Control and elimination of vector-borne diseases ...€¦ · • Modelling vector-borne diseases:...

Control and elimination of vector-borne diseases:

Thresholds, breakpoints & strategies

International Conference on Mathematical and Theoretical Biology, Pune, 23-27 January, 2012

Hans-Peter Duerr Institute for Medical Biometry

University of Tübingen, Germany

http://www.uni-tuebingen.de/modeling/Mod_Duerr_Intro_en.html

Expectation Observation

Overview • Neglected tropical diseases: Research

questions, problems & modelling

• Modelling vector-borne diseases: Transmission threshold in the example of microparasites / deterministic model: Visceral Leishmaniasis, Indian subcontinent

• Non-linearities: Breakpoints & density-dependence in the example of macroparasites / stochastic model: Onchocerciasis, Africa

• Interventions & elimination from the perspective of transmission thresholds & breakpoints

• Interventions suffering from 'chain limitation'

Number of vectors

Prevalence or intensity of infection

Number of vectors

Prevalence or intensity of infection

Control of vector-borne diseases

http://apps.who.int/tdr/

s1 to s4: developmental stages of the parasite

s4

Definite host

Intermediate host

s2

s1

s3

Vector

Common interventions: • Treatment of humans • Control of vectors

The modellers' dilemma

Richard Levins. The strategy of model building in population biology. American Scientist 1966;54(4):421-431

"... It is of course desirable

to work with managable models which maximize

generality, realism

and precision

toward the overlapping but not identical goals of

understanding, predicting,

and modifying nature.

But this cannot be done. ..."

Deterministic models Stochastic models Stochastic and precise?

Biology of disease Strategies / Planning Intervention

Aim: determination of thresholds

Example 'micro'parasites: Leishmaniasis

• Protozoa, Flagellata, Trypanosomatida

• Vector: Sandflies (Phlebotomus)

• 'Kala Azar', lethal

Visceral Leishmaniasis

Post-kala-azar dermal Leishmaniasis (PKDL)

Culex

Phlebotomus (Sandmücke)

Visceral Leishmaniasis: deterministic model

First line treatment

Symptomatic Kala Azar

Second line treatment

Treatment failures

PKDL

PKDL treatment

t3

dHL

SH

RHD

IHP

fHS gHD

lH

gHP

fHRgHD

rHD

aHNH

rHC

p1t1

RHC

lF

sF

SA

IAD

RAD

IAP

RAC

lA

gAP

gAD

rAD

aANA

rAC

IHT2 IHL

RHL

p3t1

EF

IF

SF

aFNF

a

26%

10%

2%

50%

12%

PCR

D

AT

LST

-/-/-

+/-/-

+/+/-

-/+/-

-/-/+

RHT rHT

p4t2

IHS gHS IHD IHT1

Stauch A, Sarkar RR, Picado A, et al., 2011. Visceral leishmaniasis in the Indian subcontinent: modelling epidemiology and control. PLoS Neglected Tropical Diseases 5: e1405.

Model prediction: transmission threshold

0.000.020.040.060.080.100.120.140.160.18

0.1 1 10 100

Bloodmeals per person per dayP

ropo

rtion

hum

ans

infe

cted

• Model prediction: An endemic state is not expected if flies take on average less than about two bloodmeals per person and day.

• Prevalence data: about 12% of the population infected, with the majority of humans showing no symptoms (asymptomatic infections).

• Elimination by vector control: if the vector population (in this region) can be reduced by factor 3 to 4.

Limitations of the deterministic model: assumptions on infinite population size, homogenious mixing, etc., thus not (yet) considering factors like clusters (villages), seasonal transmission, stochastic extinction, etc.

Example 'macro'parasites: Onchocerciasis

Definite host

Vector

bloo

d-

mea

l blood- m

eal

• Helminths ('Worms': are countable) • Vector: Blackflies (Simulium) • 15 Mio. infected, 120 Mio. at risk • Worst outcome: blindness ('river blindness')

Wishes, demands & constrictions

Deterministic model

Stochastic model

understanding Biology of disease

predicting Intervention success

generality

realism

Parasite distributions often cannot be ade-quately characterized by the mean only:

Rel

. fre

quen

cy

Ingested mi per fly

Heterogeneities among parasites, vectors & hosts

Discrete & limited distributions

In a finite world, limitation is the rule:

Inge

sted

m

i per

fly

500

Stage before mi

Number of hosts with v premature (developing L4) and w mature worms (adult worms). v+: maximum no. of premature worms in the host w+: maximum no. of mature worms in the host

Attempt: stochastic with deterministic properties

Hv,w

u+: maximum no. of L3 that a bite can contain. We assume: umax< vmax

H0,0 H0,1 mH

sH sH

H1,0 sH

H1,0 sH

mW mW

mW mW

L4 mature to adult worms within 1 year

Hosts are born

sW sW

sW sW

Worms die

Hosts die

uwv,L

Parasites are acquired

at rate L, depending

on • u: the no. of L3 released per bite

• v: the no. of L4 in the host

• w: the no. of adult worms in the host

Acquisition, maturation & death of parasites

H0,0 H0,1 H0,2

H1,0

H2,0

H10,0

H1,1 H1,2

H2,1 H2,2

H10,1 H10,2

mH

1sW 2sW

1mW

2mW

10mW

H0,199

H1,199

H2,199

H10,199

199sW H0,200

H1,200

H2,200

H10,200

200sW

sH sH sH sH sH

sH sH sH sH sH

sH sH sH sH sH

sH sH sH sH sH

iwv,L

No. of adult worms [0…w+ =200]

No. of im

mature w

orms [0…

v+ =10]

w

v Assumption: Hosts cannot harbour more than 10 immature worms at the same time.

Acquisition & death of parasites: model

Hv,w sH

wsW (w+1)sW

uwv,L

uwv,L

Hosts die

Parasites die

Parasites acquired

Parasites mature

( )

( )

( ) 1,1

,

1

,0,,

1,,

1,

,

,

,

1

1

-+

-

-=

-

=

+

++

-

L+

L-

++

-

-

=

å

å

+

+

wvv

wvv

v

uvMaxi

ivwiwi

u

u

uwvwv

wvW

wvW

wvH

wv

HvHv

H

H

HwHw

Hdt

dH

m

m

s

s

s

What type of model is this?

Property Model type

Same initial conditions → same output Deterministic

Distributions are modelled, not means Stochastic

The model is based on transition rates, not on transition probabilities Deterministic

The model assumes infinite population size Rather deterministic

My suggestion: "Stochastically structured deterministic model"

Frequency of people with w adult worms

å+

=

=v

vwvw HH

0,

w

v

w adult worms per person

Rel

. fre

quen

cy

'Deterministic' redistribution of parasite numbers in the life-cycle

(here: stage before transmission: s1='w', s2='ms')

0 50 100 150 200

100

200

300

400

500Data-based Distribution of ms conditional on w

ms /

mg

skin

sni

p



Distribution of ms

Marginal distribution

Rel. frequency

ms /

mg

skin

sni

p

0 50 100 150 2000

100

200

300

400

500

2-dimensional distribution

Weighting with distribution of w

ms /

mg

skin

sni

p


'Deterministic' redistribution of parasite numbers in the life-cycle

(here: stage at transmission: s2='ms', s3='mi')

Prevalence of infected flies: 99%

0 100 200 300 400 500

100

200

300

400

500

Inge

sted

mi p

er fl

y

ms / mg skin snip

Data-based Distribution of mi conditional on ms

ms / mg skin snip

Rel

. fre

quen

cy

0 100 200 300 400 5000

100

200

300

400

500 2-dimensional distribution

Weighting with MF distribution

ms / mg skin snip

Inge

sted

mi p

er fl

y

Marginal distribution

Rel. frequency

Distribution of mi

Inge

sted

mi p

er fl

y

Fitting the model to data

0

1

10

100

1000

10000

1 100 10000 1000000ABR

ATP

Garms 1983Renz 1987Pedersen 1985Dietz 1982Model

020406080

100120140

1 10 100 1000 10000 100000ATP

MF

/ mg

ss

Basanez 2002 (Mexico)Basanez 2002 (Guatemala)Basanez 2002 (Cameroon)Basanez 2002 (BF, CI)Model

0

20

40

60

80

100

1 10 100 1000 10000ATP

MF

prev

alen

ce

Basanez 2002 (Mexico)Basanez 2002 (Guatemala)Basanez 2002 (Cameroon)Basanez 2002 (BF,CI)Renz 1987Model

0

1

10

100

1000

10000

1 100 10000 1000000ABR

ATP

Garms 1983Renz 1987Pedersen 1985Dietz 1982Model

0

20

40

60

80

100

0 20 40 60 80 100 120 140MF/mg ss

MF

prev

alen

ce

Basanez 2002 (Mexico)Basanez 2002 (Guatemala)Basanez 2002 (Cameroon)Basanez 2002 (BF,CI)Model

Duerr HP, Eichner M, 2010. Epidemiology and control of onchocerciasis: the threshold biting rate of savannah onchocerciasis in Africa. Int J Parasitol 40: 641-650.

Annual Biting Rate (ABR) [ ] year · host bloodmeals

Para

site

den

sity

[

]

host

pa

rasi

tes

infection cannot persist

infection can persist, elimination is possible

infection persists, elimination is difficult

Elimination thresholds for vector-borne diseases


Para

site

den

sity

[

]

host

pa

rasi

tes




Vector-related threshold: Threshold biting rate

intervention

If vector control does not reach the threshold biting

rate, the parasite population will re-establish at its pre-

control equilibrium

intervention

If vector control under-runs the threshold biting rate, the parasite popula-

tion dies out without further interventions


Para

site

den

sity

[

]

host

pa

rasi

tes




Parasite-related thresholds: breakpoints

intervention

intervention Below a breakpoint, the parasite population dies

out without further interventions

If the intervention does not reduce the parasite

population below a breakpoint, it will re-establish at its pre-control equilibrium

Vector-host contacts (bloodmeals/year)

0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Persistence graph


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Endemic parasite burden


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Breakpoints


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Endemic states

endemic


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Non-endemic states

endemic

Non-endemic


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Sub-critical transmission

endemic

Non-endemic Sub-critical


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Infection-free state

endemic

Non-endemic Sub-critical In

fect

ion-

free


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Threshold biting rate TBR

endemic

Non-endemic Sub-critical In

fect

ion-

free

TBR


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s TBR


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Two villages

TBR


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Village A (ABR=2000)

TBR A


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Village B (ABR=6000)

TBR A

B


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s TBR A

B


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

10 years control by CDTI

TBR A

B


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s TBR

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Years

CDTI

A

B


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s TBR

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Years

CDTI No control

A

B

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Years

CDTI No control

A

B

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Years

CDTI No control

B

A


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

B

A

Termination of CDTI

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Years

CDTI No control

B

A

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Years

CDTI No control

A

B

Elimination

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


0.1

1

0.5

0.2

10

5

3

1,000 10,000 3,000

Par

asite

den

sity

in h

uman

s

Years

CDTI No control

B

A Elimination

Relapse

Non-linearities & thresholds

• Non-linearities: – e.g. density-dependence: if the parasite induces an

immuno-suppression, host properties like the immunological responsiveness can depend on the number of parasites in the host. Such processes can feedback negatively or positively.

• Two types: – Limitation processes (negative feedback mechanisms)

limit chances to eliminate the parasite – Facilitation processes (positive feedback mechanisms)

facilitate changes to eliminate the parasite

Density-dependent regulation in a host-parasite relationship

nx+i

nx

No. of parasites

in the vector

n2

n4

No. of parasites

in the human host

n1

n5

n6 Limitation

(negative feedback)

Facilitation (positive feedback) No regulation n3

Number of parasites of stage x

Density-dependent processes modifying eradicability

limitation no regulation no regulation no regulation

s1→s2: s2→s3: s3→s4: s4→s1:

Adu

lt fe

mal

e

para

site

s pe

r hos

t

ABR

10

20

30

0

15

25

5

1000 2000 3000 0

limitation no regulation limitation no regulation

ABR 1000 2000 3000

limitation no regulation no regulation limitation

ABR 1000 2000 3000

limitation facilitation limitation limitation

ABR 1000 2000 3000

facilitation

Facilitation "facilitates" the eradicability of an infection, whereas limitation "limits" the prospects of elimination.

Interventions under chain limitation

s4

Definite host

Vector

s2

s1

s3

Life cycle stages s1, s2, s3, s4:

Chain limitation: Equilibrium before intervention Equilibrium after intervention

s4

s3 1 2 3 4 5 6

1 2 3 4 5 6

s2

s3

1 2 3 4 5 6

1 2 3 4 5 6

s2

s1 1 2 3 4 5 6

1 2 3 4 5 6

s1

s4 1 2 3 4 5 6

1 2 3 4 5 6

Conclusions • Deterministic models are not necessarily bad, or too simple. At least

in the case of microparasitic infections they may remain the better tool when 'understanding' stands in the foreground.

• Stochastic models or hybrid approaches are worth to be considered when macroparasitic infections are to be studied. Advantages when fitting the model to data can recompense higher modelling efforts.

• Two types of thresholds determine the eradicability of a vector-borne disease: transmission thresholds and breakpoints.

• Non-linearities and feedback mechanisms like density-dependence can substantially impact on the value of a threshold.

• 'Chain limitation' is a very potent mechanism to stabilize the persistence of pathogens in host populations

Hans-Peter Duerr Institute for Medical Biometry, University of Tübingen, Germany

http://www.uni-tuebingen.de/modeling/Mod_Duerr_Intro_en.html

Abstract Control and elimination of vector-borne diseases:

thresholds, breakpoints and strategies

Vector-borne diseases receive increased attention since terms like ‘neglected tropical diseases’ have been established. The two major strategies to control these infections are vector control and treatment of humans. Mathematical models to describe transmission and control of these infections must consider the role of the vectors as these act as ‘engine’ of transmission. The chances for successfully eliminating these infections depend largely on the roles of the vectors, and they can be substantially altered by non-linear or density-dependent functions linking the transmission between vectors and definite hosts, or, regulating the parasite’s persistence within these hosts. Two types of thresholds determine the elimination profile of a parasite: transmission thresholds (vector density below which the infection cannot persist) and breakpoints (parasite density below which the infection cannot persist).

These two measures can be best illustrated by persistence graphs showing the average number of parasites per human host (or other host populations) dependent on the average number of vectors per definite host (often called biting rate when flies are vectors). In this talk, concepts of persistence, control and elimination of parasites are illustrated together with the concepts of transmission thresholds and breakpoints in the examples of two indirectly transmitted diseases, i. e. the microparasitic infection visceral leishmaniasis (‘Kala Azar’) in the Indian subcontinent and the macroparasitic infection onchocerciasis (‘river blindness’) in Africa. Limitations of simple deterministic modeling approaches are discussed, and options for stochastic models approaching the master equation are shown.

International Conference on Mathematical and Theoretical Biology, Pune, 23-27 January, 2012

Control and elimination of vector-borne diseases ...€¦ · • Modelling vector-borne diseases:...

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