Post on 18-Oct-2020
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
w adult worms per person
w adult worms per person
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
w adult worms per person
'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
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
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
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
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
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
Endemic parasite burden
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
Breakpoints
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
Endemic states
endemic
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
Non-endemic states
endemic
Non-endemic
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
Sub-critical transmission
endemic
Non-endemic Sub-critical
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
Infection-free state
endemic
Non-endemic Sub-critical In
fect
ion-
free
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
Threshold biting rate TBR
endemic
Non-endemic Sub-critical In
fect
ion-
free
TBR
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 TBR
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
Two villages
TBR
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
Village A (ABR=2000)
TBR A
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
Village B (ABR=6000)
TBR A
B
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 TBR A
B
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
10 years control by CDTI
TBR A
B
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 TBR
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Years
CDTI
A
B
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 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
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
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
B
A
Termination of CDTI
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
B
A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
Years
CDTI No control
A
B
Elimination
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
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