BioModels Database logo

BioModels Database


BIOMD0000000249 - Restif2006 - Whooping cough


 |   |   |  Send feedback
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.  [more]
Original Model: BIOMD0000000249.origin
Submitter: Lukas Endler
Submission ID: MODEL1003290000
Submission Date: 29 Mar 2010 02:46:18 UTC
Last Modification Date: 18 May 2017 12:12:50 UTC
Creation Date: 21 Apr 2010 00:54:23 UTC
Encoders:  Lukas Endler
set #1
bqbiol:isVersionOf Gene Ontology defense response, incompatible interaction
Gene Ontology symbiosis, encompassing mutualism through parasitism
set #2
bqbiol:hasProperty Mathematical Modelling Ontology ordinary differential equation model
set #3
bqmodel:isDerivedFrom PubMed 460424
PubMed 460412
set #4
bqbiol:hasTaxon Taxonomy Bordetella pertussis
Taxonomy Homo sapiens
set #5
bqbiol:hasProperty Human Disease Ontology pertussis
Restif2006 - Whooping cough

This model is described in the article:

Restif O, Grenfell BT.
Proc. Biol. Sci. 2006 Feb; 273(1585): 409-416


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)

To the extent possible under law, all copyright and related or neighbouring rights to this encoded model have been dedicated to the public domain worldwide. Please refer to CC0 Public Domain Dedication for more information.

Publication ID: 16615206 Submission Date: 29 Mar 2010 02:46:18 UTC Last Modification Date: 18 May 2017 12:12:50 UTC Creation Date: 21 Apr 2010 00:54:23 UTC
Mathematical expressions
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)
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
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
Compartment: environment
Initial concentration: 1.0
Compartment: environment
Initial concentration: 0.0588912
Compartment: environment
Initial concentration: 0.003775
Compartment: environment
Initial concentration: 1.0E-6
Compartment: environment
Initial concentration: 0.93733
Compartment: environment
Initial concentration: 0.0
Compartment: environment
Initial concentration: 0.0
Compartment: environment
Initial concentration: 0.0
Compartment: environment
Initial concentration: 0.0
Global Parameters (21)
Value: NaN   (Units: per_year)
   life expectancy
Value: 50.0   (Units: years)
Value: NaN   (Units: per_year)
Value: 17.0   (Units: dimensionless)
Value: NaN   (Units: per_year)
Value: NaN   (Units: per_year)
Value: 17.0   (Units: dimensionless)
Value: NaN   (Units: per_year)
   infectious period 1
Value: 21.0   (Units: days)
   infectious period 2
Value: 21.0   (Units: days)
Value: NaN   (Units: per_year)
   immune period
Value: 20.0   (Units: years)
Value: NaN   (Units: per_year)
Value: NaN   (Units: per_year)
Value: NaN   (Units: dimensionless)
Value: NaN   (Units: dimensionless)
Value: NaN   (Units: dimensionless)
Value: NaN   (Units: dimensionless)
Value: NaN   (Units: dimensionless)
Value: NaN   (Units: dimensionless)
Value: 0.2   (Units: dimensionless)
Representative curation result(s)
Representative curation result(s) of BIOMD0000000249

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

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.