Chapter2: Example of Stock-flow diagram (Infection)

Question 1

What are the steps that should be made to model the breakout of an epidemy in a last part of a population ?

Answer 1

I- Formulate verbal model:

1) Verbal Model: „Healthy host organisms can be infected by a pathogene. The infection depends on the probability of an encounter between pathogene and host and on the virulence of the pathogene Infected hosts fall ill at once and become sources of infection. After a certain time they die.“

2) Set assumptions: • No recover / immunisation possible, • The number of infected hosts is important, not the number of pathogenes, • Occurences of new healthy organisms are neglected.

3) Set the variables by which the system is expressed: -Number of healthy hosts -Number of diseased hosts, -Number of dead hosts.

II- Choose the Graphical model There are many possibilities to design graphical models. Among the best are Causal-Loop-Diagrams, and System Dynamics Diagramm (SDD, syn. Stock-and-Flow-Diagram)

Question 2

We chose SSD. What are the three state variables of this epidemy model ?

Answer 2

We define three state variables: H: Number of healthy hosts I: Number of infected hosts D: Number of dead hosts

Question 3

We chose SSD. What are the model parameters ?

Answer 3

a: The pathogene‘s virulence b: Mortality rate of infected hosts and t: Time

Question 4

How does the number of healthy hosts change ?

Answer 4

The change dH/dt depends on: H: Number of healthy hosts I: Number of infected hosts • The greater H, I (number of infected) and a (virulence) the more often healthy hosts will encounter infected ones an the more often contagions will occur. dH/dt = -a.H.I

Question 5

How does the number of dead hosts change?

Answer 5

• The number of dead hosts D can only change by deaths of infected hosts. This change is dD/dt. • The greater the mortality rate b, the more infected hosts will die during one unit of time. dD/dT = b.I

Question 6

How does the number of infected hosts change ?

Answer 6

The number of infected hosts I increases by new infections ( –> Decrease in healthy hosts -dH/dt) and decreases by deaths. The change of I is dI/dt. dI/dt= -dH/dt - dD/dt = a.H.I - b.I

Question 7

The dynamics of a simple pathogen-host system can be explained by :

Answer 7

1. Simple considerations about probabilites of encounters of pathogen and host, 2. The pathogene‘s virulence, 3. The time span, during which an infected host is contagious. f the pathogen is not too virulent, and if infected hosts are not contagious for too long, then the host population will not die out completely. There is a threshold for the size of the healthy host population. If this value can be undercut, it is possible to avoid or to confine the epidemic.

Question 8

Systems Analysis-Limits of the Model simple pathogen-host?

Answer 8

1. Simulations for longer time spans are not reasonable. We ignore the population‘s growth. 2. Parameter a (virulence) depends on many factors: Habits of the population, biology of Y.pestis, Environmental influences… Therefore, a is difficult to estimate. However, it seems reasonable to assume the factors above to be constant during the time span of interest.