Contact Number for disaggregated SIR Model

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Jim Duggan
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Contact Number for disaggregated SIR Model

Post by Jim Duggan » Wed Oct 28, 2009 8:59 am

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 (R-Zero) 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.

Jean-Jacques Lauble

Re: Contact Number for disaggregated SIR Model

Post by Jean-Jacques Lauble » Wed Oct 28, 2009 3:16 pm

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.
Jean-Jacques Laublé

Jim Duggan
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Location: Galway, Ireland.
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Re: Contact Number for disaggregated SIR Model

Post by Jim Duggan » Thu Oct 29, 2009 3:54 am

Hi Jean-Jacques,

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.

Robert Eberlein
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Re: Contact Number for disaggregated SIR Model

Post by Robert Eberlein » Thu Oct 29, 2009 5:07 am

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.

Jean-Jacques Lauble

Re: Contact Number for disaggregated SIR Model

Post by Jean-Jacques Lauble » Thu Oct 29, 2009 9:18 am

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.
Jean-Jacques Laublé

Jim Duggan
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Location: Galway, Ireland.
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Re: Contact Number for disaggregated SIR Model

Post by Jim Duggan » Thu Oct 29, 2009 3:54 pm

Thanks for posting the model Jean-Jacques, 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 x-axis, 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.

Jean-Jacques Lauble

Re: Contact Number for disaggregated SIR Model

Post by Jean-Jacques Lauble » Sun Nov 01, 2009 3:49 pm

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.
Jean-Jacques Laublé

Rosemarie Sadsad
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Re: Contact Number for disaggregated SIR Model

Post by Rosemarie Sadsad » Sun Nov 01, 2009 6:37 pm

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.
Do you mean something like:

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 - child-to-child contact
ca - child-to-adult contact
aa - adult-to-adult contact
ac - adult-to-child 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:
Bunimovich-Mendrazitsky, S and Stone, L. (2005). Modeling polio as a disease of development. Journal of Theoretical Biology. 237 : 302–315

Jean-Jacques Lauble

Re: Contact Number for disaggregated SIR Model

Post by Jean-Jacques Lauble » Mon Nov 02, 2009 3:08 am

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.
Jean-Jacques Laublé

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