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© 2010 Dr. Juliet Pulliam
Title: Dynamics of Directly-Transmitted 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
directly-transmitted 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
18 May 2009
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
Sexual contact
(STD’s)
Casual contact
(OID’s)
Directly-transmitted pathogens
“Typical” natural history
Infection Onset of symptoms
Incubation
Clinical disease
Latent period
Infectious period
Onset of shedding
Directly-transmitted pathogens
“Typical” natural history
Infection Onset of symptoms
Acute:
timecourse of infection <<
normal lifespan of host
Incubation
Clinical disease
Immunizing:
Latent period
Infectious period
Onset of shedding
infection stimulates
antibody production,
preventing future infection
Directly-transmitted pathogens
Examples
Whooping cough
Chicken pox
Measles
Smallpox
Various food-borne
illnesses
very!
Directly-transmitted pathogens
A^simple view of the world
Infection
Infectivity = 1
Infected=Infectious
Onset of shedding
Don’t worry about
symptoms and disease!
Directly-transmitted pathogens
A simple view of the world
Infection Onset of symptoms
Infectivity = 1
Incubation
Clinical disease
Latent period
Infectious period
Infected
Infectious =
Diseased
Onset of shedding
more
Directly-transmitted pathogens
A^realistic view of the world
Infection Onset of symptoms
Infectivity < 1
Incubation
Clinical disease
Latent period
Infectious period
Infected
Infectious =
Diseased
Onset of shedding
more
Directly-transmitted pathogens
A^realistic view of the world
Infection Onset of symptoms
 = infectivity x per capita
contact rate
Infectivity < 1
Incubation
Clinical disease
Latent period
Infectious period
Infected
Infectious =
Diseased
Onset of shedding
more
Directly-transmitted pathogens
A^realistic view of the world
Infected
Infectious =
(not infectious)
Diseased
more
Directly-transmitted pathogens
A^realistic view of the world
Susceptible
Infected
Infectious =
(not infectious)
Diseased
Recovered
more
Directly-transmitted pathogens
A^realistic view of the world
I
Susceptible


Infected
Infectious =
(not infectious)
Diseased


Recovered
Directly-transmitted pathogens
nearly
A^realistic view of the world
I
Susceptible


Infected
Infectious =
(not infectious)
Diseased


Recovered

Directly-transmitted pathogens
nearly
A^realistic view of the world
I
Susceptible



Infected
Infectious =
(not infectious)
Diseased



Recovered

Directly-transmitted pathogens

nearly
A^realistic view of the world
I
S



E
exposed &
infected


I

R

Directly-transmitted pathogens
nearly
A^realistic view of the world
dS
   S  SI
dt
dE
 SI  E  E
dt

dI
 E  I  I
dt

dR
 I  R 
dt


mortality rate

1/latent period

1/infectious period

transmission coefficient
birth rate
R
Directly-transmitted pathogens
nearly
A^realistic view of the world
dS
   S  SI
dt
dE
 SI  E  E
dt
dI
 E  I  I
dt
dR

 I  R
dt
Assume constant
population size
  N
R
Directly-transmitted pathogens
nearly
A^realistic view of the world
dS
   S  SI
dt
dE
 SI  E  E
dt
dI
 E  I  I 
dt
dR
 I  R
dt
R0  ?
Directly-transmitted pathogens
nearly
A^realistic view of the world
dS
   S  SI
dt
dE
 SI  E  E
dt

dI
 E  I  I
dt
dR
 I  R
dt
R0 
Rate at which an infected
individual produces new
infections in a naïve population
X
Proportion of new infections
that become infectious
X
Average duration of
infectiousness
Directly-transmitted pathogens
nearly
A^realistic view of the world
dS
   S  SI
dt
dE
 SI  E  E
   1 
dt
R0  N 



dI

     

 E  I  I
dt
dR
 I  R
dt

Directly-transmitted pathogens
nearly
A^realistic view of the world
dS
   S  SI
dt
dE
 SI  E  E
dt
dI
 E  I  I
dt
dR
 I  R
dt
Equilibria…
Disease free
equilibrium
Endemic equilibrium
Directly-transmitted pathogens
nearly
A^realistic view of measles
dS
   S  SI
dt
dE
 SI  E  E
dt

dI
 E  I  I
dt

dR
 I  R 
dt
  N
individuals/year
  0.02
years-1
 1/8
days-1
 1/5
days-1

(unknown)
Directly-transmitted pathogens
nearly
From: Grenfell & Harwood 1997 Trends in Ecology and Evolution
A^realistic view of measles
Directly-transmitted pathogens
nearly
From: Grenfell & Harwood 1997 Trends in Ecology and Evolution
A^realistic view of measles
   1 
R0  N












~ 500,000
~ 7,200
~ 300,000
Directly-transmitted pathogens
nearly
A^realistic view of measles
From: Anderson & May 1982 Science
Directly-transmitted pathogens
nearly
A^realistic view of measles
N = 7,200


N = 300,000
N = 500,000
R0  13
R0  13
  3.6 104
  5.2 106
E * I*  4.8
E * I*  330
R0  18
R0  18
  5.0 104
5
  1.2 10
E * I*  4.8
E * I*  200

Directly-transmitted pathogens
nearly
From: Grenfell & Harwood 1997 Trends in Ecology and Evolution
A^realistic view of measles
Critical community
size
Directly-transmitted pathogens
A realistic view of measles
dS
   S  SI
dt
dE
 SI  E  E
dt

dI
 E  I  I
dt

dR
 I  R 
dt
  N
individuals/year
  0.02
years-1
 1/8
days-1
 1/5
days-1

seasonal (school
terms)
Directly-transmitted pathogens
A realistic view of measles
From: Earn et al. 2000 Science
Directly-transmitted pathogens
A realistic view of measles
From: Earn et al. 2000 Science
Directly-transmitted pathogens
A realistic view of measles
From: Earn et al. 2000 Science