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BIOMD0000000249 - Restif2006_Whooping_Cough

 

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Reference Publication
Publication ID: 16615206
Restif O, Grenfell BT.
Integrating life history and cross-immunity into the evolutionary dynamics of pathogens.
Proc. Biol. Sci. 2006 Feb; 273(1585): 409-416
Cambridge Infectious Diseases Consortium, Department of Veterinary Medicine, University of Cambridge, Madingley Road, Cambridge CB3 0ES, UK. olivier.restif@m4x.org  [more]
Model
Original Model: BIOMD0000000249.origin
Submitter: Lukas Endler
Submission ID: MODEL1003290000
Submission Date: 29 Mar 2010 01:46:18 UTC
Last Modification Date: 01 Apr 2014 16:38:53 UTC
Creation Date: 20 Apr 2010 23:54:23 UTC
Encoders:  Lukas Endler
set #1
bqbiol:hasProperty Mathematical Modelling Ontology MAMO_0000046
set #2
bqbiol:isVersionOf Gene Ontology symbiosis, encompassing mutualism through parasitism
Gene Ontology immune response
Gene Ontology defense response, incompatible interaction
set #3
bqmodel:isDerivedFrom PubMed 460424
PubMed 460412
set #4
bqbiol:hasTaxon Taxonomy Bordetella pertussis
Taxonomy Homo sapiens
set #5
bqbiol:hasVersion ICD Whooping cough
Human Disease Ontology pertussis
Notes

This is the model described in the article:
Integrating life history and cross-immunity into the evolutionary dynamics of pathogens.
Restif O, Grenfell BT. Proc Biol Sci. 2006 Feb 22;273(1585):409-16. PMID:16615206, doi:10.1098/rspb.2005.3335;
Abstract:
Models for the diversity and evolution of pathogens have branched into two main directions: the adaptive dynamics of quantitative life-history traits (notably virulence) and the maintenance and invasion of multiple, antigenically diverse strains that interact with the host's immune memory. In a first attempt to reconcile these two approaches, we developed a simple modelling framework where two strains of pathogens, defined by a pair of life-history traits (infectious period and infectivity), interfere through a given level of cross-immunity. We used whooping cough as a potential example, but the framework proposed here could be applied to other acute infectious diseases. Specifically, we analysed the effects of these parameters on the invasion dynamics of one strain into a population, where the second strain is endemic. Whereas the deterministic version of the model converges towards stable coexistence of the two strains in most cases, stochastic simulations showed that transient epidemic dynamics can cause the extinction of either strain. Thus ecological dynamics, modulated by the immune parameters, eventually determine the adaptive value of different pathogen genotypes. We advocate an integrative view of pathogen dynamics at the crossroads of immunology, epidemiology and evolution, as a way towards efficient control of infectious diseases.

This version of the model can be used for both the stochastic and the deterministic simulations described in the article. For deterministic interpretations with infinite population sizes, set the population size N = 1. The model reproduces the deterministic time courses. Stochastic interpretation with Copasi UI gave results similar to the article, but was not extensively tested. The initial conditions for competition simulations can be derived by equilibrating the system for one pathogen and then adding a starting concentration for the other.

Originally created by libAntimony v1.3 (using libSBML 4.1.0-b1)

Model
Publication ID: 16615206 Submission Date: 29 Mar 2010 01:46:18 UTC Last Modification Date: 01 Apr 2014 16:38:53 UTC Creation Date: 20 Apr 2010 23:54:23 UTC
Mathematical expressions
Reactions
Birth Death in S Death in I_1 Death in I_2
Death in R_1 Death in R_2 Death in I_1p Death in I_2p
Death in R_p Primary Infection with strain 1 Primary Infection with strain 2 Secondary Infection with strain 1
Secondary Infection with strain 2 Recovery (I_1) Recovery (I_2) Recovery (I_1p)
Recovery (I_2p) Loss of Immunity (R_1) Loss of Immunity (R_2) Loss of Immunity (R_p)
Rules
Assignment Rule (variable: mu) Assignment Rule (variable: beta_1) Assignment Rule (variable: gamma_1) Assignment Rule (variable: beta_2)
Assignment Rule (variable: gamma_2) Assignment Rule (variable: sigma) Assignment Rule (variable: Lambda_1) Assignment Rule (variable: Lambda_2)
Assignment Rule (variable: I_1_frac) Assignment Rule (variable: I_2_frac) Assignment Rule (variable: S_frac) Assignment Rule (variable: R1_frac)
Assignment Rule (variable: R2_frac) Assignment Rule (variable: Rp_frac)    
Physical entities
Compartments Species
environment N S I_1
I_2 R_1 R_2
I_1p I_2p R_p
Global parameters
mu life expectancy beta_1 R0_1
gamma_1 beta_2 R0_2 gamma_2
infectious period 1 infectious period 2 sigma immune period
Lambda_1 Lambda_2 I_1_frac I_2_frac
S_frac R1_frac R2_frac Rp_frac
psi      
Reactions (20)
 
 Birth  → [S];   {N}
 
 Death in S [S] → ;  
 
 Death in I_1 [I_1] → ;  
 
 Death in I_2 [I_2] → ;  
 
 Death in R_1 [R_1] → ;  
 
 Death in R_2 [R_2] → ;  
 
 Death in I_1p [I_1p] → ;  
 
 Death in I_2p [I_2p] → ;  
 
 Death in R_p [R_p] → ;  
 
 Primary Infection with strain 1 [S] → [I_1];   {I_1} , {I_1p} , {N}
 
 Primary Infection with strain 2 [S] → [I_2];   {I_2} , {I_2p} , {N}
 
 Secondary Infection with strain 1 [R_2] → [I_1p];   {I_1} , {I_1p} , {N}
 
 Secondary Infection with strain 2 [R_1] → [I_2p];   {I_2} , {I_2p} , {N}
 
 Recovery (I_1) [I_1] → [R_1];  
 
 Recovery (I_2) [I_2] → [R_2];  
 
 Recovery (I_1p) [I_1p] → [R_p];  
 
 Recovery (I_2p) [I_2p] → [R_p];  
 
 Loss of Immunity (R_1) [R_1] → [S];  
 
 Loss of Immunity (R_2) [R_2] → [S];  
 
 Loss of Immunity (R_p) [R_p] → [S];  
 
Rules (14)
 
 Assignment Rule (name: mu) mu = 1/l_e
 
 Assignment Rule (name: beta_1) beta_1 = R0_1*gamma_1
 
 Assignment Rule (name: gamma_1) gamma_1 = 365/tInf_1
 
 Assignment Rule (name: beta_2) beta_2 = R0_2*gamma_2
 
 Assignment Rule (name: gamma_2) gamma_2 = 365/tInf_2
 
 Assignment Rule (name: sigma) sigma = 1/tImm
 
 Assignment Rule (name: Lambda_1) Lambda_1 = beta_1*(I_1+I_1p)/N
 
 Assignment Rule (name: Lambda_2) Lambda_2 = beta_2*(I_2+I_2p)/N
 
 Assignment Rule (name: I_1_frac) I_1_frac = (I_1+I_1p)/N
 
 Assignment Rule (name: I_2_frac) I_2_frac = (I_2+I_2p)/N
 
 Assignment Rule (name: S_frac) S_frac = S/N
 
 Assignment Rule (name: R1_frac) R1_frac = (R_1+R_p)/N
 
 Assignment Rule (name: R2_frac) R2_frac = (R_2+R_p)/N
 
 Assignment Rule (name: Rp_frac) Rp_frac = R_p/N
 
 environment Spatial dimensions: 3.0  Compartment size: 1.0
 
 N
Compartment: environment
Initial concentration: 1.0
 
 S
Compartment: environment
Initial concentration: 0.0588912
 
 I_1
Compartment: environment
Initial concentration: 0.003775
 
 I_2
Compartment: environment
Initial concentration: 1.0E-6
 
 R_1
Compartment: environment
Initial concentration: 0.93733
 
 R_2
Compartment: environment
Initial concentration: 0.0
 
 I_1p
Compartment: environment
Initial concentration: 0.0
 
 I_2p
Compartment: environment
Initial concentration: 0.0
 
 R_p
Compartment: environment
Initial concentration: 0.0
 
Global Parameters (21)
 
  mu
Value: NaN   (Units: per_year)
 
   life expectancy
Value: 50.0   (Units: years)
Constant
 
  beta_1
Value: NaN   (Units: per_year)
 
 R0_1
Value: 17.0   (Units: dimensionless)
Constant
 
  gamma_1
Value: NaN   (Units: per_year)
 
  beta_2
Value: NaN   (Units: per_year)
 
 R0_2
Value: 17.0   (Units: dimensionless)
Constant
 
  gamma_2
Value: NaN   (Units: per_year)
 
   infectious period 1
Value: 21.0   (Units: days)
Constant
 
   infectious period 2
Value: 21.0   (Units: days)
Constant
 
  sigma
Value: NaN   (Units: per_year)
 
   immune period
Value: 20.0   (Units: years)
Constant
 
  Lambda_1
Value: NaN   (Units: per_year)
 
  Lambda_2
Value: NaN   (Units: per_year)
 
   I_1_frac
Value: NaN   (Units: dimensionless)
 
   I_2_frac
Value: NaN   (Units: dimensionless)
 
   S_frac
Value: NaN   (Units: dimensionless)
 
   R1_frac
Value: NaN   (Units: dimensionless)
 
   R2_frac
Value: NaN   (Units: dimensionless)
 
   Rp_frac
Value: NaN   (Units: dimensionless)
 
 psi
Value: 0.2   (Units: dimensionless)
Constant
 
Representative curation result(s)
Representative curation result(s) of BIOMD0000000249

Curator's comment: (updated: 21 Apr 2010 03:50:11 BST)

Reproduction of fig 2 of the original publication using Copasi 4.5.31. The model was started from the steady state of a population infected with pathogen 1 by adding fraction of individuals infected with pathogen 2, I_2 = 1e-6.

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