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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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http://lalashan.mcmaster.ca/theobio/mmed/index.php/IntroIDData

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)

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

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

Vector-borne pathogens

“Typical” natural history

Onset of symptoms

Onset of shedding

Incubation Clinical disease

Infectious periodLatent period

Infection


“Typical” natural history

Onset of symptoms

Onset of shedding



Infection

Onset of shedding

InfectiousLatent

DeathInfection

HOST

VECTOR


“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


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

A simple view of the worldVector-borne pathogens

^

not so

Exposed & Infected

Diseased

Infectivity < 1

Infectious

Onset of symptoms

Onset of shedding



Infection

HOST


Don’t worry about symptoms and disease!

^

not so

Exposed & Infected

Infectivity < 1

Infectious

Onset of shedding


Infection

HOST

H = infectivity to humans x per capita (vector) biting

rate


^

not so

Exposed & Infected

Infectivity < 1

Infectious

Onset of shedding


Infection

HOST


^

not so

Exposed & infected (not infectious)

Infectious

Recovered

Susceptible

HOST

V = infectivity to vectors x per capita (vector) biting

rate


^

not so

Exposed & Infected

Infectivity < 1

Infectious

Infectious period

Onset of shedding

InfectiousLatent

DeathInfection

VECTOR


^

not so

EH IH

RH

SH EVIV

SVVECTOR

HOST


^

not so

EH IH

RH

SH

VECTOR

EVIV

SV

HOST

€

σ H

€

σV

€

γH

€

νV

€

μV

€

μV€

μV

€

σ H ,σ V

€

γH

birth rate

per capita mortality rate


^

not so

€

νV per capita birth rate

per capita mortality rate

1/latent period

1/infectious period

€

νV

€

μV


^

not so

EH IH

RH

SH

VECTOR

EVIV

SV

HOST

€

σ H

€

σV

€

γH

€

νV

€

μV

€

μV€

μV

infectivity = proportion of susceptible individuals that become infected, given

exposure

per capita (vector) biting rate = bites by one individual vector per time unit


^

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


exposure

per capita (vector) biting rate = bites by one individual vector per unit time


^

not so


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


exposure

per capita (vector) biting rate = bites by one individual vector per unit time


^

not so


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


^

not so

EH IH

RH

SH

VECTOR

EVIV

SV

HOST

€

H IVNH

€

σ H

€

σV

€

γH

€

νV

€

μV

€

μV€

μV

€

V IHNH


^

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


^

not so

€

dSHdt

=−βSH IVNH

dEHdt

=βSH IVNH

−σEH

dIHdt

=σEH − γIH

dRHdt

= γIH

€

dSVdt


dEVdt

=βVSV IHNH

− μV +σ V( )EV

dIVdt


HOSTVECTOR

€

R0 = ?

A simple method for complex models


€

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)

A simple method for complex models


€

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


^

not so

€

dSHdt

=−βSH IVNH

dEHdt

=βSH IVNH

−σEH

dIHdt

=σEH − γIH

dRHdt

= γIH

€

dSVdt


dEVdt

=βVSV IHNH

− μV +σ V( )EV

dIVdt


For our system, we have

€

x1 = EH ,x2 = IH ,x3 = EV ,x4 = IV


^

not so

€

dSHdt

=−βSH IVNH

dEHdt

=βSH IVNH

−σEH

dIHdt

=σEH − γIH

dRHdt

= γIH

€

dSVdt


dEVdt

=βVSV IHNH

− μV +σ V( )EV

dIVdt


€

F =

0 0 0 β H0 0 0 0

0 βV 0 0

0 0 0 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥


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

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥


^

not so

€

dSHdt

=−βSH IVNH

dEHdt

=βSH IVNH

−σEH

dIHdt

=σEH − γIH

dRHdt

= γIH

€

dSVdt


dEVdt

=βVSV IHNH

− μV +σ V( )EV

dIVdt


€

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

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥


which gives

€

x1 = EH ,x2 = IH ,x3 = EV ,x4 = IV


^

not so

€

dSHdt

=−βSH IVNH

dEHdt

=βSH IVNH

−σEH

dIHdt

=σEH − γIH

dRHdt

= γIH

€

dSVdt


dEVdt

=βVSV IHNH

− μV +σ V( )EV

dIVdt


€

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

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥


“next generation matrix”

€

x1 = EH ,x2 = IH ,x3 = EV ,x4 = IV


^

not so

€

dSHdt

=−βSH IVNH

dEHdt

=βSH IVNH

−σEH

dIHdt

=σEH − γIH

dRHdt

= γIH

€

dSVdt


dEVdt

=βVSV IHNH

− μV +σ V( )EV

dIVdt


€

R0 = ρ FV −1( ) =

β HβVσ HμV +σ V( )

2γ H


and

€

x1 = EH ,x2 = IH ,x3 = EV ,x4 = IV

€

R02 =

β HβVσ HμV +σ V( )

2γ H

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