Contact Number for disaggregated SIR Model
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This forum supports discussion on topics of specific interest to the Health Policy Special Interest Group of the System Dynamics Society. It is currently unmoderated, and anybody who is signed in may post to the forum.
This forum supports discussion on topics of specific interest to the Health Policy Special Interest Group of the System Dynamics Society. It is currently unmoderated, and anybody who is signed in may post to the forum.

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Contact Number for disaggregated SIR Model
Hi,
I'm preparing a lecture that involves a simple disaggregation of the homogeneous SIR model (John Sterman's Business Dynamics, Chapter 9), where the population is now in two "buckets", ADULT and CHILD.
For this example the only parameter that is different between the two groups is the Infectivity, which for adults in 0.25, and for children is 0.75.
The contact number (RZero) for the homogeneous case is:
contact rate * infection duration * infectivity
I am formulating the contact rate for the heterogeneous case as follows:
contact rate * infection duration * Expected Infectivity
Where Expected Infectivity = [(Number Adults/Population) * Infectivity Adults] + [(Number Children/Population) * Infectivity Children]
Is this is good way to formulate the overall infectivity, and so allow the contact number for the heterogeneous case to be calculated?
thanks in advance,
Jim.
I'm preparing a lecture that involves a simple disaggregation of the homogeneous SIR model (John Sterman's Business Dynamics, Chapter 9), where the population is now in two "buckets", ADULT and CHILD.
For this example the only parameter that is different between the two groups is the Infectivity, which for adults in 0.25, and for children is 0.75.
The contact number (RZero) for the homogeneous case is:
contact rate * infection duration * infectivity
I am formulating the contact rate for the heterogeneous case as follows:
contact rate * infection duration * Expected Infectivity
Where Expected Infectivity = [(Number Adults/Population) * Infectivity Adults] + [(Number Children/Population) * Infectivity Children]
Is this is good way to formulate the overall infectivity, and so allow the contact number for the heterogeneous case to be calculated?
thanks in advance,
Jim.
Re: Contact Number for disaggregated SIR Model
Hi Jim
I do not quite understand what you mean by contact nmuber. In which model in BD is it mentionned?
The contact number should be constant with the aggregated model.
But in the desaggregated model, the average infectivity is not constant, see the model that I have posted on the Vensim forum, System dynamics discussion thread.
The model calculates separetely the behaviour of eacn population, and one sees that the infectivity is not constant.
Maybe I am missing something?
Vensim Forum:
http://ventanasystems.co.uk/forum/
Regards.
JeanJacques Laublé
I do not quite understand what you mean by contact nmuber. In which model in BD is it mentionned?
The contact number should be constant with the aggregated model.
But in the desaggregated model, the average infectivity is not constant, see the model that I have posted on the Vensim forum, System dynamics discussion thread.
The model calculates separetely the behaviour of eacn population, and one sees that the infectivity is not constant.
Maybe I am missing something?
Vensim Forum:
http://ventanasystems.co.uk/forum/
Regards.
JeanJacques Laublé

 Posts: 42
 Joined: Sun Feb 22, 2009 8:16 am
 Location: Galway, Ireland.
 Contact:
Re: Contact Number for disaggregated SIR Model
Hi JeanJacques,
Thanks for your response.
The contact number for epidemics is described on page 308 of Business Dynamics, and is the product (c*i*d), where c = contact rate, i = infectivity, and d = recovery delay
This number represents the average number of people an infected person will infect during the period when they are infectious. My understanding is that this ratio is the same as "R0", for example, see:
http://en.wikipedia.org/wiki/Basic_reproduction_number
This number varies for different diseases, and can then be used to estimate what percentage of the population need to be vaccinated in order to provide herd immunity, whereby an infection will "fizzle out" because the recovery rate is greater than the infection rate (bathtub dynamics!).
This fraction is = (1  1 /R0), assuming all vaccinations are successful.
An interesting presentation on the web, that discusses smallpox eradication is:
http://www.bt.cdc.gov/agent/smallpox/tr ... istory.pdf
For the problem I am looking at, for a mixed population the infectivity of each subgroup is different.
I would like to confirm what ratio to use in order to calculate the contact number. [Assuming the contact rate and recovery delay remain the same for Adults and Children, and that Adults and Children are "perfectly mixed".]
best regards,
Jim.
Thanks for your response.
The contact number for epidemics is described on page 308 of Business Dynamics, and is the product (c*i*d), where c = contact rate, i = infectivity, and d = recovery delay
This number represents the average number of people an infected person will infect during the period when they are infectious. My understanding is that this ratio is the same as "R0", for example, see:
http://en.wikipedia.org/wiki/Basic_reproduction_number
This number varies for different diseases, and can then be used to estimate what percentage of the population need to be vaccinated in order to provide herd immunity, whereby an infection will "fizzle out" because the recovery rate is greater than the infection rate (bathtub dynamics!).
This fraction is = (1  1 /R0), assuming all vaccinations are successful.
An interesting presentation on the web, that discusses smallpox eradication is:
http://www.bt.cdc.gov/agent/smallpox/tr ... istory.pdf
For the problem I am looking at, for a mixed population the infectivity of each subgroup is different.
I would like to confirm what ratio to use in order to calculate the contact number. [Assuming the contact rate and recovery delay remain the same for Adults and Children, and that Adults and Children are "perfectly mixed".]
best regards,
Jim.

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 Posts: 179
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Re: Contact Number for disaggregated SIR Model
Hi Jim,
Just a quick thought, I have not worked this specific disaggregation issue but it strikes me that you may want to separately identify the different types of contacts  infected adults with healthy adults, infected adults with healthy children, infected children with healthy children and infected children with healthy adults. Then apply separate infection rates to all 4 cases.
Just a quick thought, I have not worked this specific disaggregation issue but it strikes me that you may want to separately identify the different types of contacts  infected adults with healthy adults, infected adults with healthy children, infected children with healthy children and infected children with healthy adults. Then apply separate infection rates to all 4 cases.
Re: Contact Number for disaggregated SIR Model
Hi Jim
I have calculated the number of contacts with a disagregated model, and it is equal to the formula you proposed.
The second model is posted in the Vensim Forum.
I have too calculated the number of contacts with another method, calculating first the average infectivity, and the problem is that I do not get the same value.
Another problem is that if one applies the desagregated model with an infectivity equal to 0.4 as calculated with the first run, and with a population of 10000 children and no adults, the aggretaged results are not the same as the first run.
The problem is not as simple as it looks.
I will have a look at it later on, when I have more time.
Regards.
JeanJacques Laublé
I have calculated the number of contacts with a disagregated model, and it is equal to the formula you proposed.
The second model is posted in the Vensim Forum.
I have too calculated the number of contacts with another method, calculating first the average infectivity, and the problem is that I do not get the same value.
Another problem is that if one applies the desagregated model with an infectivity equal to 0.4 as calculated with the first run, and with a population of 10000 children and no adults, the aggretaged results are not the same as the first run.
The problem is not as simple as it looks.
I will have a look at it later on, when I have more time.
Regards.
JeanJacques Laublé

 Posts: 42
 Joined: Sun Feb 22, 2009 8:16 am
 Location: Galway, Ireland.
 Contact:
Re: Contact Number for disaggregated SIR Model
Thanks for posting the model JeanJacques, and for the comments Bob. I will get back again with more comments.
A further point: I think there is another way to view the formulation. Let's assume that the infectivity of adults in 0.20, while the children's value is 0.80.
If we plot (a dimensionless) C/N on the xaxis, where C is the total number of children, and N is the total population. Therefore the range of C/N is [0,1].
Taking our two extreme points of infectivity, which arise when C=0 (Adult value applies) and C=N (Child value applies), we get two possible points:
(0,0.20) and (1,0.80)
It is intuitive to conclude that these two points represent a straight line (i.e. as C>N, the overall infectivity of the population moves proportionately to 0.80, and as C>0, the overall population infectivity tends towards 0.20).
Based on this (Y=MX+C), we can formulate the combined infectivity as:
I = 0.60 * (C/N) + 0.20
or, more generally:
I = (IC  IA) (C/N) + IA
Where IC  Infectivity of Children, IA = Infectivity of Adults
This value of I can then be used to calculate the contact number?
regards,
Jim.
A further point: I think there is another way to view the formulation. Let's assume that the infectivity of adults in 0.20, while the children's value is 0.80.
If we plot (a dimensionless) C/N on the xaxis, where C is the total number of children, and N is the total population. Therefore the range of C/N is [0,1].
Taking our two extreme points of infectivity, which arise when C=0 (Adult value applies) and C=N (Child value applies), we get two possible points:
(0,0.20) and (1,0.80)
It is intuitive to conclude that these two points represent a straight line (i.e. as C>N, the overall infectivity of the population moves proportionately to 0.80, and as C>0, the overall population infectivity tends towards 0.20).
Based on this (Y=MX+C), we can formulate the combined infectivity as:
I = 0.60 * (C/N) + 0.20
or, more generally:
I = (IC  IA) (C/N) + IA
Where IC  Infectivity of Children, IA = Infectivity of Adults
This value of I can then be used to calculate the contact number?
regards,
Jim.
Re: Contact Number for disaggregated SIR Model
Hi Jim
I have verified my calcullations. The aggregated infectivity is not constant during the epidemic period.
I have posted a new model on the Vensim forum that proves it definitely.
But i have not yet found how to calculate the contact number that needs some sort of tricky integration.
The model posted needs only Vensim PLE to be run or modified that can be downloaded freely from Vensim.com.
Regards.
JeanJacques Laublé
I have verified my calcullations. The aggregated infectivity is not constant during the epidemic period.
I have posted a new model on the Vensim forum that proves it definitely.
But i have not yet found how to calculate the contact number that needs some sort of tricky integration.
The model posted needs only Vensim PLE to be run or modified that can be downloaded freely from Vensim.com.
Regards.
JeanJacques Laublé

 Posts: 1
 Joined: Sun Nov 01, 2009 6:21 pm
Re: Contact Number for disaggregated SIR Model
Do you mean something like:Robert Eberlein wrote:Hi Jim,
Just a quick thought, I have not worked this specific disaggregation issue but it strikes me that you may want to separately identify the different types of contacts  infected adults with healthy adults, infected adults with healthy children, infected children with healthy children and infected children with healthy adults. Then apply separate infection rates to all 4 cases.
dS_c/dt =  (b_cc/N_c * I_c + b_ca/N_c * I_a) * S_c
dS_a/dt =  (b_aa/N_a * I_a + b_ac/N_a * I_c) * S_a
dI_c/dt = (b_cc/N_c * I_c + b_ca/N_c * I_a) * S_c  g_c * I_c
dI_a/dt = (b_aa/N_a * I_a + b_ac/N_a * I_c) * S_a  g_a * I_a
dR_c/dt = g_c * I_c
dR_a/dt = g_a * I_a
S  susceptible
I  infectious
R  recovered
a  adult
c  child
cc  childtochild contact
ca  childtoadult contact
aa  adulttoadult contact
ac  adulttochild contact
b  transmission coefficient
g  recovery rate
N  total population
R0_cc = b_cc/g_c
R0_aa = b_aa/g_a
R0_ca = b_ca/g_a
R0_ac = b_ac/g_c
The reproductive number R0 is the 'spectral radius of the next generation matrix'.
An example of calculating R0 can be found here:
BunimovichMendrazitsky, S and Stone, L. (2005). Modeling polio as a disease of development. Journal of Theoretical Biology. 237 : 302–315
Re: Contact Number for disaggregated SIR Model
Hi Jim
Your initial formula is finally right with some more thinking.
The infectivity that I calculated previously was an opportunistic one, that only matches the specific parameters of the disaggreagted model.
I join a model that definitively demonstrates it on the Vensim Forum. But the expected infectivity that you calculate cannot be used in an aggreagted model, because it only represents how the weight of the Child and Adult population influences both infectivity.
See the comments on the Vensim Forum.
Regards.
JeanJacques Laublé
Your initial formula is finally right with some more thinking.
The infectivity that I calculated previously was an opportunistic one, that only matches the specific parameters of the disaggreagted model.
I join a model that definitively demonstrates it on the Vensim Forum. But the expected infectivity that you calculate cannot be used in an aggreagted model, because it only represents how the weight of the Child and Adult population influences both infectivity.
See the comments on the Vensim Forum.
Regards.
JeanJacques Laublé
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