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© 2010 Dr. Juliet Pulliam
Title: Dynamics of Vector-Borne Pathogens
Attribution: 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
(juliet.mmed.clinic@gmail.com).
Dynamics of
vector-borne pathogens
Dr. Juliet Pulliam
RAPIDD Program
Division of International Epidemiology
Fogarty International Center
National Institutes of Health (USA)
Topics in Biomedical Sciences
BSc Honours Course in Biomathematics
African Institute for the Mathematical Sciences
Muizenberg, South Africa
20 May 2010
Infectious diseases
Transmission
Mode of transmission
Direct transmission
Direct contact
Droplet spread
Indirect transmission
Airborne
Vehicle-borne (fomites)
Vector-borne (mechanical or biological)
Portal of entry
Portal of exit
Infectious diseases
Transmission
Mode of transmission
Direct transmission
Direct contact
Droplet spread
Indirect transmission
Airborne
Vehicle-borne (fomites)
Vector-borne (mechanical or biological)
Portal of entry
Portal of exit
Mosquitoes
Ticks
Sandflies
Tsetse flies
Reduviid bugs
Vector-borne pathogens
“Typical” natural history
Infection
Onset of symptoms
Incubation
Latent period
Clinical disease
Infectious period
Onset of shedding
Vector-borne pathogens
“Typical” natural history
HOST
Infection
VECTOR
Onset of symptoms
Infection
Incubation
Latent period
Death
Clinical disease
Infectious period
Onset of shedding
Latent
Infectious
Onset of shedding
Vector-borne pathogens
“Typical” natural history
Often 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
Vector-borne pathogens
Examples
Mosquitoes
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
not so
Vector-borne pathogens
A^simple view of the world
HOST
Infection
Onset of symptoms
Infectivity < 1
Exposed &
Infected
Incubation
Clinical disease
Infectious
Latent period
Infectious period
Diseased
Onset of shedding
not so
Vector-borne pathogens
A^simple view of the world
Don’t worry about
symptoms and disease!
Infectivity < 1
Infection
HOST
Exposed &
Infected
Latent period
Infectious period
Infectious
Onset of shedding
not so
Vector-borne pathogens
A^simple view of the world
H = infectivity to humans x
per capita (vector) biting
rate
Infectivity < 1
Infection
HOST
Exposed &
Infected
Latent period
Infectious period
Infectious
Onset of shedding
Vector-borne pathogens
not so
A^simple view of the world
HOST
Susceptible
Exposed &
infected (not
infectious)
Infectious
Recovered
not so
Vector-borne pathogens
A^simple view of the world
V = infectivity to vectors x
per capita (vector) biting
rate
Infectivity < 1
VECTOR
Infection
Death
Exposed &
Infected
Latent
InfectiousInfectious
period
Infectious
Onset of shedding
not so
Vector-borne pathogens
A^simple view of the world
VECTOR
SV
HOST
SH
IV
EH
IH
RH
EV
not so
Vector-borne pathogens
A^simple view of the world
VECTOR
SH
IV
V
EH
IH
H


H
RH

SV

HOST
EV
V

V

V
V
Vector-borne pathogens
not so
A^simple view of the world
V
birth rate
V
per capita mortality rate

V
per capita birth rate


per capita mortality rate
 H ,V
1/latent period
H
1/infectious period
not so
Vector-borne pathogens
A^simple view of the world
VECTOR
SH
IV
V
EH
IH
H


H
RH

SV

HOST
EV
V

V

V
V
not so
Vector-borne pathogens
A^simple view of the world
 = infectivity x per capita
contact rate
 = infectivity x per capita
(vector) biting rate
infectivity = proportion of susceptible
individuals that become infected, given
exposure
HOST
exposure = bite by IV
VECTOR
exposure = bite on IH
per capita (vector) biting rate = bites by one
individual vector per time unit
not so
Vector-borne pathogens
A^simple view of the world
HOST
 = infectivity x per capita
contact rate
 = infectivity x per capita
biting rate
infectivity = proportion of susceptible
individuals that become infected, given
exposure
per capita (vector) biting rate = bites by one
individual vector per unit time
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
not so
Vector-borne pathogens
A^simple view of the world
VECTOR
 = infectivity x per capita
contact rate
exposure = bites on IH
 = infectivity x per capita
biting rate
infectivity to vector = vector
infections produced per bite by SV on
IH
infectivity = proportion of susceptible
individuals that become infected, given
exposure
per capita (vector) biting rate = bites by one
individual vector per unit time
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
not so
Vector-borne pathogens
A^simple view of the world
VECTOR
V I H
SV
NH

HOST
SH
IV
V
 H IV

EV
V
NH
EH
IH
H


H
RH


V

V
V
not so
Vector-borne pathogens
A^simple view of the world
HOST
dS H S H IV

dt
NH
dE H S H IV

 E H
dt
NH
dI H
 E H  I H
dt
dR H
 I H
dt
VECTOR
dSV
V SV I H
  V  V SV 
dt
NH
dE V V SV I H

 V   V E V
dt
NH
dIV
  V E V  V   V IV
dt
not so
Vector-borne pathogens
A^simple view of the world
HOST
dS H S H IV

dt
NH
dE H S H
IV

 E H
dt
NH
dI H
 E H  I H
dt
dR H
 I H
dt
R0  ?
VECTOR
dSV
V SV I H
  V  V SV 
dt
NH
dE V V SV I H

 V   V E V
dt
NH
dIV
  V E V  V   V IV
dt
Vector-borne pathogens
A simple method for complex models
R0  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

dx i
 f i (x)  F i (x)  V i- (x)  V i+(x)
dt
where
F i (x)
V i- (x)
+ 
V i (x)
= 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
Vector-borne pathogens
A simple method for complex models
R0  FV
1

FV-1 = is the “next generation matrix”
F and V are then the square matrices defined by

F (x )
F   i 0 
 x j 
where
V i =V i-  V i+(x)


and
V (x )
V   i 0 
 x j 
not so
Vector-borne pathogens
A^simple view of the world
dS H S H IV

dt
NH
dE H S H IV

 E H
dt
NH
dI H
 E H  I H
dt
dR H
 I H
dt
For our system, we have
x1  E H , x2  I H , x3  EV , x4  IV

dSV
 S I
  V  V SV  V V H
dt
NH
dE V V SV I H

 V   V E V
dt
NH
dIV
  V E V  V   V IV
dt
not so
Vector-borne pathogens
A^simple view of the world
dS H S H IV

dt
NH
dE H S H IV

 E H
dt
NH
dI H
 E H  I H
dt
dR H
 I H
dt
For our system, we have
x1  E H , x2  I H , x3  EV , x4  IV
and we find

dSV
 S I
  V  V SV  V V H
dt
NH
dE V V SV I H

 V   V E V
dt
NH
dIV
  V E V  V   V IV
dt
0 0

0 0

F
0 V

0 0
 H

 H

V 
 0

 0
0
H
0
0
0  H 

0 0 
0 0 

0 0 
0
0
V   V
 V
0 

0 
0 

V   V 
not so
Vector-borne pathogens
A^simple view of the world
dS H S H IV

dt
NH
dE H S H IV

 E H
dt
NH
dI H
 E H  I H
dt
dR H
 I H
dt
For our system, we have
x1  E H , x2  I H , x3  EV , x4  IV
which gives

dSV
 S I
  V  V SV  V V H
dt
NH
dE V V SV I H

 V   V E V
dt
NH
dIV
  V E V  V   V IV
dt

 0

FV 1   0
V
 H

 0
0
0
V
H
0
 H V
2
V   V 
 H 
V   V 
0
0
0
0
0
0






not so
Vector-borne pathogens
A^simple view of the world
dS H S H IV

dt
NH
dE H S H IV

 E H
dt
NH
dI H
 E H  I H
dt
dR H
 I H
dt
For our system, we have
x1  E H , x2  I H , x3  EV , x4  IV
“next generation matrix”

dSV
 S I
  V  V SV  V V H
dt
NH
dE V V SV I H

 V   V E V
dt
NH

dIV
  V E V  V   V IV
dt
 0

0
FV 1   E H EV 

R0
 0
0
0
I H EV 
R
0
0
EV E H 
R
0
I V E H 
R
0
0
0
0
0
0
0





not so
Vector-borne pathogens
A^simple view of the world
dS H S H IV

dt
NH
dE H S H IV

 E H
dt
NH
dI H
 E H  I H
dt
dR H
 I H
dt
For our system, we have
x1  E H , x2  I H , x3  EV , x4  IV
and

dSV
 S I
  V  V SV  V V H
dt
NH
dE V V SV I H

 V   V
EV
dt
NH
dIV
  V E V  V   V IV
dt
 H V  H
R0  FV 
2
V   V   H
1
R0 
2
 H V  H
2
V   V   H