge then becomes to one of incorporating thesecomplexities whilst maintaining a model that:

1 is representative of the transmission cycle

2 can be used with field data which will provide meaningful estimates ofthe parameters

3 has a minimal number of parameters to ensure the model can beadequately 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 casesproduced 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 atinfection i caused by one individual at state at infection j , during its wholeinfectious period.

For example given 2 host types i and j there are four possibilities:

K = (kij) =

(k11 k12k21 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 350Day

Mean nymph burden

Mean larvae burden

Rep

rese

ntat

ive

mea

n tic

k co

unt p

er m

ouse

5

20

50

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

Larv

al ti

ck b

urde

nLa

rval

tick

bur

den

Nym

phal

tick

bur

den

Nym

phal

tick

bur

den


Modelling emergence

Brunner and Ostfeld (2008)

Z̄N(t) =

{HNe

− 12

[ln(

(t−τN )

µN

)/σN

]2if t ≥ τN ;

0 otherwise

Z̄L(t) =

HEe− 1

2

(t−τEµE

)2if t ≤ τL;

HLe− 1

2

[ln(

(t−τL)µL

)]2+ HEe

− 12

(t−τEµE

)2otherwise


Modelling emergence

Internal functions of R0

Efficiency of transmissionInfectivity

Days

H

μ

p(t) = HPe− 1

2

[ln(

tµP

)/σP

]2


Modelling emergence

Global Sensitivity Analysis of R0

Ranks the parameters by their contribution to the variation of R0 usingSobol’s indices:

Main effect: calculates the effect of parameter xi on R0 fixing allother variables

Total effect: includes the main effect for xi plus all other interactioninvolving 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

Sob

ol’s

fIndi

ces

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 0.50.5

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 (SN

)

R0

Threshold R0=1

Fixed point estimate


Modelling emergence

Implications for control

Given, R̄0 = 1.57

Vaccination requirements (Roberts, 2003)

V = 1 − 1

R20

≈ 60%


Modelling emergence

The Coinfection Story

J.M. Dunn et al. Borrelia burgdorferi enhances the enzootic establishment ofBabesia microti in the northeastern United States, PLOS ONE (2014).


Modelling emergence

Modification of R0

k13

k31

k13

k31

k32

k23

k32

k23

White-footedmouse-Bb free,infectedawithBm1:

White-footedmouse-infectedawith Bb & Bm2:

TickainfectedawithaBmaduringa1stabloodmeal3:

Ka= 0 0

0 0

0

k13

k31

k23

k32R =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

0.4 0.6 0.8 10.3

0.4

0.5

0.6

Low8prevalence

c

s N

0.4 0.6 0.8 10.3

0.4

0.5

0.6

High8prevalence

c

s N

0.4 0.6 0.8 10.3

0.4

0.5

0.6

0.7

c

s N

0.4 0.6 0.8 10.3

0.4

0.5

0.6

0.7

c

s N

Babesia8microti

Coinfection

Block8Island

Connecticut

30w8B88 b8prevalence8in8mice 80w8Bb8prevalence8in8mice

Student8Version8of8MATLAB

B. microti

B. microti C8B. Burgdorferi BL2068

fade8out fade8out

fade8outfade8out

emergence emergence

emergenceemergence

30w8B. burgdorferi8BL2068prevalencein8mice

80w8B. burgdorferi8BL2068prevalencein8mice


Modelling emergence

Timing is everything!120 140 160 180 200 220 240 260 280 300

0

5

10

15

20

25

30

35

120 140 160 180 200 220 240 260 280 3000

5

10

15

Rep

rese

ntat

ive3

mea

n3tic

k3co

unt3p

er3m

ous

e Prop

ortion3of3infected3larval3ticks3per3m

ouseP

roportion3of3infected

3larval3ticks3per3mouseR

epre

sent

ativ

e3m

ean3

tick3

coun

t3per

3mo

use

Block3Island

Connecticut

3333333333333333333333333333330.93333330.83333330.73333330.63333330.53333330.43333330.333333330.233333330.1

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

54

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) coinfectionand (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)

dSkdt

= −βk(t)Sk

dIkdt

= βk(t)Sk − γI

dCk

dt= γI


Modelling risk

Force of Infection

The force of infection is related to the unfed nymphs from the previousyear k − 1

βk(t) =1

DNνk Z̄N(t)qN .

with the proportion of infected unfed nymphs, νk , in year k is given by

νk =

∫ 365

0aL(t)p̄

Ik−1

Nk−1dt


Modelling risk

Accounting for infectivity of hosts p(t)


Modelling risk

dSkdt

= −βk(t)Sk + b(t)Nk − (µ+Nk

K)Sk

dIk,1dt

= βk(t)Sk − (µ+Nk

K)Ik,1 − γI1

dIk,2dt

= γI1 − (µ+Nk

K)Ik,2 − γI2

dIk,3dt

= γI2 − (µ+Nk

K)Ik,3 − γI3

dIk,4dt

= γI3 − (µ+Nk

K)Ik,4 − γI4

dIk,5dt

= γI4 − (µ+Nk

K)Ik,5 − γI5

dIk,6dt

= γI5 − (µ+Nk

K)Ik,6 − γI6

dCk

dt= γI6 − (µ+

Nk

K)C


Modelling risk

νk =

∫ 365

0

aL(t)

{p̄1

Ik−1,1

Nk−1+ p̄2

Ik−1,2

Nk−1+ p̄3

Ik−1,3

Nk−1+ p̄4

Ik−1,4

Nk−1+ p̄5

Ik−1,5

Nk−1+ p̄6

Ik−1,6

Nk−1

}dt

0 5 10 15 20 25 300

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

time (years)

Pro

port

ion

of in

fect

ed fe

d la

rvae

Student Version of MATLAB


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