Sesi 5. Dynamic of Disease Transmission and Reproductive Rate
10/3/16
Dynamic of disease
transmission
Riris Andono Ahmad
1
10/3/16
Epidemic curve
Basic reproduction number
R0 = 3
2
10/3/16
Measles in Iceland
Measles in England and Wales
3
10/3/16
Immunity
1.
Measles?
2.
Influenza?
3.
Tuberculosis?
4.
HIV\aids?
5.
Malaria?
6.
Worm diseases?
The SIR-model with birth and death
SIR - model
birth and death
(population size: N)
a
µ
b
µ
g
nr. of new borns per year
= aN
nr. of deaths per year
= µ(S+I+R)=µN
µ
4
10/3/16
The SIR-model with birth and death
equations:
SIR - model
(population size: N)
(fraction susceptible: s)
a
µ
N =S+I+R
s =S/N
b
µ
S’ = a N – b I s – µ S
g
I’ = b I s – g I – µ I
µ
R’= g I – µ R
Reproductive Number, R0
A measure of the potential for transmission
The basic reproductive number, R0, the mean number of individuals directly
infected by an infectious case through the total infectious period, when
introduced to a susceptible population
probability of transmission per contact
R0 = p • c • d
duration of infectiousness
contacts per unit time
Infection will …..
disappear, if
become endemic, if
become epidemic, if
R1
(www)
5
10/3/16
Endemic - Epidemic - Pandemic
R > 1
R = 1
R < 1
Time
vEndemic
v Transmission occur, but the number of cases remains
constant
vEpidemic
v The number of cases increases
vPandemic
v When epidemics occur at several continents – global
epidemic
(www)
Immunity and R0
(www)
6
10/3/16
Reproductive Number, R0
Use in STI Control
R0 = p • c • d
p
c
condoms, acyclovir, zidovudine
D
case ascertainment (screening,
partner notification), treatment,
compliance, health seeking behaviour, accessibility of
health education, negotiating skills
services
(www)
What determines R0 ?
p, transmission probability per exposure – depends on the infection
v HIV, p(hand shake)=0, p(transfusion)=1, p(sex)=0.001
v interventions often aim at reducing p
v use gloves, screene blood, condoms
c, number of contacts per time unit – relevant contact depends on infection
v same room, within sneezing distance, skin contact,
v interventions often aim at reducing c
v Isolation, sexual abstinence
d, duration of infectious period
v may be reduced by medical interventions (TB, but not salmonella)
(www)
7
10/3/16
(Anderson &
May, 1991)
Critical vaccination level for eradication
infection
required level
malaria
99%
(P. falciparum, hyperendemic region)
measles
90 – 95%
rubella
82 – 87%
poliomyelitis
82 – 87%
diphteria
82 – 87%
scarlet fever
82 – 87%
smallpox
70 - 80%
SARS
67%
(Anderson & May, 1991)
(SARS: self)
8
10/3/16
Immunity – herd immunity
vIf R0 is the mean number of secondary cases in a susceptible population, then
R is the mean number of secondary cases in a population where a proportion, p,
are immune
R = R0 – (p • R0)
vWhat proportion needs to be immune to prevent epidemics?
If R0 is 2, then R < 1 if the proportion of immune, p, is > 0.50
If R0 is 4, then R < 1 if the proportion of immune, p, is > 0.75
vIf the mean number of secondary cases should be < 1, then
R0 – (p • R0) < 1
p > (R0 – 1)/ R0 = 1 – 1/ R0
v If R0 =15, how large will p need to be to avoid an epidemic?
p > 1-1/15 = 0.94
vThe higher R0, the higher proportion of immune required for herd immunity
However ...
1.
No heterogeneity ?? ...
2.
100% vaccine efficacy ?? ...
3.
Time to establish eradication ...
– childhood diseases
– adulthood diseases
4.
Tuberculosis ...
– role of BCG?
– BCG efficacy decreases with age?
– even if BCG would be effective: time scale?
9
10/3/16
Herd immunity
• a type of immunity that occurs when the
vaccination of a portion of the population (or
herd) provides protection to unvaccinated
individuals.
• If a large percent of the population is immune, the
entire population is likely to be protected, not just
those who are immune.
Populations are heterogeneous ...
10
10/3/16
Why do we have to think about
heterogeneity?
Measles outbreak (almost 3000 cases) despite coverage of 96%
Host heterogeneity
• Disease independent (can be measured also for
non-infected individuals):
• Age, sex, other demographic variables
• Behaviour (e.g. number of contacts, compliance with
vaccination)
• Disease dependent (only for infected individuals):
• Transmission route
• Disease stage; primary versus secondary infection
• Clininal symptoms or asymptomatic
11
10/3/16
Pathogen heterogeneity
• Heterogeneity between strains:
• Virulence (defined as host mortality or severity of
disease)
• Vulnarability to host immune response
• Competition via cross-immunity
• Within host heterogeneity:
• Immunogenic variability (HIV)
• Different location within host leads to different effects
(invasive infection versus carrier)
12
Dynamic of disease
transmission
Riris Andono Ahmad
1
10/3/16
Epidemic curve
Basic reproduction number
R0 = 3
2
10/3/16
Measles in Iceland
Measles in England and Wales
3
10/3/16
Immunity
1.
Measles?
2.
Influenza?
3.
Tuberculosis?
4.
HIV\aids?
5.
Malaria?
6.
Worm diseases?
The SIR-model with birth and death
SIR - model
birth and death
(population size: N)
a
µ
b
µ
g
nr. of new borns per year
= aN
nr. of deaths per year
= µ(S+I+R)=µN
µ
4
10/3/16
The SIR-model with birth and death
equations:
SIR - model
(population size: N)
(fraction susceptible: s)
a
µ
N =S+I+R
s =S/N
b
µ
S’ = a N – b I s – µ S
g
I’ = b I s – g I – µ I
µ
R’= g I – µ R
Reproductive Number, R0
A measure of the potential for transmission
The basic reproductive number, R0, the mean number of individuals directly
infected by an infectious case through the total infectious period, when
introduced to a susceptible population
probability of transmission per contact
R0 = p • c • d
duration of infectiousness
contacts per unit time
Infection will …..
disappear, if
become endemic, if
become epidemic, if
R1
(www)
5
10/3/16
Endemic - Epidemic - Pandemic
R > 1
R = 1
R < 1
Time
vEndemic
v Transmission occur, but the number of cases remains
constant
vEpidemic
v The number of cases increases
vPandemic
v When epidemics occur at several continents – global
epidemic
(www)
Immunity and R0
(www)
6
10/3/16
Reproductive Number, R0
Use in STI Control
R0 = p • c • d
p
c
condoms, acyclovir, zidovudine
D
case ascertainment (screening,
partner notification), treatment,
compliance, health seeking behaviour, accessibility of
health education, negotiating skills
services
(www)
What determines R0 ?
p, transmission probability per exposure – depends on the infection
v HIV, p(hand shake)=0, p(transfusion)=1, p(sex)=0.001
v interventions often aim at reducing p
v use gloves, screene blood, condoms
c, number of contacts per time unit – relevant contact depends on infection
v same room, within sneezing distance, skin contact,
v interventions often aim at reducing c
v Isolation, sexual abstinence
d, duration of infectious period
v may be reduced by medical interventions (TB, but not salmonella)
(www)
7
10/3/16
(Anderson &
May, 1991)
Critical vaccination level for eradication
infection
required level
malaria
99%
(P. falciparum, hyperendemic region)
measles
90 – 95%
rubella
82 – 87%
poliomyelitis
82 – 87%
diphteria
82 – 87%
scarlet fever
82 – 87%
smallpox
70 - 80%
SARS
67%
(Anderson & May, 1991)
(SARS: self)
8
10/3/16
Immunity – herd immunity
vIf R0 is the mean number of secondary cases in a susceptible population, then
R is the mean number of secondary cases in a population where a proportion, p,
are immune
R = R0 – (p • R0)
vWhat proportion needs to be immune to prevent epidemics?
If R0 is 2, then R < 1 if the proportion of immune, p, is > 0.50
If R0 is 4, then R < 1 if the proportion of immune, p, is > 0.75
vIf the mean number of secondary cases should be < 1, then
R0 – (p • R0) < 1
p > (R0 – 1)/ R0 = 1 – 1/ R0
v If R0 =15, how large will p need to be to avoid an epidemic?
p > 1-1/15 = 0.94
vThe higher R0, the higher proportion of immune required for herd immunity
However ...
1.
No heterogeneity ?? ...
2.
100% vaccine efficacy ?? ...
3.
Time to establish eradication ...
– childhood diseases
– adulthood diseases
4.
Tuberculosis ...
– role of BCG?
– BCG efficacy decreases with age?
– even if BCG would be effective: time scale?
9
10/3/16
Herd immunity
• a type of immunity that occurs when the
vaccination of a portion of the population (or
herd) provides protection to unvaccinated
individuals.
• If a large percent of the population is immune, the
entire population is likely to be protected, not just
those who are immune.
Populations are heterogeneous ...
10
10/3/16
Why do we have to think about
heterogeneity?
Measles outbreak (almost 3000 cases) despite coverage of 96%
Host heterogeneity
• Disease independent (can be measured also for
non-infected individuals):
• Age, sex, other demographic variables
• Behaviour (e.g. number of contacts, compliance with
vaccination)
• Disease dependent (only for infected individuals):
• Transmission route
• Disease stage; primary versus secondary infection
• Clininal symptoms or asymptomatic
11
10/3/16
Pathogen heterogeneity
• Heterogeneity between strains:
• Virulence (defined as host mortality or severity of
disease)
• Vulnarability to host immune response
• Competition via cross-immunity
• Within host heterogeneity:
• Immunogenic variability (HIV)
• Different location within host leads to different effects
(invasive infection versus carrier)
12