Transcript SIR Model with Reed Frost assumption Rate of infection of

Population dynamics of
infectious diseases
Arjan Stegeman
Introduction to the population
dynamics of infectious diseases
• Getting familiar with the basic models
• Relation between characteristics of the
model and the transmission of pathogens
Modelling population dynamics
of infectious diseases
• model : simplified representation of
reality
• mathematical: using symbols and
methods to manipulate these symbols
Why mathematical modeling ?
Factors affecting infection have a nonlinear dependence
• Insight in the importance of factors that
affect the spread of infectious agents
• Provide testable hypotheses
• Extrapolation to other situations/times
SIR Models
Population consists of:
Susceptible
Infectious
individuals
Recovered
SIR Models
• Dynamic model :
S, I and R are variables (entities that
change) that change with time,
parameters (constants) determine how
the variables change
Greenwood assumption
• Constant probability of infection (Force
of infection)
SIR Model with Greenwood assumption
*I
IR*S
S
dS
  IR * S
dt
I
dI
 IR * S   * I
dt
R
dR
 *I
dt
IR = Incidence rate
 = recovery rate parameter (1/infectious period)
Transition matrix Markov chain
To
S
I
R
S PSS PSI PSR
From I
PIS
PII
PIR
R PRS PRI PRR
P = probability to go from a state at time t to a state at time t+1
Markov chain modeling
Starting vector*
S
 
I
R
 
To
S
I
R
S PSS PSI PSR
From I
PIS
PII
PIR
R PRS PRI PRR
* number of S, I and R at the start of the modeling
Example Markov chain modeling
Starting vector*
 99 
 
1
0
 
To
S
I
R
S 0.90 0.10 0.00
From I
0.00 0.80 0.20
R 0.00 0.00 1.00
* number of S, I and R at the start of the modeling
Results of Markov chain model
Time step S
I
R
0
99
1
0
1
=99*0.9+1*0+0*0=
=99*0.1+1*0.8+0*0= 99*0+1*0.2+0*1=
89.1
10.7
0.2
Example Markov chain modeling
Starting vector *
 89.1


10.7 
 0 .2 


To
S
I
R
S 0.90 0.10 0.00
From I
0.00 0.80 0.20
R 0.00 0.00 1.00
* number of S, I and R at the end of time step 1
Results of Markov chain model
Time step S
I
R
0
99
1
0
1
99*0.9+1*0+0*0=
99*0.1+1*0.8+0*0=
99*0+1*0.2+0*1=
89.1
10.7
0.2
2
89.1*0.9+10.7*0+0.2 89.1*0.1+10.7*0.9+
*0=80.2
0.2*0=17.5
89.1*0+10.7*0.2+0.2
*1=2.3
Course of number of S, I and R animals
in a closed population (Greenwood
assumption)
Number of animals
100
80
60
40
20
0
0
5
10
15
20
25
30
Number of time steps
S
I
R
35
40
45
50
Drawback of the Greenwood
assumption
• Number of infectious individuals has no
influence on the rate of transmission
SIR model with Reed Frost
assumption
• Probability of infection upon contact (p)
p
• Contacts are with rate e per unit of time
• Contacts are at random with other
individuals (mass action assumption), thus
probability that an S makes contact with an
I equals I/N
SIR Model with Reed Frost assumption
Rate of infection of susceptibles depends on the number of
infectious individuals
S
SI/N
I
I
R
 = infection rate parameter (Number of new
infections per infectious individual per unit of time)
 = recovery rate parameter (1/infectious period)
N = total number of individuals (mass action)
SIR Model with Reed Frost
assumption
I t 1  St (1  q )
It
It+1= number of new infectious
individuals at t+1
q = probability to escape from
infection
• (formulation in text books, pseudomass action)
I t 1  S t (1  e
 I t / N
)
• (formulation according to mass action)
Example: Classical Swine Fever virus
transmission among sows housed in crates
•  = 0.29; Susceptible has a probability of:
I
(1  e

N
)
to become infected in one time step
•  = 0.10; Infectious has a probability of
(1  e

)
to recover in one time step
Course of number of S, I and R animals
in a closed population (reed Frost
assumption with mass action)
80
60
40
20
0
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
10
0
Number of animals
100
Number of time steps
S
I
R
Deterministic - Stochastic
• Deterministic models: all variables have
at each moment in time for a particular
set of parameter values only one value
• Stochastic models: stochastic variables
are used which at each moment in time
can have many different values each with
its own probability
Course of number of S, I and R animals
in a closed population (reed Frost
assumption with mass action)
80
60
40
20
0
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
10
0
Number of animals
100
Number of time steps
S
I
R
Course of number of S, I and R animals
in a closed population (1 run stochastic
SIR model)
Number of animals
100
80
60
40
20
0
0
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Number of time steps
S
I
R
Course of number of S, I and R animals
in a closed population (1 run stochastic
SIR model)
Number of animals
100
80
60
40
20
0
0
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Number of time steps
S
I
R
Stochastic models
• Preferred above deterministic models
because they show variability in
outcomes that is also present in the real
world. This is especially important in the
veterinary field, because we often work
with populations of limited size.
Transmission between
individuals
/ = Basic Reproduction ratio, R0
Average number of secondary cases caused
by 1 infectious individual during its entire
infectious period in a fully susceptible
population
Reproduction ratio, R0
R0 = 3
R0 = 0.5
Stochastic threshold theorem
prob major
The probability of a major outbreak
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
Prob major = 1 - 1/R0
0
1
2
3
4
5
R0
6
7
8
9
10
Final size distribution for R0 = 0.5
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
probability
infection fades out after infection of 1 or a
few individuals (minor outbreaks only)
final size
Final size distribution for R0 = 3
R0 > 1 : infection may spread extensively
(major outbreaks and minor outbreaks)
0,3
0,2
0,15
0,1
0,05
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
probability
0,25
final size
Final size
Deterministic threshold theorem:
Final size as function of R0
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
 ln( 1  p)
R
p
0
0,5
1
1,5
2
2,5
3
3,5
Reproduction ratio
4
4,5
5
5,5
6
Transmission in an open
population
m*N
I
S
*S*I/N
m*S
R
*I
m*I
 = infection rate parameter
 = recovery rate parameter
m = replacement rate parameter
m*R
Courses of infection in an open
population
2: Major outbreak (R0 > 1)
I
3: Endemic infection (R0 > 1)
1: Minor outbreak (R0 < 1 of R0 > 1)
S
Infection can become endemic when the
number of animals in a herd is at least:
m
R0
N
*(
)
m
R0  1
M. paratuberculosis:  = 0.003; m = 0.0009; R0 = 10
Nmin = 5
BHV1 :  = 0.07; m = 0.0009, R0 = 3.5
Nmin = 110
Transmission in an open
population
R0 

 m
At endemic equilibrium (large population)
N
R0 
S
Assumptions
• Mass action (transmission rate depends
on densities)
• Random mixing
• All S or I individuals are equal
(homogeneous)
SIR model can be adapted to:
•
•
•
•
•
SI model
SIS model
SIRS model
SLIR model
etc.
Population dynamics of
infectious diseases
• Interaction between agent - host &
contact structure between hosts
determine the transmission
• Quantitative approach: R0 plays the
central role