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This presentation is made available through a Creative Commons Attribution-Noncommercial license. Details of the license and permitted uses are available at
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© 2010 Dr. Juliet Pulliam
Title: Dynamics of Vector-Borne PathogensAttribution: Dr. Juliet Pulliam, Topics in Biomedical Sciences
Source URL: http://lalashan.mcmaster.ca/theobio/mmed/index.php/Honours Course
For further information please contact Dr. Juliet Pulliam ([email protected] ).
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Dynamics of vector-borne pathogens
Topics in Biomedical Sciences
BSc Honours Course in Biomathematics
African Institute for the Mathematical Sciences
Muizenberg, South Africa
20 May 2010
Dr. Juliet Pulliam
RAPIDD Program
Division of International Epidemiology
Fogarty International Center
National Institutes of Health (USA)
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TransmissionInfectious diseases
Mode of transmissionDirect transmission
Direct contact
Droplet spread
Indirect transmissionAirborne
Vehicle-borne (fomites)
Vector-borne (mechanical or biological)
Portal of entry
Portal of exit
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TransmissionInfectious diseases
Mode of transmissionDirect transmission
Direct contact
Droplet spread
Indirect transmissionAirborne
Vehicle-borne (fomites)
Vector-borne (mechanical or biological)
Portal of entry
Portal of exit
MosquitoesTicks
SandfliesTsetse flies
Reduviid bugs
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Vector-borne pathogens
“Typical” natural history
Onset of symptoms
Onset of shedding
Incubation Clinical disease
Infectious periodLatent period
Infection
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Vector-borne pathogens
“Typical” natural history
Onset of symptoms
Onset of shedding
Incubation Clinical disease
Infectious periodLatent period
Infection
Onset of shedding
InfectiousLatent
DeathInfection
HOST
VECTOR
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Vector-borne pathogens
“Typical” natural historyOften acute:
timecourse of infection << normal lifespan of host
BUT
timecourse of infection ~ normal lifespan of vector
Sometimes immunizing:
infection may stimulate antibody production,
preventing future infection…
or may not…
or somewhere in between
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Vector-borne pathogens
ExamplesMosquitoes
Anopheles spp., malaria vectors
Culex spp., West Nile vectors
Other biting flies
Phlebotomus papatasi, Leishmania vector
Glossina spp., African trypanosomiasis vectors
True bugs
Triatoma infestans, Chagas vector
Ticks
Amblyomma spp., heartwater vectors
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A simple view of the worldVector-borne pathogens
^
not so
Exposed & Infected
Diseased
Infectivity < 1
Infectious
Onset of symptoms
Onset of shedding
Incubation Clinical disease
Infectious periodLatent period
Infection
HOST
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A simple view of the worldVector-borne pathogens
Don’t worry about symptoms and disease!
^
not so
Exposed & Infected
Infectivity < 1
Infectious
Onset of shedding
Infectious periodLatent period
Infection
HOST
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H = infectivity to humans x per capita (vector) biting
rate
A simple view of the worldVector-borne pathogens
^
not so
Exposed & Infected
Infectivity < 1
Infectious
Onset of shedding
Infectious periodLatent period
Infection
HOST
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A simple view of the worldVector-borne pathogens
^
not so
Exposed & infected (not infectious)
Infectious
Recovered
Susceptible
HOST
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V = infectivity to vectors x per capita (vector) biting
rate
A simple view of the worldVector-borne pathogens
^
not so
Exposed & Infected
Infectivity < 1
Infectious
Infectious period
Onset of shedding
InfectiousLatent
DeathInfection
VECTOR
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A simple view of the worldVector-borne pathogens
^
not so
EH IH
RH
SH EVIV
SVVECTOR
HOST
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A simple view of the worldVector-borne pathogens
^
not so
EH IH
RH
SH
VECTOR
EVIV
SV
HOST
€
σ H
€
σV
€
γH
€
νV
€
μV
€
μV€
μV
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€
σ H ,σ V
€
γH
birth rate
per capita mortality rate
A simple view of the worldVector-borne pathogens
^
not so
€
νV per capita birth rate
per capita mortality rate
1/latent period
1/infectious period
€
νV
€
μV
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A simple view of the worldVector-borne pathogens
^
not so
EH IH
RH
SH
VECTOR
EVIV
SV
HOST
€
σ H
€
σV
€
γH
€
νV
€
μV
€
μV€
μV
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infectivity = proportion of susceptible individuals that become infected, given
exposure
per capita (vector) biting rate = bites by one individual vector per time unit
A simple view of the worldVector-borne pathogens
^
not so
= infectivity x per capita contact rate
exposure = bite by IV
HOST = infectivity x per capita
(vector) biting rate
VECTOR
exposure = bite on IH
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infectivity = proportion of susceptible individuals that become infected, given
exposure
per capita (vector) biting rate = bites by one individual vector per unit time
A simple view of the worldVector-borne pathogens
^
not so
= infectivity x per capita contact rate
exposure = bite by IV
infectivity to host = host infections produced per bite by IV on SH
H = bites (potentially infectious to host) by one individual vector per unit
time
HIV = bites (potentially infectious to host) per unit time
HIV/NH = bites (potentially infectious to host) per host per unit time
HSHIV/NH = infectious bites per unit time
HOST
= infectivity x per capita biting rate
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infectivity = proportion of susceptible individuals that become infected, given
exposure
per capita (vector) biting rate = bites by one individual vector per unit time
A simple view of the worldVector-borne pathogens
^
not so
= infectivity x per capita contact rate
exposure = bites on IH
infectivity to vector = vector infections produced per bite by SV on
IH
V = bites (potentially infectious to vector) by one individual vector per
unit time
VSV = bites (potentially infectious to vector) per unit time
VSV/NH = bites (potentially infectious to vector) per host per unit time
VSVIH/NH = infectious bites per unit time
VECTOR
= infectivity x per capita biting rate
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A simple view of the worldVector-borne pathogens
^
not so
EH IH
RH
SH
VECTOR
EVIV
SV
HOST
€
H IVNH
€
σ H
€
σV
€
γH
€
νV
€
μV
€
μV€
μV
€
V IHNH
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A simple view of the worldVector-borne pathogens
^
not so
€
dSHdt
=−βSH IVNH
dEHdt
=βSH IVNH
−σEH
dIHdt
=σEH − γIH
dRHdt
= γIH
€
dSVdt
= ν V −μVSV −βVSV IHNH
dEVdt
=βVSV IHNH
− μV +σ V( )EV
dIVdt
=σ V EV − μV + γV( )IV
HOSTVECTOR
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A simple view of the worldVector-borne pathogens
^
not so
€
dSHdt
=−βSH IVNH
dEHdt
=βSH IVNH
−σEH
dIHdt
=σEH − γIH
dRHdt
= γIH
€
dSVdt
= ν V −μVSV −βVSV IHNH
dEVdt
=βVSV IHNH
− μV +σ V( )EV
dIVdt
=σ V EV − μV + γV( )IV
HOSTVECTOR
€
R0 = ?
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A simple method for complex models
Vector-borne pathogens
€
R0 = ρ FV −1( )
FV-1 = is the “next generation matrix”
For all compartments xi containing infected individuals (ie, EH , IH, EV, IV), the time derivative can be rewritten as
where
= the rate of appearance of new infections in compartment xi
= the rate of transfer out of compartment xi
= the rate of transfer of individuals into compartment xi, other than new infections
€
dx idt
= f i(x) = F i(x) − Vi
- (x) + Vi
+(x)
€
F i(x)
€
Vi
- (x)
€
Vi
+(x)
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A simple method for complex models
Vector-borne pathogens
€
R0 = ρ FV −1( )
FV-1 = is the “next generation matrix”
F and V are then the square matrices defined by
where
€
F =∂F i(x0)
∂x j
⎡
⎣ ⎢
⎤
⎦ ⎥
€
Vi
=Vi
- − Vi
+(x)
€
V =∂V i(x0)
∂x j
⎡
⎣ ⎢
⎤
⎦ ⎥and
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A simple view of the worldVector-borne pathogens
^
not so
€
dSHdt
=−βSH IVNH
dEHdt
=βSH IVNH
−σEH
dIHdt
=σEH − γIH
dRHdt
= γIH
€
dSVdt
= ν V −μVSV −βVSV IHNH
dEVdt
=βVSV IHNH
− μV +σ V( )EV
dIVdt
=σ V EV − μV + γV( )IV
For our system, we have
€
x1 = EH ,x2 = IH ,x3 = EV ,x4 = IV
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A simple view of the worldVector-borne pathogens
^
not so
€
dSHdt
=−βSH IVNH
dEHdt
=βSH IVNH
−σEH
dIHdt
=σEH − γIH
dRHdt
= γIH
€
dSVdt
= ν V −μVSV −βVSV IHNH
dEVdt
=βVSV IHNH
− μV +σ V( )EV
dIVdt
=σ V EV − μV + γV( )IV
€
F =
0 0 0 β H0 0 0 0
0 βV 0 0
0 0 0 0
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
For our system, we have
and we find
€
x1 = EH ,x2 = IH ,x3 = EV ,x4 = IV
€
V =
σ H 0 0 0
−σ H γ H 0 0
0 0 μV +σ V 0
0 0 −σ V μV +σ V
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
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A simple view of the worldVector-borne pathogens
^
not so
€
dSHdt
=−βSH IVNH
dEHdt
=βSH IVNH
−σEH
dIHdt
=σEH − γIH
dRHdt
= γIH
€
dSVdt
= ν V −μVSV −βVSV IHNH
dEVdt
=βVSV IHNH
− μV +σ V( )EV
dIVdt
=σ V EV − μV + γV( )IV
€
FV −1 =
0 0β Hσ VμV +σ V( )
2
β HμV +σ V
0 0 0 0βVγ H
βVγ H
0 0
0 0 0 0
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
For our system, we have
which gives
€
x1 = EH ,x2 = IH ,x3 = EV ,x4 = IV
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A simple view of the worldVector-borne pathogens
^
not so
€
dSHdt
=−βSH IVNH
dEHdt
=βSH IVNH
−σEH
dIHdt
=σEH − γIH
dRHdt
= γIH
€
dSVdt
= ν V −μVSV −βVSV IHNH
dEVdt
=βVSV IHNH
− μV +σ V( )EV
dIVdt
=σ V EV − μV + γV( )IV
€
FV −1 =
0 00
EV →EH( )
R 0
IV →EH( )
R0 0 0 0
0
EH →EV( )
R 0
I H →EV( )
R 0 0
0 0 0 0
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
For our system, we have
“next generation matrix”
€
x1 = EH ,x2 = IH ,x3 = EV ,x4 = IV
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A simple view of the worldVector-borne pathogens
^
not so
€
dSHdt
=−βSH IVNH
dEHdt
=βSH IVNH
−σEH
dIHdt
=σEH − γIH
dRHdt
= γIH
€
dSVdt
= ν V −μVSV −βVSV IHNH
dEVdt
=βVSV IHNH
− μV +σ V( )EV
dIVdt
=σ V EV − μV + γV( )IV
€
R0 = ρ FV −1( ) =
β HβVσ HμV +σ V( )
2γ H
For our system, we have
and
€
x1 = EH ,x2 = IH ,x3 = EV ,x4 = IV
€
R02 =
β HβVσ HμV +σ V( )
2γ H