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 non-linear 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 Recovered

individuals

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

S I R

IR*S *I

IR = Incidence rate

= recovery rate parameter (1/infectious period)

Idt

dR*ISIR

dt

dI** SIR

dt

dS*

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

To

S I R

S PSS PSI PSR

From I PIS PII PIR

R PRS PRI PRR

R

I

S

Starting vector*

* number of S, I and R at the start of the modeling

Example Markov chain modeling

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

0

1

99

Starting vector*

* 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=

89.1=99*0.1+1*0.8+0*0=

10.799*0+1*0.2+0*1=

0.2

Example Markov chain modeling

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

2.0

7.10

1.89

Starting vector *

* 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=

89.199*0.1+1*0.8+0*0=10.7

99*0+1*0.2+0*1=

0.2

2 89.1*0.9+10.7*0+0.2

*0=80.289.1*0.1+10.7*0.9+

0.2*0=17.589.1*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)

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40 45 50

Number of time steps

Num

ber

of a

nim

als

S I R

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)

• Contacts are with rate 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

p

SIR Model with Reed Frost assumption

Rate of infection of susceptibles depends on the number of infectious individuals

SI/N I

= 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)

S I R

SIR Model with Reed Frost assumption

• (formulation in text books, pseudomass action)

• (formulation according to mass action)

)1(1tI

tt qSI

)1( /1

NItt

teSI

It+1= number of new infectious individuals at t+1

q = probability to escape from infection

Example: Classical Swine Fever virus transmission among sows housed in crates

= 0.29; Susceptible has a probability of:

to become infected in one time step = 0.10; Infectious has a probability of

to recover in one time step

)1( N

I

e

)1( e

Course of number of S, I and R animals in a closed population (reed Frost

assumption with mass action)

0

20

40

60

80

100

Number of time steps

Num

ber

of a

nim

als

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)

0

20

40

60

80

100

Number of time steps

Num

ber

of a

nim

als

S I R

Course of number of S, I and R animals in a closed population (1 run stochastic

SIR model)

0

20

40

60

80

100

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Number of time steps

Num

ber

of a

nim

als

S I R

Course of number of S, I and R animals in a closed population (1 run stochastic

SIR model)

0

20

40

60

80

100

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Number of time steps

Num

ber

of a

nim

als

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 theoremThe probability of a major outbreak

00,10,20,30,40,50,60,70,80,9

1

0 1 2 3 4 5 6 7 8 9 10

R0

prob

maj

or

Prob major = 1 - 1/R0

Final size distribution for R0 = 0.5

00,10,20,30,40,50,60,7

prob

abil

ity

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

final size

infection fades out after infection of 1 or a few individuals (minor outbreaks only)

Final size distribution for R0 = 3

0

0,05

0,1

0,15

0,2

0,25

0,3

prob

abil

ity

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

final size

R0 > 1 : infection may spread extensively (major outbreaks and minor outbreaks)

Deterministic threshold theorem: Final size as function of R0

00,10,20,30,40,50,60,70,80,9

1

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6

Reproduction ratio

Fin

al s

ize

p

pR

)1ln(

Transmission in an open population

S I R

*S*I/N *I

= infection rate parameter

= recovery rate parameter

= replacement rate parameter

*S *I *R

*N

Courses of infection in an open population

S

I

1: Minor outbreak (R0 < 1 of R0 > 1)

2: Major outbreak (R0 > 1)

3: Endemic infection (R0 > 1)

Infection can become endemic when the number of animals in a herd is at least:

)1

(*0

0

R

RN

M. paratuberculosis: = 0.003; = 0.0009; R0 = 10

Nmin = 5

BHV1 := 0.07; = 0.0009, R0 = 3.5

Nmin = 110

Transmission in an open population

S

NR 0

At endemic equilibrium (large population)

0R

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